physical geometry - prof.usb.veprof.usb.ve/ggonzalm/invstg/pblc/phsclgmtr.pdf · first attempts...

PhPhPhPhPhysical Geometrysical Geometrysical Geometrysical Geometrysical Geometryyyyy

iii

Physical Geometry A Unified Theory

Third Edition

Gustavo R. González-Martín

Professor of Physics

Simón Bolívar University

Caracas

iv

© Gustavo R. González-Martín 2000, 2010, 2014

A revised english translation of “Geometría Física, Segunda Edición”,© Gustavo R. González Martín 1999, 2010The third edition includes new contentsby the author.

First English edition, first published in 2000

Departamento de FísicaUniversidad Simón BolívarValle de Sartenejas, Baruta, Estado MirandaApartado 89000, Caracas 1080-A, Venezuela

v

To Lourdes

vi

Note on the Second and Third Editions

New results are incorporated making it necessary to introduce additional chaptersand restructure the treatment of various topics. In the latest edition I added a chapter onneutrino masses and oscillations produced by gravitational curvature. I also included asection on the geometric nuclear potential barrier indicating that the connection poten-tial is strongly reduced in a condensed matter lattice allowing possible low energy nuclearreactions. Errata were corrected in the text and in some equations.

Preface

The idea of this book was born from a series of lectures over unified theories given atUniversidad Simón Bolívar. Really, it is a coherent recollection of scattered publicationson the geometric unification of physics, including unpublished works. The objective is toestablish the foundations for this unification in order to give an answer to the followingquestion: Is there a Physical Geometry? The fundamental ideas and some results are pub-lished in the references.

It is recognized that the action of matter defines certain concepts and their relations,all of them capable of geometrical representation. The main aspects of the theory are thefollowing:

1. The physical universe is described using nonlinear geometrical equations asso-ciated to a structure group.

2. The group of relativity, the group of automorphisms of flat space time, is gener-alized to the group of automorphisms of the geometric algebra of space time.

3. The classical field equations and the equations of motion are represented interms of a geometric connection and matter frames which determine the ge-ometry.

4. Microscopic physics is seen as the study of linear geometric excitations, whichare representations of the group, characterized by a set of discrete numbers.

5. The geometry determines a mass spectrum, characterized by background pa-rameters calculable from the self energy in terms of group quotients.

The results obtained indicate that gravitation and electromagnetism are unified in anontrivial manner. There are additional generators that may represent non classical inter-actions. Multipole equations of motion determine the geodesic motion with the Lorentzforce term. If we restrict to the even part, we obtain the Einstein field equation and apurely geometric energy momentum tensor which indicate the possibility of a geometricinternal solution. The constant curvature parameter (geometric energy density) of a hy-perbolic symmetric solution may be related, in the newtonian limit, to the gravitationalconstant G. If the nonriemannian connection fields contribute to the scalar curvature, theparameter G would be variable, diminishing with the field intensity. This effect may beinterpreted as the presence of dark matter. In vacuum, the known gravitational solutionswith a cosmological constant are obtained.

The electromagnetic part is related to an SU(2)Q subgroup. If we exclusively restrict toa U(1) subgroup we obtain Maxwell’s field equations. In general, the equation of motionis a geometric generalization of Dirac’s equation. In fact, this geometry is the germ ofquantum physics including its probabilistic aspects. Statistics determine the quantum in-terference effects which also apply to classical excitations. The geometric nature of Planck’sconstant h and of light speed c is determined by their respective relations to the connec-tion and the metric. The mass may be defined in an invariant manner in terms of energy,

vii

depending on the connection and matter frames. The geometry shows a triple structurewhich determines various physical triple structures. The geometric excitations have quantaof charge, flux and spin which determine the fractional quantum Hall effect. The quotientof bare masses of the three stable particles are calculated and leads us to a surprisinggeometric expression for the proton electron mass ratio, previously known but physi-cally unexplained. There are massive connection excitations whose masses correspondto the weak boson masses and allow a geometric interpretation of Weinberg’s angle. Thegeometric equation of motion (a generalized Dirac equation) determines the anomalousbare magnetic moments of the proton, the electron and the neutron. The “strong” elec-tromagnetic SU(2)Q part, without the help of any other force, generates short range at-tractive potentials which are sufficiently strong to determine the binding energy of thedeuteron and other light nuclides, composed of protons and electrons. The nuclear po-tential barrier produced by the SU(2)Q connection is strongly reduced in a lattice be-cause the connection boson is then determined by the lower mass of the electron fieldand allows low energy nuclear reactions.

The bare masses of the leptons in the three families are calculated as topological exci-tations of the electron. The masses of these excitations increase under the action of astrong connection (relativity of energy) and are related to meson and quark masses. Theneutrinos acquire a mass-energy in the presence of curvature which determines their os-cillations. The geometry determines the geometric excitation mass spectrum, which forlow masses, essentially agrees with the physical particle mass spectrum. The proton showsa triple structure which may be related to a quark structure. The combinations of thethree fundamental geometric excitations (associated to the proton, the electron and theneutrino), forming other excitations, may be used to represent particles and show a sym-metry under the group SU(3)SU(2)U(1). The alpha coupling constant is also deter-mined geometrically.

The first two chapters represent an introduction. In chapters 3 to 10 the fundamentalgeometric ideas are developed. In chapters 11 to 19 the theory is applied to concretecases.

Caracas, 6/20/2000; Miami, 10/10/2014. Gustavo R. González-Martín

Acknowledgments

I have tried to give credit to those whose work give support to the ideas expressed in this book.Nevertheless, it appears impossible to accomplish this completely. At the moment of writing, it ismuch what I owe to those from which I have learned throughout the years. In this sense, I amgrateful to the members physics community of the Boston area, in particular to my professor,John Stachel.

Specially I want to thank the colleagues with whom I discussed these topics, even if I amunable to precisely determine the contribution to the germination and formation of the ideas; inparticular, the senior faculty of the Caracas Relativity and Fields Seminar: Luis Herrera Cometta,Alvaro Restuccia, Sebastián Salamó and Carlos Aragone (R.I.P.). I also acknowledge the collabo-ration of research assistants and some students in my relativity courses and special lectures onunification at the Simón Bolívar University, who served as stimulating test in the presentation anddiscussion of the geometric hypothesis of physics: G. Salas, G. Sarmiento, V. Villalba, V. Varela, A.Mendoza, O. Rendón, E. Valdeblánquez, I. Taboada, V. Di Clemente, J. Díaz, J. González T., A. deCastro, A. Hernández and M. A. Lledó.

Caracas, Venezuela, June 20, 2000. Gustavo R. González Martín

viii

Notation.

Lower case Greek indices, corresponding to space time, vary from 0 to 3.

Lower case Latin indices, correspond to a Lie algebra dimension, usually from 1 to 15;occasionally indicate the three dimensional space, from 1 to 3.

Upper case Latin indices correspond to dimensions of matrices or spinors, usually varyingfrom 1 to 4 or 1 to 2.

Repetition of indices indicates summation over the dimension of the corresponding space.

The partial derivative is denoted by ¶ , the covariant derivative by , the exterior de-rivative by d and the covariant exterior derivative in a fiber bundle by D .

The physical units are chosen geometrically, so that c, and e are equal to 1.

The space time metric signature is 1, -1, -1, -1.

The majority of the specialized mathematical notation is defined in the appendices.

ix

First Part: Fundamentals

1. GEOMETRIC PHYSICAL PRINCIPLES 11.1. The Geometry of General Relativity. 11.2. Review of Electro-Gravitation Unification. 21.3. First Attempts towards a Physical Geometry. 31.4. The Structure Group. 4

2. ULTRA RELATIVITY. 92.1. Introduction. 92.2. Extension of Relativity. 92.3. Relativity of Interactions. 112.4. Summary. 13

3. A UNIFIED THEORY. 153.1. Geometrical Objects of the Theory. 153.2. Variational Principle. 163.3. Some Algebraic relations. 203.4. Equations of Motion for a Matter Frame. 213.5. Relation with Quantum Theory. 23

3.5.1. Agreement with Standard Quantum Mechanics. 233.5.2. Differences in Nonabelian Coupling. 25

3.6. Electromagnetic Sector. 273.7. Other Interactions. 283.8. Summary. 30

4. CLASSICAL THEORIES. 324.1. Geometric Classical Particles. 324.2. Relation of the Bundles. 334.3. The Classical Fields. 354.4. Motion of Classical Particles. 374.5. Lorentz Equations of Motion. 39

4.5.1. Even Subalgebra Inclusion. 414.5.2. Interpretation. 42

4.6. Summary. 425. THE GRAVITATIONAL FIELD. 44

5.1. Introduction. 445.2. An Equation for the Einstein Tensor. 45

5.2.1. The Energy Equation. 455.2.2. The Einstein Equation. 48

5.3. Equations for a Geometric Internal Schwarzschild Solution. 505.4. The Newtonian Limit. 525.5. Summary. 55

x

6. FIELD QUANTIZATION. 566.1. Introduction. 566.2. Linearization of Fields. 566.3. Frame Solutions. 586.4. Connection Solutions. 616.5. Bracket as Derivation. 616.6. Geometric Theory of Quantum Fields. 626.7. Summary. 64

7. QUANTIZED CHARGE AND FLUX. 657.1. Introduction. 657.2. Induced Representations of the Structure Group G. 657.3. Cartan Subalgebras. 677.4. Relation Among Quantum Numbers. 697.5. Physical Interpretation. 727.6. Representations of the Subgroup P. 747.7. Applications. 777.8. Summary. 78

8. MEASUREMENT OF GEOMETRIC OBSERVABLES. 808.1. Introduction. 808.2. Measurement of Geometric Currents. 818.3. Geometric Spin. 848.4. Geometric Charge. 868.5. Summary. 88

9. A GEOMETRIC DEFINITION OF MASS. 909.1. Introduction. 909.2. The Concept of Mass. 919.3. Invariant Mass. 929.4. Momentum Operator. 949.5. Summary. 96

10. QUANTUM GEOMETRODYNAMICS. 9710.1. Introduction. 9710.2. Geometric Relations. 97

10.2.1. Product of Jacobi Operators. 9710.2.2. Commutation Relations. 98

10.3. Geometric Field Dynamics. 9810.3.1. “Free” Particles and Fields. 9810.3.2. Quantum Electrodynamics. 9910.3.3. Statistical Interpretation. 100

10.4. Physical Particles and Waves. 10210.5. Summary. 106

xi

11. FRACTIONAL QUANTUM EFFECTS. 10811.1. Introduction. 10811.2. Magnetic Flux Quanta. 10811.3. The Fractional Quantum Hall Effect. 11111.4. Summary. 114

12. THE SUBSTRATUM AND ITS PHYSICAL SIGNIFICANCE. 11512.1. Introduction. 11512.2. The Field Equation. 11512.3. A Substratum Solution. 117

12.3.1. The Substratum Connection. 11712.3.2. The Substratum Curvature. 12012.3.3. Relation with the Newtonian Limit. 12112.3.4. Physical Significance of the Substratum. 122

12.4. General Equation of Motion. 12512.5. General Substratum 12612.6. Summary. 129

13. MASS RATIOS AND ENERGY. 13113.1. Introduction. 13113.2. Bare Masses. 13313.3. Symmetric Cosets. 138

13.3.1. Volume of C Space. 13813.3.2. Volume of K space. 14013.3.3. Ratio of Geometric Volumes 143

13.4. Physical Mass Ratios. 14413.5. Summary. 144

14. CONNECTION EXCITATION MASS. 14614.1. Introduction. 14614.2. The General Form of the Excitation Equation. 14614.3. An Excitation Solution. 14714.4. Massive SU(2) Excitations. 149

14.4.1. Mass Values on Free Space. 15414.4.2. Connection Excitations on a Lattice. 155

14.5. Equations for Massless Fields. 15614.5.1. Restrictions to Possible Solutions. 158

14.6. Summary. 15815. WEAK NUCLEAR INTERACTION. 160

15.1. Introduction. 16015.2. Geometric Weak Interaction. 16115.3. Relation with Fermi’s Theory. 16315.4. Summary. 167

Second Part: Applications

xii

16. STRONG NUCLEAR INTERACTION. 16916.1. Introduction. 16916.2. Motion of an Excitation in a Nonrelativistic Approximation. 16916.3. Magnetic Moments. 17116.4. The Modified Pauli Equation. 17516.5. The Proton-Electron-Proton Fields for the Deuteron. 17616.6. Binding Energy of the Deuteron. 17916.7. The Electron-Proton Model for the Neutron. 18116.8. The many Deuteron Model. 183

16.8.1. Nuclear Barriers and Reactions. 18416.9. Summary. 185

17. THE GEOMETRIC STRUCTURE OF PARTICLES AND INTERACTIONS. 18717.1. Introduction. 18717.2. Geometric Classification of the Connection. 19017.3. Excitations Corresponding to Subgroups. 19117.4. Algebraic Structure of Particles. 19217.5. Physical Interpretation in Terms of Particles and Interactions. 19517.6. Topological Structure of Particles. 19617.7. Geometric Excitation Masses. 198

17.7.1. Leptonic Masses. 20017.7.2. Mesonic Masses. 202

17.8. Barut’s Model. 20617.9. Relation with Particle Theory. 20917.10. Summary. 211

18. THE NEUTRINO EQUATION AND OSCILLATIONS. 21418.1. Introduction. 21418.2. The Action of the Even Subgroup L. 21518.3. An even L-Spinor Neutrino Equation. 21618.4. A Lorentzian-section Curvature Scalar. 21718.5. Neutrino Mass-Energy. 21918.6. Topological Squared Energy Effects. 22118.7. Summary. 224

19. THE ALPHA CONSTANT. 22619.1. Introduction. 22619.2. A Geometric Measure. 226

19.2.1. Symmetric Space K. 22619.2.2. Realization of the Symmetric Space K as a Unit Polydisc. 22719.2.3. Invariant Measure on the Polydisc. 229

19.3. Wyler’s Measure on the K Space. 23019.4. Value of the Geometric Coefficient. 23319.5. Summary. 234

xiii

Third Part: Appendices

A. GEOMETRIC ALGEBRA. 236A.1. Introduction. 236

A.1.1. Clifford Algebras and Spinors. 236A.2. A Representation for the Algebra A. 238A.3. Correlated Space of the Group G. 243A.4. Relation of R3,1 with R2,0. 246A.5. Relation between Spinors of Groups G and L. 247A.6. Even Spinor Frames and Vector Frames. 249A.7. Derivative of the Orthonormal Set. 250

B. GROUPS AND SYMMETRIC SPACES. 252B.1. Lie Groups. 252

B.1.1. The Differential of a Map. 252B.1.2. The Lie Algebra of a Group. 253

B.2. Cartan Subspace. 255B.3. The Group G. 258B.4. Symmetric Spaces. 261

C. CONNECTIONS ON FIBER BUNDLES. 265C.1. A Fundamental Field. 265C.2. The Ehresmann Connection. 266C.3. Tensorial k-forms. 268C.4. Curvature and Torsion. 269C.5. Induced Connection, Curvature and Torsion Forms. 271

D. JET BUNDLES. 275D.1. Jet Bundles. 275D.2. Critical Sections And Jacobi Vectors. 277

E. SOME PROPERTIES OF FIBER BUNDLES. 280E.1. Manifolds. 280E.2. Fiber Bundles. 281E.3. Homotopic Product. 282E.4. Third Homotopy Group. 283

F. NEWTON’S GRAVITATION AND GEOMETRIC THEORIES. 285F.1. Spacetime Limits. 285

F.1.1. Instantaneous Propagation. 285F.1.2. Local Limit. 287F.1.3. Global Limit. 288F.1.4. Postulates. 289

F.2. Geometric Rigidity Condition. 290F.3. Boundary Geometric Connection. 292F.4. Newtonian Connection. 294

INDEX ................................................................................................................................. 297A Bibliography on Geometry and Relativity 314

xiv

E E iE- = 1 2

q 1

q 2

q 3

E E+ = 3

EQ

Geometric Quantization of the generators E of the SU(2) elec-tromagnetic connection and the electric charge. ¿Universal geo-metric action?

xv

…Mach felt that there was something important about this concept of avoiding aninertial system… Not yet so clear in Riemann’s concept of space. The first to see thisclearly was Levi-Civita: absolute parallelism and a way to differentiate…

…The representation of matter by a tensor was only a fill-in to make it possible to dosomething temporarily, a wooden nose in a snowman…

…For most people, special relativity, electromagnetism and gravitation are unimportant,to be added in at the end after everything else has been done. On the contrary, we have totake them into account from the beginning…

Albert Einstein

from Albert Einstein’s Last Lecture,3

Relativity Seminar,Room 307, Palmer Physical Laboratory, Princeton University,April 14, 1954,according to notes taken by J. A. Wheeler.

3 J. A. Wheeler in: P. C. Eichelburg and R. U. Sexl (Eds.), Albert Einstein (Friedr.Vieweg & Sohn, Braunschweig) p. 201, (1979).

1. GEOMETRIC PHYSICAL PRINCIPLES

1.1. The Geometry of General Relativity.Acknowledging Mach’s criticism of preferred inertial systems, Einstein constructed

general relativity using the principle of covariance and the principle of equivalence. Nev-ertheless, it was pointed out, as early as 1917, by Kretschmann [1 ] and ratified by Einstein[2 ] that the principle of covariance, as usually stated, is devoid of physical content andany theory whatever may be formulated in a general covariant form. Covariant formula-tions of Newton’s theory of gravitation were given by Cartan [3 ,4 ] and Friedrichs [5 ] anddiscussed in review articles by Havas [6 ], Trautman [7 ] and Kilminster [8 ]. These formu-lations also conform with the equivalence principle. For this reason it is necessary toclarify the meaning of these principles as stated by Einstein in the physical reasoning thatlead to general relativity.

Any theory of gravitation specifies the motion of idealized test particles. A test particlehas only monopole gravitational structure, in other words, zero nongravitational charges,no multipole structure, etc. In the absence of gravitation, Newton’s first law states theinertial principle: there exists a privileged class of motions, called “free motions”, fol-lowed by bodies not acted on by any force.

In order to incorporate the principle of equivalence to a theory of gravitation, we seeka new privileged class of motions. The best we can do is to use the trajectories of testparticles acted only by gravitational forces, called “free falls”. The motion of test particlesdefines a set of curves, free falls, at each point and in each timelike direction in the mani-fold and a physical parameter on each curve which is a measure of the displacement of theparticle along the curve. If we have a preferred family of curves in a manifold, one througheach point and in each direction through that point, and a preferred parameter along eachcurve, we define a connection in the manifold M, or equivalently in its tangent bundle TM.The connection is determined by requiring the curves to be geodesics and the parameter tobe the affine parameter. Therefore, we can say that a theory of gravitation incorporatingthe equivalence principle defines a connection on space-time.

If we introduce a connection in this way there is no assurance that there is compatibil-ity of the connection with a metric in space-time. Einstein made the implicit assumptionthat there is an organic relation between the affine and metric structures of space-time.The affine structure is required to be minimal in relation to the metrical structure, that is,the connection coefficients are the Christoffel symbols of the metric. This implies that thecovariant derivative of the metric vanishes and that standard rods and clocks are indepen-dent of the gravitational field. This minimal relation excludes the possibility of adding asymmetric tensor to the Christoffel symbols. It seems that this requirement of compatibil-ity is the content of the principle of covariance.

If we put the previous comments in a geometric language we may state the two prin-ciples as follows:

1. A theory of gravitation should be represented by a nontrivial connection on thetangent bundle TM to space-time M.

2. The associated principal bundle of TM has the Lorentz group SO(3,1) as struc-ture group.

Chapter 1PHYSICAL GEOMETRY2

The first statement guarantees the principle of equivalence. The second statement may betaken as the content of the principle of covariance. It should be noted that the possibilityof introducing local coordinates is a property of any manifold whatsoever and the use ofthese coordinates and their transformations does not introduce any additional geometricstructure apart from the manifold assumption.

1.2. Review of Electro-Gravitation Unifica-tion.

It is well known that Einstein’s field equations of gravitation imply the equations ofmotion of test particles [9 ] and, on the contrary, that Maxwell’s field equations of electro-dynamics do not imply the corresponding equations of motion. In this case, it is necessaryto postulate the Lorentz force equation, or derive it from some separate postulated varia-tional principle. This fact is related to the nonlinearity of gravitation and the linearity ofelectrodynamics [10 ].

If a unified theory of gravitation and electrodynamics is constructed with nonlinearfield equations, it should be possible to derive the Lorentz equation of motion from thefield equations of the unified theory.

Within the Einstein-Maxwell theory, the desired equations were obtained by Infeldand Wallace [11 ]. In this theory, if we properly choose the stress energy tensor of electro-magnetism TE, the conservation of the total stress energy tensor T implies the Lorentzforce equation. This should be no surprise because the TE of electromagnetism is con-structed precisely to conserve the energy and momentum of a system of electromagneticfields and electric charges moving according to the Lorentz force equation. In other wordsthe structure of TE assumes the validity of the Lorentz equation. In the Einstein-Maxwelltheory, apart from the geometric gravitational postulates, we have to postulate separatelythe exact form of the stress energy tensor of electrodynamics which contains the assump-tion of motion according to the Lorentz force. With this postulate, the motion of chargeparticles is determined by conservation, even in flat space, without the use of Einstein’sequations [12 ,13 ,14 ].

It may also be claimed that the Einstein-Maxwell theory is not a truly geometricallyunified theory. Einstein [15 ] himself was unsatisfied by the nongeometrical character ofTE and spent his later years looking for a satisfactory unified theory.

The need of theories of weak and strong interactions may revive the idea of a geo-metrically unified theory. The Einstein-Maxwell theory is incomplete, in the sense that itdoes not provide a geometrical structure capable of representing additional interactingfields.

Most of the work done on the motion of charged particles [16 ,17 ,18 ,19 ], includingthe Infeld and Wallace calculation, accounts for the motion with the assumptions indi-cated before. In line with the ideas expressed by Einstein for a unified theory, rather thanstudy the motion of matter under the forces of some unified theory, we should take note ofthe similarities between the Lorentz force expression and the expression for the force onspinning particles in general relativity and assume that the similarities are not a merecoincidence. This leads us to represent gravitation and electromagnetism by the same geo-metric object. By requiring the prediction of the correct equation of motion, including theLorentz force, at least in a certain limit, we should be able to narrow down possible theo-ries.

In the so called “already” unified field theory, it is required that the curvature tensor

3Geometric Physical Principles

satisfies the Rainich [20 ], Misner and Wheeler [21 ] conditions, which are equivalent tothe existence of the electromagnetic stress energy tensor TE. With these conditions, theLorentz force follows in the same way as in the Einstein-Maxwell theory, but in reality itis assumed in the extra requirements for the curvature.

In Weyl’s unified theory [22 ], the equations of motion are subject to an objection, firstraised by Einstein [23 ], that they imply that the frequencies of the atomic spectral linesshould depend on the location and past histories of the atoms.

In the Kaluza theory [24 ], the Lorentz equations are obtained from the geodesic equa-tion in a five dimensional space with a Killing vector along the fifth dimension. He inter-preted the component of the five velocity along the fifth direction as the electric chargeand certain components of the symmetric connection as the electromagnetic field tensor.We can object that, since it is possible to make the connection zero along any given curve,the electromagnetic tensor can be made zero in properly chosen coordinates. The physicalmeaning of this coordinate system is not compatible with known experimental facts ofelectromagnetism.

Derived equations of motion were discussed by Johnson [25 ] within the Einstein theoryof the antisymmetric field [26 ]. In the nongravitational limit, the electrodynamics of thistheory is not the conventional Maxwell theory [27 ], although the resultant equations arecompatible with the experimentally known interaction of charged particles over labora-tory distances.

1.3. First Attempts towards a Physical Ge-ometry.

As indicated in the first section gravitation is associated to a geometric structure, aconnection in an SO(3,1) principal bundle associated to the tangent bundle of space-time.It is also known that electromagnetism may be described as connection in a U(1) principalbundle over space-time. Therefore, we chose a connection as the geometric object repre-senting the unified interaction.

We proposed [28 ,29 ] a geometric unified theory where electromagnetism does notenter as part of the metric or the stress-energy tensor of matter, but as part of a connectionas originally intended by Weyl [30 ]. Similarly, gravitation also enters as part of the con-nection. This was accomplished by enlarging the structure group of the theory to incorpo-rate both gravitation and electromagnetism as part of a unified Ehresmann connection,making the theory clearly different from Einstein-Maxwell’s, Weyl’s, Kaluza’s [31 ] andantisymmetric field theories [32 ]. The physical motivation for this speculation arose fromthe fact that the gravitational curvature enters in the equations of motion for spinningbodies in the same manner as the electromagnetic curvature in the Lorentz equations forcharged particles and the fact that a theory of gravitation that incorporates naturally theequivalence principle should be represented by a connection, not necessarily by a metric.

If we desire to represent the unified interaction as a connection in a principal bundlewe are faced with the selection of the structure group of the bundle. We choose a geomet-ric criteria for this selection: The group in question should be associated geometrically tospace-time. Later we shall deal with the relation of the group to other areas of physics.

Because of this possibility and the well-known relation of Maxwell’s theory of elec-tromagnetism to U(1) connections on a principal bundle, it was felt that in order to ap-proach unification it was desirable to work with the group SL(2,C), rather than the Lorentzgroup itself SO(3,1). A gravitation theory related to SL(2,C) was discussed by Carmelli


[33 ]. It follows then that the simplest way to enlarge the group, apparently, is to use thegroup U(1)SL(2,) which is the group which preserves the metric associated to a tetradinduced from a spinor base.

It was known to Infeld and Van der Waerden [34 ,35 ,36 ] when they introduced aspinorial connection, that there appeared arbitrary components which admitted interpreta-tion as electromagnetic potentials because they obeyed the necessary field equations. Toadmit this interpretation we further required that the Lorentz force equation be a conse-quence of the field equations. Otherwise the equation of motion, necessarily implied bythe theory, contradicts the experimentally well-established motion of charged particlesand the theory should be rejected.

A first attempt [28,37 ], using U(1) SL(2,) as the group, led to a negative result,because the equations of motion depend on the commutators of the gravitational and elec-tromagnetic parts which commute. This means that a charged particle would follow thesame geodesic followed by a neutral particle. This proves that it is not possible, withoutcontradictions, to consider that the U(1) part represents electromagnetism as suggested byInfeld and Van der Waerden. This also means that to obtain the correct motion we mustenlarge the chosen group in such a way that the electromagnetic generators do not com-mute with the gravitational ones. It is not true that any structure group which containsSL(2,) U(1) as a subgroup gives a unified theory without contradicting the Lorentzequation of motion. The correct classical motion is a fundamental requirement of a uni-fied theory.

According to the criteria indicated, we desire a group associated to space-time. SinceClifford algebras and their spinors are associated geometrically to orthonormal spaces,they represent a generalization of the concept of metric. There are transformations of thisalgebra which preserve its product and consequently the metric structure of its associatedorthonormal space.

If we put these ideas in a geometric language we may state two principles:1. A unified theory should be represented by a nontrivial connection on a principal

bundle E with space-time M as the base space of the bundle.2. The structure group G of the principal bundle E is the group of correlated

automorphisms of the spinor space of the geometric algebra of the tangent spaceTM of space-time.

1.4. The Structure Group.In accordance with the second postulate we use the group SL(2,) over a ring ,

which is the group of spinor automorphisms of the universal Clifford algebra of flat space-time. It turns out that there are two not isomorphic universal algebras for space-time,according to the chosen signature for the metric (see appendix A). In correspondencethere are also two not isomorphic simple groups SL(2,), one where is the field ofquaternions and the other where is the nondivision ring (2), which we shall call thepseudoquaternions . The latter group is homomorphic to SL(4,).

The group SL(2,) is known not to preserve the corresponding metric. But, if we thinkof general relativity as linked to general coordinate transformations changing the form ofthe metric, it would be in the same spirit to use such a group. Instead of coordinate trans-formations whose physical meaning is associated with a change of observers, we havetransformations belonging to the group SL(2,) whose physical meaning should be asso-ciated with a change of spinors related to observers. Representations of this group wouldbe linked to matter fields. If we restrict to the even part of either one of the SL(2,)


groups, taken as a subset of the Clifford algebra, we get the group SL1(2,), used in spinorphysics. Since SL(2,) is larger (higher dimensional) than SL(2,) it gives us an oppor-tunity to associate the extra generators with interactions apart from gravitation and elec-tromagnetism. The generator which plays the role of the electromagnetic generator mustbe consistent with its use in other equations of physics. The physical meaning of theremaining generators should be identified.

The field equations should relate the connection to a geometric object representingmatter. We expect that matter is represented by a tensorial n-form valued in the group Liealgebra, rather than the nongeometric T. The simplest object of this type, constructed fromthe connection is the curvature W which obeys the Bianchi identity,

DW = 0 . (1.4.1)The next simple object is constructed using Hodge duality *W, if we have a metric in

space-time. In similarity with electromagnetism, we postulate the field equation,

D k JW* *= , (1.4.2)where *J must be a 3-form valued in the algebra and k a constant to be identified later.Because of the geometrical structure of the theory the source current must be a geometri-cal object compatible with the field equation and the geometry. The structure of J, ofcourse, is given in terms of some geometric objects acted upon by the connection. Thegeometrical properties of the curvature and the field equations determine that J obeys anintegrability condition,

[ ],DDX XW= , (1.4.3)

,DD W W W* *é ù= =ê úë û 0 , (1.4.4)

D J* = 0 . (1.4.5)This relation, being an integrability condition on the field equations, includes all self

reaction terms of the matter on itself. A physical system would be represented by matterfields and interaction fields which are solutions to this set of nonlinear simultaneous equa-tions. There should be no worries about infinities produced by self reaction terms. As inthe EIH method in general relativity [38 ], when a perturbation is performed on the equa-tions, for example to obtain linearity of the equations, the splitting of the equations intoequations of different order bring in the concepts of field produced by the source, forceproduced by the field and therefore the self reaction terms. These terms, not present in theoriginal nonlinear system, are a problem introduced by this particular method of solution.In the zeroth order a classical particle moves as a test particle without self reaction. In thefirst order the field produced by the particle produces a self correction to the motion.

Enlarging the group of the connection not only unifies satisfactorily gravitation andelectromagnetism, but requires other fields and it appears to give a gravitational theorywhich differs, in principle, from Einstein’s theory and resembles Yang’s theory [39 ]. Thismay be seen from the field equations of the theory, which relate the derivatives of theEhresmann curvature to a current source J.

As in general relativity, the integrability conditions imply the equations of motion fora classical particle, without knowledge of the detailed form of the source J, if we assumethat J has a multipole structure. The desired classical Lorentz equations of motion are


obtained in chapter 4, satisfying in this manner the criteria of acceptability for the pro-posed unified theory.

Nevertheless the main objective at present, is not to describe the classical motion ofmatter exhaustively but rather to construct the geometrical theory and to require that it becompatible with the classical motion of the sources and possibly to require compatibilitywith modern ideas in quantum theory. In particular, it appears, as first objective, to exploitthe opportunity provided by the theory to give a geometrical interpretation to the sourcecurrent in terms of fundamental geometric field objects. If a geometric structure is givento J, the first stage in the construction of the unified theory is completed.

A theory of connections without any other objects is incomplete from a geometricalpoint of view. A connection on a principal bundle is related to the structure group and thebase space of the bundle. Representations of the group provide a natural vector fiber foran associated vector bundle on which the connection may be made to act. The geometricmeaning of the connection is related to parallel translation of the elements of the fiber atdifferent points throughout the base space. This is, essentially, a process of comparison ofelements at different events.

A vector fiber space of this type has a base and the effect of the connection is naturallydefined on the base. From a geometrical point of view the connection should be comple-mented by a vector base. It is well known that Einstein’s gravitation theory may be ex-pressed using an orthonormal tetrad instead of the metric [40 ]. In this theory we havetaken this idea one step further, introducing a spinor base e on the fiber space of an asso-ciated vector bundle S, in addition to the base on the fiber of the tangent space. In otherwords, we work with the base of the “square root space” of the usual flat space. Theconnection, which represents the gravitational and electromagnetic fields, depends on acurrent source term. We also assume that this source current is built from fundamentalfields which have the geometrical interpretation of forming a base e on the fiber of theassociated vector bundle and defines an orthonormal subset k of the geometric Cliffordalgebra,

!J e edx dx dxm a b g

abgme k* = 13

. (1.4.6)

or equivalently

!J e e dx dx dxa b c

a b ck k k* = 13 . (1.4.7)

This base e, when arranged as a matrix with the vectors of the base as columns, isrelated to an element of the group of the principal bundle. It is natural to expect that abase field e (a section in geometric language), which we shall call a frame e, should obeyequations of motion which naturally depend on the connection field. In fact, it will beseen that a particular solution of the integrability condition, the covariant conservation ofJ, is obtained from the equation

emmk = 0 . (1.4.8)

This equation may be interpreted as a generalized Dirac equation if in the structure group,SL(2,), the ring is chosen to be the nondivision ring (2)= of pseudo quaternions,which determines that the structure group of the theory is SL(2,) or its covering group[41 ]. The equation for the frame using the other group SL(2,) does not lead to Dirac’s


equation for a free particle. This will be discussed in chapter 3. From this point, we shalluse the notation SL(2,) or SL(4,) to indicate these groups or the covering group, un-less otherwise explicitly stated when it is convenient to distinguish them.

We should note that whenever we have an sl(2,) connection, there is a canonicalcoupling of standard gravitation to spin ½ particles obtained by postulating a Dirac equa-tion which depends on a spin frame [42, 43, 44 ]. Nevertheless, strictly this does notrepresent a real unification. Our field equation implies integrability conditions in terms ofJ. Together with the geometric structure of J, these conditions imply the generalized Diracequation which, therefore, is not required to be separately postulated, as in the previouslymentioned nonunified case. On the contrary, the theory under discussion is not a merepasting together of canonical gravitation and canonical electromagnetism for spin ½ par-ticles. Rather, it is the introduction of a generalized geometric structure which nontrivi-ally modifies both canonical theories and their coupling. Actually, the nonlinear field equa-tion for the connection and the simplest geometric structure of the current are sufficient topredict this generalized Dirac equation

It appears then that the theory in question may contain both the classical and the quan-tum equations of matter. In particular, the classical equations of motion obtained are amultipole approximation. The quantum equation (1.4.8) is the geometric equation of mo-tion, in accordance with the structure of the current.

A classical particle itself may not be the appropriate idealization of the physical world.The theory should provide relations between interactions fields (gravitation, electromag-netism, etc.) and matter fields (masses, charges, spinors, etc.) and specify how these fieldsevolve. A modern definition of particle and its properties should rest on the fundamentalgeometrical fields. It would be desirable that both classical and quantum aspects of thephysical particle could be obtained from a geometrical theory.

References

1 E. Kretschmann, Ann. Phyik, 53 , 575 (1917).2 A. Einstein, Ann. Physik, 55, 241 (1918).3 E. Cartan, Ann. Ecole Norm. 40, 325 (1923).4 E. Cartan, Ann. Ecole Norm. 41, 1 (1924).5 K. Friedrichs, Math. Ann. 98, 566 (1927).6 P. Havas, Rev. Mod. Phys. 36, 938 (1964).7 A. Trautman, Brandeis Summer Institute in Theoretical Physics (Prentice Hall,

Englewoods Cliffs) Vol. 1, p. 101 (1964).8 C. J. Kilminster, J. Math. and Mech., 12 , 1 (1963).9 A. Einstein, J. Grommer, Sitzber. Preuss. Akad. Wiss. 1, 2 (1927).10 L. Infeld, J. Plebansky, Motion and Relativity, (Pergamon Press, New York), p. 16

(1960).11 L. Infeld, P. R. Wallace, Phys. Rev. 57, 797 (1940).12 M. Mathisson, Z. Phys. 67, 270 (1931).13 P. A. M. Dirac, Proc. R. Soc. London, A167, 148 (1938).14 P. A. M. Dirac, Ann. Institut Poincaré 9, 13 (1938).15 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton), p.

98 (1956).16 L. Infeld, A Schild, Rev. Mod. Phys. 21, 408 (1949).17 L. Infeld, J. Plebansky, Bull. Acad. Pol. Sci. 4, 757, (1956).


18 R. P. Kerr, Nuovo Cimento 13, 469 (1959).19 R. P. Kerr, Nuovo Cimento 13, 492 (1959).20 G. Y. Rainich, Trans. Am. Math. Soc. 27, 106 (1925).21 C. W. Misner J. A. Wheeler, Ann. Phys. 2, 525 (1957).22 H. Weyl, Sitzber. Preuss. Akad. Wiss. Berlin, 465 (1918).23 H. Weyl, Space, Time, Matter, tranlated by H. L. Brose (Methuen, London), ch. 4

(1922).24 T. Kaluza, Sitzungsber. Berl. Akad., 966 (1921).25 C. R. Johnson, Phys. Rev. D15, 377 (1977).26 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton),

p. 133 (1956).27 C. R. Johnson, Phys. Rev. D31, 1252 (1985).28 G. González-Martín, Phys. Rev. D35, 1215 (1987).29 G. González-Martín, Phys. Rev. D35, 1225 (1987). See chapter 3.30 H. Weyl, Sitzber. Preuss. Akad. Wiss. Berlin, 465 (1918).31 T. Kaluza, Sitzungsber. Berl. Akad., 966 (1921).32 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton),

p. 98, 133, 166 (1956).33 M. Carmelli, Ann. Phys. 71, 603 (1972).34 L. Infeld, B. L. van der Waerden, Sitzber. Preuss. Akad. Wiss. Physik Math. K1, 380

(1933).35 W. L. Bade and H. Jehle, Rev. Mod. Phys. 25, 714 (1953).36 H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York). (1931).37 G. González-Martín, Act. Cient. Ven. 37, 130 (1986).38 A. Einstein, L. Infeld B. Hoffmann, Ann. Math. 39, 65 (1938).39 C. N. Yang, Phys. Rev. Lett. 33, 445 (1974).40 C. Moller, K. Dan. Vidensk Selsk. Mat. Fys. Medd., 39, 1 (1978).41 See appendix B.3.42 E. Schrödinger, Sizber. Akad. Phys 57, 261 (1929).43 A. Lichnerowicz, Compt. Rend. 252, 3742 (1961).44 A. Lichnerowicz, Compt. Rend. 253, 940 (1961).

2. ULTRA RELATIVITY.

2.1. Introduction.Previous results seem to indicate that there is a geometry which may have a broad

physical significance. Before continuing the development of the theory, we shall discussthis significance, leading to an extension of relativity and therefore setting the proposedtheory on first principles

The use of Clifford geometric [1 ] algebras introduces in the geometry of space-time, agroup of dimension higher than the Lorentz group. Since the latter is the group of relativ-ity, we may consider that use of the larger group indicates an extension of relativity. Origi-nally the group was introduced to accomplish a nontrivial unification of gravitation andelectromagnetism. This unified theory, where electromagnetism is associated to an SU(2)subgroup, was shown to imply the quantization of electric charge and magnetic flux [2 ],providing a plausible explanation for the fractional quantum Hall effect. Furthermore, thetheory leads to a geometrical model for the process of field quantization [3 ], implying theexistence of fermionic and bosonic operators and their rules of quantization.

Special relativity [4 ] is related to a tetradimensional Minkowski metric space withsignature (1,-1,-1,-1). A physical observer is associated to a tetrad in this space, defininga time direction and three space directions. The signature is fixed by requiring that, for aphysical observer at rest, the metric interval be the real timelike parameter of the observerworldline.

The principle of special relativity defines an equivalence among observers movinguniformly with respect to each other, a relativity with respect to velocity. If the metric isthe same for these observers, the transformations among them form the Lorentz groupwhich preserves the metric and the observer tetrads may be chosen orthonormal.

2.2. Extension of Relativity.Associated to any orthonormal space there is a Clifford geometric algebra A [5 ]. There

are inclusion mappings k of the orthonormal space into the algebra, mapping orthonormalvector bases to orthonormal sets of the algebra. The different images of a base determinea subspace of the algebra. The geometrical reason for the introduction of these algebras isto obtain geometric objects whose square is the negative of the scalar product of a vectorx with itself,

( )( ) ( ). ,x x x I g x x Ik = - = -2

. (2.2.1)

In a sense, this is a generalization of the introduction of imaginary numbers for the realline. These algebras are useful in defining square roots of operators.

For tridimensional euclidian space, the even Clifford subalgebra also has the structureof the Lie algebra of the SU(2) group, 2 to 1 homomorphic to the rotation group. SU(2)transformations by 2p and 4p are different but associated to a rotation by 2p. Further-more, it is known that a rotation by 4p is not geometrically equivalent to a rotation by 2pwhen its orientation entanglement relation with its surrounding is considered [6 ]. To pre-serve this geometric difference in a space-time subspace we must require the use of, at


least, the even geometric subalgebra for the treatment of a relativistic space-time.When the complete algebra is defined for Minkowski space-time, the observer orthonor-

mal tetrads are mapped to orthonormal sets of the algebra. Now the number of possibleorthonormal sets in the algebra is much larger than the possible orthonormal tetrads inspace-time. There are operations, within the algebra, which transform all possible orthonor-mal sets among each other. These are the inner automorphisms of the algebra. Geometri-cally this means that the algebra space contains many copies of the orthonormal Minkowskispace. A relativistic observer may be imbedded in the algebra in many equivalent ways. Itmay be said that the space-time observer is algebraically “blind”. Usually, the algebra isrestricted to its even part when the symmetry is extended from the Lorentz group,automorphisms of space-time, to the corresponding spin group SL(2,) [7 ], automor-phisms of the even subalgebra. In this manner a fixed copy of Minkowski space is chosenwithin the geometric algebra. This copy remains invariant under the spin group.

This similarity allows the extension of the principle of relativity [8 ] by taking as sym-metry, the group of correlated automorphisms of the space-time geometric algebra spinorsinstead of the group of automorphisms of space-time itself or only the automorphisms ofthe even subalgebra spinors. A relativistic observer carrying a space-time tetrad is imbed-ded in the geometric algebra space in a nonunique way, depending on some bias related tothe orientation of a tetradimensional space-time subspace of the sixteen dimensional alge-bra. We should let the geometrical structure itself, as Dirac pointed out, lead to its physi-cal meaning.

The situation is similar to the imbedding of a three dimensional observer carrying aspatial triad into tetradimensional space-time. This imbedding is not unique, dependingon the state of motion of the observer. There are many spatial three dimensional spaces inspace-time, defining hyperplanes of simultaneity which are different for observers withdifferent velocities. All possible physical observers may be transformed into each otherby the group of automorphisms of space-time, the Lorentz group.

We may conceive complete observers which are not algebraically “blind”. These ob-servers should be associated to different but equivalent orthonormal sets in the algebra.Transformations among complete observers should produce algebra automorphisms, pre-serving the algebraic structure. This is the same situation of special relativity for space-time observers and Lorentz transformations.

In particular the inner automorphisms of the algebra are of the form

a a-¢ = 1g g , (2.2.2)where g is an element of the largest subspace contained in the algebra A which constitutesa group. This action corresponds to the adjoint group acting on A.

For Minkowski space, denoted by R3,1 , the Clifford algebra R3,1 is (2), where isthe ring of pseudoquaternions [9 ] and the corresponding group is GL(2,). The adjoint ofthe center of this group, acting on the algebra, corresponds to the identity. The quotient byits normal subgroup R+ is the simple group SL(2,). Therefore, the simple groupnontrivially transforming the complete observers among each other is SL(2,). This groupis precisely the group G of correlated automorphisms of the spinor space associated to thegeometric algebra. We may associate a spinor base to a complete observer. A transforma-tion by G of a complete observer into another produces an adjoint transformation of thealgebra and, consequently, a transformation of an orthonormal set onto an equivalent setin a different Minkowski subspace of the algebra. The metric in the equivalent Minkowskispaces in the algebra A is the same. These transformations preserve the scalar products ofspace-time vectors mapped into the algebra. The subgroup of G whose Ad(G), in addition,

11Ultra Relativity

leaves invariant the original Minkowski subspace, is known as the Clifford group. TheSpin subgroup of the Clifford group is used in physics, in a standard manner, to extend therelativity principle from vectors to spinors. Since there are many copies of the spin groupL in SL(2,), in our extension we have to choose a particular copy by specifying an inclu-sion map i. Apart from choosing an element of L, a standard vector observer, we mustchoose i, thus defining a complete spinor observer

This complete observer, associated to a spinor base, carries, not only space-time infor-mation but also some other internal information related to the algebra. The group SL(2,)of transformations of these complete observers transforms the observations made by them.The observations are relative. The special relativity principle may be extended to thissituation. We shall designate the extension as ultra relativity in order to distinguish it fromspecial and general relativity.

We state the generalized principle as follows, by replacing the Spin group by SL(2,):All observers, defined by spinor bases associated to the geometric algebra of space-time,are equivalent, up to a SL(2,) transformation, for stating the physical laws of naturalsystems.

The nonuniqueness of the orthonormal set has been known in geometry for a longtime. We have given physical meaning to the orthonormal sets by associating them tophysical observers. This implies that the physically allowable transformations are thosemapping the algebra to itself by its own operations. We also have given a relativisticmeaning to these transformations.

Furthermore, we should point out that our algebra is isomorphic to the usual Diracalgebra as a vector space but not as an algebra. Both algebras correspond to space-timesof opposite signature. The requirement to use a timelike interval to parametrize the timelikeworld line of an observer determines that the appropriate algebra is not Dirac’s algebraR1,3 but instead the algebra R3,1, indicated here. The practical differences will be seen inthe next chapter.

2.3. Relativity of Interactions.In general relativity [10 ] the space-time manifold is permitted to have curvature, spe-

cial relativity is required to be valid locally and local observer frames are introduced,depending on their positions on space-time. In this manner we get fields of orthonormaltetrads on a curved manifold. The geometry of the manifold determines the motion, intro-ducing accelerations of inertial and gravitational nature.

Similarly in our case, in order to include accelerated systems, we let space-time havecurvature and introduce local complete observers which depend on their positions. Butnow, these observers are represented by general spinor frames in accordance with ultrarelativity, which is taken to be valid locally. In this manner we get fields of spinor bases(frames) which are local sections of a fiber bundle with a curved base space. The geom-etry determines the evolution of matter, but now we have, in addition to inertial and gravi-tational accelerations, other possible accelerations due to other fields of force. In otherwords we now get a geometrically unified theory whose properties must be investigated.

The relation of the tangent space TM of the base space, the space-time M, with thegeometric algebra determines a hyperbolic manifold structure on M. The algebra deter-mines the minkowskian metric and the corresponding Levi-Civita connection in M. If inthe normal neighborhood U of a point m in M we construct all the geodesics leaving m,corresponding to this connection, we may define normal coordinates. These coordinatesare constructed using the exponential map in M. The inclusion i determines, for each vec-


tor in TMm, its image in the A subspace spanned the orthonormal subset im. These algebravectors may be exponentiated to a hyperbolic subspace C of a symmetric space K, defin-ing, in turn, normal coordinates in this symmetric space. The correspondence betweenboth coordinate systems defines a homeomorphism h from the neighborhood U in M to anopen subset in the hyperbolic space K. In this manner, we may construct an atlas over Mwith local homeomorphisms,

:i ih U K¾¾ , (2.3.1)

valued in open subsets of K. When two charts overlap, the change of coordinates corre-sponds to an element g of the group G of automorphisms of A,

( ) ( ):j i i i j j i jh h h U U h U U- Ç ¾¾ Çg1 . (2.3.2)

The action of the group G preserves the lorentzian structure of K. In particular it preservesthe lorentzian inner product on the tetradimensional subspace KU of K which is the imageof the neighborhood UM. Therefore, the group G also preserves the hyperbolic distancedefined for two timelike vectors (points) x, y in K by

( ), coshH

x yd x y

x y-

æ ö÷ç ÷= ç ÷ç ÷÷çè ø1

. (2.3.3)

Since this distance provides a metric on a manifold H4, we have that G is an isometry onH4. We conclude, then, that this atlas is a hyperbolic structure [11 ] over M as defined inappendix E. We say that the manifold M is a hyperbolic manifold modeled on K.

The curvature of this geometry is a generalized curvature associated to the groupSL(2,). Since it is known that the even subgroup of SL(2,) is the Spin group related tothe Lorentz group, we look for a limit theory to get this reduction. When ultra relativisticeffects are small, we expect that we can choose bases so that the odd part is small of ordere. This is accomplished mathematically by contracting the SL(2,) group with respect toits odd subspace [12 ]. In the contracted group this odd subspace becomes an abelian sub-space. Then the SL(2,) curvature reduces to

( )W W e+= +O , (2.3.4)

where W+ is the curvature of the even subgroup SL1(2,).The result is that the curvature reduces to an SL(2,) curvature and a separate (com-

muting) U(1) curvature. It is known that an SL(2,) curvature may represent gravitation[13 ] and a U(1) curvature may represent electromagnetism [14, 15, 16 ].

If we take this U(1) as representing the standard physical electromagnetism we mustaccept that, in the full theory, electromagnetism is related to an SU(2) subgroup of SL(2,)obtained using the inner automorphisms. Similarly the SL(2,) of gravitation may be trans-formed into an equivalent subgroup by an automorphism. This ambiguity of the subgroupswithin G represents a symmetry of the interactions. Since the noncompact generators areequivalent to space-time boosts, their generated symmetry may be considered external.The internal symmetry is determined by the compact nonrotational sector.

It is well known in special relativity, that motion produces a relativity of electric andmagnetic fields. We find, since SL(2,) acts on the curvature, an intrinsic relativity of theunified fields, altering the nonunified fields which are seen by an observer. Given theorthonormal set corresponding to an observer, the SL(2,) curvature may be decomposed

13Ultra Relativity

in terms of a base generated by the set. The quadratic terms correspond to the SL(2,)curvature and its associated Riemann curvature seen by the observer. A field named gravi-tation by an observer, may appear different to another observer. These transformationsdisguise interactions into each other.

The algebra associates some generators to space-time and simultaneously to some in-teractions. This appears surprising, but on a closer look this is a natural association. In anexperiment, changes due to an interaction generator are interpreted by an observer as timeand distance which become parameters of change. Then, it is natural that a reorientation,a gyre, of space-time within the algebra corresponds to a rearrangement of interactions. Acomplete observer has the capacity to sense forces not imputable to his space-timeriemannian curvature. He senses them as nongravitational, nonriemannian, forces. Thiscapacity may be interpreted as the capacity to carry some generalized charge correspond-ing to the nongravitational interactions. Ultra relativity is essentially interpreted as anintrinsic relativity of interactions.

Of course, to complete a theory, a field equation for the generalized curvature must begiven as discussed in the first chapter. Previously, there arose the need to use a group likeSL(2,) to avoid contradictions in the unified equations of motion of a charged particle.In fact, it was in the process of pursuing the physical significance of this group that thispossible extension of relativity was found. Now we may turn things around. We have setthe geometry of the theory using relativity as foundation. Our former results are then aconsequence of applying these first principles.

Some of the features of the theory depend only on its geometry and not on a particularfield equation and may be seen directly. For example, matter must evolve as a representa-tion of SL(2,) instead of the Lorentz group. It follows that matter states are character-ized by three quantum numbers corresponding to the discrete numbers characterizing thestates of a representation of SL(2,). One of these numbers is spin, another is associatedto the electromagnetic SU(2). This gives us the opportunity to recognize the latter as theelectric charge.

Perhaps we should realize that the idea of a quantum entered Modern Physics by theexperimental determination of the discreteness of electric charge. Later atomic measure-ments were explained by quantum theory by assuming the quantum of spin, but quantumtheory was not given the burden to quantize electric charge. The possibility of obtainingthe quantum of charge as explained before, may indicate that present day quantum theoryis an incomplete theory as Dirac indicated [17 ].

As a bonus, this theory provides a third quantum number for a matter state, which maybe recognized as a quantum of magnetic flux, providing a plausible fundamental explana-tion to the fractional quantum Hall effect.

2.4. Summary.We have seen that the use of Clifford algebras allows the extension of the principle of

special relativity. When this principle is generalized to curved spaces with connections,there arises the geometry of a unified theory. Furthermore, the geometry alone implies theexistence of quanta of charge, spin and magnetic flux. The quantum implications suggestthat this geometry is the germ of quantum physics.

Of course we can always try to avoid the introduction of this idea of ultra relativity. Inorder to do that we must take one of the following positions: 1- Do not use the Cliffordalgebra at all. Pauli and Dirac matrices are too dear to physicist to take this position seri-ously; 2- Use only the even subalgebra of space-time. This actually postulates that gravi-


tation is uncoupled to the rest of physical theories and denies the possibility of unifica-tion; 3- Use the complete algebra but a) deny its geometrical relation to orthonormal spaces,or b) deny any physical meaning to this geometrical relation. In this case we would not befollowing Einstein and Dirac in their advice of obtaining physics from the geometricalstructures; 4- Use the complete algebra, but deny the particular meaning given here byproviding a different physical meaning.

References

1 G. González-Martín, Phys. Rev. D35, 1225 (1987). See chapter 3. 2 G. González-Martín, Gen. Rel. Grav. 23, 827 (1991). See chapters 7 and 11. 3 G. González-Martín, Gen. Rel. Grav. 24, 501 (1992). See chapter 6. 4 A. Einstein, Ann. Physik 17, 891 (1905). 5 Y. Porteous, Topological Geometry, (Van Nostrand Reinhold Co., London), Ch 13

(1969). 6 W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco,

p. 1148 (1973). 7 P. A. M. Dirac, Proc. R. Soc. London, 117, 610 (1928). 8 G. González-Martín, Ultrarelativiy, USB preprint, 97c (1997). 9 See appendix A10 A. Einstein, Ann. Physik, 49, 769 (1916).11 J. G. Ratcliffe, Foundations of Hyperbolic Manifolds (Springer-Verlag, New York)

(1994).12 R. Gilmore, Lie Groups, Lie Algebras and some of their Aplications (John Wiley and

Sons. New York), ch. 10 (1974).13 M. Carmelli, Ann. Phys. 71, 603 (1972).14 H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York). (1931).15 L. Infeld, B. L. van der Waerden, Sitzber. Preuss. Akad. Wiss. Physik Math. K1, 380

(1933).16 W. L. Bade and H. Jehle, Rev. Mod. Phys. 25, 714 (1953).17 P. A: M. Dirac, Directions in Physics (John Wiley & Sons, New York), p.20 (1978).

3. A UNIFIED THEORY.

3.1. Geometrical Objects of the Theory.In order to incorporate gravitation and electromagnetism as a physical geometry we

first present the geometrical objects. The structure group G of the principal bundle E overspace-time M is selected to be the group of automorphisms of the geometric algebra spinors[1 ] of TMm. This tangent space is isomorphic to R3,1 with geometric algebra R3,1 and thegroup is SL(2,). The group has a natural action as automorphisms on the spin space Vassociated with the algebra R3,1. Geometrically we construct a vector bundle VM overspace-time M, an associated bundle to E with V as fiber. The local sections of E provide uswith local bases for spinor spaces VM, lying above point m of the base space.

Dual group elements g-1 and g correspond to inequivalent representations of G. Asindicated in appendix A, to define a correlation in the spinor space, we should includeboth representations in a double dimension space S. We put both the frame and coframetogether as elements of the fiber G´G of a principal fiber bundle 2E. The objects of inter-est are the local spinor matter frames 2e which represent matter distributions over thespace-time manifold.

The relation of the frames with the group elements may be seen as follows. The pro-cess of measurement is essentially the comparison of an unknown physical system with a“standard” physical system set up with a certain degree of arbitrariness. The notion of ameasurement on the “standard” system is logically empty because it relies on a compari-son with itself. The same problem arises in geometry when choosing a base in a givenspace. Somehow a definition of standards is a necessary step. It is natural to identify thephysical systems chosen as standards with another reference frame constructed, in prin-ciple, by observers from matter of the same type. The matter frame is related to the refer-ence frame by its components which form an element of the group.

We also construct E’, the bundle of orthonormal frames on TM, with SO(3,1) as struc-ture group and the homomorphic E’’ with SL(2,) as structure group. We may includeSL(2,) in SL(2,) and consequently E’’ in E. A vector base u in E’ may be mapped to aframe in E. We can pullback an element of the dual bundle, a coframe, to define a space-time base of 1-forms in E’.

It is natural to require that matter, represented by the matter frame 2e in certain regionsof space-time, determines the interaction field represented by the connection. In this man-ner an SL(2,) connection is physically determined. This connection has an even partwhich is an SL(2,) connection over M in E’’ and, therefore, induces an SO(3,1) connec-tion over M in E’. This last connection in E’ is pseudoeuclidian, in the sense used byLichnerowicz [2 ]. Then there is a metric on M associated with this connection which maybe defined by compatibility, using the relation

g = 0 . (3.1.1)

It is convenient to summarize here the definitions of the geometrical objects that enterin the theory, with a notation differing slightly from previous use. The symbol * is the Hodgedual in *TM and LV is the dual spinor space to V. We have

1. e-1, a frame section of the principal bundle of frames E. It may be identified with


the homeomorphism e-1 : LVM LV.2. e, a coframe section of

LE. It may be identified with homeomorphisms of thecharts e : VMV.

3. 2e the matter frame, a section of 2E, corresponding to the pair e, e-1.4. w, the connection 1-form in E, of adjoint type, w : TE sl(2,)ÌR3,1 and the

corresponding 2w.5. W, the curvature 2-form, of adjoint type W : 2TM LVM Ä VM or equiva-

lently W :2TE R3,1 and the corresponding 2W.6. k, orthonormal subset of the geometric algebra R3,1, k : R 3,1 LV Ä V and the

corresponding 2k.7. i*w, the connection in TM, pulled back from the connection 1-form w,

i*w:TE’so(3,1)8. g, the metric tensor in M induced from i*w , g: T2M R.9. u, the orthonormal frame, g compatible, a section of E’, u : *TM R3,1.10.*J, current tensorial density 3-form, of adjoint type, *J :3TM LVMÄVM.

It should be noted that conjugation in the algebra does not provide an invariant prod-uct on V unless we restrict the transformations to a subgroup. This fact imposes limita-tions on the source current proposed in [3 ]. Using the geometrical meaning implied bythe previous definitions, we may expect that the simplest current source in the field equa-tions should then be of the form

J e u ei-= 1 , (3.1.2)where the symbol represents the scalar product in R3,1 and i is an orthonormal set, de-fined up to an equivalence under the group SL(2,).

The fibers of the principal bundles E, E’, have the structure of bases. Since there is aClifford mapping k, using the inclusion from E’ to E it is possible to define a form in Erelated to the soldering form [4 ] in E’.

3.2. Variational Principle.In many cases it is convenient to have a variational principle for the equations of the

theory, thus assuring compatibility of the geometric equations. By using the invariantcorrelation preserved by the group SL(2,), in the double dimension spinor space S, dis-cussed in appendix A, it is possible to introduce a 4-form l composed of two parts. Thefirst part depends on the SL(2,) connection. It is constructed from the curvature usingthe exterior product and the Cartan-Killing metric associated with the structure group,giving a canonical 4-form. The second part is given by a 4-form expressed in terms of aframe, its exterior derivative and an algebra valued 3-form, *i, representing the orthonor-mal set, up to equivalence by the group that preserves the correlation on the spinor spaceS. The expression is

( ) ( ){ }tr k e u D e D e u el W W i i* ** - -é ù= - + + ê ú

ë û2 2 2 2 1 2 2 2 1 21

4 . (3.2.1)

In terms of components, the lagrangian is

( ) ( ){ }trL g k e u emmn

mn mW W i-é ùé ù= - + ê úê úê úë ûë û

12 2 2 2 2 21 1

4 4 . (3.2.2)

17A Unified Theory

To obtain this expression we used the fact that the operation ~ corresponds to the inverseand transferred the derivative to the variable 2e. Then we used the properties of the trace tomove variables at the end of the last monomial to the front of the monomial

In the variational procedure we shall take the connection and the matter frame as thedynamical variables of the problem, determining in this way the geometric connection interms of the matter frame. The orthonormal frame u should be introduced to have welldefined maps among the spaces. Its dual coframe q may be defined from the coframe e asshown in a following chapter. Since this relation is complicated, we prefer to treat theorthonormal frame u as an independent variable, thus obtaining an additional equationfrom the variational principle. By permitting independent variations of u we keep it sepa-rate from the connection, allowing the existence of torsion as the link between these twostructures. The metric tensor is not a dynamical variable. Eq. (3.1.1) implies that theholonomy group of the induced connection in E’ leaves the metric tensor invariant [2].The metric is invariant under SO(3,1). The expression for the metric, within the orthonor-mal frames used in the theory, remains equal to the Lorentz metric.

If we vary with respect to the connection, we obtain the equation

( ) ( )g k e u ennm

m W ié ù - =ê úë û1

2 2 2 2 2 , (3.2.3)

or

( )D k e u eW i** -=2 2 2 1 2 , (3.2.4)

which implies, breaking the expression into the V subspaces of S,

eD e uk

eD e u

W iW i

* - * -

-* * -

é ù é ù é ù é ùê ú ê ú ê ú ê ú=ê ú ê ú ê ú ê ú- - ë ûë û ë û ë û

1 1

11

00 0 000 0 0

, (3.2.5)

or an equation in V and its conjugate

( )D ke u eW i** - -= 1 1 , (3.2.6)

( )D ke u eW i** - -= 1 1 , (3.2.7)

which are the proposed field equations [5 ].If we vary the frame e2 , since the operation ~ corresponds to the inverse and the

chosen group of variations is precisely the one that preserves this correlation [6] in theassociated spin spaces, the total variation of the product of e2 by e2 is zero. If we use thetechnique of Lagrange multipliers to link e2 with e2 , since the link has zero total varia-tion, the resultant equations are equivalent to those obtained by independent variations.Hence, under variations of e2 under the group, we obtain the equation

u e u em mm mi i + =2 2 2 22 0 (3.2.8)


u e u em mm mi i + =2 0 (3.2.9)


and its conjugate equation. This equation determines the motion density of the frame eand has the structure of the Dirac equation.

Alternately, instead of independent variations, we may use that e2 is the inverse of e2 .The complete variation gives

e u e e e u e2 2 2 2 2 2 2 0 , (3.2.10)

e u e2 2 2 0 , (3.2.11)

which is the conservation equation for the current density J. This equation is also obtainedgeometrically from the first eq. (3.2.4),

( ){ } ,D e u e DDk k

i W W W* *- * é ù= = =ê úë û

2 2 1 2 2 2 21 1 0 . (3.2.12)

This conservation equation leads to

e u e e u e e u em m mm m mi i i + + =2 2 2 2 2 2 2 2 2 0 , (3.2.13)

which may be written in the form

( )~

e u e e u e e u e

e u e

m m mm m m

mm

i i i

i

+ = +

2 2 2 2 2 2 2 2 212

2 2 212

. (3.2.14)

On the other hand, since the expression in the parenthesis is an element of the algebra, theoperation ~ reverses its sign and we must have eq. (3.2.9), indicating the consistency ofthe variational equations.

We recall that the lagrangian form l was constructed from the current 3-form J whichdepends on the orthonormal set k. This introduces, in the representative algebraic 3-form*i, a dependence on the orthonormal frame u, as follows,

ˆˆ ˆˆ ˆ ˆ! !

u u umn n n mn n n aa amn n n n n n a a ai e k k k e k k k= =1 2 3 1 2 3 31 2

1 2 3 1 2 3 1 2 3

1 13 3

. (3.2.15)

The complete explicit dependence of the matter part of the lagrangian on u is contained inthe term

( )ˆˆ ˆˆ ˆˆ ˆ ˆ!

u u u u umn n n aa am r rm m n n n a a ai e k k k= 1 2 3 31 2

1 2 3 1 2 3

13

. (3.2.16)

The term enclosed in parenthesis is proportional to the volume element 4-form S,

( ) ( )*ˆ ˆˆ ˆˆ ˆ ˆ ˆ! !u u u u u u u u

u u umn n n a aa a a ab b

m n n nr r r

d d d Se

d d d

*

= =1 2 3 3 31 2 1 2

1 2 34 4 . (3.2.17)

The variation of S with respect to the orthonormal form u is a 3-form related to the inte-rior product uS. We obtain, for the variation of this term,

19A Unified Theory

uu rr

dSS

d= 4 (3.2.18)

and we may write

( )ˆ ˆ ˆ

!u u umn n nm r r m r

m n n n m md i e k k k d i dæ ö÷ç= =÷ç ÷çè ø

1 2 3

1 2 3

14 43

. (3.2.19)

The dependence of the lagrangian on the metric, which is quadratic in u, determines anadditional implicit variation. The resultant variational equation,

(ˆ ˆ ˆ ˆ)tr tru k e u e u e u emn m kl m m nrn r kl r r nW W W W i ié ù é ù- = - ê ú ê úë û ë û

2 2 2 2 2 2 2 2 2 24 4 (3.2.20)


()ˆ ˆ ˆˆtr tru k e u e u e u emmn m kl m n

rn r kl r nrW W W W i i- -é ùé ù- = - ê ú ê úë û ë û1 14 4 (3.2.21)

and its conjugate equation. This expression, a function of the 2-form W and the vectordensity i, has the structure of a sum of terms of the form corresponding to the electromag-netic stress-energy tensor. A tensor with the structure of stress-energy may be definedusing this equation. It is clear that this tensor would not represent the source term of thefield equations but only the total stress energy of the connection and frame fields.

We expect that there are solutions for which any vector u of the orthonormal space-time tetrad be a sourceless vector field, at least in a finite region R, because of its physicalinterpretation as a physical reference system. Then the flux of the vector fields in u aroundany closed hypersurface S around a region R in space-time should equal zero,

R

u gdSmm

¶

- =ò 0 (3.2.22)

and

R R R

u gdS gu dV u gdV

0 (3.2.23)

for any arbitrary region R of space-time M. This implies that

aumm = 0 (3.2.24)

and the equation of motion for the frame is simplified under this condition,

u emmi = 0 . (3.2.25)

Since i is any orthonormal subset, we may specialize for example to the explicit matri-ces km, k5k0km or e-1ke obtaining equivalent equations to those indicated previously.

When we consider the external problem, that is space-time regions where there is nomatter, and consider only the even gravitational part, the equations obtained are similar tothose of Yang’s gravitational theory. Fairchild [7 ] has shown that, for Yang’s theory, [8 ]


the eq. (3.2.21) is sufficient to rule out the spurious vacuum solutions found by Pavelle [9,10 ] and Thompson [11 ]. For the internal problem where the source J is nonzero, thetheory is essentially different from Yang’s theory. Since the metric and connection remaincompatible, the base space remains pseudoriemannian with torsion avoiding the difficul-ties discussed by Fairchild and others.

Yang’s gravitational theory may be seen as a theory of a connection in the principalbundle of linear frames TMm with structure group GL(4,). In Yang’s theory the group istaken to act on the tangent spaces to M and therefore is different from the theory underdiscussion. The connection in Yang’s theory is not necessarily compatible with a metricon the base space M which leads to the known difficulties pointed out in the cited papers.

3.3. Some Algebraic relations.There is an SL(2,) action on the matter coframes of the spaces. This is really a change

in the chart homeomorphisms which is accomplished by changing a section e -1 in thebundle E by action on the group from the right or by changing the section e in the bundleLE by action from the left. These changes of bases may be understood as an SL(2,)action on the vertical spaces of the bundles VM and SM. Under this action the e’s trans-form as vectors,

ˆˆ ˆˆ

a a bb

e e= g , (3.3.1)

( ) ( ) ( )ˆ

ˆˆ ˆ

b

a b ae e- - -=1 1 1g . (3.3.2)

The rows and columns of the corresponding matrices transform as vectors under a changeof either the matter frame (active transformation) or the reference frame (passive transfor-mations. The equations are covariant under these transformations.

A similar situation occurs in the tangent spaces of M,

ˆˆˆ ˆu u Lb

a ab= (3.3.3)

and is the normal point of view taken in quantum mechanics where the ¶a, are associatedto the components of the momentum vector and its modulus is given by

ˆˆˆˆ

aba b

¶ h ¶ ¶=2 . (3.3.4)

This situation is in contrast with the point of view held in differential geometry where the¶a, form a set of four vectors (tetrad) on M.

Within our geometric approach, therefore, we shall understand the quantum mechani-cal transformations of the vector ¶a and spinor ea as active transformations of vector framesor matter coframes respectively, which is consistent with our representation of matter byframes. The theory should be covariant under an SL(2,) change of either the mattercoframe e or the reference frame.

Given a connection on the principal bundle E we have a connection form which actsinfinitesimally on the components of elements of SM from the left (and on elements ofLSM from the right), representing the interaction.

In terms of index manipulation, we may say that the connection, with a fixed referenceframe, operates on the vector component indices of the matter frames. The change of

21A Unified Theory

reference frames operates on the indices indicating the vectors of the reference frame. Theexpression for the covariant derivative in terms of the connection coefficients G of thelocal 1-form i*w relative to a reference frame,

( ) ( )ˆ ˆ ˆˆ ˆ ˆˆˆ ˆ ˆ

a a b b nm n mbm

u e u e em a m aa a a m mk k ¶ G = - , (3.3.5)

shows the indices a, a as frame indices and the indices m, n, m as component indices.Since our formalism is based entirely on spinor coframes and orthonormal vector frames,

the derivatives indicated in expressions such as eq. (3.2.9) may be anholonomic deriva-tives.

Any element of the algebra may be written in terms of its even and odd parts as

a a ak+ -= + 0 (3.3.6)

and may be represented by a matrix of twice dimensions with even components,

†

†

a aa

a a+ -

- +

é ù-ê ú ê úë û

. (3.3.7)

Using this technique we represent the various objects as follows:

†

†eh xx h


, (3.3.8)

†

†

G GG

G G+ -

- +


(3.3.9)

and, since†m m mk k k k k s= =0 0 0 , (3.3.10)

mm

m

sk

s


00

. (3.3.11)

3.4. Equations of Motion for a MatterFrame.

If we select the vector density im=km we obtain an explicit expression for the equationsof motion density of a frame e,

( ) ˆˆe e u em a n

m m n ak ¶ G k- + =12 0 , (3.4.1)

which has for even and odd parts,

( ) ˆ†ˆum a n

m m m n as ¶ x xG h G s x+ - +- - + =12 0 , (3.4.2)


† 12 0

u , (3.4.3)

that may be written,

12

u † , (3.4.4)

ˆ †ˆum a n m

m n a ms h s h s x G+ + -- - =12 . (3.4.5)

A particular solution of these equations corresponds to

†x x= , (3.4.6)

†h h= . (3.4.7)

If in addition the SL(2,) part of the connection is zero and

A iA Im m mG k+ = =5 , (3.4.8)

mm mG d- = 0 , (3.4.9)

letting,

ix x , (3.4.10)

we obtain

( )i A mmm ms ¶ x h+ = , (3.4.11)

( )i A mmm ms ¶ h x+ = , (3.4.12)

which are Dirac’s equations with the standard electromagnetic coupling if we identify theconnection with the minimal coupling eA. These expressions should be compared withthose presented in [3 ], where the electromagnetic coupling is nonstandard. In the nextsection we discuss the origin of the difference.

The 4´4 real matrices are mapped into 2´2 complex matrices. The resultant columnsmay be taken as a pair of 2 component spinors and we see that the original frame decom-poses into complex spinors

( )A Bx x x , (3.4.13)

( )A Bh h h . (3.4.14)

These equations are pairs of Dirac equations, in the standard 2 component form. Thenatural interpretation is to say that the Dirac fields may be represented geometrically by aspinor frame in the associated vector bundle. The corresponding Dirac “field equations”are the equations of motion or covariant transplantation equations for the spinor frame.

23A Unified Theory

3.5. Relation with Quantum Theory.

3.5.1. Agreement with Standard Quantum Mechanics.We shall consider first the quantum mechanics of free particles. By this we mean that

there is no explicit coupling to an external or interacting field, except possibly that selfinteraction giving rise to the mass term in Dirac’s equations. This implies, in our theory,that the connection is zero except for the mass parameter.

The equations of the theory are essentially relations between matrices which representgeneralized spinor frames. We have introduced 2-spinors matrices h, x equal to the evenand odd parts of the frame, respectively.

We have the following equations for the h, x parts:

i mmms ¶ x h= , (3.5.1)

i mmms ¶ h x= , (3.5.2)

implying that a frame for a massive particle must have odd and even parts. In our case, ifwe set the odd part equal to zero we obtain also that m is zero,

mms ¶ h = 0 . (3.5.3)

It should be clear that a wave moving along the positive z axis according to this equa-tion only admits eigenfunctions with negative eigenvalues of s3. This means that the zeromass field associated to an even frame has negative helicity. In other words, this equationis the one normally associated with a neutrino field. The nonexistence of a positive helicityneutrino is due, within the theory, to the impossibility of having a pure odd frame, withinthe theory. The geometric reason is that a principal bundle can not be defined with onlythe odd components because they do not form a subgroup.

If a field excitation corresponds to a representation of a subgroup with specific quan-tum numbers, it may be associated to only one of the spinor columns of the frame, the onewith the corresponding quantum numbers. Accordingly, we restrict the excitations or fluc-tuations of frames to matrices which have only one column in each of the two parts of theframe, the even h and the odd x.

ˆ

ˆ

hh

h

é ùê ú= ê úê úë û

11

21

0

0 , (3.5.4)

ˆ

ˆ

xx

x


11

21

0

0 . (3.5.5)

We now restrict to the even simple subgroup SL(2,), homomorphic the Lorentz group.As shown in the appendix, the h, x parts have inequivalent transformations under thisgroup,


lh h¢ = , (3.5.6)

*lx x¢ = . (3.5.7)

These columns are spinor representations of the group.We may form a four dimensional (Dirac) spinor by adjoining the two spinors, where

the components h, x are two complex 2-spinors. We may combine the 2 columns into asingle column Dirac 4-spinor,

ˆ

ˆ

ˆ

ˆ

x

xy

h

h

é ùê úê úê úê ú=ê úê úê úê úë û

11

21

11

21

. (3.5.8)

We now show that the even and odd parts of a frame are related to the left and righthanded components of the field. We calculate the left handed and right handed compo-nents, and obtain, omitting the indices,

( ) ( )L

xy g y g

h h

æ ö æ ö÷ ÷ç ç= + = + ÷ = ÷ç ç÷ ÷ç ç÷ ÷ç çè ø è ø5 51 1

2 2

01 1 , (3.5.9)

( ) ( )R

x xy g y g

h

æ ö æ ö÷ ÷ç ç= - = - ÷= ÷ç ç÷ ÷ç ç÷ ÷ç çè ø è ø5 51 1

2 21 10

. (3.5.10)

We have that the left handed component is equivalent to the h field which in turn isdefined in terms of the even part of the frame field. Similarly, we see that the right handedcomponent is equivalent to the x field and consequently to the odd part of the field. There-fore, an even frame corresponds to a left handed particle, as should be for a neutrino. ThisLorentz frame excitation has neutrino properties.

It is easy to check that the trace of the corresponding matrices gives

( )† † †try y h h x x= + . (3.5.11)

It is clear that the equations combine, with the usual definitions, to produce the standardform of the Dirac equation,

i mmmg ¶ y y= . (3.5.12)

Originally, quantum mechanics introduced the Schrödinger differential operator act-ing on scalar wave functions. Later it was realized that the wave functions have a spinorialvector structure and the Pauli and Dirac differential matrix operator were introduced. Thestandard minimal coupling, j.A, was originally a space-time vector coupling essentiallythrough the complex structure of the abelian U(1) group. As shown in eqs. (3.4.11, 3.4.12),there is essentially an agreement of our theory with this U(1) coupling. We shall comeback to this question in chs. 12 and 16.

25A Unified Theory

3.5.2. Differences in Nonabelian Coupling.The standard couplings, although valid physical assumptions, do not easily lend them-

selves to a space-time geometric interpretation and generalization. On the contrary, ourapproach is geometric from the beginning. The differences of our theory with the standardtechniques appear, not in quantum mechanics proper but in the way nonabelian couplingterms are introduced, when it is necessary to describe interactions. There are two differ-ences:

1. The use of spinor frames instead of single spinors;2. The use of the structure group SL(2,) instead of internal symmetry groups.

3.5.2.1. Spinor frames.Nonabelian couplings require a definition of a mechanism to introduce the coupling.

When a standard coupling to gravitation (space-time curvature) was introduced, the con-nection was made to act on the spinors in the same manner as the g matrices. For otherinteractions, the j.A, was generalized in gauge theories to include an internal action onmatter (use of additional internal indices). This standard coupling does not allow a fullspace-time geometric coupling to other interactions as described here. By recognizing theframe structure of e, as a set of generalized quantum mechanics wave functions, we sepa-rated the action of the connection from the action of the k matrices, essentially modifyingthe possible geometric coupling mechanisms in the theory.

To illustrate this point consider the dual formulation of our theory, in particular, theequation of motion for the dual matter frame e-1. As indicated in eq (3.3.5) the action ofthe k matrices is on frame indices while the action of the connection is on spinor compo-nent indices. The equation is

( )e em mm m mk ¶ G k- - = + =1 1 0 , (3.5.13)

where e is a frame in the principal bundle and k is an orthonormal subset of the geometricalgebra. We shall split e-1 into its even part and its odd (complementary) part inside thealgebra A. Similarly we split the connection G into its even part G+ and its odd part G- Theprevious equation then becomes

( ) ( )m mm m m m m m¶ h ¶ xk k G h G xk G h G xk k+ + - -+ + + + + =0 0 0 0

, (3.5.14)

which can be separated into even and odd parts leading to a pair of equations,

m mm mh k G xk k+ - = - 0 , (3.5.15)

m mm mxk k G hk+ - =-0 , (3.5.16)

where the covariant derivative is understood with respect to the even part of the connec-tion G+.

If we assume that G+ is zero and that

mm mG d- = 0 , (3.5.17)

the pair of equations becomes


†mmm¶ hk x k= 0 , (3.5.18)

†mmm¶ xk k h=0 . (3.5.19)

If we take as before, the components to be hermitian,

mmm¶ hk xk= 0 , (3.5.20)

mmm¶ xk k h=0 , (3.5.21)

written using complex numbers, as

mmm¶ hs x=

, (3.5.22)

mmm¶ xs h- = , (3.5.23)

we get, letting

ix x -

,

i mmm¶ hs x=

, (3.5.24)

i mmm¶ xs h= , (3.5.25)

which are equivalent Dirac’s equations. This distinction between the dual formulations isnot apparent in standard relativistic quantum mechanics.

3.5.2.2. The group of automorphisms.The automorphisms do not form an internal group but one which is associated to the

geometric Clifford algebra. There are two geometric algebras associated to flat space-timeand the choice was made, as discussed in chapter 2, by requiring the use of a timelikeinterval (rather than a spacelike interval) to parametrize the timelike world line of anobserver. It should be noted here that if the other group, SL(2,Q) where the Q is the quater-nion field, is used, eq. (3.2.9) leads to free quantum equations in 2-component form simi-lar to Dirac’s with an extra minus sign. The reason for the extra sign may be seen in thefollowing manner. First note that the assumptions made imply that the equation becomes

e memmk ¶ = - (3.5.26)

and eliminating e from the last equation and substituting back into itself we get a Klein-Gordon equation, as it should be. If we use the g matrices as the orthonormal set of R1,3,we would be repeating the calculations replacing the k by the g finally making the re-placement

e m e e m em n m nm n m nk k ¶ ¶ g g ¶ ¶= =2 2 , (3.5.27)

which is impossible to avoid because we are working with a real algebra over the k matri-

27A Unified Theory

ces (or g matrices) and therefore m must be real. We do not get Klein-Gordon’s equation.Of course we could take ig as the orthonormal set, but this would mean we are reallyworking in the same Clifford algebra R3,1 and the same SL(2,), only with a trivial changeof notation for the orthonormal set. On the contrary, in standard quantum mechanics wework with the algebra R1,3 generated by the g and, in addition, an i is introduced inDirac’s operator to fit the physics needs. The point is that the introduction of this extra imeans, geometrically, the need to use the other algebra and a different group. The ad-equate algebra and group are in terms of the ring instead of the field Q. Although theuse of either ring makes no difference in the derivation of the classical Lorentz equationsof motion, it does make a difference when deriving the quantum equations of motion fromthe geometric theory. The presence of the odd sector, fundamental in the definition ofmass, breaks the isomorphism of the algebras.

3.6. Electromagnetic Sector.The matter source current is, in this theory, the current J which is constructed from

fundamental geometrical objects. Now we shall find an expression for the electric currentdensity j which is one component of the generalized matter current J. We know that theelectric current is the source of Maxwell equations. The latter equations correspond to ourfield equations when the connection is zero except for the part corresponding to the elec-tromagnetic generator. This generator is already determined by the requirement the Lorentzequation follow from the field equations [12] . This implies that the electromagnetic gen-erator may be one of the three generators of the SU(2) group not identified with the rota-tions. We have seen that Dirac’s equation with the standard electromagnetic coupling is aparticular solution to our equation of motion, when the electromagnetic generator is alongk5. In principle the electromagnetic current may be associated to the k5 component of thegeneralized current J. Such a component arises if e is generated, for example, byexponentiating k1k2k3. Since k5 is equivalent to k0 there is an easier way to find the elec-tromagnetic current. In order to find it we shall restrict the group to a particular subgroupwhich has only one of the electromagnetic generators.

The conjugation operator on the algebra A has eigenvalues 1. The subspace corre-sponding to the negative eigenvalue, spanned by ka and k[akb], defines a subalgebra Ap .The exponentiation of this algebra gives a subgroup P of G whose elements have the form

( )exp pe a= . (3.6.1)

The inverse in this subalgebra is given by conjugation,

( )exp pe a e-= - = 1 . (3.6.2)

If the element i is just k we have for the generalized current,ˆ

ˆJ e u em a mak= . (3.6.3)

If we choose k0 as the electromagnetic generator, the only one of the three contained inthe subgroup P, we are interested in the k0 components of eq. (3.2.6),

( ) ( )tr trD Jk W k* *- = -0 01 14 4 . (3.6.4)

If there are no additional connection components, apart from k0, this equation reduces to


the Maxwell field equations,

* *D F d F k j*= = , (3.6.5)

and of course, the Bianchi identity,

DF dF= = 0 . (3.6.6)The expression for the electric current should be given by the k0 component of J. Sinceonly the odd part makes a contribution to the trace, we may write, using eq (3.6.4),

( )( )trj e e e em m mk k k+ + - -= - + 014 (3.6.7)

and recognizing the relation

e h k x= + 0 , (3.6.8)

we have

( )† †trj m m mx s x h s h= +14 . (3.6.9)

Finally since the matrices x and h are pairs of 2-component spinors, we may write

( )† † † †j m m m m mx s x x s x h s h h s h= + + +11 1 2 2 1 1 2 24 . (3.6.10)

This is the usual expression for the electric current density in 2-spinor form, for thetwo fields associated to the two spinors of a given frame. This fact may be seen using theexpression in terms of Dirac spinor y and g matrices,

j m mYg Y= , (3.6.11)

† † † †jm

m m mm

xsh x x s x h s h

hs

é ù é ùé ù ê ú ê ú= = +ê úë û ê ú ê úë ûë û

00

. (3.6.12)

If we take into consideration our result, electromagnetism may be represented by thegenerators k0, k5, k0k5 of a subalgebra of sl(4, ), isomorphic to su(2). Electromagnetismis represented by a sector corresponding to a subgroup of SL(2,), isomorphic to SU(2),rather than by a single U(1) subgroup.

In the same manner as a single parameter rotation may be represented by a linear com-bination of the three rotation generators, depending on the choice of a spatial referenceframe, a single parameter electromagnetism may be represented by a linear combinationof the three generators, k0, k5, k0k5 depending on the choice of an internal reference frame.

The quantization of electric charge may be seen as a consequence of the relation ofelectromagnetism with the SU(2) subgroup in the same manner as the quantization of spinis a consequence of rotation.

3.7. Other Interactions.It should be clear that in a truly unified theory the distinction between different types

of fields is not clear-cut, in the same manner as the distinction between electric and mag-netic fields is not clear-cut in relativity. Once a reference frame is given, it is possible to

29A Unified Theory

talk about the different fields measured by an observer associated with the given frame.The extra generators required by the theory in order to avoid contradicting the motion ofcharged particles, may be associated with other interactions, for example nuclear forces,etc. To consider this conjecture seriously we should, in first place, discuss whether thetheory contains the ingredients that may allow us to claim that it does not contradict, atfirst sight, the main features of strong and weak nuclear phenomenology.

It is known that connections in a bundle may be classified using the holonomy groupof the connection. If this group is one of the two SU(2) compact subgroups, we may saythe connection describes interactions which are, respectively, spin dependent or electriccharge dependent. If we take the 6-parameter even subgroup L, we may say that the con-nection describes the gravitational interaction. If we take the subgroup P generated ka andk[akb] we may say that the connection represents an interaction nontrivially coupling gravi-tation with electromagnetic forces. Finally, if the holonomy group is the whole group Gwe may say that the connection represents additional interactions, possibly related to nuclearforces. Since L Ì P Ì G, this scheme allows us to classify interactions geometrically inincreasing order of complexity.

This group chain has a symmetry because there is no unique way of identifying LGand LP. The coset G/L represents how many equivalent L subgroups are in G. There isalso an equivalence relation R among the noncompact generators of G, all of them equiva-lent to a boost generator or space-time external symmetry. The subspace obtained as thequotient of G/L by this relation is the internal symmetry group of LG,

/ ( )G LSU

R= 2 . (3.7.1)

Similarly the coset P/L gives, as the internal symmetry of LP, the group

/ ( )P LU

R= 1 . (3.7.2)

The total internal symmetry of the chain LPG is the product of the two groupsSU(2)U(1) which coincides with the symmetry group of the weak interactions. There isno geometrical reason to identify the structure group of the theory with the symmetrygroup because they are different geometrical concepts.

In chapters 15 and 16 we shall come back to these geometric questions in detail. Herewe limit ourselves to introductory general remarks. Strong and weak nuclear interactionswere historically introduced to account for physical phenomena not explained by electro-magnetic and/or gravitational fields. We may say that nuclear effects are residual, in thesense that they form the set of unexplained phenomena after taking account of Maxwell’selectromagnetism and Newton’s gravitation.

In a consistent electromagnetic-gravitational nonlinear theory like the one proposed, itis clear that the residual set of unexplained phenomena is different from the set mentionedabove. Then, the assumptions required for the extra interaction fields may differ fromthose of standard strong and weak nuclear interactions. Only after resolving the nonlineareffects of the proposed theory, are the residual set of unexplained phenomena and theproperties of the extra interactions defined. In fact, it may be possible, and of paramountimportance, that this residual set be the null set, in which case no additional interactionswould be needed to explain the experimental facts.

An example of a similar situation is known in the history of science. The system pro-


posed by Ptolemy to describe astronomical motion, in terms of circles, had residual unex-plained motions which required more and more circles (epicycles) to account for the ob-served motion. The recognition that the ellipse was the adequate orbit curve eliminatedthe residual effects and the large amount of parameters and circles required.

Because of these reasons it may be naive to think that the theory should completelycontain present-day theories of weak and strong nuclear interactions. We should not re-quire that the standard model [13 ] symmetry group SU(3)SU(2)U(I), be a subgroupof the structure group of the theory. Rather, we should verify the theory directly withrespect to experimental results. Furthermore there is no reason to associate a particle toeach 1-form corresponding to each generator of the group. Possible transformations amongthe generators indicate that they represent different aspects of the same geometrical ob-ject. Within this theory, this would be the same as associating separate particles to anelectric field quantum and to a magnetic field quantum.

The properties of the nuclear forces lead to the concept of strong and weak interac-tions and the works of Yukawa [14 ] and Fermi [15 ]. Both theories were modeled on theelectromagnetic interaction. Nowadays, the corresponding gauge theories are consideredeffectively to reduce, at low energies, to these theories.

Apart from the indicated symmetry, the theory under discussion has features resem-bling nuclear forces. Because of the nonlinear equations we may expect the self interac-tion to be very strong at certain regions and display saturation. Due to the spin generatorsand the electromagnetic generators, it is possible to expect that there are spin dependentforces which additionally would show the noncentral character similar to the magneticinteraction. Because of the algebraic structure, there are generators associated to properand axial vectors permitting a description of processes violating P and C invariance. Atlow energies, it may be possible to reduce the source current associated to a pair of par-ticles to a form compatible with the current-current interaction of the Fermi theory.

3.8. Summary.The group SL(2,), instead of preserving the Lorentz metric, preserves a correlation

in the spinor spaces. This correlation is expressed in a space of double dimension, formedof spinors of two kinds (as in standard Dirac theory) which transform as conjugate of eachother.

This preserved correlation allows us to present a variational formulation of the theorywhere the connection and frames play the role of dynamical variables. The given principleleads to three equations:

1. The field equation proper, a differential equation for the connection with a sourceexpressed in terms of the frames;

2. The equation of motion density of the frame, a generalized Dirac equation interms of the covariant derivative of the frame;

3. The energy momentum equation, which suggests the definition of a generalizedstress-energy tensor and serves as an additional handle to determine the orthonor-mal frame and its torsion.

For empty space the third equation obtained, when the algebra is reduced to its evenpart, is similar to an equation given by Fairchild in Yang’s theory and shown by him torule out those solutions to the vacuum equation which are not also solutions of Einstein’svacuum equations with the cosmological constant.

With the geometrical definitions given in Section 1 the equation of motion includes, asa particular case, the Dirac equation with the standard minimal electromagnetic coupling.

31A Unified Theory

References

1 I. Porteous, Topological Geometry, (Van Nostrand Reinhold, London), p. 201, ch 13(1969).

2 A. Lichnerowicz, Théorie Globale des Connexions et des Groupes d’Holonomie, (Ed.Cremonese, Roma), p. 62, 101 (1972).

3 G. González-Martín, Phys. Rev. D35, 1225 (1987).4 A. Trautman, Geometrical Aspects of Gauge Configurations, preprint IFT/4/84, War-

saw University, (1981).5 G. González-Martín, Gen. Rel. and Grav. 22, 481 (1990).6 See appendix A.7 E. Fairchild, Phys. Rev. D14, 384 (1976).8 C. N. Yang, Phys. Rev. Lett. 33, 445 (1974).9 R. Pavelle, Phys. Rev. Lett. 34, 1114 (1975).10 R. Pavelle, Phys. Rev. Lett. 37, 961 (1976).11 A. H. Thompson, Phys. Rev. Lett. 34, 507 (1975).12 See chapter 4.13 A. Zee, Unity of Forces in the Universe, (World Scientific, Singapore) Vol. 1, p. 4

(1982)14 H. Yukawa, Proc. Phys. Math. Soc. Japan 17, 48 (1935).15 E, Fermi, Z. Physik 88, 161 (1934).

In addition, the standard spinor transformation in quantum mechanics is seen as a spinorframe transformation.

The conjecture that electromagnetism may be linked to a subalgebra of the algebrasl(4,), isomorphic to su(2), with only one u(l) subalgebra seen classically and in stan-dard quantum mechanics, may offer an explanation of charge quantization. Implicationsof this idea will be discussed in later chapters.

The features expected from the generators in the complementary sector appear to becompatible with main features of nuclear interactions. The fact that nuclear effects haveresidual character allows us to waive the usual requisite of containing the standard modelsubgroup.

We conjecture that the foundations of classical euclidian geometry and mechanics aretoo restrictive and unnecessary for constructing physical theories. A more fundamentalphysical geometry of interacting matter frames, or “kosmetry” (measurement of the“kosmos”, Greek word for order, universe), does not have these restrictions and may beuseful as a foundation for physical theories. Furthermore, it appears that this approach issufficient to eliminate the introduction of a separate quantum physics. The quantum na-ture of modern theories may lie in a proper setting of the rules of this physical geometry.

Within this context, particles may be associated to linear fluctuations or excitations ofthe nonlinear frame e and the connection w, which describe matter and its interaction.

4. CLASSICAL THEORIES.

4.1. Geometric Classical Particles.In the first chapter, in our quest to obtain a unified theory of gravitation and electro-

magnetism consistent with a geometric dynamics for classical point particles, we intro-duced a principal bundle E with SL(2,) as the structure group. Since the classical theo-ries of gravitation and eletromagnetism are associated, respectively, to principal bundleswith SO(3,1) and U(1) groups, we are faced with the task of recovering these two bundlesand the associated classical theories from the bundle E. This is necessary to obtain theconsistent field equations for both theories and the correct Lorentz equation for classi-cal particles.

By an equation of motion of a singular classical point particle we mean a differentialequation determining the evolution of the tangent vector to the world line of the particle.

We could idealize a test particle by associating an orthonormal tetrad to it, the time-like vector being the 4-velocity and the three spacelike vectors related to the rotatingproperties of the particle. At two different events on the world line, the correspondingtetrads should differ by a Lorentz transformation, at most. The evolution of the tetradalong the world line is determined by a Lorentz transformation which evolves as a func-tion of the parameter on the world line

Consider the principal bundle E’ constructed by taking the Lorentz group as fiber andthe space-time manifold as base space. If we are given a curve in this bundle, we haveprecisely an element of the group evolving as a function of the curve parameter. It isclear that the given curve determines, by projection, a curve in the base space of thebundle. The projected curve may be taken as the path of a particle in space-time by im-posing certain conditions which identify the tangent to the path with the timelike vectorof the tetrad. These conditions are the differential equations of motion, expressed by theaction of the Lie algebra so(3,1) of the Lorentz group on the tangent to the path. If werepresent the evolution of the particle by the original curve in the principal bundle (andits projection) we can obtain the space-time path and additional information related tothe spacelike triad associated to each point of the path.

On the other hand, we have introduced in the previous sections a principal bundle witha larger group. Nevertheless, the integrability conditions determine equations for a curvein this bundle, if we associate a classical multipole structure to the current J and identifythe tangent to the path. If we are able to solve the equations for this curve and its projec-tion, we would have the path of the particle and additional information about the evolu-tion of internal elements representing the particle.

If we have a way to relate a curve in E to a curve in E’ we could think that the curve inE represents the evolution of a complete idealized observer. This observer carries a com-plete basis of the fiber, which allows him to measure external (base space) magnitudesand internal (fiber space) magnitudes. The curve in E’ would represent an observer carry-ing a space-time tetrad whose evolution is determined by the curve. A classical measure-ment may be interpreted as the measurement of only the projected path in space-time,which is determined by the curve in E. In other words, classically we only notice the“shadow” of the particle. We should point out that similar projections are used normally

33Classical Theories

in physics, for example, when we work with the field of complex numbers and restrict tothe field of real numbers.

It is clear that we could project the ring in the algebra sl(2,) to the complex field leading to the even subalgebra sl1(2,) from which we could pass to the Lorentz alge-bra using the known homomorphism. This should permit us to find a differential equationfor the timelike vector of the space tetrad under the group action, which is, a classicalequation for the motion of the particle [1 , 2 ].

4.2. Relation of the Bundles.The group SL(2,), indicated by G, has a subgroup determined by the even generators

of the algebra, homomorphic to U(1)SL(2,) which we shall indicate by L in this chap-ter. We want to project from the bundle E, to another bundle E’ with SO(3,1) as structuregroup. To specify this projection, it is convenient to do it in two steps. First we passfrom E to a bundle E’’ with L as structure group and then we pass from E’’ to E’.

Let us consider the first step. We define the inclusion mapping i from the ring tothe ring as the even subring and a corresponding retraction r from to ,

r i I= . (4.2.1)This determines an inclusion of the Lie algebra sl(2,) of L into the Lie algebra sl(2,)of G. Let us indicate the left coset G/L by K and write the coset decomposition,

G KL= . (4.2.2)The group G also has a principal fiber bundle structure over K, with structure group L,which we shall indicate as (G,K,L). We may establish that the image of the inclusion of Linto G is a vertical subspace of (G,K,L) at point k. We may write for an element lL

( )i l kl= º g . (4.2.3)

In this case the retraction which sends the vertical space at point kK to the group L maybe written in terms of the element k,

( )kr k l-= =1g g . (4.2.4)

There are corresponding induced inclusions, from the principal bundle E’’ to the bundleE and from the associated spinor bundle V’’M to VM. In particular we have for the tangentbundles,

:i TE TE*¢¢ , (4.2.5)

:i TV M TV*¢¢ . (4.2.6)

There is a pullback connection on TE´´ induced from the connection w on TE,

: ( , )i TE A slw* ¢¢ ¢¢ = 1 2 , (4.2.7)

( ) ( ) ( ) ( )i t i t r t tw w w w** +¢¢ ¢¢= = = , (4.2.8)

which corresponds to the even part of w, which we indicate by the + sign.Consider also the matter coframe e, a section of the dual bundle

LE. It may be iden-


tified with homeomorphisms of the charts e : VMV. As a matrix it may be identified as theset Ja of covectors forming a base in the dual space LV in 1 to 1 correspondence with theelements of G. In general we have the pullback frame,

:i V M VJ* ¢¢ ¢¢ , (4.2.9)

( ) ( ) ( )= base Li v i v r v v V Va a aJ J J J**¢¢ ¢¢ ¢¢ ¢¢= Î Î , (4.2.10)

which provides a way to define a set of covectors i*Ja in LV’’ from a given set of covec-tors Ja in V. Since these are in 1 to 1 correspondence with their respective groups wehave a relation that allows a definition of a curve in LE’’ from a curve in LE. The path ofthe frame i*Ja is obtained from the retraction of to . Clearly the retraction gives theeven subalgebra and does not correspond to simply taking the even part of an element ofthe group G. The action of i* is the first step for obtaining bundle E’.

Consider now the second step. Note that the bundle is not a direct product. There is aknown homomorphism hf between the SO(3,1) and SL1(2,) fiber matrices expressed inmatrix notation by

( ) ( ) ( )†tr tr fh G Lb b ba a ak k s s-¢¢ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢= = Î º1 1

4 2g g g g g g . (4.2.11)

Define a map on M´G’’ composed of a pair of maps, on the base and the fiber,

:h M G M G¢¢ ¢´ ´2 , (4.2.12)

{ }, fh I h=2 , (4.2.13)

( ) ( ), , ; ;h m m m M G G¢¢ ¢ ¢¢ ¢¢ ¢ ¢= Î Î Î2 g g g g . (4.2.14)

Define the mapping,

:h E E¢¢ ¢ , (4.2.15)by the relation in terms of bundle chart maps f

h hf f-¢ ¢¢= 12 . (4.2.16)

This mapping, which is the second step, preserves the bundle projection

hp p¢ ¢¢= (4.2.17)and its action on the fibers is a homomorphism,

( )( ) ( ) ( )( ) ( )( ) , fh m h m m h g mp f f p f p- - - - -¢¢ ¢ ¢¢ ¢¢ ¢ ¢¢ ¢= = Î1 1 1 1 12 . (4.2.18)

Consider now the differential h* of the bundle homomorphisms h,

( ): ( , ) ,f Ih sl so* 1 2 3 1 , (4.2.19)

( ) ( ) ( )†tr ,Ih a a a a slb b ba a as s s s* = + Î1

2 4 . (4.2.20)

These relations allow us to obtain equations for tangent forms in *TE’, from the knowl-edge of equations for tangent forms in *TE. We see that the projection m needed to go from


the bundle E to bundle E’ is the composition of the two previous maps,

h im *= . (4.2.21)

The differential of m may be written, with the chosen trivialization, as

( ) ( ) ( )( ) ( )†tr ,I a r a r a a slb b ba a am s s s s* = + Î1

2 4 , (4.2.22)

in terms of the retraction of a, according to the inclusion map.

4.3. The Classical Fields.We have separated the equations with respect to the even subalgebra or subgroup be-

cause the even part represents the classical fields. Before dealing with the Lorentz equa-tions we should take some time to develop the significance of the even field equations.The inclusion map allows the possibility of defining the pullback connection i*w and itscomposition with the homomorphisms h provides an so(3,1) valued connection which iscompatible with a metric in space-time. In other words, we have the sl1(2,) forms, asfunctions of the generators i, E,

aai A I Ew i G* = + , (4.3.1)

n n aai F I R EW i* = + . (4.3.2)

It should be noted that the even curvature does not just arise from the even part of theconnection because it depends on the product of odd parts.

The curvature form R of the G connection corresponds to the Riemann curvature withtorsion, in standard spinor formulation. They obey the equations

*D R k J* += , (4.3.3)

DR = 0 , (4.3.4)which are not Einstein’s equations but represent a spinor gravitation formulation equiva-lent to Yang’s [3] theory restricted to its SO(3,1) subgroup. All vacuum solutions ofEinstein’s equations are solutions of this equation for J=0. In particular the Schwarz-child solution is a solution to these equations and the newtonian motion under a 1/rgravitational potential is obtained as a limit of the geodesic motion under the proposedequations. It should be pointed out that the existence of a full newtonian limit is notobvious [4, 5]. Later we shall come back to this question.

The curvature F of an SU(2) connection A corresponds to the Maxwell curvature ten-sor in electromagnetism and obeys

* *D F d F k j*= = , (4.3.5)

DF dF= = 0 , (4.3.6)the standard Maxwell’s equations.

An element of the sl(2,) algebra canonically defines, by the homomorphism h ex-pressed by equations (4.2.11) and (4.2.22), an element a of the algebra so(3,1),


( ) ( )†trIh a a ab b ba a as s s s* = +1

2 . (4.3.7)

It may be verified that if we define the components of the connection

( )†trb b bma m a a mG s G s s s Gº +

12 (4.3.8)

for each of the six generators (rotations and boosts) of SO(3,1) which correspond iso-morphically to the six generators of SL(2,), the following relation is obtained,

†amb a m b b mG s G s s Gº + . (4.3.9)

Thus we can calculate the derivative of the s matrices as a section of a product bundleobtained by the product of the dual spinor bundles and the cotangent bundle,

† am n m n m n n m a mns ¶ s G s s G s G = + + - = 0 (4.3.10)

and it follows that the metric defined in TM from a metric e in spinor space,

AC X YA CXY

gmn m ne e s s=

, (4.3.11)

satisfies the compatibility condition,

g = 0 . (4.3.12)

We may define a canonical soldering form in E’ by pulling back the forms Ja. In orderto do this it is necessary, since h-1 is not a well defined mapping, to consider an inclusionh of E’ into E’’ where the fiber SO(3,1) is included in SL(2,)/Z2. In particular e corre-sponds to the inverse of the inclusion of some tetrad u which is a base in TM. We maywrite

( ) ( )( )i h u l k i h uJ-- - -= = =

11 1 1g . (4.3.13)

In this manner we have the form

:TE RQ ¢ 4 , (4.3.14)

( )u i h uh ia aQ J* *º , (4.3.15)

which satisfies the relations

( ) ( ) ( ) ( ) ( )( )( )( )

u i h u i h uu t h i u t t i h u

t u t t TM

a a b ab

ab ab

Q J J

d

* * **

-

= =

= = Î1

, (4.3.16)

( ) ( ) Y u Y Y TEQ p-* ¢= Î1 . (4.3.17)

The existence of this soldering form allows us to define torsion in the fiber bundle E’ asusual by means of


DS Q= . (4.3.18) We should note that the necessary retraction makes the orthonormal vector base u of

space-time depend, with certain degree of arbitrariness, on the spinor base e in the asso-ciated fiber bundle. In the previous chapter we obtained the equations from a variationalprinciple taking u independent of e. Before proceeding it is necessary to verify that thisdependence does not affect the variational equations.

The dual base may be expressed as a coset decomposition, eq. (4.2.3),

( )e kl u- =1 , (4.3.19)

which allows us to add to the lagrangian a term with a Lagrange multiplier and the linkexpressed by the last equation,

( )( )L e kl ul -+ -1 . (4.3.20)

When choosing the variation groups, we shall establish that a variation of e generates avariation on the fiber bundle (G,K,L), which is a Lorentz transformation on the fiber L anda translation on the base space K. The element of group L which corresponds to a varia-tion under an element of G is chosen so that the total variation of the set of three trans-formations e, l, k,

( )( ) ( ) ( ) ( )e kl u e k l u k l ud d d d- -- = - - =1 1 0 , (4.3.21)

be zero. Therefore, under these groups, the added link does not generate variations whosecombined total changes Euler equations. These equations are equivalent to those ob-tained making independent variations.

4.4. Motion of Classical Particles.We define a singular classical point particle by the multipole structure, in terms of

delta functions, of the source current J which is the dual of a 3-form valued in the Liealgebra of the structure group G of the bundle. This structure is expressed along a trajec-tory curve x’(s) in the base space with tangent vector xm by the following equation [6 , 7 ],

( )( ) ( )( )( ) ( )( )1

A A AB A ABB B A B B A B

... A ... AB! B ... A B ... n n

n n

n

nn

J x x s ds x x s ds

x x s ds

m m m a m amam a m

a a a a a aa a a a a a

t d t d

t d

¢ ¢ ¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ ¢ ¢

- ¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢

é ù¢ ¢= - - - +ê úë û

é ù¢+ -ê úë û

ò òå ò 1 2 2

1 2 1 2

1 ,(4.4.1)

where the capital Latin letters indicate sl(4,) matrix indices.The equations of motion for a classical test particle are determined by the integrability

condition of the field equations, which is the conservation law,

D J* = 0 , (4.4.2)using a method given by Tulczyjew [8 ], for the case of general relativity, to relate thedifferent algebraic multipole terms and the tangent to the path. We shall do the calcula-tion for the first two multipole terms. We decompose these two terms as follows,

A A AB B Bm mm m mt x= + , (4.4.3)


A A A A AB B B B Bmn m n m n n m mnt x x h x h x h h= + + +1 2 , (4.4.4)

where the following relations hold among the Lie algebra elements,

mmx x = 1 , (4.4.5)

A AB Bm m

mt x= , (4.4.6)

A AB B

mnm nh t x x= , (4.4.7)

A A AB B Bn mn n

mh t x x h= -1 , (4.4.8)

A A AB B Bm mn m

nh t x x h= -2 , (4.4.9)

so that m, h1, h2, are orthogonal to xm.The method to get the equations of motion is based on the following lemmas which

are still valid in the present context.Lemma 1:

( )( )

( )( )

A ... B A ...B BA ...

A ... B A ...B BA ...

ds a x x s

Dds a x x s

ds

m a b u m ab um m a b u

a b u ab ua b u

x d

d

¥¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢ ¢

-¥

¥¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢

-¥

é ù¢ - =ê úë û

é ù ¢-ê úë û

ò

ò . (4.4.10)

An expression with d has meaning after integration over its argument. Contracting witharbitrary yA

Bab and integrating, the expression can be verified.

Lemma 2: If we have

( )( )... A B ...N ... AB...NAB...N... A B ...N... k k

k k

n

k

ds x x sm m m m m mm m m m m mn d

¥¢ ¢ ¢ ¢ ¢ ¢

¢ ¢ ¢ ¢ ¢ ¢= -¥

é ù= -ê úë ûå ò 1 2 1 2

1 2 1 20

, (4.4.11)

......

AB NAB NK d x =ò 4 0 , (4.4.12)

where the expressions n are symmetric in the m indices and orthogonal to xm, then there isa necessary condition,

... AB...Nkm m mn =1 2 0 . (4.4.13)To verify this lemma we contract with arbitrary KAB... and integrate by parts,

( ) ...... ...kK mn

m n- =å 1 0 (4.4.14)

and then go over to the rest system and make use of the arbitrariness of K, assuming that


all space derivatives of order lower than k vanish.Using lemma 1, the conservation equation,

ABJ m

m* = 0 , (4.4.15)

may be transformed to obtain,

( )( ) ( )( )

( ) ( )( ) }

,

-

Dm DNS x x s m x x s

ds ds

x x s ds

mmn m

mn m

mnm n

W d d

h d

é ùæ öìé ùï ÷ï çé ù ê ú¢ ¢÷ê ú- - + - -çí ê ú ÷ë û ê úç ÷çïê ú è øë ûïî ë ûé ù¢ - =ê úë û

ò

0 , (4.4.16)

where D is the absolute derivative along the curve with respect to the SL(2,) group and

A A A AB B B BN m m m mx h h h= + +1 2 , (4.4.17)

A A[ ]B 1BS mn m nh x= . (4.4.18)

From eq (4.4.16) we obtain the desired multipole equations, using lemma 2,

( )AB

mnh = 0 , (4.4.19)

A AB B

Dm N

dsm m- = 0 , (4.4.20)

AB ,

A

B

Dm S

dsmn

mnWé ù- =ê úë û 0 . (4.4.21)

Eq. (4.4.21) determines the evolution of an sl(4,) matrix element mAB of the Lie algebra

sl(2,)of G in terms of a parameter s associated with a given curve xm in the base spacemanifold M. The object of the calculation is to obtain, precisely, the curve x in M. If thiscurve is unknown a priori, this equation is not sufficient to determine the complete evolu-tion of mA

B. Additional information is needed as indicated before. If we impose the physicalrequirement of identifying the tangent to the curve x with the timelike vector of the tetradproduced by the evolution of mA

B , then this equation expresses the evolution of the alge-braic element along the integral curve tangent to the timelike vector of the tetrad. In par-ticular, this requirement should be sufficient to obtain an equation for the evolution of thetimelike vector along its own direction. The integral curve determined in this form is thespace-time trajectory curve desired, describing the motion of the particle.

4.5. Lorentz Equations of Motion.Since E is a principal bundle, there is a natural action of the structure group G, from the

right, on the bundle E, which produces a vertical displacement along the fiber. In particulara curve in the Lie algebra sl(2,) of G induces a vertical curve on E,

( ) ( )s s mJJ J= , (4.5.1)


where we represent by a dot the absolute covariant derivative with respect to the tangentvector x. Nevertheless, we are interested in a curve on the dual bundle LE’ by left actionof the algebra at some point e’ÎLE’ which should be the path of a coframe q,

( ) ( )s a sqq q= . (4.5.2)

The retraction of the tangent vector m defined by the equation (4.4.21) gives pre-cisely the equation for the tangent vector to this curve. The relation m gives the curve inLE’ and the differential m* defines an equation for the tangent vector to the path of thecoframe q in LE’,

( ),I Ia m S mnmnm m W* *

é ù= = ê úë û , (4.5.3)

where W is the SL(2,) curvature and S is the tensor defined by eq. (4.4.18).We obtain then the desired equation for q,

( ),I S mnmnq m W q*

é ù= ê úë û , (4.5.4)

where the dot indicates the absolute derivative, now with respect to the SO(3,1) connec-tion. In other words, m* allows us to find an equation for a curve in LE’.

We are interested only in the evolution of the 4-velocity of the particle associated withthe timelike 1-form of the cotetrad q. In other words, we should make the tangent vector xto the path in space-time, correspond to the timelike form of coframe tetrad q0. We have,

ˆoD Dx xx q q= = 0 . (4.5.5)

Using eq, (4.2.22) we obtain

( ) ( )†tr r a r aa b a ab bq q s s s sé ù= +ê úë û

12

. (4.5.6)

Considering the equation for the tangent vector we get

( )†ˆ tr , ,r S r Sb mn mnmn b b mnq q W s s Wé ù é ù= +ê ú ê úë û ë û

0 12

, (4.5.7)

where the retraction of the commutator should be obtained in accordance with the inclu-sion map i.

In order to obtain the Lorentz equation of motion, the commutator [W,S] must satisfycertain requirements. For this purpose we expand the curvature W in terms of the genera-tors of the Lie algebra of G. We shall single out one generator as the one related to classi-cal electromagnetism and indicate it by E. The curvature 2 form associated to E will beidentified as the electromagnetic curvature tensor F. Additionally we express the tensorS in terms of its defining multipole terms h as indicated in the previous section. The explicitcalculation of this equation leads to various terms. We shall assume that the terms notrelated to E are small and therefore we shall keep only the electromagnetic terms of interest.We obtain

( )†ˆ ˆ tr , , + F r E r Eb n m mmn b bq q x h s s hé ù é ù= +ê ú ê úë û ë û

0 12

. (4.5.8)

The additional terms, not related to E, are generated by the sl(2,) subalgebra and repre-


sent rotational phenomena like angular momentum including spin terms.We note that E should not be a quadratic element to avoid physical interpretation prob-

lems with the quadratic generators associated to the Lorentz group.

4.5.1. Even Subalgebra Inclusion.If the image of the inclusion is the even part of the algebra A the retraction corresponds

to taking the even part of the R3,1 Clifford algebra which corresponds to sl(4,). In order toobtain the Lorentz equation we desire that

, p p aaE h d s

+é ù =ê úë û . (4.5.9)

In this case, there are only two possibilities for both E and h: they may be simple or tripleproducts of k.

Taking the first possibility,

E mk= , (4.5.10)

[ ]p p pa abg

a a b gh h k h k k k= + , (4.5.11)

obtaining,

[ ], , ,p p pa abgm m a m a b gk h h k k h k k k ké ù é ù é ù= +ê ú ê úê ú ë û ë ûë û , (4.5.12)

from which we see that it is not possible to obtain eq. (4.5.9) unless

E k= 0 , (4.5.13)

p paaqh d k= , (4.5.14)

where q is a constant.If we take the second possibility, we get essentially the same result of the last two

equations, but with the additional factor of

ek k k k

e

é ùê ú= ê ú-ë û

0 1 2 3

00

, (4.5.15)

which implies the possibility that

E k k= 0 5 . (4.5.16)

Choosing then, for example, the value given by eq. (4.5.10), it follows that

( )ˆ ˆ tr a p a pa p a pF q i q ib n m m m m

mn b bq q x d s d h s s d s d h+ +

é ù é ù= - + -ê ú ê úë û ë û0 30 321

2 , (4.5.17)

( )ˆ ˆ tr a aa aqF b n m m

mn b bq q x d s s d s s= +0 12

, (4.5.18)

( )ˆ ˆ ˆqF q Fn b m m b n

mn b b bnq x q d d d q x= - =0 00

, (4.5.19)


which is equivalent to

DqF

dsn

mnm

xx

æ ö÷ç =÷ç ÷çè ø , (4.5.20)

the desired Lorentz equation of motion in terms the four velocity x, the electromagnetictensor F and a constant q, which may be identified with a constant proportional to a chargeobtained from the electromagnetic multipole.

4.5.2. Interpretation.Since the generator k5 is equivalent to k0 and k1k2k3, under the group of automor-

phisms of the algebra, let us consider the case when this generator is taken as E. Wenotice that this even generator commutes with the complete even subalgebra and anti-commutes with the complete odd part and therefore we must have

[ ],A Ak -Ì5 . (4.5.21)

Nevertheless, under the transformations which takes the other two generators to k5,the even part of the algebra is not invariant. In fact, there is not a unique copy of theLorentz group in the algebra A. Therefore, the expression for the Lorentz force is notobtained only in terms of these even matrices but also in terms of those that would beeven for some other equivalent inclusion.

We have obtained the Lorentz equation for a singular particle with electric chargefrom the conservation law for the current. The geometric conservation of J which deter-mines the evolution of the frame is compatible with the conservation of a stress energytensor T. The latter is related to the energy equation as will be shown in the next chapter.It is known [6, 7, 8] that a multipole expansion of T leads to the equations of motion ofspinning particles. The associated momentum vector equals the timelike vector of thetetrad times the mass. The constant q in eq. (4.5.20) is the charge per unit mass.

The identification of the electromagnetic generator depends on the inclusion mapping.The complex subalgebra (2) may be included in A in different ways according to theinclusion mapping i. Now we understand the situation. Within sl1(2,), the generator k5

does not give the Lorentz equation because it commutes with all generators. Withinsl(2,), the three generators k0, k5, k0k5 generating an SU(2) subgroup, lead to the Lorentzequation and are valid electromagnetic generators. The even F does not just arise fromthe even part of A because it depends on the product of odd parts. In principle, we couldorient the electromagnetic generator in any of the three axes and obtain the same finalresult. Of course, the actual calculations to obtain this invariant result may be different.

The same situation should be valid with regard the electromagnetic coupling in Dirac’sequation, discussed in the previous chapter. If we orient the electromagnetic generatoralong k5 we obtain the usual electromagnetic coupling. If we orient the generator along k0

or k0k5 we may obtain a nonstandard coupling but final observable results should, never-theless, be the same.

4.6. Summary.The classical Maxwell field equations follow from the geometric field equation. The

classical Lorentz equations of motion also follow from the conservation of the current


source. In general, the motion of a singular particle within the geometric theory is deter-mined by the even multipolar structure, which corresponds to the known general relativ-istic terms [8] depending on the gravitational curvature and particle angular momentum,and by additional odd parts which include the Lorentz force for charged particles. Formatter in general the geometric theory requires the complete equation of motion of thematter current which corresponds to a geometric Dirac equation as indicated in the pre-vious chapter.

References

1 G. González-Martín, Phys. Rev. D35, 1215 (1987). 2 G. González-Martín, Phys. Rev. D35, 1225 (1987). 3 C. N. Yang, Phys. Rev. Lett. 33, 445 (1974). 4 G. v. Bicknell, J. Phys, A7, 1061 (1974). 5 P. Havas, Gen. Rel. Grav. 8, 631 (1977). 6 M. Mathisson, Z. Phys. 67, 270 (1931). 7 M. Mathisson, Acta Phys. Pol. 6, 163 (1937). 8 W. Tulczyjew, Acta Phys. Pol. 18, 393 (1959).

5. THE GRAVITATIONAL FIELD.

5.1. Introduction.Physical geometry determines a gravitation theory which under certain conditions reduces

to Einstein’s gravitation. The standard approach of general relativity requires the constructionof an energy momentum or stress energy tensor T determined by the classical equations ofmacroscopic matter and fields. In general these matter fields are described by the classicalfluid equations. It may be claimed that these theories are not truly geometrically unifiedtheories. Einstein [1] himself was unsatisfied by the nongeometrical character of T and spenthis later years looking for a satisfactory unified theory.

The geometric unified theory we are discussing [2, 3, 4] provides three fundamentalequations for the objects of the theory. The first or field equation is a differential equationfor the generalized connection with a current matter source. The second or equation ofmotion determines the evolution of source objects in terms of the covariant derivative. Thethird equation (3.2.15)

ˆ ˆtr u Xmn m klrn r klW W W Wé ù- =ê úë û4 (5.1.1)

has the structure of a sum of terms, each of the form corresponding to the electromagneticstress energy tensor. Using this equation we may define a tensor with the structure ofstress energy. It is clear that this tensor would not represent the source term of the fieldequations. It should represent the energy of the interaction connection and matter framefields.

In the previous chapter it has been shown that the even parts of the equations representthe classical interactions of gravitation and electromagnetism. We separate the equationswith respect to the even subalgebra or subgroup, because this part represents the classicalfields. The inclusion map allows the possibility of defining the pullback connection i*wand its composition with the homomorphisms h provides a so(3,1) valued connectionwhich is compatible with a metric g in space-time.

The even curvature form R corresponds to the Riemann curvature in standard spinorformulation. It obeys field equations different from Einstein’s which represent a spinorgravitation formulation equivalent to Yang’s [5] theory restricted to its SO(3,1) subgroup.The restriction to only the even part gives vacuum solutions.

Yang’s gravitational theory may be seen as a theory of a connection in the principalbundle of linear frames TMm with structure group GL(4,). In Yang’s theory the group istaken to act on the tangent spaces to M and therefore is different from the theory underdiscussion. The connection in Yang’s theory is not necessarily compatible with a metricon the base space M, which leads to the known difficulties pointed out in the citedpapers. Nevertheless when Yang’s theory is restricted to its SO(3,1) subgroup its metricalproblems are eliminated. The well known homomorphism between this group and oureven subgroup SL(2,) establishes a relation between these two restrictions of thetheories.

It may be claimed that the Einstein equation of the geometric unified theory is equation(5.1.1) rather than the field equation. When we consider the external problem, that is,space-time regions where there is no matter, the gravitational part of the field equations

45The Gravitational Field

for J=0 are similar to those of Yang’s gravitational theory. All vacuum solutions ofEinstein’s equation are solutions of these equations. In particular the Schwarzchild metricis a solution and, therefore, the newtonian motion under a 1/r gravitational potential isobtained as a limit of the geodesic motion under the proposed equations. Nevertheless,there are additional spurious vacuum solutions for the field equation in these theories,which are not solutions for Einstein’s theory. Fairchild [6] has shown that, for Yang’stheory, the equation (5.1.1), the restricted geometric equation, is sufficient to rule outthese spurious vacuum solutions found by Pavelle [7, 8] and Thompson [9].

Nevertheless, the interior problem [10] provides a situation where there are essentialdifferences between the unified physical geometry and Einstein’s theory. For the internalproblem where the source J is nonzero, the theory is also essentially different fromYang’s theory. Since in physical geometry the metric and the so(3,1) connection remaincompatible, the base space remains pseudoriemannian with torsion avoiding thedifficulties discussed by Fairchild and others for Yang’s theory.

The presence of a matter current term in the equation for the stress energy mayafford the possibility of getting a geometric stress energy tensor which could enter ina fully geometric Einstein equation for the metric, thus resembling Einstein’s theory.

5.2. An Equation for the Einstein Tensor.

5.2.1. The Energy Equation.The equation (3.2.21)

()ˆ ˆ ˆˆtr tru k e u e u e u emmn m kl m n

rn r kl r nrW W W W i i- -é ùé ù- = - ê ú ê úë û ë û1 14 4 (5.2.1)

defines a tensor field on M, a section in the tensor bundle over M. It is clear that thetrace in the equation introduces a scalar product, the Cartan-Killing metric Cg, in thefiber of the bundles and the result is valued in the tensor bundle. This Killing scalarproduct allows us to write the left-hand side of this equation in terms of a summationover all components along the 15 generators of a base in the Lie algebra. We shallseparate away the terms due to the even sl(2,) subalgebra of sl(4,) as follows,

tr

L a L b L a L bab

c a c b c a c bab

g g

g

n kl n klrn m rm kl rn m rm kl

n klrn m rm kl

W W W W W W W W

W W W W

é ù é ùê ú ê ú- = - +ê ú ê úë û ë û

é ùê ú-ê úë û

C

C

1 1 1g4 4 4

1g4

,(5.2.2)

where the summation over the latin indices of LW is restricted to the six componentsof the sl(2,) even subalgebra of sl(4,) and the summation over latin indices withtilde is over the nine components of the coset algebra. The latter terms, coming fromthe odd generators and the even k5 generator, correspond to the stress energy tensorcomponent due to the additional nonriemannian “coset” fields present in the theory,including the standard electromagnetic field. They all have the familiar quadraticstructure in terms of the curvature components,


c c a c b c a c bab

gn klrm rn m rm klQ W W W W

pé ù-ê úº -ê úë û

C1 1g4 4

, (5.2.3)

and define a coset field stress energy tensor cQ. Since the electromagnetic generatoris compact, the Killing metric introduces a -1 and we must define cQ as shown so thatthe standard electromagnetic energy is positive definite.

The right hand side of the main equation (5.2.1) is interpreted as a stress energytensor jQ related to the matter current source, in terms of the vector density i,

()( )

()

ˆ ˆˆ

ˆˆ tr

j ke u e u e u e

e u e u e u e

mm m lr r lr

m m lr lr

Q i ip

ai i

- -

- -

-º - =

é ù- - ê úë û

C 1 114

1 114

g4

4

. (5.2.4)

Thus, we may write equation (5.2.1) in this manner

L a L b L a L b c jab gn kl

rn m rm kl rm rmW W W W p Q p Qé ùê ú- - = -ê úë û

C 1g 4 44

. (5.2.5)

The even sl(4,) generators corresponding to the left side in the last equation maybe expressed in terms of the (2 to 1) homomorphic so(3,1) generators, 4´4 matrices Xa(Lorentz rotation generators), acting on the tangent bundle TM. For these two algebrasL and L’ the respective Cartan-Killing metrics, defined by the trace over the dimension,differ by a factor of 2. Thus we may also write

( )

( )

tr

L L a L b L a L bab

L a L b L a L bab

L a L b L a L ba b

g

g

X X g

n klrm rn m rm kl

n klrn m rm kl

n klrn m rm kl

p Q W W W W

W W W W

W W W W

¢ ¢ ¢ ¢

¢ ¢ ¢ ¢

é ùê ú= -ê úë û

é ù¢ ê ú= -

ê úë ûæ ö÷ç= - ÷ç ÷çè ø

C

C

14 g4

1g 24

1 124 4

tr .L a L b L a L ba bX X gn kl

rn m rm klW W W W¢ ¢ ¢ ¢

é ùê úê úë û

æ ö÷ç= - ÷ç ÷çè ø1 12 4

(5.2.6)

Furthermore, the curvature tensor has even sl(4,) components that arise from thequadratic product of components in the quotient sector of the connection and are clearlynonriemannian. The riemannian part of the curvature, designated by RW, is defined by themetric connection preserving the sl(4,) quadratic component and we may write in terms ofa complementary nonriemannian part nW

L n Ra a abkl bkl bklW W W¢ = + . (5.2.7 )

Consequently, there is a contribution to the even stress energy tensor corresponding to


this nonriemannian part of the curvature.The connection associated to the SO(3,1) group, responsible for RW, admits the

possibility of torsion. We can further split away the torsion S from the Levi-Civitaconnection,

a abm bm

aG S

bm

ì üï ïï ï= +í ýï ïï ïî þ(5.2.8)

and express the RW curvature in terms of the Riemann tensor Rabmn, defined by the space-

time metric, and an explicit dependence on the torsion,

( )L n nR Z R Za a a a a abkl bkl bkl bkl bkl bklW W¢ = + + º + , (5.2.9)

where nRabmn is defined as a nonriemannian curvature including the Riemann tensor and

Z a a a a g a gbkl k bl l bk gk bl gl bkS S S S S S= - + - . (5.2.10)

It should be noticed that we define the Riemann tensor R in the strict sense usedoriginally by Riemann for metric spaces and denote by nR an even curvature whichincludes a nonriemannian part nW .

Substitution in equation (5.2.6) gives an expression in terms of the Riemann tensor ofthe symmetric metric connection,

ˆ ˆˆ ˆˆ ˆˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ

L n n n n

n n

gR R R R

g gZ Z Z Z Z R Z R

rma b n a kl brm am aklbrn b

rm rma b n a kl b a b n a kl bam akl am aklbrn b brn b

p Qæ ö÷ç ÷= -ç ÷ç ÷çè ø

+ - + -

142 4

1 12 8 2 8

ˆ ˆˆ ˆˆ ˆˆ ˆ n ng

R Z R Zrma b n a kl bam aklbrn b

+ -12 8

.

(5.2.11)The first term in parenthesis in the right hand side is similar to an expression previouslydeveloped by Stephenson [11] within the Yang [5] theory of gravitation,

ˆ ˆˆ ˆˆ ˆˆ ˆH R R g R Ra b n a kl b

rm am rm aklbrn b= -

14

. (5.2.12)

We may define a stress energy tensor associated to the torsion

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ

ˆ ˆˆ ˆˆ ˆˆ ˆ

n n

t

n n

g gZ Z Z Z Z R Z R

gR Z R Z

rm rma b n a kl b a b n a kl bam akl am aklbrn b brn b

rmrma b n a kl b

am aklbrn b

Qp

æ ö÷ç ÷- + -ç ÷ç- ÷ç ÷º ç ÷ç ÷ç ÷÷ç + - ÷ç ÷çè ø

1 4 48

4 .

(5.2.13)


In this manner we may write equation (5.2.5) as

( )c t n jHrm rm rm rmp Q Q p Q- - + = -8 8 (5.2.14)

and we may consider H as the stress energy tensor of gravitation. The mattercurrent energy momentum tensor jQ is equivalent to the total sum of geometricf ie ld energy momentum contr ibut ions . The f ie ld energy momentum tensorincludes terms which are equivalent to the energy momentum associated tothe motion of the matter current (particles). I t is a question of conveniencein the classical theories to consider this energy momentum to be either inthe field or in the matter. In the geometric theory this equation is consideredas an energy momentum balance equation rather than the proper field equation.

5.2.2. The Einstein Equation.There i s an a l t e rna t e exp re s s ion fo r H ob ta ined by decompos ing the

Riemann tensor in terms of the Weyl tensor, the Ricci tensor and the scalarR, using the expression

[ ] [ ][ ] [ ]

n n n nR C R Rba ba b a b amn mn m n m nd d d= + -

123

. (5.2.15)

Again we should note that we are dealing with the so(3,1) curvature tensor rather thanthe more general gl(4,) curvature tensor also called Riemann by others. The quadraticterm in C vanishes [6],

n n n nC C g C Cb a l b a nlakl bm mk anl b- =

1 04

, (5.2.16)

and makes no contributions to H. The other contributions for the first term of nH become

n n n n n n n n nR R R R g R R g R R Cb a n am al

amn bk mk mk am mk amlk= - - + -21 1 1 23 2 6

(5.2.17)obtaining finally

nn n n n nRH R g R C Rk l

rm rm rm mlr k

æ ö- ÷ç= - -÷ç ÷çè ø1 2

3 4 . (5.2.18)

The last expression implies that it may be formally written in terms of the Einstein tensorGmn.

nn n n n nRH G g R g C Rk l

rm rm rm rm rm mlr kW Wæ ö- ÷ç= + + - -÷ç ÷çè ø

1 1 23 4 4

. (5.2.19)

Because of the nonlinearity of the equations there is a contribution of gravitationto its own source. We may split H into a G dependent part, representing gravitation,and a complementary part, representing a geometric energy-momentum tensor contri-bution,


n ng n n nR R

g g C Rk lmr rm rm rm mlr k

WQ W

p

é ùæ ö÷çê ú÷º + - +ç ÷ê úç ÷çè øë û

1 28 3 4 4 (5.2.20)

and represent equation (5.2.14) as Einstein’s equation. The different Q field termsmay be grouped together; defining a total generalized geometric field energy momentumtensor for the external fields designated by Qmn. Equation (5.2.1) may be written

( )n

g t c jRGrm rm rm rm rmp Q Q Q p Q+ + + =8 8

3 . (5.2.21)

We have a generalized Einstein equation with geometric stress energy tensors.Nevertheless, as in equation (5.2.1), the energy momentum tensor jQ of the matter currentis equivalent to the total of geometric field energy momentum contributions includingthe Einstein tensor. If nR is nonzero we may write formally this Einstein equation,

( )j c g tn n

G GTR Rrm rm rm rm rm rm rmp Q Q Q Q p Q p= - - - º º3 38 8 8 , (5.2.22)

which defines two stress energy tensors: the geometrical Q and the classical T. The geo-metric energy momentum tensor has terms which indicate the energy and motion of matterand interaction potentials in a similar way to known physical situations. Nevertheless, it ispossible that it includes some geometric terms which may be related to the so called darkmatter and dark energy. In certain phenomenological macroscopic situations it is also pos-sible that this tensor approaches only a combination of the tensors normally used in astro-physics. In any case, the fundamental difference rests in the presence of nR in place of thegravitational constant G in the equation, caused by its quadratic structure.

It should be kept in mind that, as we said before, that this equation should beconsidered an energy momentum balance equation rather than the proper field equation.The conservation of the Einstein tensor G with respect to the induced Levi-Civitaconnection in the bundle TM implies the conservation of a tensor defined by

nG

R

rmrm

r r

Qp

æ ö÷ç ÷ = =ç ÷ç ÷çè ø38 0 . (5.2.23)

There should be compatibility of the resultant equations with those obtained from theconservation of the current J, which determine equations of motion, [2, 4]. If the stressenergy tensor is decomposed in terms of a multipole expansion we find the usualequations of motion [12]: geodesic equation for a monopole, equations for a spinningparticle and other multipole equations of motion.

For the case of a pure metric gravitation theory there are zero coset fields and zerotorsion. We have then

j g mGR R 3 38 8 (5.2.24)

where we have defined the matter current energy momentum tensor mQ.In vacuum, additionally there is no matter current. Only gQ remains and we get back

the Stephenson-Yang equation for vacuum,


RH R g R C Rk l

rm rm rm mlr k

æ ö- ÷ç= - - =÷ç ÷çè ø1 2 0

3 4 . (5.2.25)

Fairchild has shown [6] that a null Hmn implies, in Yang’s theory, that the only emptyspace solutions are the Einstein spaces, ruling out the exceptional static sphericallysymmetric solutions given by Thomson [9] and Pavelle [7, 8]. A simple way to prove thishere is to decompose Rmn into its trace and traceless parts.

RR g Prm rm rm= +

4(5.2.26)

and substitute into (5.2.25)

RH P C Pk l

rm rm m r lk-

= - =2 03

. (5.2.27)

This equation may be written as an eigenvalue equation for an operator C witheigenvector P

RCP P

æ ö- ÷ç= ÷ç ÷çè ø6 . (5.2.28)

The traceless symmetric P is spanned by a 9 dimensional linear space. The set of diagonalcomponents of the 99 C matrix operator vanishes in any real coordinate system becauseof the properties of the Weyl tensor. On the other hand, in order to reproduce a nonzeroeigenvector by the action of the operator, it is necessary that there exists a real coordinatesystem where C is a nonzero diagonal matrix. Since no such system exists the onlysolution of (5.2.28) is that the eigenvector P is the zero vector. The expression in equation(5.2.26), with zero P defines the Einstein spaces. These spaces clearly satisfy equation(5.2.25) and therefore are the only possible pure gravitational vacuum solutions in thistheory. They correspond to the Einstein equation with a cosmological constant.

5.3. Equations for a Geometric InternalSchwarzschild Solution.

If we assume spherical symmetry, the metric takes the form,

d e dt e dr r dF jt W= - -2 2 2 2 2 2 . (5.3.1)Under this condition, an internal solution for Einstein’s theory may be determined

by solving the following equations:1. The time-component of the field equations G00;2. The radial component of the field equations G11;3. The conservation of the energy momentum tensor;4. The equations of state of matter.

In our theory we have essentially the same requirements.First we may calculate the necessary values of Gmn with respect to an orthonormal

frame [13]. For the time component we have,


( )( ) ˆ ˆˆ ˆ n

e de dG r e

r r r dr r dr R

j jj Q

p- -

-æ ö÷ç ÷= - - = - =ç ÷ç ÷çè ø

2 2002

2 2 200

31 1 1 1 8 , (5.3.2)

where the right-hand side is only a function of r and may be integrated,

( )( ) ( )ˆ ˆrr

n

rdr der e Gdr r

R drjjQ

p --æ ö÷ç =÷ = -- ºç ÷ç ÷çè ø òò 222 00

00

3114

22 , (5.3.3)

defining the Schwarzschild mass which is the macroscopic active gravitational massof the solution, using a gravitational constant G which we keep arbitrary. The newtonianspherical potential inside the matter distribution may be taken as

( )G r

rj º -0

, (5.3.4)

obtaining the following expression inside the matter distribution,

( )G re

rj j- = - = +2

0

21 1 2

. (5.3.5)

In second place we have for the radial component,

ˆ ˆˆ ˆ n

e e dG

r r r dr R

j j QFp

- -æ ö÷ç ÷= - + =ç ÷ç ÷çè ø

2 211

2 211

31 2 8 , (5.3.6)

which may be solved for the derivative of F,

( )

( )( )

ˆ ˆnG r rRd

dr r r G r

Qp

Fæ ö÷ç+ ÷ç ÷çè ø

=-

3 1134

2

, (5.3.7)

where we have use the previous definition of .In third place we have the conservation of the Einstein tensor, with respect to the

induced Levi-Civita connection in the bundle TM, which implies the similar conservationof the right hand side of equation (5.2.22)

nR

rm

m

Qæ ö÷ç ÷ =ç ÷ç ÷çè ø0 . (5.3.8)

Finally, instead of equations of state, we have that the expression for the geometricQ is determined by solutions of three coupled nonlinear geometric equations, includingthe one which reduces to the quantum Dirac equation. Since all the equations may bederived from a variational principle, we expect that they are consistent with each otherand together with appropriate boundary condition would determine a fully geometricinternal Schwarzschild solution. It should be noted here that this is a consequence ofhaving a geometric energy momentum as Einstein aspired for a fully geometric theory.


Alternately, we may assume that although the source is fully geometric, it should havethe phenomenological properties known for macroscopic matter, for example the knownequations of state for a fluid, and in this phenomenological approximation, reduce theproblem to the usual general relativistic macroscopic internal problem.

This internal spherical solution should be matched with the external Schwarzschildsolution at the spherical boundary. Due to the vacuum equations outside the sphericalmatter distribution, it is known that, in the exterior

g e eF j j-= = = +2 200 01 2 , (5.3.9)

which relates the external Schwarzschild metric and the spherical newtoniangravitational potential, where G is a constant related to the total mass inside thespherical boundary.

The relevant nonzero classical connection coefficients G (Christoffel symbols) maybe calculated from the metric. The Cartan connection 1-forms w which correspond one-to-one to the SO(3,1) Lorentz generators are obtained from the orthonormal tetrad.

ˆˆ ˆˆd a a bb

Q w Q+ = 0 (5.3.10)

and the so(3,1) Lorentz curvature forms are

D dW w w w w= = + . (5.3.11)The only nonzero components of the external so(3,1) Lorentz curvature are

rj

W W= = -0 20110 3322

2 , (5.3.12)

rj

W W W W= = = - = -0 0 1 30220 330 221 1132 . (5.3.13)

These equations show that the corresponding matching exterior solution satisfies the physi-cal requirements for the gravitational field of a massive spherical body.

5.4. The Newtonian Limit.It is usual to assume, as a newtonian approximation, that the characteristic parameters

of a newtonian solution of Einstein’s equation v/c1, j1 are of order e2 [14] in a smalldimensionless parameter e. This small parameter e may be related to the orthonormaltetrad u, thus characterizing the propagation of gravitational disturbances. The newtonianlimit of Einstein type theories of gravitation is discussed in detail in appendix F. There itis shown that, in the limit e0, the metric becomes singular in e. Nevertheless theconnection remains regular in the limit and defines a newtonian affine connection notrelated to a metric. Since we have taken the connection as the fundamental representationof gravitation, the gravitational limit may be defined appropriately. The correspondingnewtonian curvature tensor is the limit of the Riemann tensor. The projection of thistensor on the tridimensional time hypersurfaces t defines a Riemann tensor on them,which is not necessarily flat.

Nevertheless, assumptions on the stress energy tensor determine that thetridimensional Newton space is flat. In this case the nonvanishing components 0G00

a of


the limit connection would give the only nonvanishing components of the curvaturetensor determining Poisson’s equation.

In the geometric theory, the newtonian limit e0 should be obtained from eq. (5.2.22).The curvature scalar R may become singular in the limit due to the singularity in themetric. It is possible to make assumptions on the geometry to avoid this singularity, butit is also possible to let the geometry be determined by the stress energy tensor, makingassumptions of regularity for the latter tensor. In Einstein’s theory this singularity maybe handled by actually moving it to the stress energy tensor

R g R g R g T Tmn mnmn mn mnk k

æ ö÷ç- = - = =÷ç ÷çè ø12

, (5.4.1)

R T g Tmn mn mnkæ ö÷ç= - ÷ç ÷çè ø

12

. (5.4.2)

In the limit the relation between the scalars R and T is broken due to singularity in themetric. When the newtonian limit is taken, it is assumed that the stress energy tensorhas such properties that the possible singularity in R is avoided.

We actually may do the same to handle the possible R singularity present in theEinstein tensor G adjoining it to Q. Equivalently it is also possible to move the term fromthe left side to the right side. We also assume, as in Einstein’s theory, that the contravariantstress energy tensor remains regular in the limit. As shown in appendix F the curvaturetensor in the limit, in terms of hypersurface orthogonal timelike vector, is

( )limR T t t t t T t t t tab abmn a b m n a b m ne

k e k

æ ö÷ç= + =÷ç ÷çè ø0 2 0 0

0

1 12 2

. (5.4.3)

As in Einstein’s theory, the curved time hypersurfaces become flat in the limit becausethe field equation

( )lim lim limmn mn m n mnR R g g T g g Tab aba b abe e e

k e

æ ö÷ç= = - = =÷ç ÷çè ø0 2

0 0 0

1 02

(5.4.4 )

determines that the corresponding tridimensional Riemann tensor R0 is zero in the limit.

Similarly the other space components of the Riemann tensor R also vanish and the timevector tm becomes orthogonal to these hypersurfaces. Thus, we get the equation for theonly geometric component of the Riemann tensor in the limit,

ˆ ˆlimn

RRe

Qp

æ ö÷ç ÷= ç ÷ç ÷çè ø00

00 0

382

. (5.4.5)

In the limit, as shown in the appendix F, we obtain Poisson’s equation,

ˆ ˆlimaa nRe

Q¶ ¶ j p

æ ö÷ç ÷= ç ÷ç ÷çè ø00

0

34 , (5.4.6)

where the newtonian limit of the expression in parenthesis must be interpreted as the


newtonian mass density rG. An essential difference with Einstein’s theory is the explicitappearance of the scalar nR in this equation. As we approach the newtonian limit the tetradi-mensional curvature scalar nR may be singular of order e-2, due to the metric as notedbefore. In other words, it increases as

limn

n RR

e e- ¹20

0 (5.4.7)

in terms of a tridimensional curvature scalar of the time hypersurfaces. Nevertheless it ispossible to have solutions such that the regular value of the scalar nR in the limit dependson the curvature of a solution outside the flat newtonian limit.

The limit of the field equations determines, using the regularity of the stress energytensor, that the tridimensional curvature tensor should be of order e2, as shown byequation (5.4.4 ). There should be a relativistic physical solution, away from thenewtonian limit, where the time hypersurfaces have an isotropic curvature of order e2.This space should be a symmetric space. In chapter 12 a solution which meets theserequirements is shown. The corresponding curvature provides an undeterminedcharacteristic length l, for example,

ˆˆˆˆ

nR e ea bmn m nab

h l-= 234

. (5.4.8)

We must have a positive definite curvature parameter, l2, in order to relate it tothe gravitational constant. In the newtonian limit the vectors of the triad, whichdetermine the space metric, are of order e and we obtain

ˆˆˆˆ

n a bmn m n mnab

vR e e h

cd l e l- -= - º -

20 0 2 2 2

23 34 4

, (5.4.9)

where v is a characteristic velocity of the solution. At very small velocities this relativisticphysical solution is approximately equal to the newtonian limit solution.

In the newtonian limit the singularity e2 in nR is canceled and nR takes the value 3l-2. Thegeometric and classic densities determine a relation between the newtonian gravitationalconstant G and the constant curvature parameter l2.

lim limn

GRe e

Ql Q r

= º200

000 0

3 . (5.4.10)

A curvature of order e2 in the original time hypersurface of the relativistic physicalsolution may produce a very large value in the limit of the nR scalar and consequently berelated to a very small gravitational constant G for the newtonian limit. This scalar isgeometrically related to the curvature parameter l of a hyperbolic hypersurface which inthe newtonian limit becomes the flat newtonian space.

The density expressed in equation (5.4.6) is in agreement with the definition of thetotal mass of the spherical solution given in the previous section, equation (5.3.3). Inother words the density of matter, corresponding to the source of Poisson’s equation inNewton´s theory, determines by integration the total mass of the Schwarzschild solution.


References

1 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton), p. 98(1956).

2 G. González-Martín, Phys. Rev. D 35, 1225 (1987). See chapter 3. 3 G. González-Martín, Gen. Rel. and Grav. 22, 481 (1990). See chapter 3. 4 G. González-Martín, Gen. Rel. Grav. 23, 827 (1991). See chapter 7. 5 C. N. Yang, Phys. Rev. Lett. 33, 445 (1974). 6 E. Fairchild, Phys. Rev. D14, 384 (1976). 7 R. Pavelle, Phys. Rev. Lett. 34, 1114 (1975). 8 R. Pavelle, Phys. Rev. Lett. 37, 961 (1976). 9 A. H. Thompson, Phys. Rev. Lett. 34, 507 (1975).10 G. González-Martín, USB preprint, 01b (2001) and ArXiv gr-qc/0007066.11 G. Stephenson, Nuovo Cimento 9, 263 (1958).12 W. Tulczyjew, Acta Phys. Pol. 18, 393 (1959).13 W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco),

p. 602 (1973).14W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco),

p. 412 (1973).

5.5. Summary.The general relativity Einstein equations with cosmological constant follow from the

stress energy equation for empty space. For nonempty space we obtain a generalizedEinstein equation (5.2.22) relating the Einstein tensor Gmn to a geometric stress energytensor Qmn. For an internal solution with spherical symmetry the mass of a body may bedefined in terms of energy-mass integrals, as in Einstein’s theory. The matching exteriorsolution is in agreement with the standard relativity tests. Further, there exists thenewtonian limit where we obtain the corresponding Poisson equation in terms of ageometric energy density related, by a characteristic length l and the gravitational constantG, to the classical energy density.

6. FIELD QUANTIZATION.

6.1. Introduction.We may raise the following question: Can we obtain quantum field theory from this

geometry without recurring to classical or quantum mechanics? This question is in linewith the ideal aspiration indicated by Schrödinger [1 ]. The answer is yes.

The results of previous work, which provide a possible geometric unification of gravi-tation, electromagnetism and Dirac’s equation, give us support in the hope of obtainingmore interesting results from further analysis of the proposed geometrical theory, in par-ticular in connection with quantum phenomena. It may be shown that the sections admitan interpretation similar to generalized wavefunctions. The procedures of quantum me-chanics to determine physical results may arise canonically from the geometric propertiesof the theory.

In quantum field theory the fields are considered operators and form an algebra. Itappears possible that this algebra arises from geometrical properties of sections in theprincipal bundle, which are properties related to group elements. Here we discuss thisquestion, which is interesting since the group is associated to a Clifford algebra with ananticommuting algebraic structure.

The postulates of Newton’s classical mechanics [2 ] are based on the concept of pointparticles and the free motion along straight lines of euclidian geometry. A classical field isusually seen as a mechanical system where classical mechanics is applied. With the ad-vent of general relativity and gauge theories, there should be a recognition that the geom-etry of the physical world is not as simple as that provided by classical greek geometry.The idea of a physical geometry determined by the distribution of mass and energy [3, 4 ]is attractive as a criterion for setting the fundamental laws of nature.

It was natural when quantum mechanics was born [5, 6 ] to base its development onclassical mechanics and euclidian geometry. Postulates disclosed the difficulties with thesimultaneous measurement of position and velocity of a point particle. Another approach,with hindsight, may be the realization that a point is not an appropriate geometrical ele-ment on which to superpose physical postulates. The concept of fields is close to moderngeometrical ideas and points in the direction of establishing the physical postulates on ageneral geometry, away from prejudices introduced by classical geometry and classicalmechanics.

In consequence, we discuss a generalization of field theory, directly to geometry, by-passing the intermediate step of mechanics. Why should we introduce these intermediateconcepts which history of physics proved need revision in relativistic and quantum me-chanics? Mechanics may be seen, rather than as a fundamental theory, as a simplificationwhen the evolution of matter can be approximated as the motion of a point particle.

6.2. Linearization of Fields.In the theory of connections and frames developed in previous chapters, where the

fundamental equations are

57Field Quantization

D k JW* *= , (6.2.1)

!J e edx dx dxm a b g

abgme k* = 13

, (6.2.2)

ˆˆe u em a m

m m ak k + =2 0 , (6.2.3)

the matter fields are represented by the coframe e which is a section in a principal bundleE. The interaction is represented by a connection w with curvature form W. Both theframe and the connection are determined by the equations, in terms of an orthonormalframe u and an orthonormal set k which generates the geometric algebra [7 ] associatedto space-time.

If we introduce the bundle of connections W, a connection may be taken as a section ofthis affine bundle. When we consider an equation relating the connection and the frame,we are dealing with differentiable manifolds of sections of the fiber bundles E, W andnonlinear differential operators which define differentiable maps between these manifoldsof sections. The technique necessary for attacking this problem is known as global nonlin-ear analysis [8 ].

Generally, it is accepted that the process of quantization requires the existence of aclassical mechanical theory which is quantized by some fundamental rules. Instead, wethink of the physically dependent geometry giving rise directly to fields and in an ap-proximate way to both classical and quantum theories.

We make the conjecture that the process of field quantization is the technique of re-placing the nonlinear problem, just indicated, by a linear problem obtained by variationsof the nonlinear maps, reducing Banach differentiable manifolds of sections to Banachlinear spaces and the nonlinear differentiable maps between manifolds of sections, to lin-ear maps between Banach spaces.

From a geometrical point of view, this means working at the tangent space of thesesection manifolds at some particular “point” (section). In order to give a rigorous defini-tion to the concept of excitations we use the geometric version of the calculus of varia-tions. The variation of a section leads to the geometric concept of the “jet” of a section.This term may physically be interpreted, in a loose manner, as a variation or excitation ofconnection and frame currents.

Certain manifolds, convenient to work with bundle sections, are the jet bundles oforder k, indicated by JkE, which are, essentially, manifolds of sections equal at a pointmodulus derivatives [9] of higher order k+1. Of interest is the manifold of solutions g,which is the submanifold of all the sections that satisfy a given nonlinear differentialequation. Sections obtained from a solution by the action of the structure group are equiva-lent solutions. The quotient of g by this equivalence relation are the physical solutions.

If we have a variational problem, its critical sections (solutions) may be characterizedgeometrically as follows: A section is critical if and only if the Euler-Lagrange form iszero on the 1-jet prolongation [9] js of the section s,

( )js

D fL h+ = 0 . (6.2.4)

The set of all the critical sections forms the differentiable manifold g of solutions ofthe variational problem. On this manifold there are vector fields whose flow generates thespace of solutions. These fields are sections of the tangent space Tg. Instead of studying g


it is possible to study the linear spaces Tgs. We may introduce in Tg a vector y repre-senting a solution of the associated varied equations. A “quantum operator” field may berelated to a Jacobi field on the bundles in consideration.

Really a solution to the equations corresponds to a mapping between two such mani-folds of solutions: one gE, corresponding to the bundle E, representing a matter framesolution and the other gW corresponding to the bundle W, representing the interaction con-nection solution. A single manifold of solutions may be obtained by combining gE and gW,within a product manifold related to E and W, but this approach leads to unnecessarymathematical complications for our purpose. Here we shall consider both manifolds sepa-rately.

6.3. Frame Solutions.For the manifold of frame solutions gE we shall consider its tangent Tg. We define a

Jacobi vector field as a section Vs, of s*TvE, induced by the section s from the verticalsubbundle of TE, such that the corresponding 1-jet prolongation satisfies

( )jV D fL h+ = 0 . (6.3.1)

Here indicates the Lie derivative. The space of all the Jacobi vector fields formsthe tangent space of g at a given section, denoted by Tgs. The last equation is a lineariza-tion of the nonlinear field eq. (6.2.4) which is eq. (6.2.1) in another form. A Jacobi fieldmay be seen as the tangent vector to a differentiable curve of solutions st of the nonlinearequations at a given solution s. When we have such a curve, we may consider it to be theintegral curve of an extended Jacobi vector field V which takes the values Vs, at eachpoint of the curve.

The fiber of the bundle E is the group SL(2,). The Lie algebra A of this group isenveloped by a geometric Clifford algebra with the natural product of matrices of theorthonormal subset of the Clifford algebra. The elements of A may be expressed as fourthdegree polynomials in terms of the orthonormal subset of the Clifford algebra.

When variations are taken, as usual, in terms of the fiber coordinates, the algebraicproperties of the fiber are not brought to full use in the theory. In order to extract theadditional information contained in the fiber which, apart from being a manifold, is re-lated to algebra A, we note that in many cases we have to work with the fiber representa-tive by the bundle homeomorphisms. We are dealing then with elements of G.

The vertical spaces of s*TvE are homeomorphic to the vertical spaces of TG. Thismeans that the fiber of s*TvE is A, taken as a vector space. The association of the algebraicstructure of A to the vertical spaces of s*TvE depends on the image of the section s in Ggiven by the principal bundle homeomorphisms.

When an observer frame is chosen, the local homeomorphism is fixed and the alge-braic structure of A may be assigned to the vertical spaces. A variation of the observer hproduces an effect equivalent to a variation of the section s, and should be taken intoconsideration. The total physical variation is due to a variation of the section and/or avariation of the observer in the composition mapping hos , defined on a local chart U

( ):h s U U G G¢ ´ ´ . (6.3.2)

A variation of hos is represented by a variation vector, a generalized Jacobi vector, valuedin the Clifford algebra.

We are working with a double algebraic structure, one related to the algebra A, of the


fiber of TE and the other to the Lie algebra of vector fields on the jet bundle JE. Thisallows us to assign a canonical algebraic structure to the Jacobi fields, which should bedefined in terms of these natural structures in each of the related spaces.

For instance, if we use both algebraic structures and calculate the following Lie bracketof vectors,

,x ym nk ¶ k ¶é ùê úë û0 1 , (6.3.3)

where m, v are coordinates on the jet bundles and x, y the components, the result is not avector because of the anticommuting properties of the orthonormal set. Nevertheless, ifwe calculate the anticommutator, the result is a vector.

In particular, it is also known [10 ] that the Clifford algebra, taken as a graded vectorspace, is isomorphic to the exterior algebra of the associated tangent space, which in ourcase is *TM. There are two canonical products, called the exterior and the interior prod-ucts, which may be defined between any two monomials in the algebra using the Cliffordproduct. For any element ka of the orthonormal set the product of a monomial a of degreep by ka gives a mapping

: p p pA A Aak + - +1 1 . (6.3.4)

The first component of the map is the exterior product ka a and the second component isthe interior product kaa. Then we may write

a a aa a ak k k= + . (6.3.5)

Due to the associative property of the Clifford product we may extend this decomposi-tion to products of monomials. We may define the grade of the product ab, indicated bygr(ab), as the number of interior products in the Clifford product. The grade is equal tothe number of common elements in the monomials.

For example, (k0k1) (k2k0k3) is of grade one. This decomposition may be applied to theproduct of any two elements of A. It is clear that the maximum grade is the number oforthonormal elements and that the product of grade zero is the exterior product.

The use of the exterior product instead of the Clifford product turns the algebra A intoa Grassmann algebra isomorphic to the exterior algebra of differential forms on the tan-gent spaces. This fact leads us to look for gradation of the Lie algebra structure [11 ].

The structure group of the theory was chosen as the simple group of inner automorphismsof the Clifford algebra. In this sense, the group, its Lie algebra and their respective prod-ucts arise from the Clifford algebra. In fact the Lie bracket is equal to the commutator ofClifford products. To avoid an unnecessary factor of 2 we may define a Lie product as theantisymmetrized product (1/2 of the commutator).

We may consider that the elements of A also satisfy the postulates to form a ring, in thesame manner as the complex numbers may be considered as an algebra over or as afield. In this manner, the elements of A play the role of generalized numbers, the Cliffordnumbers. The candidates for the ring product are naturally the geometric Clifford product,the exterior Grassmann product, the interior product and the Lie product. As mentioned,all these products arise from the Clifford product. In fact, for any two monomials a, b Î A,all the other products a⋅b are either zero or equal to the Clifford product ab. The geomet-ric Clifford product is more general and fundamental, and the others may be obtained asrestrictions of the Clifford product. Besides, the Lie product is not associative and mayneed generalizations of the algebraic structures. On the other hand, the fiber bundle of


frames, E, is a principal bundle and its vertical tangent bundle TE has for fiber a Liealgebra structure inherited from the group, it would be more natural that the chosenproduct be closed in the algebra so that the result be also valued in the Lie algebra. Forthe moment, we take the more general geometric Clifford product as the product of thering structure of A and we shall specialize to other products when needed.

It is convenient, then, to work with the universal enveloping associative algebra of theLie algebra of vector fields. In this manner we have an associative product defined andwe may represent the brackets as commutators of the elements. The vertical vectors on JEmay be taken as a module over the ring A. Using the canonical algebraic structure we maydefine a bracket operation on the elements of the product A´. For any monomial a,b,g Î Aand V,W,Z Î , define

{ } [ ], ,V W V W Za b a b g= · ´ = , (6.3.6)

turning the module into an algebra . This bracket satisfies

{ } ( ) { }, ,V W W Vaba b b a= - -1 , (6.3.7)

( ) { }{ } ( ) { }{ }( ) { }{ }

, , , ,

, ,

V W Z W Z V

Z V W

ga ab

bg

a b g b g a

g a b

- + - +

- =

1 1

1 0 , (6.3.8)

where |ab| is the gradation of the product ab or the bracket, equal to the number of permu-tations given by

( )grab a b ab= - , (6.3.9)

in terms of the degree |a| of the monomials in A and the grade of the product ab. For anytwo elements y, F of the algebra which are polynomials in A, the bracket is defined asthe sum of the brackets of the monomial components,

{ } ( ) ( ){ },

, ,n m

n m

f y f y= å . (6.3.10)

Another way of looking at this bracket operation is to consider that the action of avector y on a scalar field f on M gives a section a of the bundle AM with A as the fiber,

( ) aaf E f aa

aY y ¶= = . (6.3.11)

With this understanding, the bracket of vector monomials in A may be defined by its ac-tion on functions, as follows:

( ) ( ){ } ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ),n m n m m nf f fFYF Y F Y Y F= + -1 . (6.3.12)

This definition agrees with the previous one, eq. (6.3.6). In order to see this, we decom-pose the vectors in terms of a base and apply the first definition.

The previous formulas are valid in general if we restrict the Clifford product to theexterior or the interior products taking the proper number of permutations for each case.For example in the case of Grassmann product, gr(ab) is always zero. If we restrict to theLie product, the bracket is always symmetric in eq. (6.3.6) and in the graded Jacobi iden-


tity, eq. (6.3.8).

6.4. Connection Solutions.For the manifold of connection solutions gW, we may proceed similarly, taking in

consideration that the bundle of connections W has a different geometrical structurethan the bundle of frames E.

A connection on the principal bundle may be defined by an splitting of the short exactsequence of vector bundles [12 ],

GAdE T E TMp ¾¾ 0 0 , (6.4.1)

where the splitting

: GTM T E H Vw = Å (6.4.2)

is a homomorphism defining the horizontal subspaces of the principal bundle. These hori-zontal subspaces define a connection form w.

A connection on E may be identified with a section of the bundle of connections p:WMdefined in appendix E. It may be seen that each point wm, of the vertical space Wm of thefiber bundle W corresponds to a vector space complement of AdE in TGE. It is known [7]that the space of linear complements of a vector subspace in a vector space has a naturalaffine structure. Therefore the fiber of the bundle W is an affine space with linear partL(TGE / AdE, AdE) » L(TM, AdE).

We define Jacobi vector fields Vs, as sections of s*TvW and associate quantum opera-tors to the prolongation of the extended Jacobi vectors. A difference arises, in this case,because the algebraic structure of the fiber of W, which is not a principal bundle, differsfrom that of the fiber of E. A connection in E is defined by giving a section in W. Thisdefines horizontal vector subspaces in TE. The fiber of the manifold of connections W isthe space of complements of the vertical space in TGEe. The fiber of W is an affine spaceand we say that W is an affine bundle.

Intuitively, we may consider an affine space as a plane P of n dimensions without adefined origin. The affine structure of the space allows that any point o Î P be defined asorigin, turning the space into a vector space V,

:P VQ 0 . (6.4.3)

The algebraic structure of the tangent space to the fiber of W is then isomorphic to TPp andconsequently isomorphic to V with the usual operation of addition of vectors. Correspond-ingly the vertical vectors on JW form a vector space over a commutative ring and thebracket defined in the previous section reduces to the Lie bracket, since no sign permuta-tion arises under commutation.

6.5. Bracket as Derivation.In both cases we have that the bracket is defined using the natural geometric product

related to the algebraic structure of the fiber of the corresponding bundles E and W.It is clear that the bracket is a derivation,

{ } { } ( ) { }, , ,XFX FY X F Y F X Y= + -1 , (6.5.1)


and we have, therefore, a generalized Lie derivative with respect to the prolongation ofJacobi fields given by

{ },jV jW jV jW= , (6.5.2)

where the bracket is the anticommutator or the commutator depending on the num-ber of permutations for frames and is always the commutator for connections. Physi-cally this means that matter fields should be fermionic and that interaction fieldsshould be bosonic. The quantum operators in quantum field theory may be identi-fied with the prolongation of the extended Jacobi vector fields.

The meaning of these brackets, eq. (6.5.2), is equivalent to the postulate of quan-tum theory that gives the change in some quantum operator field F produced by thetransformation generated by some other operator Y. In the present context, this equa-tion is not a separate postulate, but rather it is just the result of taking the deriva-tive with respect to a direction tangent to a curve in the manifold of solutions g and itis due to the geometry of the bundles.

The derivative may be generalized to tensorial forms valued over the ring A.These derivatives represent the variation of sections along some direction in Tg, whichcorresponds to a generator of some transformation on the jet bundle along a verticaldirection.

The complete geometrical formulation of this physical variational problem maybe carried by constructing the extension of the vertical vector space TvEe to a moduleover the ring A. In a manner similar to the complexification of a real vector space weconsider the dual (TvEe)

* and define mappings

( ): veV T E A

* . (6.5.3)

These mappings are elements of a right A-module which we designate by ATvEe. If weform the union of these spaces over the manifold E we get a fiber bundle ATvEe overE.

Further, we may construct the bundles ATvJEe over JE and s*TvE over M. The stan-dard geometric version of variations may be generalized to these physical variationsby substituting appropriately these bundles for the bundles TvE, TvJ and s*TvE re-spectively, using the derivative and keeping track of the noncommutative prod-ucts. [9]

6.6. Geometric Theory of Quantum Fields.The operator bracket leads to the quantization relations of quantum field theory using

Schwinger’s action principle. The generator of a variation F may be written in the jetbundle formalism by the appropriate term in eq. (2.9), appendix D,

F jV Pµ . (6.6.1)

It may be seen that the elements entering in the expression are tensorial forms on the jetbundle which inherit the algebraic properties of the fiber and therefore have the propertiesof operators.

The generating function F determines the variations as indicated in eq. (6.5.2). Weshall write the generator F in the equivalent standard expression [13 ] used in quantumtheory obtaining


( ) { } ( ), ,x F x dmm

s

dY Y y P dY sì üï ïï ï= = í ýï ïï ïî þ

ò , (6.6.2)

for the field operator Y variations. In this expression we recognize that the elements arequantum operators, as defined above, which obey the defined bracket operation. If weexpress the bracket in this relation as in eq. (6.5.1), we get

( ) ( ) ( ){ } ( ) ( ) ( ) ( ){ }, ,x x y y d y x y dm mm m

s s

dY Y P dY s P Y dY s= ò ò . (6.6.3)

It is clear that the resultant commutators for the connection operators and anticommutatorsor commutators for the frame operators are none other than the quantization relations inquantum field theory for boson fields and fermion fields.

The last equation implies that

( ) ( ){ },x yY dY = 0 , (6.6.4)

( ) ( ){ } ( ), ,x y x ym mY P d= - , (6.6.5)

where the bracket is interpreted as the commutator or anticommutator depending on thegradation of the bracket according to the type of field.

From this formulation, the geometrical reasons for the existence of both bosonic andfermionic fields also become clear. Another advantage of this geometric formulation isthat the operator nature of the fields may be explicitly given in terms of the tangent vectoroperators on the space of solutions and the Clifford algebra matrices.

A tangent vector may be considered as an infinitesimal action on functions on a mani-fold. In particular for g, the manifold of sections which are solutions, the quantum op-erator may be considered to act on a solution section, called the substratum or back-ground section, producing solution perturbations, called quantum excitations. The sub-stratum or background section may take the place of the vacuum in conventional quantumfield theory.

If the manifold of sections is taken as a Hilbert manifold its tangent spaces are Hilbertlinear spaces. This means that the quantum operators, which act on tangent spaces, wouldact on Hilbert spaces as usually assumed in quantum theory.

If the Hilbert manifolds in question admit local harmonic sections we may introducefundamental harmonic excitations. Any arbitrary excitation may be linearly decomposedin terms of these harmonic excitations. The eqs. (6.6.4, 6.6.5) determine commutation rulesfor the harmonic excitation amplitudes which correspond to the creation, annihilation andparticle number operators of harmonic field excitations at each point, with the same prop-erties of those of the quantum harmonic oscillator. The energy of the harmonic field exci-tations may be defined in the usual manner leading to the concept of a quantum particle.

Nevertheless, if this geometric model is taken seriously, field (second) quantizationreduces to a technique for calculating field excitations or perturbations to exact solutionsof theoretical geometrical equations. There may be other techniques for calculating theseperturbations. In fact, it is known that quantum results may be obtained without quantiza-tion of the electromagnetic field. See, for example, an alternate technique to QED [14 ]where the self-field is taken as fundamental.

In general relativity, the self-field reaction terms do not appear as separate features in


the nonlinear equations but they appear in the linearized equations obtained using a per-turbative technique. Once the geometrical nonphenomenological structure of the sourceterm J is known, the exact equations of motion for the fields describing matter, includingparticles, would be the conservation law for J. This relation, being an integrability condi-tion on the field equations, includes all self-reaction terms of the matter on itself. Thereshould be no worries about infinities produced by-self reaction terms. A physical sys-tem would be represented by matter fields and interaction fields which are solutions tothe set of simultaneous equations. When a perturbation is performed on the equations,for example to obtain linearity of the equations, the splitting of the equations into equa-tions of different order bring in the concepts of field produced by the source, forceproduced by the field and therefore self-reaction terms. We should not look at self-reaction as a fundamental feature but, rather, as an indication of the need to use nonlinearequations.

6.7. Summary.It was shown [15] that if we take into consideration the geometrical structure of the

fiber bundles E or W, related to the algebraic structure of their fibers, a process of varia-tions of the equations of the theory leads to an interpretation of the extended Jacobi fieldsas quantum operators. It is possible to define a bracket operation which becomes the com-mutator for the Jacobi fields associated to the connection and becomes the anticommutatoror commutator, according to the gradation, for those associated to the frame. This bracketoperation leads the quantization relations of quantum field theory for bosonic interactionfield and fermionic matter fields. The process of field quantization is the technique ofreplacing the nonlinear geometric problem by a linear problem obtained by variations ofthe nonlinear maps.

References

1 E. Schrödinger, Space-time Structure, 1st ed. (University Press, Cambridge), p. 1 (1963). 2 I. Newton, in Sir IsaacNewton’s Mathematical Principles of Natural Phylosophy and

his System of the World, edited by F. Cajori (Univ. of California Press, Berkeley andLos Angeles) (1934).

3 A. Einstein, M. Grossmann, Zeit. Math. Phys. 62, 225 (1913). 4 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton), p.

55, 98, 133, 166 (1956). 5 E. Schrödinger, Ann. Physik 79, 489 (1926). 6 W. Heisenberg, Z. Physik, 33, 879 (1925). 7 I. Porteous, Topological Geometry, (Van Nostrand Reinhold, London), ch 13 (1969). 8 R: S. Palais, Foundations of Global Non linear Analysis, (W. A. Benjamin, New York)

(1968). 9 See appendix D10 M. Atiyah, Topology Seminar Notes, (Harvard University, Cambridge) p. 23 (1962).11 R. Hermann, preprint HUTP-77/AO12 (1977).12 P. L. Garcia, Rep. on Math. Phys. 13, 337 (1978).13 J. Schwinger, Phys. Rev. 82, 914 (1951).14 A. O. Barut, Phys. Rev. 133B, 839 (1964).15 G. González-Martín, Gen. Rel. Grav. 24, 501 (1992).

7. QUANTIZED CHARGE AND FLUX.

7.1. Introduction.The geometric excitations (particles) must carry quanta of angular momentum, electric charge

and magnetic flux. This is a consequence of the theory because the necessary association ofelectromagnetism to an SU(2) subgroup of the structure group of the theory.

The geometry of the theory, in terms of a connection describing the interaction and a framedescribing matter, is determined by the field equation which relates the generalized curvature tothe matter distribution. The integrability condition leads to a geometric equation which is ageneralized Dirac equation coupled to an electromagnetic potential. Some aspects of the theorymay be discussed by group techniques without necessarily solving the field equations.

The structure group of the theory, SL(2,), the group of spinor automorphisms of theuniversal Clifford algebra of the tangent space at a space-time point, acts on associated spinorbundles and has generators that may represent other interactions, apart from gravitation andelectromagnetism.

In order to give a complete picture, the frame fields should represent matter and, conse-quently, the frame excitations should represent particles. The connection acts on the frameexcitations which may be considered linear representations of the group. Within this context itis of interest to consider the irreducible representations of this group and to discuss its physi-cal interpretation and predictions [1 ]. Representations of related groups have been discussedpreviously [2 ].

7.2. Induced Representations of the Struc-ture Group G.

The irreducible representations of a group G may be induced from those of a subgroupH. These representations act on the sections [3 ] of a homogeneous vector bundle over thecoset space M = G/H with fiber the carrier space U of the representations of H.

It is then convenient to consider the subgroups contained in SL(2,), in particular themaximal compact subgroup. The higher dimensional simple subgroups are as follows:

1. The 10-dimensional group P, generated by ka, k[akb]. This group is isomorphicto subgroups generated by k[akbkg], k[akb] and by k[akbkc], k[akb], k5;

2. The 6-dimensional group L, corresponding to the even generators of the alge-bra, k[akb]. This group is isomorphic to the subgroups generated by ka, k[akb] andby k0k[akb], k[akb];

3. There are two compact subgroups, generated by k[akb] and by k0, k5, k1k2k3.The P subgroup is, in fact, Sp(2,), homomorphic to Sp(4,), as may be verified by

explicitly showing that the generators satisfy the simplectic requirements [4 ]. This groupis known to be homomorphic to SO(3,2), a De Sitter group. The L subgroup is isomorphicto SL(2,). The compact subgroups are both isomorphic to SU(2) and therefore the maxi-mal compact subgroup of the covering group G is SU(2)ÄSU(2) homomorphic to SO(4).As shown in section 3.7, the LPG chain internal symmetry is SU(2)U(1) and coin-cides with the symmetry of the weak interactions.


Irreducible representations of the covering group of SL(2,) may be induced fromits maximal compact subgroup. The representations are characterized by the quantumnumbers associated to irreducible representations of both SU(2) subgroups. These rep-resentations may be considered sections of an homogeneous vector bundle over the cosetSL(4, ) /(SU(2)ÄSU(2)) with fiber the carrier space of representations of SU(2)ÄSU(2).

One of the SU(2), acting on spinor space, is associated to the rotation group, acting onvector space. Its irreducible representations are characterized by a quantum number l ofthe associated Casimir operator L2, representing total angular momentum squared. Theother SU(2), as indicated in previous chapters, may be associated to electromagnetism. Ithas a Casimir operator C2, similar to L2 but representing generalized total charge, withsome quantum number c. The irreducible representations of SL(2,) have a third labelwhich may be associated to one of the Casimir operators of SL(2,), for example, thequadratic operator M2 on the symmetric space SL(4,)/SO(4).

The states of these irreducible representations of SL(2,) should be characterized byintegers m, q corresponding to physical particles with z component of angular momentumm/2 and charge qe. These quantum numbers are studied in detail in sections 4 and 5. Theprice paid to obtain this charge quantization is only the association of electromagnetism tothe second SU(2) in SL(2,). In fact, this may not be a disadvantage in a unified theory ofthis type, where it may lead to new geometric representations of physical phenomena. Itshould be noted that Dirac’s charge quantization scheme [5 ] requires the existence ofmagnetic monopoles. This requirement is not part of this theory.

Representations of SL(2,) may also be induced from L, naturally including the spinrepresentations of its SU(2) subgroup. Since L is a subgroup of P, it is more interesting tolook at the representations induced from P which include, in particular, those of L. Insome situations the holonomy group of the connection may be P= Sp(2,) and we mayexpect that representations induced from it should play an interesting role. It should beclear that the coset P/L is the De Sitter space S- [6 ]. The points of S- may be seen astranslation operators on the space S- itself, in the same manner as the Minkowski space.The Laplace-Beltrami operators on S- have eigenvalues that should correspond to the con-cept of mass in this curved space. The isotropy subgroup at a translation (point) in S- is therotation subgroup SO(3) for translations along lines inside the null 3-hypercone and theSO(2,1) subgroup for translations along lines in the null 3-hypercone itself. The SL(2,)representations induced from the De Sitter group P are characterized by mass and spin orhelicity and expressed as sections (functions) of a bundle over the mass shell 3-hyperbo-loid or the null 3-hypercone, respectively for massive or null mass representations as thecase may be. These representations are solutions of Dirac’s equation in this curved spaceand geometrically correspond to sections of bundles over subspaces of the De Sitter spaceS-.

The group SO(3,2), homomorphic to Sp(2,), is known [4] to contract to ISO(3,1)which is the Poincaré group and we may have approximate representations of SL(2,)related to the Poincaré group. Then, the representations of the direct product of Wigner’slittle group [7 ] by the translations are introduced approximately in the theory. The SO(2,1)isotropy subgroup of P contracts to the ISO(2) subgroup of ISO(3,1). Thus, we may workwith representations characterized by mass and spin or helicity expressed as sections (func-tions) of a bundle over the mass shell 3-hyperboloid or the light cone, as the case may be.It should be clear from these considerations of groups that the standard Dirac equation,which is an equation for a representation of the Poincaré group, plays an approximaterole in the geometric theory.

67Quantized Charge and Flux

We should point out that the isotropy subgroup of P at a De Sitter translation (point)in a null subspace is the group of transformations which leaves invariant the null tangentvectors at the given translation. The isotropy subgroup is SO(2,1) acting on the curvedDe Sitter translations space or its contraction ISO(2) acting on the flat Minkowski trans-lations space. In either case there is only one rotation generator, which corresponds tothe common compact SO(2) subgroup in both isotropy subgroups. Therefore, the nullmass representations induced from P are characterized by the corresponding eigenvalueor SO(2) quantum number which is known as helicity. For the rest of the chapter we shallrestrict ourselves to massive representations.

7.3. Cartan Subalgebras.We should consider the relation of induced representations with those obtained in

Cartan’s approach [8 ], in terms of the generators of a canonical subspace of the algebra,formed by a maximal set of commuting elements of the algebra (Cartan’s subspace).

In order to study the irreducible representations of the group SL(2,), we need tointroduce the complex extension of this group, which is SL(4,). The Cartan subspace ofthe complex algebra sl(4,), also known as the root space A3, describes the commutationrelations of the canonical Cartan generators of this complex algebra and all its real formssl(4,), su(4), su(3,1), su(2,2) and su*(4) [9] . In particular, the real form sl(4,) is the leastcompact real form of sl(4,).

In the defining representation, the elements of sl(4,) are traceless complex matrices.On the restriction to real variables the normal real form has the structure of traceless realmatrices.

The maximal compact subalgebra of sl(4,) consists of all real matrices which are anti-symmetric, leading to a Cartan decomposition,

( )sl , t ip= Å4 , (7.3.1)

where t is compact, real antisymmetric, and ip is noncompact, real symmetric.Under the Weyl unitary trick [4], we construct the Lie algebra for su(4), with all compact

matrices,

( )su t p= Å4 , (7.3.2)

which corresponds to the algebra of all traceless antihermitian matrices. Under similarinvolutive automorphisms the other three real forms may be obtained.

This relation, of the sl(4,) and su(4) algebras, implies that their irreducible representa-tions are described by equivalent sets of quantum numbers. The reason for this may beseen by studying the irreducible representations of their common complex extension. Thisfact, together with those indicated in the previous section, gives us hope that these repre-sentations may be used to describe physical particles. The canonical Cartan generatorsare:

H

é ùê úê ú-ê ú¢ = ê úê úê úë û

1

1 0 0 00 1 0 010 0 0 040 0 0 0

, (7.3.3)


H

é ùê úê úê ú¢ = ê ú-ê úê úë û

2

1 0 0 00 1 0 010 0 2 04 30 0 0 0

, (7.3.4)

H

é ùê úê úê ú¢ = ê úê úê ú-ë û

3

1 0 0 00 1 0 010 0 1 04 60 0 0 3

, (7.3.5)

a bE

é ùê úê úê ú= ê úê úê úë û

0 0 0 00 0 1 010 0 0 02 20 0 0 0

. (7.3.6)

The roots are the following:

[ ]r¢¢ =11 1 0 02 , (7.3.7)

ré ù¢¢ = ê úê úë û

231 1 02 2 2 , (7.3.8)

ré ù¢¢ = ê úê úë û

31 1 1 2

2 2 32 3 , (7.3.9)

ré ù¢¢ = - -ê úê úë û

41 1 1 2

2 2 32 3 , (7.3.10)

ré ù¢¢ = -ê úê úë û

51 1 202 33

, (7.3.11)

ré ù¢¢ = -ê úê úë û

631 1 02 2 2 , (7.3.12)

and the weight vectors as follows:


wé ù¢¢= ê úê úë û

11 1 114 3 6

, (7.3.13)

wé ù¢¢= -ê úê úë û

21 1 114 3 6

, (7.3.14)

wé ù¢¢= -ê úê úë û

31 2 104 3 6

, (7.3.15)

wé ù¢¢= -ê úë û4

31 0 04 2 . (7.3.16)

Another base of the Lie algebra may be introduced by taking as generators the matri-ces generated by an orthonormal subset of sl(4,) taken as a Clifford algebra. Using theprevious notation these generators are explicitly given in appendix A. The use of thesegenerators imply a change of basis in the Cartan subspace expressed by the followingrelations:

G H H H k¢ ¢ ¢= - + =1 1 2 3 21 1 1 12 6 3 4 2

, (7.3.17)

G H H k k¢ ¢= + =2 2 3 0 32 1 13 6 4 2

, (7.3.18)

G H H H k k k¢ ¢ ¢= - - + =3 1 2 3 2 3 01 1 1 12 6 3 4 2

. (7.3.19)

The base introduced in eqs. (7.3.17)-(7.3.19) has the merit that there is a subset whichspans a Lie subalgebra which generates the even Clifford subalgebra. This even Cliffordalgebra is related to SL(2,) and correspondingly to Lorentz transformations, which areusually associated to the subgroup of external symmetries. The remaining generatorsspan a space which generates the coset

( )( )1

SL 4,RK

SL 2,C= . (7.3.20)

This coset represents the internal transformation symmetries. It is clear that we havereplaced the usual internal group of transformations, that enters as a factor in the group ofsymmetries of the theory, by a coset space.

7.4. Relation Among Quantum Numbers.The relation of the quantum numbers associated to these Gi generators with the quan-

tum numbers of the representations induced from the maximal compact subgroup may beseen by considering the corresponding Cartan subalgebras. Of course, the Gi generators


span the same Cartan subalgebra generated by the H´i. There are other Cartan subalgebrasin sl(4,). It is known that a Cartan subalgebra is not uniquely determined but depends onthe choice of a regular element in the complete algebra. Nevertheless, it is also knownthat there is an automorphism of the complex algebra which maps any two Cartan subal-gebras [8]. This implies there is a relation between the sets of quantum numbers associ-ated to two different Cartan subalgebras within sl(4,). We may take the quantum num-bers linked to spin and charge as the physically fundamental quantum numbers and con-sider the others that arise by use of different Cartan subalgebras as numbers which arefunctions of the fundamental ones.

For physical reasons, since electromagnetism and electric charge are associated to theSU(2) subgroup generated by k0, k1k2k3, k0k1k2k3 within our geometric theory, it is of inter-est to consider an algebra decomposition with respect to one of these generators. Since allthree of them commute with all the generators of the spin SU(2), k1k2, k2k3, and k3k1, wehave different choices at our disposal. For convenience of interpretation we choose k1k2,and k0k1k2k3 as our starting point. The only other generator that commutes with them isk0k3. It is shown now that neither k0k1k2k3 nor k1k2 are regular elements of the total sl(4,)algebra in spite of being regular elements of the two su(2) subalgebras. The expression forthese generators in the adjoint (regular) representation is, up to a normalization constant,

( )ad I

I

S

k k

é ùê úê úê úê ú= ê úê ú-ê úê úê úë û

3

1 2 3

3

0 0 0 00 0 0 0 00 0 0 00 0 0 00 0 0 0 0

, (7.4.1 )

( )adS

k k k k SS

S

é ùê úê úê úê ú= ê úê úê úê úê úë û

3

0 1 2 3 3

3

3

0 0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

, (7.4.2)

where each entry in the matrices is a tridimensional matrix, giving the total 15 dimensions ofthe adjoint representation. The ordering of columns and files is chosen as follows:

1. The three indices i of rotation,2. The three indices a of electromagnetism, and3. The nine products ia (11,12, etc.).

The matrix I3 is the tridimensional unit matrix and S3 is

S

é ùê úê ú= -ê úê úë û

3

0 1 01 0 00 0 0

. (7.4.3)


It is clear that each of the two matrices, k1k2 and k0k1k2k3, generates a subspace Vo,corresponding to eigenvalues l=0,

( )ad kH X = 0 , (7.4.4)

which is seven-dimensional. Since the Cartan subalgebra of sl(4,) is tridimensional, itfollows that neither of the two generators is a regular element. Nevertheless,

( )ad I

I

SS

k k k k k k SS

S

é ùê úê úê úê ú+ = ê úê ú-ê úê úê úë û

3

3

1 2 0 1 2 3 3 3

3 3

3

0 0 0 00 0 0 00 0 00 0 00 0 0 0

(7.4.5)

has a VO subspace, for l=0, which is tridimensional. Therefore, the sum generator is aregular element of the Lie algebra.

The VO space generated by this regular element is a Cartan subalgebra, which is spannedby the generators

X k k=1 1 2 , (7.4.6)

X k k k k=2 0 1 2 3 , (7.4.7)

X k k=3 0 3 , (7.4.8)

where the product is understood in the enveloping Clifford algebra.It is clear that X1, and X2 are compact generators and therefore have imaginary eigenval-

ues. Because of the way they were constructed, they should be associated, respectively, to z-component of angular momentum and electric charge. Both may be diagonalized simulta-neously in terms of the imaginary eigenvalues, leaving X3 invariant. When dealing with com-pact elements of a real form, as spin, it is usual to introduce the standard notation in terms ofthe corresponding noncompact real matrices of the real base in the complex algebra,

i

iX i iH

i

i

é ù é ùê ú ê úê ú ê ú- -ê ú ê ú= = ºê ú ê ú- -ê ú ê úê ú ê úë û ë û

1 1

0 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

, (7.4.9)

i

iX i iH

i

i

é ù é ùê ú ê úê ú ê ú- -ê ú ê ú= = ºê ú ê úê ú ê úê ú ê ú- -ë û ë û

2 2

0 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

, (7.4.10)


X H

é ùê úê úê ú= ºê ú-ê úê ú-ë û

30 3

1 0 0 00 1 0 00 0 1 00 0 0 1

. (7.4.11)

The Clifford algebra matrices provide a geometric normalization of roots and weightsin the Cartan subspace. The weight vectors, in the bases Xi and Gi have the same structureexcept for a standard normalization factor of (32)-1/2:

[ ]w = + + +0 1 1 1 , (7.4.12)

[ ]w = + - -1 1 1 1 , (7.4.13)

[ ]w = - + -2 1 1 1 , (7.4.14)

[ ]w = - - +3 1 1 1 . (7.4.15)

Similarly, the roots in both bases have the same structure, differing by the standard normal-ization factor:

[ ]r = - -01 0 2 2 , (7.4.16)

[ ]r = - -02 2 0 2 , (7.4.17)

[ ]r = - -03 2 2 0 , (7.4.18)

[ ]r = -12 2 2 0 , (7.4.19)

[ ]r = -23 0 2 2 , (7.4.20)

[ ]r = -31 2 0 2 . (7.4.21)

7.5. Physical Interpretation.It may be seen that, in the fundamental representation, one of the generators in the

Cartan subalgebra may be expressed as a Clifford product (not a Lie product) of the othergenerators of the subalgebra. This is true for both the Gi and the Xi generators,

X X X=1 2 3 . (7.5.1)

This implies that, within the Clifford algebra, there is a multiplicative relation among thequantum numbers in the theory. In particular, the z-component angular momentum genera-


tor, spin, is the Clifford product of the electric charge generator and the X3 generator.The quantum numbers associated to X3 must have the physical meaning of angularmomentum divided by electric charge, or equivalently, magnetic flux. Then the fun-damental quantum of action should be the product of the fundamental quantum ofcharge times the fundamental quantum of flux,

( )h he e=2 2 . (7.5.2)

We may intuitively interpret the last equation as a quantum betatron effect, when aquantum change in magnetic flux is related to a quantum change in angular momentum.

We have taken the quantum of action in terms of h rather than because thenatural unit of frequency is cycles per second rather than radians per second. Thenthe quantum of flux f0 is

he e

pf = =0 2 . (7.5.3)

The four members of the fundamental irreducible representation form a tetrahedronin the tridimensional A3 Cartan space as shown in figure 1. They represent the combi-nation of the two spin states and the two charge states of an associated particle,which we shall call a G-particle. One charge state represents a physical particle stateand the other represents a charge conjugate state. The G-particle carries one quantumeach of angular momentum, electric charge and magnetic flux and may be in one of thefour state whose quantum numbers are:

(-1, -1, 1)

(-1, 1, -1)

(1, 1, 1)

(1, -1, -1)

k k k k0 1 2 3 charge

k k0 3 flux

k k1 2 spin

Figure 1.Irreducible representation

of SL(4,).


charge spin flux negative charge with spin up

negative charge with spin down positive charge with spin up

positive charge with spin down

f

f

f

f

-

-

+

+

- + -- - ++ +++ --

1 1 11 1 11 111 11

The dual of the fundamental representation, defined by antisymmetric tensor productsof 3 states of the fundamental representation or triads, corresponds to the inverted tetrahe-dron. A conjugate state or a dual state may be related to an antiparticle. This may be usefulbut is not a necessary interpretation. It is better to keep these mathematical conceptsphysically separate. We consider, on one hand, fundamental excitations and dual excita-tions and, on the other hand, excitations with particle and conjugate states.

The states of an irreducible representation of higher dimensions, built from the funda-mental one, are also characterized by three integers: angular momentum m, electric charge qand magnetic flux f. We may conjecture then that the magnetic moment is not as fundamen-tal as the magnetic flux when describing particles.

7.6. Representations of the Subgroup P.It is known that the SL(2,) subgroup has a one dimensional Cartan subspace of type

A1 associated to the quantized values of angular momentum. The Sp(4,) subgroup has atwo dimensional Cartan subspace of type C2. In the same way we handled SL(4,), we maychoose a regular element associated to the spin generator k1k2, in particular,

( )ad k k k k

é ù-ê úê ú-ê úê úê úê úê úê úê ú+ = ê úê úê ú-ê úê ú- -ê úê úê úê úê úë û

1 2 0 3

0 1 0 0 0 0 0 1 0 01 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 0 00 1 0 0 0 0 0 1 0 00 0 0 0 1 0 0 0 0 01 0 0 0 0 1 0 0 0 0

0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

(7.6.1)which annihilates the two dimensional subspace, with zero eigenvalues, spanned by thek1k2 y k0k3 generators. The corresponding weight vectors are the following, with thesame previous normalization:


[ ]w ¢ = - -3 1 1 , (7.6.2)

[ ]w ¢ = - +2 1 1 , (7.6.3)

[ ]w ¢ = + -1 1 1 , (7.6.4)

[ ]w ¢ = + +0 1 1 , (7.6.5)

and the roots are the following:

[ ]r¢ = -02 2 0 , (7.6.6)

[ ]r¢ = -01 0 2 , (7.6.7)

[ ]r¢ = - -03 2 2 , (7.6.8)

[ ]r¢ = -12 2 2 . (7.6.9)

Nevertheless we may also choose as regular element one element related to the charge,

( )ad k k k

é ùê úê ú-ê úê úê úê úê úê úê ú+ = ê ú-ê úê ú-ê úê ú

- -ê úê úê úê úê ú-ë û

0 1 2

0 1 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 00 0 0 0 1 0 0 1 0 00 0 0 0 1 0 0 1 0 00 0 0 0 0 1 1 0 0 00 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 1 0

(7.6.10)

which annihilates the two dimensional subspace, with zero eigenvalues, spanned by thegenerators k1k2 and k0.

In both cases the four members of the fundamental representation form a square in atwo dimensional C2 Cartan space, but in different Cartan subspaces of the A3 root space.We may visualize the relation of these vectors with those of the full SL(4,) group recog-nizing that the tridimensional Cartan A3 space collapses to a bidimensional C2 subspace asindicated in figure 1. The tetrahedron which represents the states of the fundamental rep-resentation collapses to a square. The 4 tetrahedron vertices project to the 4 square verti-


ces. The set of 4 weight vectors in C2 may be obtained by projecting the 4 weight vectorsin A3, which correspond to the tetrahedron vertices, on the plane spanned by the vectorsk1k2 y k0k3 or k1k2 and k0, forming squares in these Cartan subspaces. The 6 tetrahedronedges project to the 4 sides and 2 diagonals of each square. The opposite sides of asquare are equivalent because they have the same directions. The set of 8 roots in C2 mayalso be obtained by projecting the set of 12 roots in A3, which correspond to the tetrahe-dron edges. In this case, 8 roots project on 4 degenerate root pairs (equivalent) corre-sponding to the square sides. The other 4 roots project to the 4 roots corresponding tothe square diagonals. A collapse of the C2 bidimensional spaces to the unidimensional A1

Cartan space of SL(2,) produces a projection of the squares to the line segment whichrepresents the 2 standard spin states and the 2 roots of the latter Cartan space, associatedto an L-particle.

The fact that the Cartan two dimensional subspace is not uniquely determined hasphysical consequences within the given interpretation. The 4 states of the irreduciblerepresentation may be labeled by spin and flux or charge depending on the chosen regularelement, eq. (7.6.1) or eq. (7.6.10). But, as indicated before, both Cartan spaces are relatedby a complex algebra automorphism implying that one set of quantum numbers may beexpressed as functions of the other set, in particular the flux quantum is a function of thespin and charge quanta. In this case, the relation is interpreted as the remnant of themultiplicative relation among spin, charge and flux in the fundamental representation of theparent group SL(4,). This indicates that a physical particle associated to this representa-tion, which we shall call a P-particle, has the three quantum numbers. There are no continu-ous variables which represent the measurable values of spin, charge and flux of the P-particle. The difference between the G-particle and the P-particle is not displayed bythese quantum numbers.

Three sp(4,) spaces may be injected as subalgebras into the three geometrically andalgebraically independent sectors of the complete sl(4,) algebra of the group G. In thismanner we may construct a section p valued in sl(4,) by injecting three independentsections (e1, e2, e3) valued in the subalgebra sp(4,). Therefore, for any state e of thefundamental representation of P, the dual e formed by the triad (e1, e2, e3) of fundamentalP-representation states determines a particle state p of the fundamental representation ofG. The corresponding states, equally charged, define a charge equivalence relation amongthese states,

( ), ,e e e e p@ @1 2 3 . (7.6.11)

We may choose any state in the fundamental representation of G to physically representthe fundamental particle p associated to this representation. The fundamental (1, 1, 1)SL(2,) states define a proton charge sign. Nevertheless, since the charges of correspond-ing e and p states are equivalent, we are free to define the charge of only one state in theG y P representations taken together. The charge corresponding to this chosen particle p(the proton) may be defined to be positive. This determines an inequivalent negativecharge for the corresponding state which physically represents the other fundamentalparticle e (the electron) in the fundamental representation of P,

( ) ( ) ( )Q p Q e Q e+ º = =-1 . (7.6.12)

In other words the charge of the electron (dual) particle must have a sign opposite to thecharge sign of the proton (dual) particle.


7.7. Applications.From the previous discussion it follows that any particle which is a fundamental rep-

resentation of SL(4,) or Sp(4,) must be a charged particle with a quantum of flux.This quantum of flux is exactly the value determined experimentally by Deaver and Fair-bank [10 ] and Doll and Nebauer [11 ], first predicted theoretically by London [12 p. 152]with an extra factor of two. The experiment consisted in making a small cylinder of su-perconductor by electroplating a thin layer of tin on a copper wire. The wire was put in amagnetic field, and the temperature reduced until the tin became a superconducting ring.Afterwards the external field was eliminated, leaving a trapped minimum of flux throughthe ring.

Our result is consistent with the experimental result if we consider that the trapped fluxthrough the cylindrical ring layer of superconducting tin is really associated to the discreteminimum intrinsic flux of a single electron inside the normal (not superconducting) copperwire which serves as nucleus for the superconducting tin. At present, it is generally be-lieved that the quantization of flux is due to the topology of the superconducting materialin the experiment and related to the charge of an electron pair inside the superconductor.The idea expressed here is that the quantum of flux is an intrinsic property of matter par-ticles (electron, proton etc.) and only the possibility of trapping the flux depends on thetopology of the superconductor.

If a particle crosses a line in a plane normal to its flux, there is a relation between thecharge and flux crossing the line,

( )hf eQ qe

DFD

= 2 . (7.7.1)

If there are no resistive losses along the line, the induced voltage leads to a transverseresistance which is fractionally quantized,

( )( )tf hR q e

= 22 . (7.7.2)

This expression leads us to conjecture that the fractional quantized Hall effect (FQHE)[13, 14 ] gives evidence for the existence of these flux quanta instead of evidence for theexistence of fractional charges. The FQHE experiment consists essentially of the measure-ment of the transverse Hall conductivity occurring at low temperature in bidimensionalelectron gas crystal interfaces in semiconductors.

If we consider that the corresponding carrier wavefunctions should be representationsof SL(2,) or Sp(2,), they are characterized by quantum integers m, q and f. If we imaginethe buildup of the field, the details are complicated by the reaction of the induced currentson the original field. Independent of these details, after a steady state is reached, the totalflux should have f flux quanta, including intrinsic, orbital and reaction fields. We have then

( )hNf eF = 2 . (7.7.3)

The 2-dimensional conductivity in terms of the electric field E, the Hall velocity V andthe surface area A is


( )( )qeNv q ef hAE

s = =22 . (7.7.4)

This expression is compatible with the general structure of the FQHE experimentaldata. For p, q random integers, odd denominator fractions of e2/h are 11 times moreprobable than even ones.

Furthermore, eq. (7.7.3) provides a simpleminded interpretation for the Meissnereffect [15 ]. This expression should still be valid if the electrons are paired. If the pairingis such that the quanta of flux cancel each other, the value of f should be zero for eachpair and consequently the total magnetic flux inside the superconductor should also bezero.

It is interesting to note that the geometrical and classical motivation for this researchled to a simple relation implying a deep quantum result. The existence of this quantum offlux appears to be physically realized in nature and is not associated at all with a quantumof magnetic charge. The fundamental character of this flux helps to explain the extraordi-nary accuracy of the quantum of resistance in the FQHE.

7.8. Summary.We may conclude that the fundamental irreducible representations of SL(2,) induced

from the rotation SU(2) and the electromagnetic SU(2) are characterized by quantum num-bers whose interpretation is associated, respectively, to angular momentum and to totalcharge, within the proposed theory.

It was shown that a Cartan subalgebra may be generated by the regular element (k1k2 +k0k1k2k3) and has for basis the compact generators k1k2 of the rotation SU(2), k0k1k2k3 of theelectromagnetic SU(2) and the noncompact generator obtained from the other two by theClifford product in the defining representation.

In this subalgebra, the states of an irreducible representation may be labeled by inte-gers m, q, f. The first two determine the possible values of the z-component of angularmomentum m/2 and of electric charge qe. Furthermore, the multiplicative relation amongthe basis generators of the subalgebra implies that the third quantum number f is related tothe other two and associated to a quantum of magnetic flux. The theory provides a mecha-nism for electric charge quantization, requiring magnetic flux quantization instead of Dirac’smagnetic monopoles.

If we require that a charged physical particle is a representation of sl(4,) or sp(4,), itmust carry quanta of angular momentum, electric charge and magnetic flux. Other physicalquantum numbers, related to other Cartan subalgebras, may be expressed in terms of thesefundamental quantum numbers.

References

1 G. González-Martín, Gen. Rel. Grav. 23, 827 (1991). 2 A. O. Barut, Phys. Rev. 133B, 839 (1964). 3 R. Hermann, Lie Groups for Physicists (W. A. Benjamin, New York) p. 56 (1966). 4 R. Gilmore, Lie Groups, Lie Algebras and Some of their Applications (John Wiley

and Sons, New York), p. 246, 453, ch. 9, 10 (1974). 5 P. A. M. Dirac, Phys Rev. 74, 817 (1948).


6 B. Doubrovine, S. Novikov, A, Fomenko, Géométrie Contemporaine, Méthodes etApplications (Ed. Mir, Moscow), tranlated by V. Kotliar, Vol. 2, p. 62 (1982).

7 E. P. Wigner, Ann. Math. 40, 149 (1939). 8 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New

York)p. 130, 137 (1962). 9 See appendix B.10 B. S. Deaver, W. M. Fairbank, Phys. Rev. Lett. 7, 43 (1961).11 R. Doll, M. Nabauer, Phys. Rev. Lett. 7, 51 (1961).12 F. London, Superfluids (John Wiley & Sons, New York), Vol. 1 p. 152 (1950).13 D. Tsui, H. Stormer, A. Gossard, Phys. Rev. Lett. 48, 1559 (1982).14 K. V. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980).15 W. Meissner, R. Ochsenfeld, Naturwiss. 21 787 (1933).

8. MEASUREMENT OF GEOMETRIC OB-SERVABLES.

8.1. Introduction.We have seen in chapters 3, 6 and 7 that the physical geometry shows some quantum

characteristics. We now continue to develop the theory. It is possible to give a geometricalrepresentation to the postulates of quantum mechanics. In most cases this would only begeometrical superstructure that may not contain new physical ideas. On the contrary, westart from a geometrical physical theory and obtain its quantum implications. In this man-ner new physical phenomena may arise. Similar aims originally guided Drechsler in dis-cussing extended objects as functions on homogeneous de Sitter fiber bundles within ageometric theory of hadrons [1, 2 ].

Drechsler incorporated electromagnetism in his theory by using a modified Weyl ge-ometry in the construction of the homogeneous De Sitter bundles and relating the fullcurvature to quantized matter currents [3, 4 ]. Our approach has been different. Originallyour aim was to unify gravitation and electromagnetism by means of the connection. Toavoid contradictions we had to introduce groups acting on the Clifford algebra of spaceand time, forcing a geometrical structure which implies quantum aspects.

Our theory considers excitations of physical matter as representations of the structuregroup of the geometrical theory. From this idea it follows that there are certain discretenumbers associated with the states of microscopic matter. It was shown that these num-bers may be interpreted as quanta of angular momentum, electric charge and magneticflux, providing a plausible explanation to the fractional quantum Hall effect (EHCF) [5 ].Furthermore, the theory also leads to a geometrical model for the process of field quanti-zation [6 ], implying the existence of fermionic and bosonic operators and their rules ofquantization. It should be clear that a process of physical measurement should displaythese discrete geometric numbers as experimental quanta, leading to a particle (atomic)description of matter. It is necessary, therefore, to discuss measurements within the physi-cal geometry [7].

We may wonder whether these results are accidental or the consequence of a deepfundamental relation of the geometrical structure of the unified field with the standardquantum structure. Here we pursue the second alternative. It may be claimed that, if theuncertainty principle is taken as fundamental, the geometry at very short distances be-comes a “fuzzy” geometry and the applications of differential geometry is questionable.Nevertheless, nothing prevents us, within our theory, to continue taking differential ge-ometry as fundamental, providing the germ of quantum physics principles and representparticles by fluctuations or excitations of a geometric nonlinear substratum.

In particular, we consider the following questions: Can we define, within our theory, ageometric operation representing the process of physical quantum measurement? Are theresults of this operation compatible with well known facts of experimental physics? Canwe define angular momentum and charge geometrically? First we shall review the mainideas of the geometrical theory.

We require the use of the spinor group of automorphisms of the universal Clifford

81Measurement of Geometric Observables

algebra associated to flat tetradimensional space-time or equivalently its highest dimen-sional simple subgroup SL(2,). This implies an extension of relativity. The connection,which represents the interaction, not only unifies gravitation with electromagnetism in-cluding the correct motion but actually gives a gravitational theory which differs in prin-ciple with Einstein’s theory and resembles Yang’s [8 ] theory. This may be seen from thefield equation of the theory, which relates the derivatives of the Ehresmann curvature to acurrent source J,

D k JW* *= . (8.1.1)Because of the geometrical structure of the theory the source current must be a geo-

metrical object compatible with the field equation and the geometry. The field equationimplies integrability conditions in terms of J. Together with the geometric structure of J,these conditions imply a generalized Dirac equation which, therefore, is not required to beseparately postulated as is normally done in nonunified theories. Actually, the nonlinearfield equation for the connection and the simplest geometric structure of the current aresufficient to predict this generalized Dirac equation. The structure of J, of course, is givenin terms of geometric objects acted upon by the connection,

( )ˆˆkJ ke u em a mai= , (8.1.2)

in terms of the coframe e, an orthonormal set of the algebra i, the correlation on the spinorspaces ~, and a space-time tetrad u.

The three compact generators k0, k5, and k1k2k3 are equivalent as electromagneticgenerators within the theory because there are automorphisms which transform any oneof them into any other. It follows that the set ia that enters in the current is defined up toan algebra automorphism. This allows us to take the i0 element as any of the three elec-tromagnetic generators or a linear combination without changing the physical content ofthe theory.

8.2. Measurement of Geometric Currents.It is possible to study the properties of fluctuations or excitations of the geometric

elements of the unified theory. Furthermore, if, as suggested before [5], a particle may berepresented as an excitation of the geometry, its physical properties may be determined byits associated fluctuations. These fluctuations may be characterized mathematically by avariational problem. From a variational principle, if the equations of motion hold, it ispossible to define the generator of the variation. There is a canonical geometric currentassociated with this generator which geometrically represents the excitation, and shouldbe considered the subject of a physical measurement (observable).

The lagrangian density, in general, has units of energy per volume and the action hasunits of energy-time. In the natural units defined by the connection (c=1, =1, e=1), theaction is dimensionless. In the standard relativistic units (c=1), the constant arises as afactor in the action W.

It is well known that the variation of an action integral along a transformation of thevariable y with parameters l is

RR

LW yd x Q d

ym

m¶

dd d d s

d= + òò 4 , (8.2.1)


where the canonical current , and the conjugate momentum P are

dy dxQ L

d d

mm m md dl P dl

l l

æ ö÷ç ÷= = +ç ÷ç ÷çè ø , (8.2.2)

,

Ly

m

m

¶P

¶= . (8.2.3)

The current is a geometrical object field which represents an observable propertyof a physical excitation, e.g., angular momentum density. In general, a measurement isnot a point process but rather it is an interaction excitation with an apparatus over a localspace-time region. In the geometric theory, the results of a physical measurement shouldbe a number depending on a variational current about a variation of a background sec-tion e, over a region R(m’) around a characteristic point m on the base space M, withsome instrumental averaging procedure over the region. Therefore, we shall make thehypothesis that a measurement on a geometrical excitation is represented mathemati-cally by a functional m of the observable geometric current defined by the associatedvariation over a complementary hypersurface s,

( ) ( ) ( ),m

R

m m m dmm

¶

s¢ ¢= ò . (8.2.4)

In some cases, excitations may be approximated as point excitations with no extendedstructure. In order to show the relation of our geometric theory to other theories, withoutusing any knowledge about the structure of excitations, we shall define a geometric mea-surement of a point excitation property by a process of shrinking the region R(m’) of thecurrent to the point m. With this procedure, the local section representing the excitationshrinks to a singular section at m. We may express this mathematically by

( ) ( )( )

lim m mR m m

d¢

= , (8.2.5)

where the functional dm is the Dirac functional,

( ) ( )m md = . (8.2.6)

This process shrinks the current to a timelike world line. We may visualize the bound-ary ¶R of region R as an infinitesimal cylindrical pillbox pierced by the current at thebottom and top spacelike surfaces S. As the pillbox is shrunken to point m, the functionalsof the current dm() at the top and bottom surfaces are equal, if the current is continuous.The functional dm() on either of the spacelike surfaces S is the geometric measurement

( ) ( ) ( )m m d x m x u d xm mm m

S

d d s d= = -ò ò3 3 3 , (8.2.7)

( ) ( )m u mmmd = , (8.2.8)

where u is the timelike velocity, orthogonal to S, of the space-time observer.


From the lagrangian given for the theory, eq. (3.2.2), the geometric current of anarbitrary matter excitation has the general form

e Xem mi= , (8.2.9)in terms of the frame e, an orthonormal set of the algebra i, the correlation on the spinorspaces ~, and the group generator X of the variation.

It should be noted that the frame e is associated with a set of states forming a base of arepresentation of the structure group. It does not represent a single physical state, but acollection of physical states. For any operator L in the algebra we may select, as the col-umn vectors of the frame e, the eigenvectors f corresponding to L. Then we may write

( ) ( ), , , , ii ie eL L f f f l f l f l f l= = =1 2

1 2 1 2 , (8.2.10)

where l is the diagonal matrix formed by the eigenvalues li.Accordingly, the results of the measurement given by eq. (8.2.9), in coordinates adapted

to the 4-velocity u, is

( )m e Xe e ed i L= = =0 0 , (8.2.11)

which defines an associated operator L.In this expression, we should note e is the group inverse of e, and the correlated

product in the spinor space is a scalar. The product e e gives a unit matrix of scalars, andthe measurement values of the current coincide with the diagonal matrix formed with itseigenvalues,

( )m

i

ee

ll

d l

l

é ùê úê úê ú= = ê úê úê úë û

1

2

. (8.2.12)

This result agrees with one of the postulates of quantum mechanics. The mathematicalcontent of the last equation is truly independent of a physical interpretation of the frame e.In particular, it does not require, but allows that e is a probability amplitude.

The result of the measurement essentially equals the value of the current at the repre-sentative point m. Equivalently it is the average over a characteristic 3-volume V of S,

VV

Qd d

V V

mm

m m

ds s

dl= = òò

1 1 . (8.2.13)

This averaging, indicated by á ñ, is similar to the operation of taking the expectationvalue of an operator in wave mechanics.

The different generators of the group produce excitations whose properties may beinvestigated by measuring the associated geometric currents. In particular, we are inter-ested here in currents associated with generators of the compact subgroups, which wereused to characterize the induced representations.


8.3. Geometric Spin.The concept of spin is related to rotations. In the geometric theory the compact even

generators form an su(2) subalgebra which is related to the rotation algebra. The grouphomomorphism between this SU(2) subgroup and rotations is

( )†tra ab bR s s= 1

2 g g , (8.3.1)

where gÎSU(2) and RÎSO(3). The isomorphism between this SU(2) and the compact evensubgroup of SL(2,) is the well known isomorphism between the complex numbers and asubalgebra of the real 2 ´ 2 matrices,

, ié ù é ù-ê ú ê ú« «ê ú ê úë û ë û

1 0 0 11

0 1 1 0 . (8.3.2)

The isomorphism given by these expressions is not accidental, but is part of the con-ceptual definition of the geometric Clifford algebras as a generalization of the complexnumbers and the quaternions. These algebras, and the spinor spaces on which they act,have well defined complex structures.

If we consider that the generators kikj that belong to su(2) are the rotation generators,the associated geometric current is the angular momentum. For example, the result of themeasurement of this current in a preferred direction 3, using as illustration eq. (8.2.13)for the expectation value is

de dxd L

V d d

mm

ms Pl l

æ ö÷ç ÷= +ç ÷ç ÷çè øò3 1 , (8.3.3)

where the variation de is generated by k1k2. Then, for a flat metric,

ˆˆde de

d e u d xeV d V d

a mm as i i

l l= =ò ò3 3 01 1

. (8.3.4)

Since i0 commutes with k1k2 and i0i0 is -1, we may use eq. (8.3.2) to identify

, iI ii k k s 0 1 2 3 , (8.3.5)

and express the variation generated by k1k2 as the differential of eq. (8.3.1),

( )tra a ab b b

iR

ld s s s s s s= -3 3

2 . (8.3.6)

The only nonzero elements are

R Rd d l q- = = =1 22 1 2 , (8.3.7)

which represents a rotation by angle q in the 1-2 plane. This rotation induces a change inthe functions on tridimensional space, giving a total variation for e of

( )y x

ie e x y ed s ¶ ¶ dl

æ ö÷ç= + - ÷ç ÷çè ø3

2 , (8.3.8)


which leads to

( )( )y xd xe i x y eV

s ¶ ¶= - -ò3 3 312

1 . (8.3.9)

The factor ½ indicates, of course, that the SU(2) parameter l is half the angular rota-tion, due to the 2-1 homomorphism between the two groups.

This calculation was done, for simplicity, with only one component. It is clear that ifwe use the three spatial components we get

( ){ }a a abcb cd xe i x e

Vs e ¶= -ò 3 1

21

, (8.3.10)

where the expression in parentheses is the angular momentum operator L in quantummechanics. If e is an eigenframe of this operator, we obtain the values of the componentsof the angular momentum associated with a fluctuation related to the spin representation.The result is

e ed x ee d xV V

L L l= =ò ò3 31 1 . (8.3.11)

As before, the product e e gives a unit matrix of scalars, and we may construct theintegral in the base space

eed x V I=ò 3 , (8.3.12)

where I is the identity and V is the characteristic volume. It is clear that we may introducea volume-normalized e by dividing by V, The measured value of the operator L coincideswith the diagonal matrix formed with its eigenvalues, as indicated above:

i

ll

L

l

é ùê úê úê ú= ê úê úê úë û

1

2

. (8.3.13)

In the case described, the frame states are pure (quantum terminology) with respect tothe angular momentum. Both the frame and operator may be diagonalized simultaneously,or equivalently they commute with each other.

In general, the frame is not pure with respect to the operator. Thus the result of themeasurements is not the diagonal elements of the operator (eigenvalues). If we designatethe columns of e by Fa, and the files of ê by Fb for the measurements we have the matrix

d xV

b ba ar F LF= ò 31

, (8.3.14)

which corresponds to the density matrix (for the observable operators L).The frame sections play the role of wave functions and the group generators play the

role of quantum operators. These similarities between our geometric theory and quantum


mechanics provide essentially equivalent results. There are differences; in particular,wave functions have a complex structure and our frame sections have a Clifford struc-ture. Rather than a contradiction, this difference is a generalization, since there are com-plex structures in different subspaces of the geometric Clifford algebra. It is possible tointroduce spaces of sections, but they certainly may have a structure more general than aHilbert space. The geometrical and group elements in the theory actually determine manyof its physical features.

8.4. Geometric Charge.The geometric source current J is a generalization of electric current. The three

compact generators k0, k5, and k1k2k3 are equivalent as electromagnetic generators withinthe theory because there are automorphisms which transform any one of them into anyother. It follows that the set ia, which enters into the current, is defined up to an algebraautomorphism. This allows us to take the i0 element as any of three electromagneticgenerators without changing the physical content of the theory.

The generalized source current J is the canonical current corresponding to a varia-tion generated by an electromagnetic generator. In order to see this, we choose the set km

for im, and look for a variation generator which results in an automorphism of the current.In other words, we look for a generator which gives a set equivalent to the set km by rightmultiplication. A generator which accomplishes this is k5,

exp expm mp pk k k k k

æ ö æ ö÷ ÷ç ç= - ÷ ÷ç ç÷ ÷ç çè ø è ø5 5 5

4 4 . (8.4.1)

It is possible to find another which results in a different automorphism but it would lie inthe electromagnetic sector. Thus it is clear that the current J corresponds to variationsgenerated by the electromagnetic sector.

When we make a measurement of this canonical current J, we are measuring the chargeassociated with the fluctuation of e related to a fundamental irreducible representation ofthe group. We repeat the same calculation done in the previous section for angular mo-mentum current. If we neglect the gravitational part, the metric is flat, the expression forthe measurement on the charged current is

ˆˆd e u ed

V Vm a m

m a ms k s= =ò ò1 1

e ed xV

k k= ò 0 5 31 , (8.4.2)

where it is understood that we are working in the bundle SM which is the Whitney sum ofthe associated spinor vector bundle VM and its conjugate, as described in appendix A.Explicitly, in terms of elements of VM the last equation is written as

eej d x

eV e

k kk k

-

-

é ù é ù é ù é ùê ú ê ú ê ú ê ú= ê ú ê ú ê ú ê ú- - ë ûë û ë û ë û

ò1 0 5

310 5

00 0 0100 0 0

. (8.4.3)

It should be emphasized that the matrices in the last equation are 8´8 real matrices,


the double-dimensional representation used in the SM bundle, as indicated in appendix A.The even generators may be written as 4´4 complex matrices using the isomorphisms of eq.(6.2.1). We may substitute any equivalent generator for k0k5.

It is possible to choose a frame section e corresponding to eigenvectors of the funda-mental representation of SU(2). In fact, since k5 commutes with k1k2, it is also possible tochoose a frame corresponding to common eigenvectors of these two antihermitian genera-tors belonging to the two su(2) subalgebras in sl(4,). The eigenvalues correspond to thequanta of spin and charge, as follows:

( )

i

i

i

i

k k

é ù-ê úê úê ú= ê ú-ê úê úë û

2 1 2 , (8.4.4)

i

i

i

i

k

é ù-ê úê ú-ê ú= ê úê úê úë û

2 5 , (8.4.5 )

which are precisely the explicit forms of these matrices when working on the bundle SM.Of course it is usual to work with the associated hermitian operators obtained by multi-plication by i, with real eigenvalues ± 1. Nevertheless the use of the antihermitian ex-pression is natural since they are generators of the two compact su(2) subalgebras

If e is an eigenframe of the generator, we get

j i= . (8.4.6)

In other words, the result of this measurement for the fundamental representation is aquantum number equal to ±1. This conserved number may be interpreted as the quantumof charge.

It is known that the electron charge plays two roles, one as the quantum of charge andthe other as the square root of the fine structure coupling constant a. In order to be able toreduce the theory to electromagnetism we must account for Coulomb’s coupling constantk/4p and understand the relationship of these constants. This k may be absorbed into thedefinition of the current in eq. (8.1.1), but at the end it must be identified. It is better to showit explicitly and to keep the frame e separate, as a section in the principal bundle, so that theconjugate e is the dual inverse of e, and the product e e is the unit matrix.

The dimensionless fine structure constant a is given by ke2/4pc. The units of thearbitrary constant k, which allows us to define the electromagnetic units from the me-chanical ones, are ml3t-2q-2. If we set k=4p, the units correspond to the gaussian system,where Coulomb’s constant is 1. If we set k=1, we obtain the Heaviside-Lorentz systemwhere Coulomb’s constant is 1/4p. If we set k=4pc210-7, we obtain the rationalized MKSAsystem of units, where Coulomb’s constant is c210-7. In these systems, the minimal cou-pling determines that the connection G corresponds to eA, in terms of the potential A andthe electron charge e. On the other hand, rather than setting the value of k, it seems better


to consider that the geometric theory introduces a natural unit of charge by defining theelectromagnetic potential equal to the connection. In this form the new geometric unit ofcharge (the “electron”) equals e Coulombs, k is determined to be 4pa and Coulomb’sconstant becomes the fine structure constant a. With our definition of current and cou-pling constant (4pa) the calculated charge of the electron is ±1, in these geometric units.In arbitrary units, the calculated quantum of charge e is ±(4pac/k)1/2.

In other words, this value is the minimum quantum of measurable charge changes.This geometric prediction explains the two roles played by the electron charge, as cou-pling constant and as quantum. In the previous section the calculation lead to the wellknown values of angular momentum. In this section we obtain a new theoretical result.

When there is only a U(1) electromagnetic field in flat space, our field equations re-duce to [9]

d d jG pa* *= 4 . (8.4.7)

A particular solution for a static spherically symmetric connection G is

qr

aG =0 , (8.4.8)

where q is the charge in “electrons”. If we now change our units to the gaussian system (1electron = e coulombs), where a=e2 ,

2 = (Gauss)q qee

e er rG a

j a= = =0 , (8.4.9)

which is Coulomb’s law in terms of the charge qe in Coulombs.

8.5. Summary.We have shown that it is possible to introduce, in the geometric unified theory, a hy-

pothesis concerning the mathematical representations of measurable properties of geo-metric excitations. Accordingly, the measurement process of an excitation property arounda geometric matter section is defined as a functional of the geometric current that is thegenerator of the excitation. For pointlike excitations (point particles) the functional re-duces to the Dirac functional, leading to the expression for expectation values. Because ofthe properties of the sections e, the results of the measurement are the eigenvalues of thegenerators (operators) of the excitation.

The angular momentum is the canonical geometric current associated with a variationof the sections generated by a rotation. This leads to the expression of total angular mo-mentum as the differential and matricial operator of quantum mechanics. Similarly, theelectric charge is represented by the canonical current associated with a variation of sec-tions generated by the electromagnetic sector.

The measurement of the angular momentum current of a fundamental excitation of asection (the fundamental representation) results in a quantum number ±½ [±(/2) in otherunits] corresponding to the eigenvalues of the spin generator. Similarly, since electromag-netism is related to other SU(2) contained in SL(2,), using its generators the measure-ment of charge results in the quantum number ±1 [(4pac/k)1/2 in other units] correspond-ing to the eigenvalues of the charge generator. We should emphasize again that the naturalunit of electric charge is the one which makes the electromagnetic potential coincide


with a component of the geometric connection (e disappears for minimal coupling).Of course, these ideas apply to the fundamental representation of the group which

corresponds to spin /2 and charge e. This representation forms a building block fromwhich higher-dimensional irreducible representations may be constructed. For such fieldsthe matrices are of higher dimensions and should have eigenvalues of n /2 and qe.

The picture that emerges from this theory is that differential geometry is the germ ofquantum physics. Through nonlinear field equations, matter determines the geometry andmust obey integrability equations of motion. The equations imply a generalized Diracequation for the sections e which play the role of wave functions representing matter. Anirreducible fluctuation (particle) is an irreducible representation of the group carryingcertain discrete numbers. In this manner the discreteness of quantum theory arises in com-patibility with the continuity of differential geometry. The numerical results of micro-scopic measurements on a geometric excitation necessarily reveal these geometric dis-crete eigenvalues of the geometric group generators.

References.

1 W. Drechsler, Found. Phys. 7, 629 (1977).2 W. Drechsler, J. Math. Phys. 26, 41 (1985).3 W. Drechsler, Class. Quantum Grav. 6, 623 (1989).4 W. Drechsler, Found. Phys. 22, 1041 (1992).5 G. González-Martín, Gen. Rel. Grav. 23, 827 (1991). See chapter 7.6 G. González-Martín, Gen. Rel. Grav. 24, 501 (1992). See chapter 6.7 G. González-Martín,.Phys. Rev. A51, 944 (1995).8 C. N. Yang, Phys. Rev. Lett. 33, 445 (1974).9 See chapter 4..

9. A GEOMETRIC DEFINITION OF MASS.

9.1. Introduction.After discussing the fundamental concepts of spin and charge quanta we are faced with the

concept of mass. This concept plays a fundamental role in relativity as shown by the relationbetween mass and energy and the principle of equivalence between inertial and passive gravi-tational mass. The fundamental action-reaction dynamical process allows us to define theconcept of mass essentially as the product J.A of the current and potential. This definitionsatisfies the relativistic point of view that the rest mass of a system should be a unique conceptdefined in terms of the self energy of the system.

In quantum theory, a parameter in Dirac’s equation is interpreted as the relativistic rest mass,by means of the correspondence principle, This parameter is an immeasurable number, baremass, and a process of renormalization is required to include self energy effects in a correctedphysical mass. Apart from the infinities that appear in renormalization, there is no clear funda-mentally derived relation between these masses and a relativistic definition based entirely interms of energy. We consider that such a relation is only possible within a unified theory.

Our physical geometry leads to the equations of relativistic quantum mechanics and wasshown to provide a mass parameter for a generalized Dirac equation [1, 2 ]. Since the nonlinearfield equation of the theory and its integrability condition determine simultaneously the evolu-tion of the field and the motion of the sources, all self interaction effects are included inprinciple in any given solution. In other words, conceptually, it is not possible to have a solu-tion for the field equation which does not satisfy the equations of motion of the sources. Noadditional self reaction forces are needed to describe the evolution of field and source.

As in general relativity, [3 ] the self interaction difficulties arise when separating the sourcesfrom the field while attempting an approximate solution. An approximation (linearization)which splits these equations in an infinite system of equations of different orders, requiresthe calculation of an infinite number of corrections due to self interactions. This renormal-ization is a consequence of the method of approximation and not due to the nonlinear theory.In particular, it may be possible to define a mass including self interaction, without introduc-ing immeasurable bare masses.

The field equation

( )D ke u eW i** - -= 1 1 (9.1.1)

implies integrability conditions in terms of J. Together with the geometric structure of J,these conditions imply a generalized Dirac equation which, therefore, is not required to beseparately postulated as is normally done in nonunified theories,

( ) ˆˆe e u em a n

m m n ak ¶ G k- + =12 0 . (9.1.2)

As Dirac once pointed out, in a new theory we should let the geometric structure itselfsuggest its possible physical interpretation. To relate the equation of motion density to thestandard Dirac equation, the mass term was interpreted as a parameter associated to theodd part of the connection. The connection defines a unit of mass in the same manner asthe metric defines units of velocity and time. In other words, the connection Gm should

91A Geometric Definition of Mass

have the same units as ¶m and therefore the mass term is naturally expressed in units ofinverse length L-1. This unit of mass, together with the natural unit of time, leads to a unit ofangular momentum which has the value 1 in our natural units. Of course, this geometric unitof angular momentum corresponds to the value of Planck’s constant in any system of units.

It is possible to study the properties of fluctuations or excitations of the geometricelements of the unified theory. Furthermore, if as suggested before, a particle may berepresented as an excitation of the geometry, its physical properties may be determined byits associated fluctuations.

9.2. The Concept of Mass.In relativity, the inertial rest mass of a particle is the norm of its four momentum. In

quantum mechanics, momentum is related to derivatives in space-time. To arrive at a con-cept of mass within our geometric theory, we shall consider the variation generated by atranslation in the base space along the integral curve of the vectors of the space-timetetrad ua. We obtain, in this way, four canonical geometric currents, as defined in theprevious chapter, whose average values over the volume V are

ˆ ˆedVm

a a mq P s= ò1 , (9.2.1)

where a indicates the four Lie derivatives with respect to the vectors ua and P is thecanonical momentum. In adapted coordinates we get

ˆ ˆ ˆe ed e ed xV V

ma a m aq k ¶ s k ¶- -= =ò ò1 1 0 31 1

. (9.2.2)

In particular, consider the trace of the time current,

ˆ e ed xV

q k ¶-= ò1 0 3

00

1 , (9.2.3)

using the equation of motion and assuming, for the moment,

ae¶ = 0 . (9.2.4)

We express the integral in terms of the connection,

e e e u em m a mm m m ak k ¶ k G k = - = - =0 1

0 2 0 , (9.2.5)

ˆ trtr tr J d xe e d xV V

mmmm Gq k G-= =ò ò 31 3

0

1 1 . (9.2.6)

It is clear from the analogy with jmAm that JmGm should have the meaning of energy and that<q0> is the corresponding measured value. Later we shall come back to indicate the mean-ing of the condition assumed, eq. (9.2.4).

This average energy leads to a concept of mass parameter. It was pointed out above,that mass may be defined as a parameter related to the connection. For geometrical rea-sons the unit of connection is inverse length, the same as the operator ¶m. This provides a


geometric (natural) unit of mass in terms of inverse length. Previously we have definedthe mass by

( )trm mmk G= 1

4 . (9.2.7)

It should be noted that since we use time (length) as units of interval, the metric and itsrelated km matrix are dimensionless. Then the mass m has units of inverse length.

We propose now a better definition for the mass parameter, which may be obtainedfrom the integrand in eq. (9.2.6),

( ) ( )tr trm e e Jm mm mk G G-= =11 1

4 4 . (9.2.8)

This definition reduces to the previous one, eq. (9.2.7) under the simplifying assumptionsgiven in chapter 3.

The mass of a particle is defined as a parameter related to a connection and currentwhich solve the nonlinear self interacting system. If the sign of the current in the sourceis changed it is conjectured that the sign of the connection also changes and the massremains positive. For an excitation around a geometric solution, the nonlinear solutionitself (substratum) provides a parameter m0 for the linear excitation equation which maybe considered the bare mass parameter of the particle associated to the excitation.

Since the only element in the algebra with nonzero trace is the unit, we find that

e e mImmk G- = +1 (9.2.9)

and we may write the equations of motion

( )e e e u em m a mm m m m ak k ¶ G k = - = - =1

2 0 , (9.2.10)

as a Dirac equation,

e memmk ¶ = + . (9.2.11)

If we used the standard unit of mass, instead of the geometric unit, a constant appearsin front of the differential operator, which is Planck’s constant . The geometric nature ofPlanck’s constant is determined by the connection in the same manner that the geometricnature of the velocity of light c is determined by the metric.

9.3. Invariant Mass.There is one difficulty with the given definition. The connection G is not a tensor and

under arbitrary change of reference frame the mass is not invariant. For example, assumewe make a transformation by an element gÎSL(4,). The new mass parameter m’ is

( )( ){ } ( )tr trm J m Jm mm m mG ¶ ¶- - - -¢ = + = +1 1 1 11 1

4 4g g g g g g gg . (9.3.1)

The change in mass is given by

( )trm J mmD ¶ -= 11

4 gg . (9.3.2)

If we let


( )exp aaEt=g , (9.3.3)

where the index a runs over all generators Ea, the change in mass may be expressed as

( ) ( )tr tra b aa a am J E J E Em m

m mD ¶ t ¶ t= =1 14 4 , (9.3.4)

aam J m

mD ¶ t= 14 . (9.3.5)

In words we may say that, in order to have an invariant mass, we must restrict the changeg of the reference frame in such a way that the generator of the transformation be orthogo-nal to the current J. For example, this would mean for standard electrostatics that thescalar potential should not change in the transformation. If the current J is odd, the massgiven by eq. (9.2.8) is invariant under SL1(2,) and is, therefore, a Lorentz invariant.

Nevertheless, we realize that the theory applies to matter in the whole universe. If aparticle is associated to an excitation on a local matter current in a dynamic cosmic geomet-ric background, we expect that the corresponding mass should be related to the part of theconnection responsible for the nonlinear local interaction with the local matter current.

It is convenient to separate the current into a part Js corresponding to a local systemrepresenting a particle and another part Jb corresponding to the cosmic background thatpermeates the system. The field equation takes the form,

( )t s bD J JW pa* * *= +4 , (9.3.6)

where Wt is the curvature of the total connection Gt. If there are no local particles, we havea background equation,

b bD JW pa* *= 4 , (9.3.7)

where Wb is the curvature of a cosmic background connection Gb.Far away from the region of the local material current we may consider that its effect is

a small perturbation with respect to the dynamic cosmic background, but this is not thecase very close to a local material frame. In fact, close to it, the dynamic cosmic backgroundmatter may be considered a perturbation, with respect to the nonlinear self interaction inthe area of the tetrad section.

With this in mind, we may define the material potential tensorial form Ls, the differ-ence between the total and background connections,

s t bL G G= - , (9.3.8)

as the element responsible for the self energy of the interaction in the system.It is clear that a difference of connections is a tensor and that the last equation is valid

even in the case where the effect of the background matter is zero. In this case Gb would bea flat inertial connection GI, which would be zero in some reference frames but may havenonzero values in arbitrary reference frames. In general this is not the case due to thepresence of far away matter. Nevertheless, we may write

b I bG G L= + , (9.3.9)

where the potential tensorial form Lb represents the effect of this far away matter. Let usdefine the total dynamic potential tensorial form L as the sum of Ls and Lb.


An invariant expression for the mass is obtained from this total potential L by defin-ing

( )( )

( )( )

( )( )tr

trtr

C

s bC

J Jm J

I

m mm m m

m m

L LL L

k kº = = +

-140

0

g

g . (9.3.10)

The mass parameter is determined by the scalar product, using the Cartan-Killingmetric, of the potential and current which solve the nonlinear self interacting sys-tem. Since the Cartan-Killing metric depends on the representation [4] of the alge-bra A used in the Dirac equation, the parameter m depends on the chosen representa-tion. In some representations the products involved are convolutions rather thanmatrix multiplication.

If the far away matter is negligible, Gb may be taken as a flat connection and thisequation reduces to the original one in those reference frames where Gb is zero. Theextra masslike terms that appear in Dirac’s equation when an improper reference frameis used are similar to the inertial effects which appear in accelerated reference sys-tems. These effects appear in eq. (9.2.11), apart from the mass, as part of a “ficti-tious” inertial interaction. In fact, the flat GI related to the background connectionis called the inertial connection since it is the part responsible for all these inertialeffects.

It should be noted here that a particle is associated to an excitation of the matterlocal frame in a very small region compared with the universe. The local materialframe is the dominant element and the cosmic background matter should be treated asa perturbation which we consider negligible. We shall say that the local matter frameand connection determine the substratum for the excitation. The substratum is domi-nant, and the excitation should be treated as a linear perturbation on the nonlinearsubstratum solution which forms its background. The value of the mass parameter forthe excitation should be related to a constant associated to the substratum solution.

9.4. Momentum Operator.The first definition of mass eq. (9.2.7) was used, in chapter 3 section 4, to show the

relation of a particular solution of the equation of motion, under certain restrictions,with Dirac’s equation. With the new definition these restrictions are not necessaryand a generalized Dirac equation is obtained, as indicated in section 2. On the otherhand, the concept of material particles was associated to frame excitations that strictlyare elements of the algebra but not elements of the group. Because of these reasons,we should revise the relation of the equation of motion with the standard Dirac equa-tion for a particle.

The generalized Dirac equation, eq. (9.2.10), is essentially a relation among ma-trices that represent the elements of the Clifford algebra. In particular, the frame e isan element of the group of automorphisms of the algebra. If particles are associatedto geometric excitations, we should represent them as fluctuations of frames ratherthan as frames themselves. The standard Dirac equation for a particle should corre-spond to a fluctuation of the generalized Dirac equation for matter. If the particlescorrespond to representations of the group with specific quantum numbers, the asso-ciated excitation should correspond to only one of the vectors of the frame, the eigen-vector with eigenvalues equal to the specified quantum numbers. Accordingly, in or-


der to represent a particle, we should restrict the fluctuation of frames to matricesof the form

ˆ

ˆ

hh

h


11

21

0

0 , (9.4.1)

ˆ

ˆ

xx

x


11

21

0

0 , (9.4.2)

where hA indicates two complex 2-spinors. We may combine the 2 columns into a singleone, which may be interpreted as a spinor which satisfies the standard Dirac equation,

ˆ

ˆ

ˆ

ˆ

x

xY

h

h

é ùê úê úê úê ú=ê úê úê úê úë û

11

21

11

21

. (9.4.3)

It is easy to check that the scalar y†y corresponds to the trace of the correspondingmatrices. In fact we have

( )† † †trY Y h h x x= + . (9.4.4)

Under this condition, all the information in the excitation equation is contained in the firstcolumn of the matrices and the matrix equation reduces to a spinor equation. If the con-nection is approximately flat, except for the mass term, the resultant spinor equation ofmotion is

mmmk ¶ Y Y= . (9.4.5)

Since the nonrelativistic limit of this equation is Schrödinger’s equation which repre-sents the free motion of a particle of mass m, it is clear that Ehrenfest’s theorem [5 ] isvalid. As a result, the measurable values obey Newton’s second law of motion. In a senseour theory says that there is a correspondence with mechanics, and we must interpret <-ih¶m> as the classical momentum and <-ih¶0> as the classical energy. It is clear that theoperators representing momentum and position satisfy the Heisenberg commutation rela-tions,

[ ],p x i= - , (9.4.6)

[ ],E t i= . (9.4.7)

Now we are in a position to indicate that eq. (9.2.4) means, in this theory, that theparticle associated to the fluctuation specified has zero lineal momentum and its energy


equals its rest mass, as should be the case in relativistic quantum mechanics.The physical uncertainty relations are a consequence of the geometry rather than its

source. As indicated in section 8.1, it may not be claimed that the geometry is “fuzzy”. Itmay be said that the association of a differential operator with momentum in quantummechanics has a geometrical character. It should also be clear that the imaginary i inSchrödinger’s (or Dirac’s) equation arises from the use of the geometric algebra to con-struct the matter current, as shown in previous chapters. Furthermore, the ¶m is present inthe equation because it arises from the conservation equation for the current J.

9.5. Summary.We presented a definition of mass, within our geometric unified theory, in terms of the

concept of self energy of the nonlinear interaction. A geometric linear excitation of a non-linear substratum solution obeys Dirac’s equation with the appropriate mass term.

References

1 G. González-Martín, Gen. Rel. and Grav. 22, 481 (1990). See chapter 3.2 G. González-Martín, Gen. Rel. Grav. 23, 827 (1991). See chapter 7.3 A. Einstein, L. Infeld, B. Hoffmann, Ann. Math. 39, 65 (1938). 4 R. Gilmore, Lie Groups, Lie Algebras and some of their applications (John Wiley and

Sons, New York), p. 248 (1974).5 A. Messiah, Quantum Mechanics, translated by G. M. Temmer (North Holland,

Amsterdam), p. 216 (1961).

10. QUANTUM GEOMETRODYNAMICS.

10.1. Introduction.Now we discuss, in particular, how the theory gives a quantum field dynamics (QFD)

including its probabilistic interpretation [1 ]. A basic feature of the proposed theory isnonlinearity. A solution cannot be obtained by the addition of two or more solutions andtherefore it is not possible to build exact solutions from small subsystems. Neverthe-less, it is possible to study its local linearized equations which represent excitations thatevolve approximately under the influence of effects inherited from the nonlinear equa-tion.

We showed in chapter 6 that if we take into consideration the geometrical structureof the principal fiber bundle E and the affine fiber bundle W, related to the algebraicstructure of their fibers, a process of variations of the equations of the theory leads to aninterpretation of the extended Jacobi fields as quantum operators. It was also shown thatit is possible to define a bracket operation which becomes the commutator for the Jacobifields associated to the connection and becomes the anticommutator or commutator forthose associated to the frame. This bracket operation leads the quantization relations ofquantum field theory for bosonic interaction field and fermionic matter fields.

If we represent particles as excitations, then an interaction between two particlescorresponds to the interaction of two of these geometric excitations, generated by thesegeometric Jacobi operators. The geometric coupling is between the connection and theframe. The interaction among particles (frame excitations) is mediated by connectionexcitations.

10.2. Geometric Relations.

10.2.1. Product of Jacobi Operators.The product ab in the ring A has been taken, in general, as the Clifford product [2].

Since the fiber bundle of frames E is a principal bundle and its vertical tangent bundle TEhas for fiber a Lie algebra structure inherited from the group, it is necessary that thechosen product be closed in the algebra so that the result be also valued in the Lie alge-bra. Geometrically we should specialize the ring product, as indicated before, to be theLie product. Then the product ab is zero whenever its gradation is even. The bracketsurvives only when the commutator ab is nonzero because its gradation is odd and corre-sponds to the commutator. In other words the ring product obeys

[ ], na b a b ab· = ¹ 2 , (10.2.1)

na b ab· = =0 2 , (10.2.2)

With this product, the bracket defined before satisfies

{ } { }, ,V W W Va b b a= , (10.2.3)


which is the anticommutator, for the matter Jacobi operator fields.The fiber bundle of connections W is an affine bundle and the ring product associated

to the fiber of TW is commutative. Therefore, the bracket is the commutator for theconnection Jacobi operator fields.

10.2.2. Commutation Relations.If we take Schwinger’s action principle [3] , we obtain the quantization relations by

requiring that the quantum operators Y be Jacobi vector field operators,

( ) ( ){ },x yY dY = 0 , (10.2.4)

( ) ( ){ } ( ), ,x y x ym mY P d= - . (10.2.5)

As discussed before, the Jacobi vectors Vs represent fluctuations of sections s of thefiber bundle E. The corresponding jet prolongation jV of an extension V of Vs is a vector fieldon JE which acts as an operator on functions on JE. A Jacobi field, as a vertical vector on thesection s(M), may be considered as a displacement of s. Similarly, its jet prolongation may beconsidered as a displacement of js, in other words a variation,

( )j

jj

d sV s

dl= . (10.2.6)

This linear action of the Jacobi fields on the substratum sections allow us to associatethem with quantum theory objects. We may consider the substratum section as the state |ñ ofthe physical field system and the Jacobi fields as physical linear operators Y which have anatural geometric action on the states.

A given physical matter section is expressed in terms of frame sections. A measurementis the projection of the observed frame on a similar reference frame which is also a physicalsystem. The projection results are scalars. Hence both frames may be considered to evolveunder the action of the same group which acts on the matter sections. This represents theknown equivalence of the active and passive views of evolution. We may consider the physi-cal system evolving in a fixed frame or, equivalently, we may consider the system fixed in anevolving reference frame.

We may choose the reference frame so that the substratum section does not evolve. Inthis condition the substratum section remains fixed and the Jacobi operators obey the firstorder linearized equations of motion. (Heisenberg picture).

The Jacobi vector fields and their jet prolongations transform under the adjoint represen-tation of the group which transforms the sections. We may use these transformations to anew reference frame where the Jacobi operators do not evolve. In this condition, the timedependence of the Jacobi operators may be eliminated and the substratum sections obeyequations of motion, indicating that the state is time dependent (Schrödinger picture).

10.3. Geometric Field Dynamics.

10.3.1. “Free” Particles and Fields.It should be clear that an exact solution of the proposed problem is not possible with

this linearized technique. The reason for this is the presence of the nonlinearity of the

99Quantum Geometrodynamics

self interaction in the equations. We must consider only approximate solutions. The in-teraction between frame excitations is mediated by connection excitations. Accordingto QFT the interaction is imposed on special fields called “free fields” which in QED arethe electron field and the radiation or photon field. Here we have to discuss which eframe and connection excitation sections correspond to “free” fields. A frame exci-tation e representing a “free” particle (or field) must be chosen. Similarly, we mustchoose a connection excitation representing a “free” field (or particle).

The equation of motion

emmk = 0 , (10.3.1)

includes self interaction terms. It is not possible to set the substratum connection tozero because it will automatically eliminate the self energy and the mass parameter. Weunderstand a “free” particle (or field) as a frame excitation with the correct mass param-eter determined by the substratum [4]. The simplest approximation is then to assume thatall self interaction effects of the substratum connection , to first order, are concen-trated in the single mass parameter. The fluctuation equation is

interactione m emmk ¶ d d= + . (10.3.2)

It is natural to consider a “free” frame field e as one which satisfies the previousequation without the interaction term due to G. A solution to this free equation providesa frame excitation section which can be used with our technique. The linearized fluctua-tion equations, with the interaction term, determine the local evolution of the connec-tion and frame excitation fields.

A similar equation for should be obtained from the field equation,

D JW pa* *= 4 , (10.3.3)by linearization at some particular point around a substratum connection solution,

( ) interactiond d JdG pa d* *+ = 4 . (10.3.4)

It is natural to consider a “free” connection field as one which satisfies the previ-ous equation with just the substratum interaction and J equal zero. Solutions to the thisequation which may provide “free” connection excitation sections are discussed in chap-ter 14.

10.3.2. Quantum Electrodynamics.In order to get the standard QED from the geometric theory, we have to reduce the

structure group to one of the 3 U(1) subgroups in the SU(2) electromagnetic sector. Thecorresponding u(1) component of the fluctuation of the generalized connection is theelectromagnetic potential Am. Similarly, an electromagnetic fluctuation of the matter sec-tion determines a fluctuation of the generalized current. This fluctuation must be gener-ated by one of the three electromagnetic generators, for example k5. The standard QEDelectric current is the electromagnetic sector component of this current fluctuation,

( ) ( ) ( )( )

tr tr , tr

tr

j J J J

J

d k k k d k k k k k k k k

k

é ù= = =ê úë û

=

1 2 3 1 2 3 5 5 1 2 31 1 1 14 4 2 4

014

. (10.3.5)


It was already shown in section 3.6 that the standard electric current in quantum theoryis related to the k0 component of the generalized current, [5]

( )tr trJ e em m mk k k Yg Y=0 01 14 4 , (10.3.6 )

which is the electric current for a particle with a charge equal to one quantum in thegeometric units.

It is known that the lagrangian for the first variation of the Lagrange equations is thesecond variation of the lagrangian. For both free fields, the corresponding second varia-tion of the lagrangian gives the free Maxwell-Dirac field lagrangian. Since the currentvariation has only a k0 component given by eq. (10.3.6 ) and the connection variation hasAm for k0 component, the second variation or interaction hamiltonian gives, in terms ofthese interaction field Jacobi operators dG, dJ,

( ) ( )tr trL J A A Hm mm md dGd k k Yg Y Yg Y= = = - º2 0 01 1

4 4 , (10.3.7)

which is the standard interaction hamiltonian in QED.The notation should distinguish the linear variations from the nonlinear sections. The

only invariant effect of the self interaction in the free motion of the fields is the mass param-eter m. The remaining effects of the total physical interaction, associated with the variationfields, correspond to an effective net interaction energy that may be written as

e e mI Hmmk G - . (10.3.8)

The first part of H is the total physical nonlinear interaction. The meaning of thesecond part is that massive self energy is not included in the variation and this method(or QED), by construction, is not adequate for a calculation of bare mass.

We may choose a reference frame so that both substratum sections, which includethe corresponding self interactions of the two “free” systems, also include an observerdependent free motion for the two excitations. The two substratum sections are not solu-tions to the interacting system, they are free background solutions. This situation corre-sponds to an interaction picture in QED. By choosing an appropriate reference frame,the substratum sections (states) have the “free motion”, including self interaction ef-fects and the excitation Jacobi operators represent the dynamics of the interaction of thetwo systems, excluding the free self interactions.

We have obtained a framework of linear operators (the prolongations of Jacobi fields)acting on sections forming a Banach space (states) where the operators obey the bracketcommutation relations, commutators for A and anticommutators for Y. Furthermore, thegeometric lagrangian of the theory reduces to a lagrangian in terms of operators which isthe standard lagrangian for QED.

From this point we can proceed using the standard techniques, language and notationof QED, equivalent to the geometric techniques, for any calculation we wish to carry inthis approximation of the geometric theory. The result of the calculation should be inter-preted physically, in accordance with the geometric ideas.

10.3.3. Statistical Interpretation.Our next task is to obtain a statistical interpretation for the geometrical variations.

The physical significance of the local sections in the nonlinear geometrical theory in-cludes the holistic influence of the total universe of matter and interactions. The geom-


etry of the theory, including the notion of excitations, is determined by all sources in theuniverse. Within this geometric theory and its interpretation, it is not appropriate to con-sider the sections to be associated with a single free particle or excitation. Rather, theyare associated to the global nonlinear geometric effects of bulk matter and radiation.

This is not the customary situation in which physical theories are set. It is usual topostulate fundamental laws between free elementary microscopic objects (particles). Inorder to translate the global universal geometry to the usual microscopic physics, sincewe associate particles to geometric excitations, we shall establish what we call a “manyexcitations microscopic regime”, distinguished from the situation described in the pre-vious paragraph.

Statistics enter in our approach in a manner different than the usual one, where funda-mental microscopic physical laws are postulated between idealized free elementary par-ticles and statistical analysis enters because of the difficulties that arise when combin-ing particles to form complex systems. Here, the fundamental laws are postulated geo-metrically among all matter and radiation in the system (the physical universe) and sta-tistical analysis arises because of the difficulties and approximations in splitting thenonlinear system into “free” elementary microscopic linear fluctuation subsystems. Asa consequence of this holistic situation, the results associated to fundamental excita-tions must have a natural (classical) statistical character corresponding to that of quan-tum theory.

We consider that we may work in two different regimes of the geometrical theory.One is the holistic geometrical holophysical regime, not representing particles, wherethere are exact nonlinear equations between the local frame sections, representing mat-ter, and a connection section representing the interaction. The other is the many-excita-tions microscopic regime where we have linearized approximate equations among exci-tation sections which are variations of the frame and connection sections and representboth Fermi-Dirac and Bose-Einstein particles and fields.

An excitation, as a linear equation, has a lowest energy solution which is a spacetangent to the substratum solution at some point. In the many-excitations regime, thenonlinear local effects of the interaction remain hidden in the substratum and may beseen as linear approximate local effects on the collection of excitations. This is theexcitation vacuum. The number of excitations is naturally very large and the cross inter-actions among them preclude an exact treatment for a single excitation. Instead it isnecessary to treat any excitation section as one among a large ensemble and use classicalstatistical theory including Fermi-Dirac and Bose-Einstein statistics.

The geometric excitation sections form statistical ensembles of population densityni. It is not possible to follow the evolution of any single one because of the previousarguments. It is absolutely necessary to use statistics in describing the evolution of theexcitations. The situation is similar to that of chemical reactions or radiation, whereparticles are statistically created or annihilated.

There are adequate classical statistical techniques for describing these processes.The mathematical methods used in physical chemistry may be applied to the geometricexcitations. In particular we may use the theory of irreversible thermodynamics [6 ] tocalculate the rates of reaction between different geometric e, excitation sections.The process is described by the flux density characterizing the flow of n excitationsbetween two systems or reaction rate,

dndt

= . (10.3.9)


It is also necessary to introduce a driving force function, , which is called affinityand represents differences of thermodynamic intensive parameters. The use of these clas-sical concepts is derived from fundamental statistical analysis. In case of equilibriumbetween two different subsystems, both the affinity and the flux are zero.

The identification of the affinity is done by considering the rate of production ofentropy s,

kkk

k

d nds sdt n d t

¶¶

= = , (10.3.10)

from which the affinity associated to any given excitation is

kk

k

s uu n T

¶ ¶ m¶ ¶

= = , (10.3.11)

where u is the energy, T is the temperature and m is the excitation potential, similar to thechemical potential used to determine the statistics of chemical reactions.

The statistical flux is a function of the affinity and we see that the statistics of reac-tion of geometric excitations should depend on the classical geometric energy of theexcitations. An excitation is an extensive parameter x which has associated an intensiveparameter F thermodynamically related with the change of energy,

du Fdx= - . (10.3.12)At a point, the excitation is a function of time which may always be decomposed in har-monic functions.

In some cases, for linear markoffian systems (systems whose future is determined bytheir present and not their past), the flux is proportional to the affinity

= ¶D D

¶ 0

(10.3.13)

and the calculation of reaction rates is simplified, indicating explicitly the dependenceof the flux on the difference of excitation potentials.

10.4. Physical Particles and Waves.A physical particle (f.e. a photon or electron) is an excitation of the corresponding

substratum field. In geometric terms, a Jacobi vector Y, associated to a variation of asection e or w, represents the particle. As indicated in the previous section, it is notpossible to follow the evolution of a single excitation Y. The classical statistical ap-proach is to treat the radiation as a thermodynamic reservoir of excitations. This ap-proach was used in blackbody radiation [7, 8, 9 ] and was the origin of Planck’s quantumtheory. In recent years, a similar idea lead to introduction of stochastic quantization [10,11 ], which has been shown to be equivalent to path integral quantization. Our approach isdifferent, for example we do not introduce evolution along a fictitious time direction asdone in stochastic quantization. We rely on the existence of a global nonlinear geometrywhich makes a practical necessity the statistical treatment of the linearized equations de-scribing the evolution of microscopic subsystems.

The particle fields form ensembles of frame excitations e and connection excita-


tions in contact with their frame and connection substrata which serve as radiationreservoirs. The equilibrium of the total system, reservoirs and fields, is determined bythe equality of the excitation potentials associated to the geometric excitations formingthe frame and connection subsystems. When there is no equilibrium there is a flow ofexcitations between the reservoir and the field. The techniques of irreversible thermody-namics relate this excitation flux density to the affinity, which depends on the differenceof the corresponding excitation potentials. We also assume that we are dealing withmarkoffian systems. This approximation has been used successfully in the quantum theoryof damping [12, 13 ] for laser systems.

In order to calculate the excitation potential we need to express the energy in termsof the number of excitations. The geometric excitations are representations of the groupG=SL(2,) induced from subgroups. As indicated in section 7.2, these representationsare characterized by quantum numbers. The Jacobi vector operators Y representing theconnection and the frame may be decomposed in its irreducible creation operators ausing a Fourier series wave decomposition.

It should be clear that the geometric commutation or anticommutation relations ofthe Jacobi vector operators Y imply similar relations for the fundamental oscillator am-plitudes (Fourier components) a. Consequently the energy of these geometric oscilla-tors is quantized, with the same results of the quantum theory. These amplitudes becomecreation or annihilation operators a. Under these conditions the particle number operator,

†N a a= , (10.4.1)has discrete eigenvalues n which indicate the number of particle quanta and determinethe energy of the oscillators. The energy quanta, for large n, are the eigenvalues

n ne n næ ö÷ç= + »÷ç ÷çè ø

12

, (10.4.2)

in geometric units where Planck’s constant equals 1. The energy of the excitations hasquanta of value proportional to the frequency n.

In general, the wave energy depends on the excitation amplitude Y. Thus, the prob-ability of occurrence of a single detection event, implicit in the reaction rate deter-mined by the excitation potential, depends on the excitation amplitude. This is the sig-nificance of the probabilistic interpretation of quantum fields. Two geometric waves ar-riving at a point by alternative paths have a well known phase difference which allow us towrite for the resultant amplitude,

i ie ef fY Y Y -= +0 0 . (10.4.3)

The geometric field has an amplitude which is modulated throughout certain spaceregions because of the interference pattern produced by the waves. This means that theenergy is not homogeneously distributed but concentrated in the regions of higher am-plitudes. For large n, when the amplitude Y0 is expressed in terms of creation operatorsa, we obtain an expression for the modulated energy u,

† † cos cosa a n uY Y f f= µ µ2 24 . (10.4.4)

This equation shows that, as a linear harmonic oscillator of any type, the conservedtotal energy of our geometric excitations also equals the maximum potential energy givenby the square of the displacement amplitude Y of the extensive parameter. This potential


energy amplitude is the physical interpretation of the Jacobi vector operator Y.We should consider a local excitation potential defined within domains of volume,

determined by a correlation distance where the density of energy u may be taken as con-stant. Then the excitation potential is

cosun

¶m f

¶= µ 2 , (10.4.5)

where n is the eigenvalue of N or particle number. This potential should also apply toweakly coupled macroscopic systems where classical statistics is required because ofdifficulties in treating a nonlinear system.

In order to illustrate the statistical character of the linearized regime within the con-cepts of the geometric theory, we discuss the wave corpuscular duality of matter and itsinteraction using Young’s double interference experiment. We have to consider the pos-sibility of an m-particle wave. To clarify the situation and avoid confusion with the con-cept of particle, we define an m-quanta excitation as the tensor product representation ofm fundamental (1-quantum) representations. The Jacobi excitation field amplitude is thenan m-product representation.

In particular we consider the case of a 2-quanta excitation experiment. We have acorrelation between alternative paths for the first quantum but also, in addition, correla-tions among the alternative paths for the 2-quanta excitation formed by the product ofthe two 1-quantum excitations.

For any particle field there may be excitations which correspond to multiple quantaand lead to Schrödinger’s “entangled states”. There are modern techniques for the ex-perimental preparation of multi-quanta entanglements [14, 15, 16, 17, 18 ]. In particular,we may consider a two-quanta interferometer illustrated in figure 2. At the center thereis a source of decaying quanta, with a vertical extension d. Two collimating screens, eachwith a pair of holes, offer alternative paths to a given pair of quanta which are finallydetected in two screens.

The energy of the excitation leads to the expression for the excitation potential,

cos sinlpm a

læ ö÷çµ ÷ç ÷çè ø

2 . (10.4.6)

We consider the case of a 2-quanta excitation experiment, in particular the case oftwo Young’s experiments side by side. We not only have a correlation between alterna-tive paths through A and B for the right quantum but also correlations among paths A or Band A’ or B’ for the 2-quanta excitation formed by the product of the two 1-quantumexcitations. If a quantum is produced at height x above the center line, a quantum may bedetected in the right screen at height y and another in the left screen at height z. Thephase difference f for the right side quantum has a contribution from the screen angle

P

lyr

pf

l= , (10.4.7)

and another from the source angle,

P

lxr

pf

l= , (10.4.8)


leading to, for small y and z,

( )P x ypq

fl

= + , (10.4.9)

where q is the angle subtended by the two holes. A similar expression is obtained for theleft side quantum,

( )P x zpq

fl¢ = + . (10.4.10)

The amplitude of the Jacobi excitation, which is a product representation of the twoexcitations, is

( ) ( )cos cosd

d

adx x y x z

dpq pq

Yl l

+

-

é ù é ùê ú ê úµ + +ê ú ê úë û ë û

ò2

2

. (10.4.11)

If d is much smaller than l/q the integral gives the product of two Young’s patterns asit should. If, on the contrary, d is much larger than l/q, this integral gives,

( )cosaz y

pqY

læ ö÷çµ - ÷ç ÷çè ø2

. (10.4.12)

We obtain for the excitation potential

( )cosaz y

pqm

l

é ùæ ö÷çê úµ - ÷ç ÷çê úè øë û

2

2. (10.4.13)

The probability for the simultaneous detection of quanta at P and P’ is given by thisexcitation potential. There are regions of high and low probabilities because of the inter-ference of the excitation amplitudes. These cases just illustrated for 1-quantum and 2-quanta interferometers give a general characteristic of section excitations in the geo-

y

P

zP’

xd/2

-d/2

AB’

A’ B

Figure 2Young’s doubleinterferometer.


metrical theory. Similarly, the probability for the detection of correlated quanta of en-ergy E and E’ or momentum k and k’ should be given by an excitation potential. There areno exact equations for physical situations with a single excitation. Any real physical situ-ation in the microscopic linear regime of the theory requires the use of statistics. Theoutcome of transition experiments depends on the excitation potentials of the systems.The requirement of statistics necessarily leads to the probability interpretation of quan-tum theory. If, within a particular experimental setup, we can physically distinguish be-tween two excitation states, there is no room to apply statistics and there follows theabsence of the interference pattern. This is the content of Feynman’s statement aboutexperimentally indistinguishable alternatives [19 ].

Fundamentally the quantum statistics essentially are the classical statistics of fermi-onic and bosonic section excitations in this physical geometry. The objections raised byEinstein [20, 21 ] to the probabilistic interpretation of quantum mechanics are resolvedautomatically because the statistics enter due to the lack of detailed knowledge of thestates of many excitations.

Equation (10.4.5) for the excitation potential also applies to certain macroscopicsystems and explains their probabilistic behavior. For example [22, 23], oil droplets mayproduce ripples on a liquid surface which react back on the droplet producing an effec-tive nonlinear interaction field similar to excitation waves on the substratum background.The excitation potential produced by the ripple waves should determine probability equa-tions for interference effects in experiments analogous to the one discussed here

10.5. Summary.The nonlinear geometric equations apply to the total universe of matter and radiation.

If we work with excitations, this implies the need to use statistical theory when consid-ering the evolution of small subsystems. The use of classical statistics, in particulartechniques of irreversible thermodynamics, determine the probability of absorption oremission of a geometric excitation through the excitation (chemical) potential, a func-tion of the classical energy density.

Emission and absorption of geometric excitations imply discrete changes of certainphysical variables because they are representations of a group, but with a probabilitydetermined by the energy density of the excitation. Hence, this geometric theory deter-mines the fundamental statistical nature of quantum theory. In addition it also geometri-cally determines discrete quanta of energy, spin, electric charge and magnetic flux. Useof harmonic analysis introduces creation and annihilation operators for the associatedparticle and field waves.

The geometric techniques reduce to the QFT techniques. Nevertheless, their use islimited to perturbative excitation and therefore exclude their application to selfinteractionand in particular to the calculation of bare masses.

In particular, excitations of the geometry determine the theory of quantum electrody-namics. The connection excitations and the frame excitations reduce, respectively, tothe electromagnetic field operator and electron field operator. Because of the inherentgeometric algebraic structure these operators obey the standard commutation rules ofQED. The energy of the electromagnetic excitations is n.

The classical nature of the wave excitation potential determines that certain macro-scopic systems also show a quantum mechanical behavior.


References

1 G. González-Martín, ArXiv physics/0009042, USB Report SB/F/272-99 (1999).2 See section 5.3.3 See section 5.6.4 See section 9.3.5 See section 3.6.6 H. B. Callen, Thermodynamics, (J. Wiley & Sons, New York), p. 289 (1960).7 M. Planck, Verh. Dtsch. Phys. Gesellschaft, 2, 202 (1900).8 S. Bose, Z. Physik 26, 178 (1924).9 A. Einstein, Preuss. Ak. der Wissenschaft, Phys. Math. Klasse, Sitzungsberichte, p,

18 (1925).10 G. Parisi, Y. S Wu, Sc. Sinica, 24, 483 (1981).11 P. H. Damgaard, H. Huffel, Physics Reports 152, 229 (1987).12 W. H. Louisell, L. R. Walker, Phys. Rev. 137B, 204 (1965).13 W. H. Louisell, J. H. Marburger, J. Quantum Electron. QE-3, 348 (1967).14 M. A. Horne, A. Zeilinger, in Proc. Symp. on Foundations of Modern Physics, P.

Lahti, P. Mittelstaedt, eds., (World Science, Singapore), p. 435 (1985).15 C. O. Alley, Y. H. Shih, Proc. 2nd Int. Symp. on Foudations of Quantum Mechanics in

the Light of New Technology, M. Namiki et al, eds. (Phys. Soc. Japan, Tokyio), p. 47(1986).

16 M. A. Horne, A. Shimony, A Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).17 R. Ghosh, L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).18 Y. H. Shih, C. O. Alley, Phys Rev. Lett. 61, 2921 (1988).19 R. P. Feynman, The Feynman Lectures on Physics, Quantum Mechanics. (Addison

Wesley, Reading), p.3-7 (1965).20 A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935).21 N. Bohr, Phys. Rev. 48, 696 (1936).22 Y. Couder, A. Eddi, J. Moukhtar, A macroscopic type of wave-particle duality, research at

Laboratoire Matiere et Systemes Complexes (Paris) (2014); http://www.msc.univ-paris-diderot.fr.

23 A. U. Oza, R. R. Rosales, J. W. M. Bush, J. Fluid Mech., 737, 552 (2013).

11. FRACTIONAL QUANTUM EFFECTS.

11.1. Introduction.As shown in chapter 7, the excitations representing particles have quantized magnetic

flux and charge. This quanta determine the quantum Hall effect. It is possible that thesemagnetic flux quanta correspond to definite total angular momentum levels and be associ-ated to electron orbits. In this case, it would be possible to observe their effects in anelectron gas in a constant intense magnetic field at low temperatures. The theory of theIQHE and the FQHE has evolved by the explicit construction of quantum state wave func-tions that describe many features of this phenomena [1, 2 , 3]. Nevertheless it may still bepossible to explain certain facts from general principles, as the extraordinary accuracy ofthe experimental results suggests. Of course, detailed microscopic analysis and wave func-tions are still necessary for a complete description. It would be like using the theory ofrepresentations of the Lorentz group to characterize the electron states in a central poten-tial instead of using actual wave functions. In this manner we expect to display the generalprinciple involved. In fact, it was the generalization of the Lorentz group (automorphismsof Minkowski space) to the group of automorphisms of the geometric algebra of Minkowskispace and the corresponding physical interpretation [4, 5], that indicated that any massiveparticle (or quasi particle) must carry not only quanta of angular momentum but alsoquanta of charge e and quanta of magnetic flux, h/2e , (one or more), both intrinsicand orbital.

11.2. Magnetic Flux Quanta.Usually the motion in a constant magnetic field is discussed in cartesian coordinates in

terms of states with definite energy and a definite linear momentum component. The re-sultant Landau energy levels are degenerate in terms of the momentum component. Addi-tionally, for the electron, the Landau energy levels are doubly degenerate except the low-est level. In order to use definite angular momentum states, the problem is expressed incylindrical coordinates and the energy of the levels is [6 ],

( )eBU n m sMc

= + + +12

, (11.2.1)

where B is the magnetic field along the positive z symmetry axis, m is the absolute valueof the orbital angular momentum quantum number, also along the positive z axis, n is anonnegative quantum number associated to the radial wave function and s is the spin. Thisexpression has the peculiarity that even for zero m there may be energy quanta associatedto the radial direction.

The equations of motion of charged particles in a constant magnetic field, in quantumor classical mechanics, do not determine the momentum or the center of rotation of theparticle. These variables are the result of a previous process that prepares the state of theparticle. We can idealize this process as a collision between the particle and the field.Since the magnetic field does not do work on the particle, the energy of the particle is

109Fractional Quantum Effects

conserved if the system remains isolated after the initial collision. The angular momentumof the particle with respect to the eventual center of motion is also conserved. The valuesof angular momentum and energy inside the field equal their values outside. This impliesthere is no kinetic energy associated to the radial wave function, as the equation allows.Therefore, among the possible values of energy we must exclude all values of n except thevalue zero. This means that the energy only depends on angular momentum, as in theclassical theory.

The degeneracy of energy levels, apart from that due to the momentum componentalong the field, is the one due to the spin direction, as in the case of cartesian coordinates.Only moving electrons make a contribution to the Hall effect, hence we disregard zeroorbital angular momentum states. Each degenerate energy level has two electrons, onewith spin down and orbital momentum m+1 and the other with spin up and orbital angularmomentum m. Each typical degenerate energy level may be considered to have a totalcharge q equal to 2e, total spin 0 and a total integer orbital angular momentum 2m+1.Using half integer units,

( )( )zL m= +2 2 1 2 . (11.2.2 )

In the classical treatment of motion in a magnetic field, it is known that the classicalkinetic momentum of a particle has a circulation around the closed curve corresponding toa given orbit, which is twice the circulation of the canonical momentum. The latter, inturn, equals the negative of the external magnetic flux enclosed by the loop [7 ]. That is,for the given number of quanta of orbital angular momentum in a typical level, we shouldassociate a number of orbital external magnetic flux quanta. In accordance with the previ-ous equation, we assign a flux FL to a degenerate level,

( )( )Lhm eF = - +2 2 1 2 . (11.2.3)

When we have an electron gas instead of a single electron, the electrons occupy anumber of the available states in the energy levels depending on the Fermi level. Theorbiting electrons are effective circular currents that induce a magnetic flux antiparallel tothe external flux. The electron gas behaves as a diamagnetic body, where the applied fieldis reduced to a net field because of the induced field of the electronic motion. Associatedto the macroscopic magnetization M, magnetic induction B and magnetic field H we intro-duce, respectively, a quantum magnetization flux FM , a quantum net B flux FB and aquantum bare H flux FH per level, which must be related by

B H MF F F= + . (11.2.4 )

The first energy level of moving electrons is nondegenerate, corresponding to oneelectron with spin down and one quantum of orbital momentum. In order for the electronto orbit, there has to be a net (orbital) flux inside the orbit. If the flux is quantized, theminimum possible is one quantum of orbital flux for the electron. In addition, anotherquantum is required by the intrinsic flux associated to the intrinsic spin of the electron.Therefore without further equations, the minimum number of net quanta in this level is 2and its minimum net magnetic flux is

( )B

he

F = 22

. (11.2.5)


It should be noted that the minimum flux attached to the electron, in this state, istwice the flux quantum. In other words, the attached flux is split into orbital andintrinsic parts.

The quantum magnetization flux is, considering its induced flux FL opposed toFH, two (-2) quanta for the orbital motion of the single electron and one (1) addi-tional quantum for its intrinsic flux FS,

M L S

h h he e e

F F Fæ ö÷ç= + = - + = -÷ç ÷çè ø

22 2 2

, (11.2.6)

which gives, for this nondegenerate level, the relation,

B MF F= 2 , (11.2.7)

indicating an equivalent magnetic permeability of 2/3.This reasoning is not applicable to the calculation of FB for the degenerate levels

in the electron gas, rather, we somehow have to find a relation with FH . To determinethis exactly we would need to solve the (quantum) equation of motion for the elec-trons simultaneously with the (quantum) electromagnetic equation determining thefield produced by the circulating electrons. Instead of detailed equations we recog-nize that, since flux is quantized, the orbital flux must change by discrete quanta asthe higher energy levels become active. As the angular momenta of the levels in-crease, a proportional increase in any two of the variables in equation (11.2.4 ) wouldmean that the magnetic permeability does not vary from the value 2/3 determined bythe nondegenerate level. Therefore, in general, the flux increases should obey thefollowing quantum condition: the flux FB may deviate discretely from the propor-tionality indicated by the last relation, by an integral number of flux quanta accord-ing to,

B MF F DF= +2 , (11.2.8)

where DF is an indeterminate quantum flux of the electron pair in the degenerate level.The magnetization flux FM has FL as upper bound,

( )( )M Lhm eF F£ + =2 2 1 2 , (11.2.9)

and we may indicate the attached quanta by the inequality

( ) ( )Bh hm e eF dé ù£ + +ê úë û

2 2 2 1 2 , (11.2.10)

where d is an integer indicating a jump in the flux associated to each electron in the pair.Whatever flux FB results, it must be a function of only the flux quanta per electron

pair. A degenerate energy level consists of the combination or coupling of two electronswith orbital momentum levels m and m+1, in states of opposite spins, with 2m orbital fluxquanta linked to the first electron and 2m+2 linked to the other electron. The pair, has2(2m+1) flux quanta, in terms of a nonnegative integer m. Hence the only independentvariable determining the jump in the flux FB is 2(2m+1). In order for FB to be only afunction of this variable, the integer d must be even so that it adds to m, giving the pos-sible values of the net flux of the level,


( )( )Bh

eF m= +4 2 1 2 , (11.2.11)

The physical interpretation of the last equation is the following: If the external magneticfield increases, both the effective net magnetic flux B and the effective magnetizationflux M per electron increase discretely, by quantum jumps, and the equivalent permeabil-ity remains constant.

The number of possible quanta, indicated by f, linked to this level called a superfluxedlevel, is

( )f m= +4 2 1 , (11.2.12)

where the flux index m has undetermined upper bound. The number of electric chargequanta per level, indicated by q, is

q = 2 . (11.2.13)

11.3. The Fractional Quantum Hall Effect.Each single electronic state is degenerate with a finite multiplicity. This gives the popu-

lation of each typical energy half-level corresponding to a single electron, indicated byN0, when it is exactly full. It is clear that if the degeneracy in the energy levels is lifted, bythe mechanism described in the references, half levels may be filled separately and ob-served experimentally.

Since there is the same number of electrons N0 in each sublevel we may associate oneelectron from each sublevel to a definite center of rotation, as a magnetic vortex. We havethen, as a typical carrier model, a system of electrons rotating around a magnetic center,like a flat magnetic atom (quasi particle?). The validity of this vortex model relies in thepossibility that the levels remain some how chained together, otherwise, the levels wouldmove independently of each other. A physical chaining mechanism is clear: If we havetwo current loops linked by a common flux, and one loop moves reducing the flux throughthe other, Lenz’s law would produce a reaction which opposes the motion of the first loop,trying to keep the loops chained together.

The total electronic population when the levels are exactly full is,

( ) oqN N N Nn n= + = +0 02 2 1 , (11.3.1)

where n doubly degenerate Landau energy levels are full and where the first N0 corre-sponds to the nondegenerate first Landau level. The charge index n is an integer if thehighest full level is a complete level, or a half integer if the highest full level is a half levelor zero if the only full level is the nondegenerate first level.

The flux linked to this vortex, whatever quantum theory wave functions characterizethe states of the electronic matter, should have quanta because the system is a representa-tion that must carry definite quanta of charge, angular momentum and magnetic flux. Inother words, the vortex carries quanta of magnetic flux. The flux linking a particular vor-tex is equal to the net flux FB chained by the orbit of the electron pair corresponding to thehighest level. The value of f, the number of net quanta linked to a vortex, is expressed byequation (11.2.12) depending on the value of the flux index m for the highest level in thesystem.

We now make the assumption that the condition for filling the Landau levels should be


taken in the sense given previously [4, 8], by counting the net flux quanta linked to theorbital motion of all N0 vortices. The filling condition is determined by conservation offlux (continuity of flux lines). The applied external flux must equal the total net fluxlinked only to the highest level. This determines a partial filling of the levels, in the ca-nonical sense (only h/e per electron). This filling condition implies, for a degenerate levelwith N0 pairs,

hf

N eF æ ö÷ç= ÷ç ÷çè ø0 2

. (11.3.2)

The general expression for the Hall conductivity, in terms of the two dimensional den-sity of carriers N/A and the number of quanta of the carriers q, is

qeNAB

s = . (11.3.3)

Substituting, we obtain for the conductivity, for values of q an f given by equations (11.2.12),(11.2.13),

( )( )

, e N e

h

n ns n

F m

æ ö+ + ÷ç ÷= = ³ç ÷ç ÷ç+ è ø

20 1

2

2 1 2 12 2 1

. (11.3.4)

This expression is not valid if the highest level is the nondegenerate first level. For thisspecial case, where n is equal to zero, there are separate values for q and f ,

hf

N eF æ ö÷ç= ÷ç ÷çè ø0

0 2 , (11.3.5)

and, we get, using equation (11.2.5),

, eN e ef h h

s nF

æ ö÷ç ÷= = = =ç ÷ç ÷çè ø

2 20

0

1 2 0 . (11.3.6)

For half integer n we may define another half integer n which indicates the number offull levels of definite angular momentum number m (not energy levels), by

n n= +12

. (11.3.7)

If we replace n by n we get for the conductivity, the equivalent expression,

( ), n e

nh

sm

æ ö÷ç ÷= ³ç ÷ç ÷ç+ è ø

2

12 1

. (11.3.8)

The highest full level is characterized by the integer flux index m, indicating the num-ber of flux quanta per electron in this highest level and the half integer charge index n,number of active degenerate energy levels, or n, number of active orbital angular momen-tum levels. As the magnetic intensity is increased, the levels, their population and flux


quanta are rearranged, to obtain full levels. Details for this process depends on the micro-scopic laws. Nevertheless, the quantum nature of the magnetic flux requires a fractionalexact value of conductivity whenever the filling condition is met independent of details.As the magnetic field increases, the resultant fractions for small integers n, m are

, / , , / , / , / , / , , / , / , / , / ,/ , 7/ , / , / , / , / , / , / , / , / ,/ 7, / ,

3 7 3 2 5 3 7 5 4 3 6 5 1 6 7 4 5 7 9 5 72 3 11 3 5 4 7 5 9 6 11 7 13 6 13 5 11 4 93 2 5

. (11.3.9)

These fractions match the results of the odd fractional quantum Hall effect [9,10]. The same expression, for integer n , indicates plateaus at fractions with an addi-tional divisor of 2. For example, if m is zero, 3/2 for n=1, 5/2 for n=2, (but not ½since n>0). These values appear to be compatible with the accepted values [11, 12,13].

A partially filled level or superfluxed level occurs at a value of magnetic field which isfractionally larger than the corresponding value for a normal level with equal populationbecause the carriers have fractionally extra flux.

The values of the conductivity are degenerate in the sense that one value correspondsto more than one set of indices m, n. The magnetic field for any two sets with a givenconductivity ratio is the same because the smaller number of carriers for one set is pre-cisely compensated by the larger number of flux per carrier. Nevertheless, the two setsdiffer electromagnetically because of the extra flux for one of the carriers, and we mayexpect small energy differences between them, lifting this degeneracy. Of course, the proofof this difference would require detailed analysis using wave functions corresponding to ahamiltonian which includes appropriate terms. If all electrons are in Landau levels, theFermi level would jump directly from one Landau level to the next and the conductivitycurve would be a set of singular points. Localized states due to lattice imperfections allowa Fermi level between Landau levels, as discussed in the references. If this is the case,their small separation would make the two sets coalesce into a plateau. Since the value ofthe conductivity, at two sets m, n with the same fraction, is exactly the same, the value ofthe conductivity should be insensitive to a small variation of the magnetic field (or en-ergy) indicating a finite width plateau at the value given by equation (11.3.2) with verygreat precision. In particular for the ratio 1 there are many low numbered sets which coa-lesce producing a very wide plateau. This accounts for the plateaus seen at the so calledfractional and integral fillings.

These results may be understood in an alternate manner. As indicated in the reference[4], eq. (11.3.2) is a consequence of the magnetic flux quanta linked and carried by theorbiting electrons. Since the carriers in the FQHE are polarized in the magnetic field di-rection, when they cross a line parallel to the electric field there is a fixed relation be-tween the charge and the linked flux crossing that line,

( )hf eQ qe

DFD

= 2 . (11.3.10)

If there are no resistive losses, the voltage induced along the line, by the carried inducedflux cutting it, leads to a transverse resistance which is fractionally quantized,


( )( )tf hR q e

= 22 . (11.3.11)

This is a fundamental general relation which only depends on the quanta carried by thecarriers. This resistance value is a high precision number depending on fundamental con-stants and integers, which is displayed when appropriate microscopic conditions are met.The FQHE is such an experiment, which essentially measures the ratio of charge quantato flux quanta of the relevant representations of the group (carriers).

11.4. Summary. We obtained the expression for the fractional quantum Hall effect conductivity from

the general principle that fundamental particles are representations of SL(2,) andcarry quanta of charge, magnetic flux and angular momentum as shown in chapter 7.

The numerical results are essentially compatible with the experimental results. In par-ticular we may claim that in the FQHE, the currents and voltages involved are producedby carriers of quanta of charge e and magnetic flux h/2e which form a vortex system ofelectrons rotating around a magnetic axis.

References

1 R. Laughlin, Phys. Rev. B23, 5652 (1981). 2 J. K Jain, Phys. Rev. B41, 7653 (1990). 3 Kamilla, Wu, J. K. Jain, Phys. Rev. Let., 76, 1332 (1996). 4 G. González-Martín, Gen Rel. Grav. 23, 827 (1991). See chapter 7. 5 G. González-Martín, ArXiv cond-mat/0009181and Phys. Rev. A51, 944 (1995). 6 L. D. Landau, E. M. Lifshitz, Mécanique Quantique, Théorie non Relativiste (Ed.

Mir, Moscow), 2nd. Ed. p. 496 (1965). 7 J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, New York), Second

Ed., p. 589 (1975). 8 See section 6.7. 9 D. Tsui, H. Stormer, A. Gossard, Phys. Rev. Lett. 48, 1559 (1982).and R. Willet, J.

Eisenstein , H. Stormer , D. Tsui, A Gossard, J. English, Phys. Rev. Lett. 59, 1776(1987).

10 K. V. Klitzing, G. Dorda, M Pepper, Phys. Rev. Lett. 45, 494 (1980)11 R. Willet, R. Ruel, M. Paalanen, K. West, L. Pfeiffer, Phys. Rev. B47, 7344 (1993).12 R. Du, H. Stormer, D. Tsui, L. Pfeiffer, K. West, Phys. Rev. Lett. 70, 2994 (1993).13 J. Eisenstein, L. Pfeiffer, K. West, Phys. Rev. Lett. 69, 3804 (1992).

12. THE SUBSTRATUM AND ITS PHYSICALSIGNIFICANCE.

12.1. Introduction.Physically the substratum provides related mass scales to its excitations (particles). The

proposed nonlinear field equation and its integrability condition have peculiar aspects whichdistinguish them from standard equations in classical physics. Normally coupled field equa-tions and equations of motion, for example Maxwell and Lorentz equations, in presence of acurrent source do not provide, by themselves, a static internal solution for a source whichmay represent a particle under the influence of its own field. Use of delta functions for pointparticles avoid the problem rather than solve it and may introduce self accelerated solutions[1, 2]. The choice of current density in the theory, together with the developed interpretation,allows a discussion on different grounds. The frame e which enters in the current representsmatter. Since a measurement is always a comparison between similar objects, a measurementof e entails another frame e’ to which its components are referred. If we choose e’ properlywe may find interesting solutions.

As seen in the previous chapters, the integrability conditions of the nonlinear equationleads to a generalized Dirac equation with a parameter that may be identified with a mass definedin terms of energy. The recognition of a single concept of mass is fundamental in GeneralRelativity and merits discussion of possible solutions of the coupled equations.

If we identify a geometrical excitation with a physical particle the linear excitation equation,which now is the equation for a particle, contains parameters provided by the curved (nonlin-ear) solution. Some of the particle properties could be determined by a substratum geometry. Inparticular a mass parameter arises for the frame excitation particle from the mass concept de-fined in terms of energy. It is clear that this parameter is not calculable from the linearizedexcitation equation but rather from a nonlinear substratum solution. This appears interesting,but requires a knowledge of a substratum solution to the nonlinear field equation. Thus it isnecessary to find a nonlinear solution, the simpler the better, which could illustrate this ideas.It is in this context that the following solution is presented [3 ].

12.2. The Field Equation.The substratum geometry satisfies the nonlinear equation,

b bD JW pa* *= 4 . (12.2.1)

The differential operator may be expanded,

( ) ( ) ( )( )( ) ( )( )

D D d d d d

d

w w w w w w w w w

w w w w w

*** ** * * * * *

** **

= + + +

- - , (12.2.2)

and the exterior product may be written in terms of differential forms and group genera-

Chapter 12 PHYSICAL GEOMETRY116

tors,

[ ]a b

a bE E dx dxm nm nw w w w = , (12.2.3)

( ) a ba bE E dx dxm n k l

mnklw w e w w* = 1

2 , (12.2.4)

( ) c a bc a bE E E dx dx dxm n r k l

mnkl rw w w e w w w* = 12 . (12.2.5)

The cubic term, which is the one responsible for the self interaction energy is

( ) ( ) [ ]

[ ]

,

, ,

c a bc a b

c a bc a b

E E E

E E E

ara m nmn r

r ar

w w w w w w d w w w

w w w

* * *é ù - =ê úë ûé ù= ë û (12.2.6)

and the derivative terms are

[ ]a

ad E dx dxmn k lkl m nw e ¶ w* = 1

2 , (12.2.7)

( )[ ] agd d gg g E dx dx dxsm tn r k l

stkl r m nw e ¶ ¶ w*-

= - 12 , (12.2.8)

( ) ( ).a b

a bgd g E E dx dx dxm n r k l

mnkl rw w e ¶ w w* = - 12 , (12.2.9)

[ ]a b

a bd g g E E dx dx dxsm tn r k lstkl r m nw w e w ¶ w* = 1

2 . (12.2.10)

The equation may be written in terms of the spacetime metric g and the connectioncomponents referred to bases in the differential form algebra and the Lie algebra. The traceof products of the Lie algebra base gives the Cartan-Killing metric g on the fiber [4 ],

tr d a daE E =14 g (12.2.11)

and the commutators in the expressions introduce the structure constants,

[ ] tr , tr nd a b ab d n abdE E E c E E c= =1 1

4 4 , (12.2.12)

[ ] tr , , tr m n mnd c a b ab cm d n abn cmdE E E E c c E E c cé ù = =ë û

1 14 4 g . (12.2.13)

Finally the field equations become, denoting the two metrics by g and g,

( )

( )

[ ]

[ ]

c a b mnabn cmd d

a b a babdd

c c gg gg

cg g J

g

r a rm anr r m n

r a an m ar m n

w w w ¶ ¶ w

¶ w w w ¶ w pa

+ - +-

é ù+ - + =ê úë û-

2g

2 4 . (12.2.14)

117The Substratum and its Physical Significance

12.3. A Substratum Solution.12.3.1. The Substratum Connection.

The nonlinear equations of the theory are applicable to an isolated physical systeminteracting with itself. Of course the equations must be expressed in terms of componentswith respect to an arbitrary reference frame. A reference frame adapted to an arbitraryobserver introduces arbitrary fields which do not contain any information related to thephysical system in question. The only not arbitrary reference frame is the frame defined bythe physical system itself.

Any excitation must be associated to a definite substratum. An arbitrary observation ofan excitation property depends on both the excitation and the substratum, but the physicalobserver must be the same for both excitation and substratum. We may use the freedom toselect the reference frame to refer the excitation to the physical frame defined by its ownsubstratum.

We have chosen the current density 3-form J to beˆ

ˆJ e u em a mak= , (12.3.1)

in terms of the matter spinor frame e and the orthonormal space-time tetrad u .Since we selected that the substratum be referred to itself, the substratum matter local

frame eb, referred to er becomes the group identity I. Actually this generalizes comovingcoordinates (coordinates adapted to dust matter geodesics) [5 ]. We adopt coordinatesadapted to local substratum matter frames (the only not arbitrary frame is itself, as are thecomoving coordinates). If the frame e becomes the identity I, the substratum current den-sity becomes a constant. Comparison of an object with itself gives trivial information. Forexample free matter or an observer are always at rest with themselves, no velocity, noacceleration, no self forces, etc. In its own reference frame these effects actually disappear.Only constant self energy terms, determined by the nonlinearity, make sense and should bethe origin of the constant mass parameter.

At the small distance l, characteristic of excitations, the elements of the substratum,both connection and frame, appear symmetric, independent of space-time. We should re-member that space-time M is, mathematically, a locally symmetric space [6 ] or hyperbolicmanifold [7] . We recognize these as the necessary condition for the substratum to locallyadmit a maximal set of Killing vectors [8 ] which should determine the space-time sym-metries of the connection (and curvature). This means that there are space-time Killingcoordinates such that the connection is constant but nonzero in the small region of par-ticle interest. A flat connection does not satisfy the field equation. The excitations mayalways be taken around a symmetric nonzero connection.

In particular the nonlinear equation admits a local nonzero constant connection solu-tion. This would be the connection determined by an observer at rest with the matter frame.Of course, this solution is trivial but since the connection has units of inverse length, ormass, this actually introduces fundamental lengths in the theory. Furthermore, a constantnonzero solution assigns a constant mass parameter to a particle excitation and may allowthe calculation of fundamental masses m related to the connection and energy in terms ofthe dimensionless coupling constant a. Therefore, we present a constant nonzero (trivial)solution which is the relative effective nonlinear background for a fluctuation and weshall call it the trivial substratum solution.

First we look into the left side of the field equation, and notice that for a constantconnection form w and a flat metric, the expression reduces to triple wedge products of w


with itself, which may be put in the form of a polynomial in the components of w. Thiscubic polynomial represents a self interaction of the connection field since it may alsobe considered as a source for the differential operator.

Rather than work with the whole group G we first restrict the group to the 10 dimen-sional Sp(4,) subgroup. Furthermore we desire to investigate the nongravitational part ofthe connection. Hence, we limit the components of the connection to the Minkowski sub-space defined by the orthonormal set. This is possible because if the connection is odd sois the triple product giving an odd current as required. Using the relations,

tr a b abk k h= -14 , (12.3.2)

( )tr , ,

tr

d c a b

d c a b d c b a d a b c d b a c

k k k k

k k k k k k k k k k k k k k k k

é ùé ù =ê úë ûë û- - +

14

14 , (12.3.3)

( )tr ,d c a b db ca da cbk k k k h h h hé ùé ù = -ê úë ûë û14 4 , (12.3.4)

we obtain the expression,

( ) ( )ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆtr , ,g am bl a m l a m l

m g a m a m ad b d dk w w w k k k w w w w w w- é ùé ù = - -ê úê úë ûë û

14 4 . (12.3.5)

We now consider the right hand side of the field equation. The material frame e referredto itself is the identity and the space-time frame u is dn

m. Hence the nonlinear differentialequation reduces to a polynomial equation which may have constant solutions for thecomponents of w,

ˆ ˆˆ ˆ ˆˆ ˆ

a m l a m l lm a m ab b b

w w w w w w pad- + = . (12.3.6)

The unknown components of w may be indicated by four 1-forms w, one for each of the fourelements k. We shall assume that one of the forms is timelike and the other three spacelike.Since we have the freedom to choose coordinates in the base space, we choose the timecoordinate adapted to the timelike form. Similarly we choose the x coordinate adapted tothe time-orthogonal part of one of the spacelike forms. As in Schmidt orthogonalization, wechoose the y coordinate adapted to the t, x orthogonal part of the second spacelike form. Inthe same manner we choose the z coordinate adapted to the orthogonal part of the lastform, so the four forms may be written as,

ˆ ˆmw wé ù= ê úë û

00 0

0 0 0 , (12.3.7)

ˆ ˆ ˆmw w wé ù= ê úë û

0 11 1 1 0 0 , (12.3.8)

ˆ ˆ ˆ ˆmw w w wé ù= ê úë û

0 1 22 2 2 2

0 , (12.3.9)

ˆ ˆ ˆ ˆ ˆmw w w w wé ù= ê úë û

0 1 2 33 3 3 3 3 . (12.3.10)

We now write the equation for component 3 of the 1-form 0 and notice that because


the third components of the forms are zero, except for 1-form 3, this equation implies

ˆˆ ˆm

mw w w =3 30 3

0 , (12.3.11)

ˆ mmw w =3

0 0 . (12.3.12)

Using this orthogonality relation, we now write the equation for component 2 of thesame 1-form and again notice that, because of the null components, we must also have,

ˆˆ ˆm

mw w w =2 20 2

0 , (12.3.13)

ˆˆm

mw w =20

0 (12.3.14)

and similarly we get,

ˆˆm

mw w =10

0 . (12.3.15)

Taking now the other equations for 1-form 1 and then the equation for 1-form 3, we getin the same manner,

ˆˆm

mw w =31

0 , (12.3.16)

ˆˆm

mw w =21

0 , (12.3.17)

ˆˆm

mw w =32 0 . (12.3.18)

Because of this orthogonality of the four 1-forms, we realize that there remain onlyfour unknown components which we shall indicate as T, X, Y, Z, and the equations become

( )X Y Z T pa- + + =2 2 2 , (12.3.19)

( )Y Z T X pa- + + =2 2 2 , (12.3.20)

( )Z X T Y pa- + + =2 2 2 , (12.3.21)

( )X Y T Z pa- + + =2 2 2 . (12.3.22)

We notice that these equations are all essentially the same and that a null connectionis not a solution. We have a unique isotropic homogeneous constant solution, propor-tional to the current, which we shall call the trivial solution. A Lorentz transformationdetermines an equivalent solution. We choose the name substratum for this equivalenceclass of solutions under SO(3,1). There is another constant solution which depends on anoninertial algebraic element [9]. Since there is no preferred direction, the trivial con-nection should not distinguish T, X, Y, Z and the components of w should be proportionalto the identity I. In this case the four equations reduce to one for a single constant, asfollows,


T pa- =33 , (12.3.23)which determines a positive constant, with units of inverse length, in terms of the finestructure constant a, a dimensionless number. In other words, together with the fundamen-tal charge we have a fundamental mass or fundamental length determined by a term in theconnection field. We can write the expression for the connection in this reference system,in terms of the bases of 1-forms and the Clifford algebras,

ˆˆT dx aaw k= - . (12.3.24)

and in an arbitrary reference system, in terms of the current form J,

( )ˆˆe T dx e e de T J e deaaw k- - -= - + = - +1 1 1 . (12.3.25)

The trivial connection is essentially proportional to the current, up to an automorphism. Itshould be noted that in the expression for w, the term containing the current J is thepotential tensorial form L, used in chapter 9 for the definition of mass. Its subtraction fromw gives an object, e-1de, which transforms as a connection.

It is convenient to introduce a parameter M determined by the definition of mass fromthe interaction energy J.G [10 ], given by eq. (9.3.10) in chapter 9. The total dynamic poten-tial tensorial form L valued in the sl(4,) algebra may be expressed as

JL-

=4M

. (12.3.26)

Using this expression, the calculated mass parameter corresponding to this substratumsolution is

( ) ( ) ( )

( )

tr tr tr

tr

m J J e e e e

I T

m m mm m mG L k k- -

æ öæ ö- ÷ç ÷ç= = = ÷÷ç ç ÷÷çç ÷è øè ø

= = =

1 11 1 14 4 4

14

4M

M M . (12.3.27)

12.3.2. The Substratum Curvature.The curvature form may be calculated in the special reference system,

ˆˆˆˆ[ ]T J J dx dxa b

a bW w w k k= = = 2 21

16M , (12.3.28)

which is a covariant expression, and therefore valid in all reference systems. The curvatureis independent of the e-1de term, in accordance with the interpretation of this term as aninertial effect [11 ].

The substratum solution determines, on space-time, a flat metric h and the even curva-ture form W of a nonmetric connection. The orthonormal set k defines an equivalent setusing an orthonormal tetrad u associated to g, the space-time metric in arbitrary coordi-nates. An so(3,1) curvature tensor may be obtained from the sl(2,) even algebra valuedform using the homomorphism corresponding to eq. (4.3.7)

( )trb b bamn a mn mn aW k W k W k k-= +1

4 (12.3.29)


which reduces to

( )tr , ,bamn a b m nW k k k ké ùé ù= ê úê úë ûë û21

64M . (12.3.30)

Therefore, the curvature tensor of the substratum solution may be written as

[ ]gb bamn m n aW d= 21

2M (12.3.31)

showing the hyperbolic nature of the solution. The constant M determines the curvatureparameter. The corresponding space to this curvature is conformally flat. The contractedcurvature tensor is

gan anW = 234

M . (12.3.32)

If we change to another reference system by a group transformation which does notpreserve the even part we obtain a different point of view, with gravitational effects. Inparticular, the w connection acquires a riemannian component when we perform an internalrotation generated by k0. If the rotation angle is p/4 we obtain

( ) ( ) ( )exp exp expi ip p pk k k k k k k- = - = -0 0 0 1 04 4 2 . (12.3.33)

The even curvature of the substratum is transformed, by this ko-rotation, to a curvaturewith odd and even parts,

ˆ ˆˆ ˆˆˆ ˆ[ ]

a a ba a b

dx dx dx dxW k k k W W- +¢ ¢ ¢= + º +2 0 21 18 16

M M . (12.3.34)

12.3.3. Relation with the Newtonian Limit.The substratum solution is compatible with the energy equation (5.2.1) and precisely

provides a symmetric hyperbolic curvature necessary to obtain a correct newtonian limit.The projected components on the hypersurfaces orthogonal to the time direction have thenewtonian limit specified in section 5.4,

lim liman anhe e

W e

-= =2 2

0 0

3 04

M . (12.3.35)

When establishing this limit, for any connection solution G, we can always define a newconnection by subtracting the tensorial potential form L corresponding to the substratumsolution,

JG G L Gæ ö- ÷çº - = - ÷ç ÷çè ø4

M . (12.3.36)

The even so(3,1) curvature component may be split into the substratum part sW, determinedby eq. (12.3.32) in terms of the new metric, the Riemann tensor of the new connectionand a complementary nonriemannian part determined only by the total odd connection,


n n sR Ra a a abkl bkl bkl bklW W= + +

. (12.3.37)

Einstein’s equation (5.2.22) may be written as

n nG GT

R Rrm rm rm rmp Q p Q pW

= = º+ +2

3 38 8 83

M

(12.3.38)

where the expression for the gravitational stress energy tensor, equation (5.2.20), is

ng R Rg C Rk l

rm mlr kQp

é ùæ ö÷çê ú÷= +ç ÷ê úç ÷çè øê úë û

1 28 3 4

. (12.3.39)

The only nonriemannian curvature contribution due to the substratum is through the sca-lar nR. All other contributions of the conformally flat curvature cancel themselves.

Since Newton’s theory is a pure gravitational theory, we assume that the new connec-tion, as we approach the limit, is strictly gravitational. As discussed in section 5.4, it isnecessary to make the usual assumptions of regularity of the matter stress energy tensor.When we take the newtonian limit of the generalized Einstein equation (5.2.22), the threedimensional space curvature vanishes as shown in appendix F, eq. (4.4). Then the onlysurviving component of the Riemann tensor satisfies Poisson’s equation (5.4.6)

ˆ ˆ ˆ ˆlim lim

n s nR

R Re e

Q Qpp

W W

æ ö æ ö÷ ÷ç ç÷ ÷= =ç ç÷ ÷ç ç÷ ÷ç ç + +è ø è ø00 00

00 0 0

3 38 42

. (12.3.40)

If in the newtonian limit we simultaneously approach the trivial substratum solution, R andnW are negligible, in the limit, with respect to the constant sW. The limit of nR is

( ) ( )lim lim lim

lim lim

n n n s

bk

bk

R R g R g

hg g h

a bk a a a bkbka bka bka bkae e e

bkbke e

W W

ee

é ù= = + +ê úë ûé ùé ù æ ö÷çê úê ú ÷= = + =ç ÷ê úçê ú ÷çè øë û ë û

0 0 0

2 22 2

20 0

3 3 1 34 4

M MM . (12.3.41)

The expression for the newtonian energy density, eq. (5.4.10), may then be written in thefollowing form:

ˆ ˆlim limaa G

e e

Q p¶ ¶ j p Q p r

æ ö÷ç ÷= = =ç ÷ç ÷çè ø00

002 20 0

44 4MM

. (12.3.42)

In the newtonian limit the geometric and classic densities are related by the constants Mand G. If the odd connection nonriemannian fields contribute to the curvature scalar nR, theparameter G would be variable, diminishing with the field strength. This effect may beinterpreted as the presence of dark matter.

12.3.4. Physical Significance of the Substratum.On the substratum there is a fundamental scalar interaction energy field characterized by

the constant mass parameter M, corresponding to the mass definition from J and w, using the


Cartan-Killing metric Cg in the defining representation. This scalar energy field is the dual ofa density 4-form w *J field also characterized by M. The current stress-energy tensor den-sity field jQ on the substratum is also related to J and w. Essentially the geometric objects onthe substratum are determined by these two A-valued forms: the current 3-form *J and theconnection 1-form w. These forms are constant on the substratum.

Since the Cartan-Killing metric depends on the particular representation [4], the fun-damental mass parameter which characterizes the substratum is determined up to a rep-resentation dependent normalization constant. For a representation (A) the mass pa-rameter is, with the normalization we are using,

( )( ) ( )( )( ) ( )( )

( ) ( )( )( )

( ) ( )( )

trtr trtr tr tr

tr tr

tr

JJ Jm

I

J J

N I N d

m mm m GG G

k k k k

G G

= = =- -

= =

0 00 0

(12.3.43)

in terms of the dimension d and a constant N, both related to the representation .Apart from the defining representation, we have to consider the induced representa-

tions [12 ] necessary to characterize particle excitations. In the next chapter we shall dis-cuss these induced representations in more detail. The value of the mass parameter in theserepresentations differs from the value M obtained in the defining representation. A funda-mental Dirac physical mass parameter, associated to a particular induced representation,should be related by a constant to the interaction energy scalar parameter M,

m N= M . (12.3.44)

In order to find this constant N it is sufficient to calculate the Cartan-Killing product forone generator in both representations [4]. The constant, that relates the parameters in thedefining matrix representation and the induced representation, is mathematically an inde-terminate number because it depends on the ratio of infinite value integrals on the symmet-ric noncompact coset spaces involved in the Cartan-Killing metric in the induced repre-sentation. The relation of the geometric stress-energy tensor density field Q with thestandard physical density r, expressed by equation (12.3.42), obtained in the newtonianlimit, provides a physical determination of this indeterminate constant. In other words, itis possible to equate M -2 to the value of the gravitational constant and m to the value ofa fundamental particle mass without contradictions. Physically we consider that the sub-stratum provides two conceptually related mass scales: in the fundamental representa-tion it determines the gravitational parameter G that characterizes a macroscopic massscale and in the induced representation it determines a fundamental particle mass m thatcharacterizes a microscopic mass scale.

We have seen that for a pure gravitational spherical solution, the Schwarzschild mass is the integral of the energy density using the nR/3 coupling, as shown in equations(5.3.3) and (12.3.38). This mass determines the Schwarzschild geometry geodesicsand the particle motion without explicit knowledge of a parameter G. In principle, similarrelations should exist for other solutions.

The riemannian curvature associated to the even part in equation (12.3.34) may corre-spond to a symmetric tridimensional space that was used by Einstein in cosmological


applications. From this cosmological gravitational point of view, the spacelike three-dimensional manifold is a symmetric spaces S3, H3 or R3. Assuming the S3 symmetry,Einstein [13 ] derived a relation between the total mass M, the radius of the universe Rand the gravitational parameter. Putting the results together we obtain an estimate for thesubstratum mass parameter M which represents the radial line mass density,

Mp= 2M R . (12.3.45)

In solutions near the newtonian limit, the scalar nR/3 should have a value near the Gconstant which would be the exact value in this limit. The introduction of the gravitationalparameter as the substratum interaction energy modifies the Einstein equation by a factorthat may be interpreted as an “apparent” increment (or decrement) of the energy momentumtensor. The Einstein tensor may be written

( ) ( )n nG G

R G Rrm rm

rm

pQ Qp

W W= =

+ + + +2

88

3 1 3

M . (12.3.46)

Therefore, the substratum interaction energy density is related to mass in differentcontexts. In particular the energy determines the following: 1- the Dirac particle mass pa-rameter m, in corresponding representations; 2- the Schwarzschild metric gravitational massparameter ; 3- the gravitational parameter G. The concept of macroscopic active gravita-tional mass is consistent with the concept of microscopic inertial mass.

The use of the substratum as a quantum vacuum is also consistent. To show this it isconvenient to evaluate other energy terms related to the substratum solution. The fieldtensor depends quadratically on the substratum curvature. Using eq. (12.3.31) we obtain

c gab n abklabrn m rm abklQ W W W W= - =

1 04

. (12.3.47)

Similarly, the current stress energy tensor is

( )

( )

tr

tr

j e e g e e

g

lmr m r mr l

m r mr

aQ k k

ak k

- -é ù= - - ê úë û

é ù= - + =ê úë û

1 1144

016M

. (12.3.48)

Therefore, the substratum solution, which has an interaction energy G.J, is a minimum forthe internal energies of both its geometry and its matter source.

The equation (12.3.46) may be responsible for the anomalous effects blamed on darkmatter and energy. If a body of mass M produces the Friedmann cosmological metric

( ) ( )( )sin

sin sinh

k

d dt a t d d d k

k

S ct c S q q f S c

S c

ì = =ïïïï= - + + = =íïï = - =ïïî

2 2 2 2 2 2 2

101

(12.3.49)


it may determine an apparent stress energy tensor which may be written as follows,using a well known result [14] for its scalar curvature R and neglecting W,

TGR a k a

Ga a

rm rmrm

Q Q= =

æ ö++ ÷ç ÷- +ç ÷ç ÷çè ø

2

21 3 1 2

. (12.3.50)

Of course, the new solutions for the radius a of the universe, using the differentknown energy momentum tensors, will differ from the presently known solutions.

If k and ä are zero, using eq. (12.3.50) the newtonian velocity of a rotating particleunder this field would be determined [15] by the relation

( )( ) ( )vMG

vG a ar G a a

= =--

22 0

22 1 21 2 , (12.3.51)

where v0 would be the particle velocity if the Friedmann metric velocity was zero. Thispure geometric effect increases the apparent rotational velocity. It is not necessary toblame this effect, which corresponds to the anomalous velocity measurement of rotat-ing galaxies, to the presence of a “dark matter”.

12.4. General Equation of Motion.Using the new connection defined by equation (12.3.36), but in induced representa-

tions, the equation of motion (3.4.1) explicitly displays a term depending on the substratummass, required by a Dirac equation

( )

me e e e e e J

e me

m m mm m m m m m

mm

k k G k G

k

æ öæ ö- ÷ç ÷ç = ¶ - = ¶ - - ÷÷ç ç ÷÷çç ÷è øè ø

= - =

0

4

0

(12.4.1)

and the equations for the even and odd parts of a G-system are, using the correspondingexpressions in section 3.4,

( ) † mm m mm m m ms x xG s x s h G h+ -¶ - º = -

, (12.4.2)

( ) † mm m mm m m ms h hG s h s x G x+ -- ¶ - º - = -

. (12.4.3)

Here we should point out that the presence of s in the odd equation determinesimportant unusual signs in the electromagnetic coupling.

We now designate different vector potentials in the connection with the follow-

ing notation: +Ai is the even part of the SU(2)Q sector ( )QG

of the connection and

corresponds to the standard electromagnetic potential; A is the complementary oddpart in the same sector; G is, in this section, the SL(2,) sector of the connectionwhich determines an L covariant derivative; is the complementary part. Both -A


and are complex matrices, respectively proportional to the identity and the Paulimatrices, along the odd direction k0. The first equation may be written as

( ) † †i A A mm m mm m m m ms x x xG s h s h ¡ h+ -¶ - - = + - . (12.4.4)

Inserting the unit -i2 in the left side of this equation

( ) ( ) ( )( )( ) † †i i A i i i A mm m mm m m m ms x x x G s h s h ¡ h+ -- ¶ + - = + - . (12.4.5)

The odd and even parts, x and h, are elements of sl(2,) and sl(2,)Åu(1) respectively. Theu(1) component contributes an overall phase and should be neglected. Therefore, the conju-gation gives

h h= - , (12.4.6)

x x= - (12.4.7)

and we get

( )( ) † †i A i A mm m mm m m ms x s h s h ¡ h+ - + = + + . (12.4.8)

Similarly we get for the second equation after multiplying by i

( )( ) ( ) ( ) ( )† †i A i A i i m im m mm m m m ms h h h G s x s x ¡ x+ -- ¶ + - = + - , (12..4.9)

( )( ) ( ) ( ) ( )† †i A A i i m im m mm m m ms h s x s x ¡ x+ - + = + + . (12.4.10)

If we define

( )ij h xº +12 , (12.4.11)

( )ic h xº -12 , (12.4.12)

we can add and subtract the even and odd equations and write the generalized Dirac’sequations in the following form

( ) ( ) † †

† †

m mm m m

mm

i A i A A A

m

j s c c s j

c ¡ s j ¡ j

+ + - - + - + = + +

+ + +

0 0 0

0 , (12.4.13)

( ) ( ) † †

† †

m mm m m

mm

i A i A A A

m

c s j j s c

j ¡ s c ¡ c

+ + - - + - + = - - +

- - -

0 0 0

0 . (12.4.14)

12.5. General SubstratumAn analysis, similar to that of section 3, may be done for the general case without

restriction to the Sp(4,) subgroup. We want a solution without gravitation, so we take the


trivial connection only in terms of the nine generators of the coset space G/L,ˆ ˆ

ˆ ˆa

aEa aa aw w k w k k wk w wk+ += + + = +5 5 5 , (12.5.1)

where the dots on top of w indicates the number of k matrices in the correspondinggenerator. For convenience, we indicate the eight K space generators by Ea. If we re-strict the Latin generator indices from 1 to 9 in equation (12.2.6), we may write theconnection triple product, here indicated by X , in the following form,

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ]

, , , ,

, , , ,

, , , ,

, , , ,

c b a c bc b a c b

c a cc a c

b a bb a b

aa

X E E E E E

E E E

E E E

E

a m a m am m

m a m am m

m a m am m

m a m am m

w w w w w w k

w w w k w w w k k

w w w k w w w k k

w w w k k w w w k k k

+

+ + +

+ + +

+ + + + +

é ù é ù= +ë û ë ûé ù é ù+ +ë û ë ûé ù é ù+ +ë û ë ûé ù+ +ë û

5

5 5 5

5 5 5

5 5 5 5[ ]é ùë û5 . (12.5.2)

The total sl(4,) odd vector subspace, spanned by ka and k5ka, is isomorphic to Kk,the tangent space of the symmetric coset K at a point k. The symmetric coset space K hasa complex structure, as shown in section 13.3.2, which allows us to define a complexmetric in its tangent space using the Cartan-Killing metric. Thus, the total sl(4,) oddvector subspace may be considered as a complex Minkowski space with metric h,

( ) ( ) ( ), , ,CC CX Y X Y X Yh* *º = -g g . (12.5.3)

In general, we are really interested in the whole class of possible substratum connec-tions. This class of complex connections is generated by the symmetric action of the U(1)group associated to the k5 generator, thus extending the substratum space to the complexspace KkÅk5. Taking in consideration the basis in the Kk space formed by the orthonormal setka over the complex numbers, the equation for the total geometric complex current, includingthe conjugate and imaginary component units, is

iX J el l f lpa pa k= = , (12.5.4)

which is different from the real equation (12.3.6) but reduces to it when f is zero. The theorymay also require the consideration of excitations around the substratum connection solution2w. The complete representation of 2w is formed by a combination of an algebraic element w andits Clifford conjugate, as indicated in a previous work [11]. The equation for this connectionexcitation would represent a massive bosonic excitation related to a pair of conjugate excita-tions.

Since the generator k5 commutes with the even sector and generates the complex struc-ture, eq. (12.5.2) becomes

[ ] { }

{ }

, , ,

,

c b a c bc b a c b

c a b ac a b a

X E E E E E

E E E E

a m a m am m

m a m a m am m m

w w w w w w k

w w w k w w w w w w

+

+ + + + +

é ù= +ë û- + -

5

5

2

2 . (12.5.5)

We may use the K complex structure to take w as a complex function and the generators aska,


[ ], ,

c b a c bc b a cb

c a b aca b a

X a m a m am m

m a m a m am m m

w w w k k k w w w k h

w w w k h w w w k w w w k

+

+ + + + +

é ù= -ë û+ + -

5

5

4

4 4 4 . (12.5.6)

Finally, we evaluate the components in the term X,

[ ]( )tr tr , ,c b a b ad d c b a b aX a m a m a m a

m m mk k w w w k k k w w w k w w w k+ + + +- - é ù= + -ë û1 1

4 4 ,

( ) ( )c cd c d d c d dX a m a m a m a m a

m m m mw w w w w w w w w w w w* * + + + += - - - -4 4 (12.5.7)

and write the field equation in the following form,

( ) ( )( )( )

c c dc d d c d d

c cc c J

m a m a m a m am m m m

m a m a am m

w w w w w w w w w w w w k

w w w w w w k pa

* * + + + +

* + * +

- + -

+ - = -5 . (12.5.8)

If we assume that the even connection component +w is zero we obtain

( )c c ic d d c dem l m l f l

m mw w w w w w pa d* * - - = (12.5.9)

and the odd complex substratum solution is

( )ˆ ˆ ˆˆˆ ˆ ˆcos sini

gT e dx m dx dxf a a aa a aw k f k f k w = - º - =

2

5 (12.5.10)

in terms a geometric mass mg, given by eq. (12.3.23) for M in the defining representation,

( )gm pa= =1 3

3M

4 . (12.5.11)

Therefore the real part is the original real solution.We may represent the odd complex substratum connections w, given by these solu-

tions, as points on the complex Minkowski space Kk, with equal moduli. The energy asso-ciated to the solution is

( ) ( ) ( )

( )ˆˆ tr .

C CJ J J J

aa

w h w w w

k k

* * = ⋅ = - ⋅ = - ⋅

æ ö÷ç= - =÷ç ÷÷çè ø

g g

14

MM

4(12.5.12)

The points define complex vectors -q+,

q i ix y za a a a a ak k k- + ¶ ¶ ¶

= + º + =¶ ¶ ¶5 (12.5.13)

and their complex conjugates -q-, which form a base on the Kk sector, related to theassociated complex coordinates za. The pair of vectors -q0 together with the vector de-termined by the static connection +w,


q k+ =35 , (12.5.14)

provide a quaternion base on a compact subspace for the subalgebra su(2)Q.If we assume that the connection even component +w is non zero we should extend the

current accordingly. The even part of eq. (12.5.8) is

( )c cc c jm a m a

m mw w w w w w k pa k* + * + +- = -5 5 . (12.5.15)

In fact, an arbitrary constant solution of this type may be added to the complex substratumsolution without changing the structure of eq. (12.5.8). The value of the odd part of thisequation is modified by the presence of an even component in accordance with the term

( )d d dY a a aw w d w w+ + += -2 (12.5.16)

which must be zero off-diagonal. The field equations determine that only any one compo-nent of the 1-form +w may be a non zero constant. This +w is either timelike or spacelike andmay be called electrostatic or magnetostatic. Thus, we obtain a series of related complexsubstratum solutions with different possible constant values of -w and +w. The modulusof the total connection solution w is not given by eq. (12.5.12). Furthermore, it can beshown that the modulus of any of these related complex solutions increases due to +w andis larger than the value given by this equation.

12.6. Summary.The proposed field equation admits constant connection fields as solutions. In particu-

lar, we displayed an isotropic homogeneous constant solution, which we called the sub-stratum solution. The constant value of any nonzero component of the connection hasdimensions of inverse length and was given in terms of the fine structure constant a, whichis a pure number. This introduces a fundamental length in the theory. Furthermore, we haveconstructed constant complex solutions, in terms of this geometrical fundamental unit,by extending the real functions of the substratum to complex functions.

The expression for the constant curvature of the substratum shows that this space is ahyperbolic, conformally flat, symmetric space. The associated fundamental mass, in thedefining representation, may be related to the gravitational constant G. From an Einsteincosmological space we obtain an estimate for the mass parameter M on the substratum.

The substratum even curvature scalar W may be considered the germ and origin of thephysical constants related to the substratum and, in general, plays this role in generalsolutions. Physically we consider that the substratum provides two related mass scales:the gravitational constant G which characterizes a macroscopic mass scale and a funda-mental particle mass m which characterizes a microscopic mass scale, in accordancewith Mach’s ideas.

The substratum is the background around which we define the excitations of frameand connection fields. It provides a mechanism for assigning constant masses to them.The equation of motion may be written as a Dirac equation in terms of these constantmasses. In general, the substratum determines the inertial properties of matter. In thissense we may say that it geometrically represents the physical concept of inertial sys-tem.


References

1 A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (MacMillan,New York), p. 197 (1964).

2 G. N. Plass, Rev. Mod. Phys., 33, 37 (1962). 3 G. González-Martín, USB preprint, 96a (1996). 4 R. Gilmore, Lie Groups, Lie Algebras and some of their applications (John Wiley and

Sons, New York), p. 248 (1974). 5 W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco),

p. 715 (1973). 6 E. Cartan, Assoc. Avanc. Sc. Lyon, p. 53 (1926). 7 See section 2.3. 8 A. Trautman, Geometrical Aspects of Gauge Configurations, IFT/4/81, Acta Phys.

Austriaca, Supl. (1981). 9 J. G. González-T., Tesis (Universidad Simón Bolívar, Caracas), p. 45 (1999). 10 G. González-Martín, Gen. Rel. Grav. 26, 1177 (1994). See chapter 9. 11 See section 9.3. 12 R. Hermann, Lie Groups for Physicists (W. A. Benjamin, New York) (1966). 13 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton), p. 104

(1956).14 W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco),

p. 715 (1973). 15 G. González-Martín, USB preprint SB/F/351.1 07 (2009).

13. MASS RATIOS AND ENERGY.

13.1. Introduction.The action of the connection on the frame is the fundamental dynamic process in the

theory and its energy determines the concept of mass. Since the connection is valued in theLie algebra of the group G and the frame is an element of G, the dynamic is realized by theaction of the group on itself. The principal bundle structure of the group, (G,K,L), providesa natural geometric interpretation of its action on itself. A particular subgroup L defines asymmetric space K, the left coset G/L, the base space of the bundle. The subgroup L, thefiber of the bundle, acts on itself on the right and also is the isotropy subgroup of the cosetK. The complementary coset elements act as translations on the symmetric coset.

This geometric interpretation may be transformed into a physical interpretation if wechoose L to be homomorphic to the spinor group, SL(2,), related to the Lorentz group.The action of L is then interpreted as a Lorentz transformation (pseudorotation) of theexternal space, the tangent space TM of the physical space-time manifold M, defining itsmetric. The action of the complementary coset K is interpreted as a translation in theinternal space, the symmetric coset K itself. There is then a nontrivial geometric relationbetween the internal and external spaces determined by the Clifford algebra structure ofthe manifold. The space K is the exponentiation of the odd sector of the Clifford algebraand is related to local copies of the tangent space TM and its dual cotangent space T*M. Itmay be interpreted as a generalized momentum space. The states of momentum k wouldcorrespond to the points of K.

It follows that the frame excitations are also acted by the connection and evolve asrepresentations of G. The constant parameter m characterizes an eigenvalue of a differen-tial operator , defined by the equation of motion, acting on excitations de of a substratumsolution over a locally symmetric space-time, seen by some definite observer,

( ) ( )e e m emmd k ¶ d d= + = , (13.1.1)

( )e em

d d=

, (13.1.2)

( )e m ed d=2 2 (13.1.3)

and therefore m also characterizes the eigenvalue of the quadratic differential operator 2

on space-time.Different observers would measure different relative momenta k for a given exci-

tation. A measurement for each k corresponds to a function in momentum space. Anabstract excitation is an equivalence class of these functions, under the relativitygroup. Since the group space itself carries its own representations, the realization ofexcitations as representations defined on the group space have a fundamental geo-metric character. The geometric action of the K sector is a translation on itself andthe functions on K are the internal linear representations that may be observed


(observables). The action of the L sector is a Lorentz transformation and sl(2,)spinors are the observable external representations. For these reasons, we must rep-resent physically observable excitations by classes of spinor valued functions onthe symmetric K space. In particular, we realize them on a vector bundle, associatedto the principal bundle (G,K,L), taking as fiber the sl(2,) representation valuedfunctions on the symmetric K space. Essentially, this is, in fact, done in particlephysics when considering representations of the Poincaré group. In mathematics,these representations are called representations of SL(4,) induced from SL(2,).

It was shown in chapter 2 that the base manifold, the space-time M, is a hyper-bolic manifold modeled on K. Given an observer, the excitations on M may be locallyidealized as functions on the symmetric space K locally tangent to M at a point u inan a neighborhood UÌM. For the moment we restrict G to its 10 dimensional P sub-group. The tangent symmetric model space is isomorphic to the hyperbolic space P/Lwhich has a LaPlace-Beltrami differential operator corresponding to the quadraticCasimir operator 2. Because of the local isomorphism between the tangent spaces ofM and P/L, we may take 2 as the corresponding space-time image of 2.

The constant m would then characterize the eigenvalues of 2. It is known that therepresentations of P induced on P/L are characterized by the eigenvalues of thisCasimir operator and therefore they would also be characterized by m. In general wewould then have that the parameter m associated to an excitation (wave) function onM, as seen by some definite observer, characterizes an induced representation of Gon G/L. Because of this pure geometric relation between the 2 and 2 the inducedrepresentations should play a significant physical role. It should be clear the geo-metrical origin of this well known physical hypothesis in elementary particle phys-ics.

Choosing a particular induced representation of G on G/L and definite local ob-servers in a neighborhood around a given point in M determines the excitation (wave)function. Since we deal with a class of relativistic equivalent observers, we actuallyhave a class of (wave) functions on M, one for each transformation linking validobservers. The section of the fiber bundle E defining a particular induced representa-tion has for support the mass hyperboloid in K characterized by the rest mass valuem. Once we have chosen observers at points u in a neighborhood U in M, correspond-ing to points k on the mass hyperboloid, we have (wave) functions on M with definitevalues of k(u) in the neighborhood U. The isotropy group of the mass hyperboloid,the one that leaves a point (momentum) invariant (“little” group), transforms the(wave) functions on the mass hyperboloid. If we have the flat trivial bundles used instandard quantum theories, for example, if M is flat Minkowski space, P is thePoincaré group and L is the Lorentz group then the quotient P/LK is a flat Minkowski(momentum) space distinct but isomorphic to (space-time) M and the little group onthe mass hyperboloid in K is SO(3) or ISO(2). The (wave) function becomes the stan-dard particle wavefunction depending on a single definite value of k throughout M.

Due to the preceding arguments, when we try to calculate the physical values m,corresponding to excitations on a substratum, we must use the induced representa-tions characterized by m, which are the ones that represent geometric excitations ofdefinite momentum k by (wave) functions on space-time, carrying geometrical andphysical significance.

From our geometric point of view, it has been claimed [1 ] that the proton, theelectron and the neutrino are representations of SL(2,) and its subgroups, inducedfrom the subgroups SL1(2,) and SL(2,). This is a generalization of particles as

133Mass Ratios and Energy

representations of the Poincaré group induced from its Lorentz subgroup. Usingour definition of mass [2, 3], it is possible to find an expression for the mass ofthese geometric excitations and compare them with the proton electron mass ratio(a not fully explained geometrical expression is known [4, 5, 6]). It should be notedthat there is no contradiction in this calculation with present physical theories, whichmay be considered as effective theories derived under certain conditions and limitsfrom other theories. The structure group G of the theory, SL(4,), has been used inan attempt to describe particle properties [7, 8, 9] in another approach.

13.2. Bare Masses.The definition of the mass parameter m, in terms of a connection w on the principal

fiber bundle (E, M, G), has been given in the fundamental defining representation of SL(4,)in terms of 4´4 matrices, but in general, may be written for other representations usingthe Cartan-Killing metric Cg, defined by the trace. The definition of this metric may beextended to the Clifford algebra A, which is an enveloping algebra of both sl(4,) andsp(4,). The Clifford algebra A is a representation and a subalgebra of the universal en-veloping algebra U of these Lie algebras. We have normalized the mass, within a fixedgroup representation, by the dimension of the vector space carrying the representation,given by the trace of the representative of the identity in A. We may write the definition ofthe mass parameter in any representation of the algebras sl(4,) and sp(4,) and the cor-responding representation of the common enveloping Clifford algebra (A) as

( )( )( )

trtr

tr CA A

JJm J

I I

mmmmm

m

GGG= = º

Cg14 g

. (13.2.1)

It is known that the Cartan metric depends on the representations, but we shall only applythis expression to find ratios within a particular fixed induced representation of the envel-oping algebra.

If we consider geometric excitations on the constant substratum which determines theequipartition of the excitation energy, this mass may be expanded as a perturbation around thesubstratum in terms of the only constant of the theory, the small fine structure parameter acharacterizing the excitation,

J J J Ja a= + + +20 1 2 , (13.2.2)

G G aG a G= + + +20 1 2 , (13.2.3)

( )( )trm J J Jm m mm m mG a G a G a= + + + 2

0 0 1 0 0 114

, (13.2.4)

indicating that the zeroth order term, which we shall call the bare mass, is given entirelyby the substratum current and connection, with corrections depending on the excitationself interaction.

The variation of J is orthogonal to J. In the same manner as the 4-velocity has unitnorm in relativity and the 4-acceleration is orthogonal to the 4-velocity, we have in thedefining representation of tetradimensional matrices,


G G H H G G H HJ J e ee e I J J J J J Jmmk k- -· · · ·¢ ¢ ¢ ¢= = - = = =1 1 4 . (13.2.5)

This implies that J1 is orthogonal to J0,

( )tr J J J Jmm d ·= =1 0

1 04

. (13.2.6)

The first order mass correction depends on the connection w given by eq. (12.5.10)

( ) ( )tr tr trgm J J m J J J Ja a a

D G G G G· · · · ·= + = + =1 0 0 1 1 0 0 1 0 14 4 4 .(13.2.7)

The first order equation for the connection is linear,

( )L JG pa=1 14 , (13.2.8)

where L is a linear differential matricial operator. If we take the projection along J0, itssource becomes zero and since J0 is constant,

( )trL J L ma

G D·æ ö÷ç = =÷ç ÷çè ø0 1 0

4 . (13.2.9)

Therefore, there are no Dm solutions which depend physically on the source J. The onlyphysical solution of this homogeneous equation, which is not an external field solution,is the solution where Dm is zero. It follows that the first order correction must be zeroand the mass correction, giving the theoretical physical mass from the bare mass, shouldbe of order a2, or 10-5.

As indicated in previous work [10 ], these corrections correspond to a geometric quan-tum field approximation (QFT). In this chapter we limit ourselves to the zeroth order termwhich we consider the bare mass of QFT.

The structure group G is SL(2,) and the even subgroup G+ is SL1(2,). The subgroupL is the subgroup of G+ with real determinant in other words, SL(2,). There is anothersubgroup H in the group chain GÉHÉL which is Sp(2,). The corresponding symmetricspaces and their isomorphisms are discussed in appendix B. We are dealing with twoquotients which we shall designate as C and K,

( , ) ( , )( , ) ( ) ( , ) ( )

G SL SOK

G SL SO SO SO+

º @ @Ä Ä

4 3 32 2 3 1 2

, (13.2.10)

( , ) ( , )( , ) ( , )

H Sp SOC

L SL SOº @ @

4 3 22 3 1 . (13.2.11)

These groups have a principal bundle structure over the cosets and themselves carry rep-resentations. The geometric action of the K generators are translations on the coset K. Thefunctions on K are the natural internal representations.

We shall consider, then, the representations of SL(2,) and Sp(2,) induced fromthe subgroups SL1(2,) and SL(2,) over the symmetric spaces SL(2,)/SL1(2,) andSp(2,)/SL(2,), respectively. These geometric induced representations may be real-ized as sections of a homogeneous vector bundle (D, K, [SL1(2,)], G+) with SL1(2,)


representations as fiber F over the coset K [11 ], as shown in figure 3. To theinduced representation of SL(2,) on D, there corresponds an induced representationof the enveloping Clifford algebra A on D [12 ]. Furthermore, to the latter also corre-sponds a representation of the subgroup Sp(2,) on D. In other words, the vectorbundle D carries corresponding representations of A, SL(2,) and Sp(2,). Thesethree representations are functions on K valued on representations of SL1(2,). Ateach point of the base space M we consider the function space S of all sections of thehomogeneous vector bundle D. Define a vector bundle Sº (S, M, S, G), associated tothe principal bundle E, with fiber the function space S of sections of D. The fiber of Sis formed by induced representations of G.

There is an induced connection acting, as the adjoint representation of G, on thebundle S. The connection w is represented by Lorentz rotations on L and translationson K. The induced connection w may be decomposed in terms of a set of basis func-tions characterized by a parameter k, the generalized spherical functions Yk on thesymmetric space [13 ]. If K were compact, the basis of this function space would bediscrete, of infinite dimensions d. The components, relative to this basis would belabeled by an infinite number of discrete indices k. The Cartan-Killing metric in theinduced representations expresses the energy equipartition and we formally wouldhave for the mass parameter, in terms of the G, J components,

,

tr k kkk

k k

m Jd

mmG ¢

¢¢

= å14

. (13.2.12 )

All states equally contribute to the mass. Since the spaces under discussion arenoncompact, the discrete indices k which label the components, become continuouslabels and the summation in matrix multiplication becomes integration over the con-tinuous parameter k or convolution of functions J(k), G(k). In addition, if we workwith tetradimensional matrices and continuous functions on the coset K, the Cartan-Killing metric in A is expressed by trace and integration, introducing a 4V(KR) dimen-sion factor for the common representation space D, giving for the mass

Figure 3. Vector bundle sections.

(S, M, , G)

M

s (L)

(D, K, , L)

K


( )( ) ( )tr , ,

R

dkm J k k k k dkV A

G= ò ò 2 2 21

4 . (13.2.13)

where V(AR) is a characteristic volume determining the dimension of the continu-ous representation of A. We interpret the value of a function at k as the componentwith respect to the basis functions Y(kx) of the symmetric space K, parametrized byk, as usually done in flat space in terms of a Fourier expansion. We may say thatthere are as many “translations” as points in K. It should be noted that these “trans-lations” do not form the well known abelian translation group.

The G-connection on E induces a SO(3,1)-connection on TM. The combined action ofthe connections, under the even subgroup G+, leaves the orthonormal set km invariant [14] ,defining a geometric relativistic equivalence relation R in the odd subspace K. Each ele-ment of the coset is a group element k which corresponds to a space-time moving frame.Physically a class of equivalent moving frames k is represented, up to an SL1(2,) transfor-mation, by a single rest frame, a point k0 which corresponds the rest mass m. The decompo-sition of G and J is into equivalence classes of state functions Y(kx). The number of classesof state functions Y(kx) (independent bases), is the volume of a subspace KRÌK of classes(inequivalent points). The current and connection components are functions over the cosetK. Physically the integral represents summation of the connection ´ current product, overall inequivalent observers, represented by a rest observer.

There is a constant substratum solution [15 ], discussed in chapter 12, for the nonlineardifferential equations which provides a trivial connection v to the principal fiber bundle(E,M,G). This SL(2,) valued 1-form on E represents the class of equivalent local connec-tion forms of the substratum solution. At some particular frame s, which we may take asorigin of the coset, the local expression for v is the constant

g gs m dx m Jaav k* = - = - . (13.2.14 )

All points of K or C may be reached by the action of a translation by k, restricted tothe corresponding subgroup, from the origin of the coset. As the reference frame changesfrom s at the origin to sk at point k of the coset, the local connection form changes,

gk s k s k k dk m J k dk k dkv v L* * - * - - -= + = - + º +1 1 1 1 , (13.2.15)

which corresponds to the equivalence class of constant solutions v. All k*s*v cor-respond to the same constant solution class v, seen in the different reference framesof the coset.

In the principal bundle the constant connection v combines with the constant J toproduce a constant product over the coset K, as long as the transformation is orthogonalto J [16]. As indicated before, the mass variation produced by the last term in the equation,due to an arbitrary choice of frame, corresponds to inertial effects. The noninertial effectsare due to the first term in the right hand side of the equation because the current J is atensorial form which corresponds to the substratum potential L. It is clear that its sub-traction from the connection, the last term in the equation which only has a k depen-dence, transforms as a connection, and corresponds to the inertial connection. The physicalcontribution to the bare mass parameter may be calculated in the special frame s giving

( ) ( ) ( )C C Cgs J J m J Jm m

m mw L·* = = -g g g , (13.2.16)


and defining an invariant expression in G in terms of L, valid for a given representa-tion.

We are interested in the representations of SL(2,) and Sp(2,) induced by thesame representation of SL1(2,) and corresponding to the same trivial substratumsolution. In the defining representation of tetradimensional matrices, the product JJ

G G H H G G H HJ J e ee e I J J J J J Jmmk k- -· · · ·¢ ¢ ¢ ¢= = - = = =1 1 4 (13.2.17)

is invariant under a SL(2,) transformation on coset K and equal to the unit in A for both theG group and any H subgroup. There is a representation of A on the bundle S correspondingto the induced representation. The invariance (equality) of the product JJ must be valid inany representation of A, although the value of the product may differ from one representa-tion to the other. For the induced function representations valued in the sl(2,) algebra(Pauli matrices), the invariant product becomes the k integration,

( ) ( ) ( )

( ) ( )

ˆˆˆ ˆ

ˆˆˆ ˆ

,, ,

, ,

FF k kJ k k J k k dk

J k k J k k dk

a m bm a b

a m bm a b

k k k k

k k k k¢

=º =

¢ ¢ ¢ ¢ ¢

ò

ò

001 301 2 2 3 2

0201 2 2 3 , (13.2.18)

which, must be a constant 4´4 matrix F0 on K, independent of k1 and k3.The expression for the mass becomes,

( )( )tr ,

R

g

KR

mF k k dkm

AV= ò4

. (13.2.19)

The integrand is the same F0 constant for both groups, but the range of integration KRdiffers. Integration is on a subspace KRÌK of relativistic inequivalent points of K for thegroup G and on a subspace CRÌCÌK for the group H. The expressions for the massescorresponding to G-excitations and H-excitations become,

( )( ) ( )

( )( )trtr ,

R

g RG g

KR R

m V Km m FF k k dkV A V A

= =ò 04 4 , (13.2.20)

( )( )

( )( ) ( )

( )( )

tr,trR

g g G RH R

C RR R

m m m V CFm F k k dk V C

V KV A V A== =ò 04 4

.(13.2.21)

The bare mass parameters m are related to integration on coset subspaces, dependingproportionally on their volume. In other words, the ratio of the bare mass parameters forrepresentations of SL(2,) and Sp(2,), induced from the same SL(2,) representation asfunctions on cosets K and CÌK should be equal to the ratio of the volumes of the respec-tive subspaces.


13.3. Symmetric Cosets.

13.3.1. Volume of C Space.First consider, the volume of the tetradimensional symmetric space Sp(2,)/SL(2,)

which coincides with the quotient SO(3,2)/SO(3,1) as shown in appendix B. In the regularrepresentation, it has the structure

[ ]

*

*

x

x

xCx

x

é ùé ùé ùê úê úê úê úê úê úê úê úê úê úê úê ú= ê úê úê úê úê úê ú ê úê úë û ë ûê úé ùê úê úê úë ûë û

0

1

2

3

4

, (13.3.1)

where x4 must be a function of the xm imposed by the group structure [17 ] determining thatthe space is a unit hyperboloid,

( )x x xm nmnh= -

1 24 1 . (13.3.2)

The euclidian volume element dV(c) given in terms of the forms dxa(c) varies over thetetradimensional coset. The invariant measure dm(c) is determined by weighing the euclid-ian element by a density equal to the inverse of the jacobian of the transformation gener-ated by a translation in the coset,

( )( )

( )dV c

d cJ c

m = , (13.3.3)

v

dx dx dx dx dVd

x x x xm m nmn mn

mh h

= =

- -

0 1 2 3

1 1 , (13.3.4)

or in R5 with w as the fifth coordinate x4,

( )w x x dx dx dx dx dwm nmnd h+ - 2 0 1 2 31 , (13.3.5)

leading to

dx dx dx dx dVd

w wm = =

0 1 2 3

, (13.3.6)

where,

x x x x wm n m nmn mnh h + 2 . (13.3.7)


Inspection of this equation allows us to physically interpret the parameter w as a mea-sure of a variation of mass energy throughout the coset C ,

w x x mm nmnh= - = - 21 1 . (13.3.8)

The presence of the jacobian means that we should use coordinates adapted to thissymmetric space, polar hyperbolic coordinates, to calculate its invariant volume density,

( )C C

dV C dVg

w

m= = -

ò ò . (13.3.9)

The quotient space must be one of the standard tetradimensional unit hyperboloids H4

[18 ]. We add a superscript a which indicates the number of negative signs in the invari-ant standard definition of Hn,a,

( ) ( )n

i ii i

w x xa

a= + =

= + -å å2 22

1 1

1 . (13.3.10)

In particular, the space corresponds to the hyperboloid H4,3 ,

w u x y z w u k= + - - - = + -2 2 2 2 2 2 2 21 , (13.3.11)

in terms of the coordinates u, k corresponding respectively to the energy and to themomentum absolute value, and an overall parameter l=1 which characterizes theunit size of a particular tetradimensional hyperboloid. We introduce a parametriza-tion in terms of arcs in the symmetric space by defining the hyperbolic coordinatesj, q, b, c, by

cos x

x yj =

+2 2 , (13.3.12)

cos zk

q = , (13.3.13)

cosh u

u kb =

-2 2 , (13.3.14)

cos wc

l= . (13.3.15)

It should be noted that the hyperbolic arc parameter b is not the relativistic velocitybut is related to it by

tanh k vu c

b = = . (13.3.16)

In particular the volume of C is obtained by an integration over this curved


minkowskian momentum space, where the coordinates x stand for u, k. We split theintegration into the angular integration on the compact sphere S2, the radial boost band the energy parameter c,

( )

( )2

sin sinh

sinh C

V C dd d

d I

b pp

b

Wc c b b

p pb b b

= =

=

ò ò ò

ò

4

23 2

0 0 0

0

16 163 3

, (13.3.17)

obtaining the result in terms of a boost integral I(b).

13.3.2. Volume of K space.For the volume of K, the integration is over an 8 dimensional symmetric space. This

space G/G+ has a complex structure and is a nonhermitian space. The center of G+, which isnot discrete, contains a generating element k5 whose square is -1. We shall designate by 2Jthe restriction of the endomorphism of A, ad(k5), to the tangent space TKk. This space,which has for base the 8 matrices ka, kbk5, is the proper subspace of A corresponding tothe eigenvalue -1 of the operator J2 ,

( ) , ,

J x y x y

x y

l l l ll l l l

l ll l

k k k k k k k k

k k k

é ùé ù+ = + =ê úê úë ûë û- -

2 15 5 5 54

5 . (13.3.18)

The endomorphism J defines an almost complex structure over K. In addition, usingthe Cartan-Killing metric, in the Clifford representation,

( ) ( ) ( )( ) ( ) ( ), tr tr tr ,Ja Jb JaJb J a b ab a b= = - = =21 1 14 4 4g g , (13.3.19)

we have that the complex structure preserves the pseudoriemannian (minkowskian) Kill-ing metric. Furthermore the torsion S vanishes,

( ) [ ] [ ] [ ] [ ], , , , ,S a b a b J Ja b J a Jb Ja Jb= + + - = 0 . (13.3.20)

In this form, the conditions for J to be an integrable, complex structure, invariant by G,are met and the space K is a nonhermitian complex symmetric space [19 ].

It is known that the hermitian symmetric spaces are classified by certain group quo-tients. The symmetric space K is a noncompact real form of the complex symmetric spacecorresponding to the complex extension of the noncompact group SU(2,2) and its quotientsas shown in appendix B. This space coincides with the quotient SO(3,3)/SO(3,1)SO(2) ofthe SO(4,2) series. In particular we have the 8 dimensional spaces

( , ) ( , ) ( )( ) ( ) ( , ) ( ) ( ) ( )SO SL SO

R KSO SO SL SO SO SO

º » @ @ »´ ´ ´4 2 4 6

4 2 2 2 4 2

(13.3.21)

which are the five real forms characterized by SO(4,2). The extreme spaces correspond tothe two hermitian spaces, compact and noncompact R.

Between the extremes we find the three nonhermitian noncompact spaces, in particular


the space of interest K. In the regular representation these quotients have the matricialstructure [17, p. 178]

*

*

x y

x y

x yK

x y

x y

x y

é ùé ùé ùê úê úê úê úê úê úê úê úê úê úê úê úê úê úê ú= ê úê úê ú ê úê úë û ë ûê úé ùê úé ùê úê úê úê úê úê úë û ë ûë û

0 0

1 1

2 2

3 3

4 4

5 5

, (13.3.22)

where the lower right submatrix determines the conditions,

x x x yx y

y x y yx y

é ù é ù+ · ·ê ú ê ú=ê ú ê ú· + ·ë ûë û

14 4 2

5 5

11

, (13.3.23)

imposed by the corresponding associated groups on the coordinates x4, x5, y4, y5 in higherdimensional spaces (d>8), expressed by the scalar product in this submatrix, in terms of thecorresponding 4-vectors x, y and respective metric, related to the euclidian metric by Weyl’sunitary trick. As in the previous case, this condition determines a unit symmetric space.

Since these conditions are difficult to analyze, it is convenient to find the volume usingthe complex structure of the manifold. We may introduce complex coordinates zm on K. Inthis way we obtain a symmetric bilinear complex metric in the complex tetradimensionalspace K,

( )tr z z za a ba abk h Î= -

214

, (13.3.24)

iz z emm m y= . (13.3.25)

The group which preserves this symmetric bilinear complex metric is the orthogonalgroup SO(4,).

There are two standard projections of the complex numbers on the real numbers , thereal part and the modulus. In a similar manner as the modulus projects the complex plane tothe real half line, we define an equivalence relation S in the points on K by defining equiva-lent points as points with equal coordinate moduli. This equivalence relation may be ex-pressed by

S S S S S= ´ ´ ´1 1 1 1 , (13.3.26)where S1 is the one dimensional phase sphere. We may define the tetradimensional quotientQ by

KQ

S= . (13.3.27)


It is then convenient to parametrize K in accordance with this equivalence relation S.Each of the 4 complex coordinates z has a modulus êzê and a phase y which will be used asparameters. The Haar measure in terms of the euclidian measure should correspond tointegration on a complex tetradimensional space. We choose the parameters defining theeuclidian volume element at the identity,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

dV I d z I d I d z I d I

d z I d I d z I d I

y y

y y

=

10 0 1

2 2 3 3 , (13.3.28)

and translate it to the point k by a coset operation,

( )( )

( )dV k

d kJ k

m = . (13.3.29)

As before, the jacobian is a measure of a variation of mass energy throughout the cosetK. We calculate the volume integral using the invariant volume density,

( )d z d z d z d z d d d d

dJ k

y y y ym

=

0 1 2 3 0 1 2 3

. (13.3.30)

The jacobian does not depend on the phases and the volume element of S is separablefrom dm defining a complementary volume element corresponding to the quotient Q. Thevolume of K will then be the product of the volumes of S and Q, as indicated also by eq.(13.3.27). From eq. (13.3.23) it may be seen that Q is a symmetric space, by taking as itsrepresentative points those with real coordinates. On the other hand Q must be noncom-pact, otherwise K would also be compact since the relation S is compact. The quotientcannot be a product of a manifold times a discrete set because then K would have discon-nected components. The SO(4,) subgroup of SO(4,), which acts on this subspace Q of Kpreserving the bilinear metric in this space, has S3 as orbit. Therefore Q is symmetric,tetradimensional, noncompact with a tridimensional compact subspace SS equal to theriemannian sphere S3,

KQ S

S= É 3 . (13.3.31)

The quotient Q must, then, be the hyperboloid H4,4, the only one with an S3 spacelikesubspace, characterized in five dimensional space by the invariant

w u x y z w u k= - + - - - = - + -2 2 2 2 2 2 2 21 . (13.3.32)

The physical boost parameter b (the one which represents a change in kinetic energy) isthe hyperbolic arc parameter along the orbit of a one parameter noncompact subgroup.The parametrization is in terms of the hyperbolic coordinates j, q, z, b defined, insteadof eqs. (13.3.13, 13.3.14), by

cosh ub

l= , (13.3.33)


cos w

w kz =

+2 2 . (13.3.34)

We split the integration into the integration on the 4 compact phase spaces S andintegration on the tetradimensional space Q, parametrized by the coordinate moduli. Thelast integration is further split into integration on the compact 3 spheres S3 and integrationon the complementary noncompact direction, which corresponds to boosts b, obtaining

( ) ( ) ( )

sinh sin

K

d d d d d V Q V Sg VV K

dd d db pp p

y y y y

yb b Wz z

´- == =

æ ö÷ç ÷ç´ ÷ç ÷÷çè ø

ò

ò ò ò ò

40 1 2 3

4243 2 3

0 0 0 0

, (13.3.35)

( ) ( )( ) ( )sinh KdV K Ib

p p b b p b= =ò42 3 5 6

0

2 2 2 , (13.3.36)

in terms of another boost integral I(b). This result shows, as should be expected, that therelativistic equivalent subspace of K is of higher dimension than the relativistic equivalentsubspace of , as indicated by the respective integrals of the hyperbolic sine of b.

13.3.3. Ratio of Geometric VolumesWe expect that the ratio of the volumes V of the inequivalent subspaces of K and C,

corresponding to fundamental representation of spin ½, should be related to the ratio ofthe corresponding geometrical masses. As indicated in section 2 there is a subset of pointsof the symmetric spaces which are relativistic equivalent. We have to eliminate theseequivalent points by dividing by the equivalence relation R under the boosts of SO(3,1).Equivalent points are related by a Lorentz boost transformation of magnitude b. Thereare as many equivalent points as the volume of the orbit developed by the parameter b.

If we do not count the equivalent points, the respective inequivalent volumes are thecoefficient in eq. (13.3.17) for C,

( ) ( )( )( )

( )

( )C

RC

IV CV C

IV R

pb p

bb= = =

16163

3 , (13.3.37)

and eq. (13.3.36) for K,

( )( )

( )( )( )

( )K

RK

V K IV K

IV R

p bp

bb= = =

5 65 62

2 . (13.3.38)

Their ratio taking in consideration the equivalence relation R has a well defined limit, ob-taining


( )( )

R

R

V K

V Cp= 56 . (13.3.39)

We actually have shown in this section a theorem that says: The ratio of the volumesof K and C, up to the equivalence relation R under the relativity subgroup, is finiteand has the value 6p5 .

13.4. Physical Mass Ratios.It is clear from the previous discussion that, for a constant solution, the masses are

proportional to the respective volumes V(KR). The constants of proportionality only de-pend on the specified inducing SL(2,) representation. In particular for any two represen-tations induced from the spin 1/2 representation of L, the respective constants are equal.

Using these results, the ratio of masses of the fundamental representations of thegroups G and its subgroup H induced from the spin 1/2 representation of L, for the constantsolution, equals the ratio of the volumes of the inequivalent subspaces of the respectivecosets G/L1 and H/L, and has the finite exact value

( )( )

. .pRG

H R e

mV Kmm V C m

p= = = » =56 1836 1181 1836 153 , (13.4.1)

which is a very good approximation for the experimental physical proton electron massratio, in confirmation of the relation of the G group to the proton and the H group to theelectron. If this were the case the only other physical subgroup L=SL(2,) of G should leadto a similar mass ratio. Previously we have related L to the neutrino. In this case thequotient space is the identity and we get

( )( )( )

( )

( )/L R

H R R e

V L Lm V I mm V C V C m

n= = = =0 , (13.4.2)

which is in accordance with the zero bare rest mass of the neutrino.Small energy terms due to interaction field energies may be interpreted as small relative

“dressed” physical rest mass corrections to be added to the bare masses. In particular, thephysical mass of the neutrino may be entirely due to this small correction.

13.5. Summary.The geometrical ratio of volumes of the quotients SL(2,)/SL1(2,) and Sp(2,)/

SL(2,)) up to the equivalence relation under SL(2,) is equal, with a discrepancy of 2´10-5,to the value of the ratio of the masses of the proton and the electron.

It is possible to take the geometrical values as the values of the rest bare masses of theproton, electron and neutrino, which necessarily need correction terms because of theinteractions of the excitations. The correction due to the self interaction of the excitationswas estimated to be of the order of a2, equal to the order of the discrepancy.


References

1 G. G. González-Martín, in Strings, Membranes and QED, Proc. of LASSF, Eds. C.Aragone, A. Restuccia, S. Salamó (Ed. Equinoccio, Caracas) p, 97 (1989).

2 G. González-Martín, Gen. Rel. Grav. 26, 1177 (1994). See chapter 9.3 G. González-Martín, ArXiv physics/0009066, USB Report SB/F/274-99 (1999).4 F. Lenz, Phys. Rev. 82, 554 (1951).5 I. J. Good, Phys. Lett. 33A, 383 (1970).6 A. Wyler, Acad, Sci. Paris, Comtes Rendus, 271A, 180 (1971).7 Y. Ne’eman, Dj. Sijacki, Phys. Lett., 157B, 267.8 Y. Ne’eman, Dj. Sijacki, Phys. Lett., 157B, 275.9 F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne’eman, Phys. Rep. 258, 1 (1995).10 G. González-Martín, Gen. Rel. Grav. 24, 501 (1992).11 R. Hermann, Lie Groups for Physicists (W. A. Benjamin, New York) (1966).12 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

p. 90 (1962).13 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

p. 360 (1962).14 See section 4.4.15 G. González-Martín, USB preprint 96a. (1996).16 See section 9.3.17 R. Gilmore, Lie Groups, Lie Algebras and Some of their Applications (John Wiley

and Sons, New York), ch. 9 (1974).18 J. G. Ratcliffe, Foundations of Hyperbolic Manifolds (Springer-Verlag, New York)

(1994).19 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

p. 285 (1962).

14. CONNECTION EXCITATION MASS.

14.1. Introduction.The discussion in the previous chapters ratifies that some quantum mechanics effects are

contained in the unified theory. A curved substratum is the geometric mechanism to givemasses to the connection excitations or bosons. This is possible for nonlinear theories likethe one under discussion.

In particular, we have found that the nonlinear field equations of the theory admit a con-stant substratum solution for the connection field in terms of a geometric fundamental unit oflength [1 ]. The nonlinear field equations also have integrability conditions which lead to ageneralized Dirac equation which introduces a fundamental unit of angular momentum anda parameter which may be identified with mass. The mass associated to the matter excitationsaround this equation or fermions was defined geometrically in terms of the interaction en-ergy of the substratum solution. The ratio of the proton mass to the electron mass was calcu-lated with great precision using this theory [2 ].

Here we further illustrate these ideas by finding particular equations for the case of aconnection field excitation around this substratum solution and their ranges or masses [3 ]. Inaddition we will discuss under what conditions the excitations are massless.

14.2. The General Form of the ExcitationEquation.

In chapter 12 the field equation of the theory,

D JW pa* *= 4 , (14.2.1)was expanded by writing the exterior product in terms of differential forms and group genera-tors. The expression for the field equation becomes

( ) ( )[ ]

[ ] [ ][ ]

[ ]

,

, , ,

a a ba a bg g

a b c a b aa b c a b a

gg g E g E E

g g E E E E E J E

rm an r ar m n r

rm an r a ar m n r

¶ ¶ w ¶ w w

w ¶ w w w w pa

- -- + - +

é ù+ =ë û

2 1

2 4 . (14.2.2)

The commutator in the expressions introduce the structure constants and the trace of productsof the Lie algebra base introduces the Cartan-Killing metric.

The holonomy groups of the connection may be used geometrically to classify the interac-tions contained in theory. The subgroup chain SL(2,) É Sp(2,) É SL(2,) characterizes achain of subinteractions with reducing sectors of interactions.

A section is related by charts (coordinates) to elements of the group, SL(2,), which arematrices that form a frame of general SL(2,) column spinors, in the defining representation.The connection is an sl(4,) valued 1-form which acts naturally on the frame e (sections).

We shall proceed formally in the defining 4´4 representation. The equation may be writ-ten in terms of components referred to these bases,

147Connection Excitation Mass

( )

( )

[ ]

[ ]

c a b mnabn cmd d

a b a babdd

c c gg gg

cg g J

g

r a rm anr r m n

r a an m ar m n

w w w ¶ ¶ w

¶ w w w ¶ w pa

+ - +-

é ù+ - + =ê úë û-

2g

2 4 . (14.2.3)

We note that the terms quadratic in the structure constant also include terms depending onthe Cartan-Killing metric. This may be seen from the trace expression,

( )tr o.t. a b c d ab cd ac bd ad bcE E E E = - + +14 g g g g g g . (14.2.4)

To obtain field excitations we perform perturbations of the geometric objects in the equa-tion. Then the linear differential equation for the perturbation of the connection takes thegeneral form

( )[ ]

c cd c d d c dg

d

gg g L L g

J

rm an r a a r amnr m n r r mn

a

¶ ¶ dw w w dw dw d

pad

-- + + + =1 22 4

4 , (14.2.5)

where 1L and 2L are linear first and second order operators, respectively, with variable coeffi-cients which are functions of w . The second term, which arises from the cubic self interactionterm, may provide a mechanism to give effective masses to the connection excitations in thecurved substratum in terms of a parameter which represents the self energy, determined usingthe Cartan-Killing metric in the defining representation,

ccr

rw w w=2 . (14.2.6)

14.3. An Excitation Solution.In section 12.5 we have constructed a constant complex substratum solution, eq. 12..5.10

in terms of the geometrical fundamental unit of length mg, by extending the real functions ofthe substratum to complex functions,

( )ˆ ˆ ˆˆ ˆ ˆ ˆgm dx ia a aa a a aw w k w k k k k= + = - 5 . (14.3.1)

In particular if we assume that we can obtain a connection solution with algebraic compo-nents only in the complex Minkowski plane Kk generated by the Clifford algebra orthonormalset k, we have that the complete base E may be replaced by the orthonormal set k. Since theconnection has no even components, the metric in the base space-time M is flat. Using the tracerelations we obtain:

tr a b abk k h= -14 , (14.3.2)

tr ,d a bk k ké ù =ë û14 0 , (14.3.3)


( )tr , ,

tr

d c a b

d c a b d c b a d a b c d b a c

k k k k

k k k k k k k k k k k k k k k k

é ùé ù =ê úë ûë û- - +

14

14 , (14.3.4)

( )tr ,d c a b db ca da cbk k k k h h h hé ùé ù = -ê úë ûë û14 4 . (14.3.5)

If the complex excitation around the complex substratum has the form

( )ˆ ˆaa ar rr

w w dw= +

, (14.3.6)

the excitation equation becomes, using the complex metric g of the complex Minkowskiplane Kk, which involves lowering the indices and taking the complex conjugate,

( )( )

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆ

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ

d d

J

n c ar bn c ar bn c ar bnr r rd

nca ddb da cb

dw dw w w w dw w w w dw

pa d

* * + + + ´

- =

4

g g g g 4 ,

ˆˆ ˆ ˆ ˆd d Ja a r a a

rd d d ddw w dw w dw d pad* * + + =

2 22 4 . (14.3.7)

Given a representation, a solution to these coupled linear equations may always be foundin terms of the Green’s function of the differential operator and the current excitation dJ,which may be an extended source. In order to decouple the equations, it is necessary to assumethat dwr

r vanishes. If this is the case then all equations are essentially the same and the solutionis simplified.

We shall restrict ourselves to consider the equation for a unit point excitation of the quan-tum of physical charge and its solution which is the Green’s function. If we assume a timeindependent excitation with spherical symmetry, the only relevant equation would be the ra-dial equation. Since mg is constant, it represents an essential singularity of the differentialequation, an irregular singular point at infinity. The corresponding solutions have the expo-nential Yukawa [4 ] behavior. Let us interpret that the curved substratum gives an effectiverange w-1 to the linear excitations. Equation (14.3.7) time independent for a point source is,designating the fluctuation as a weak field W,

( ) ( ) ( )W x W x j g j g x xw pa d pa d pa d- ¢ - = - = - = - -2 2 2 2 2 2 24 4 4 , (14.3.8)

where we explicitly recognize that the current is of order a or equivalently of order chargesquared. Furthermore, we realize that the assumed excitation is the odd part of an su(2)Q

representation and should explicitly depend on the formal odd part of the charge. The chargewhich enters in this first order perturbation equation is the su(2)Q charge, eq. (8.4.5), definedby the original nonlinear unperturbed equation. This is the quantum of the only physical charge:the electric charge quantum e=1. We should recognize a geometric odd projection factor gwhich determines the odd component of the current and expresses this geometric charge quan-tum in units of a formal odd charge (or weak charge). The (ag)2 factor is not part of the dfunction which represents a unit odd charge. This equation may be divided by (ag)2, obtainingthe equation for the odd excitations produced by the unit odd point charge or equation for theGreen’s function


( )( ) ( ) ( )W x W x x x

gg

wpd

aa

æ ö÷ç ¢ - = - -÷ç ÷÷çè ø

22

21 4 . (14.3.9)

We may define a new radial coordinate r=agx’, rationalized to a new range -m-1 for oddexcitations, which incorporates the ag constant. Taking in consideration the covariant trans-formations of W and d, the radial equation is then

( ) ( )rW W r rr r

¶m pd

¶- ¢- = - -

22

21 4 . (14.3.10)

Since -m is constant, the Green’s function for this differential operator is

x x re ex x r

m m

p p

- -¢- - ¢-

= =¢ ¢-

1 14 4

. (14.3.11)

whose total space integration introduces, in general, a range factor,

r ddr r e

p

m Wp m

-

¥

¢--

¢¢ ¢ =òò4

22

00

1 14

, (14.3.12)

The range -m may be evaluated using eqs. (14.3.7) and (12.5.10),

( )CC gmm mm mw w w w w*= = =2 2g 8 , (14.3.13)

obtaining

gm

g gw

ma a

- = =2 2

. (14.3.14)

In order to obtain a complete excitation, instead of its odd part, set g equal to 1.

14.4. Massive SU(2) Excitations.According to physical geometry we may consider excitations of the SU(2)Q subgroup of G

around the substratum. The complex substratum solution indicates that this excitation acquiresan effective mass. Let us consider the three components of the SU(2)Q, connection as the threeclassically equivalent electromagnetic potentials A defined in section 12.4.

The electromagnetic subgroup is SU(2)Q, similar to the spin subgroup SU(2)S. The groupitself, as a fiber bundle (SU(2),S2,U(1)) carries its own representations [5]. The base space isthe coset SU(2)/U(1) which is the bidimensional sphere S2. The fiber is an arbitrary evensubgroup U(1). The action of this electromagnetic SU(2)Q is a multiplication on the fiber by anelement of the U(1) subgroup and a translation on the base space S2 by the action of the SU(2)Q

group Casimir operator, representing a squared total SU(2)Q rotation.This action is not as simple as translations in flat spaces, but rather has complications

similar to those associated with angular momentum due to the SU(2) group geometry. Theorientation of directions in SU(2) is quantized. In particular only one component of the elec-tromagnetic rotation generator E commutes with the group Casimir operator E2. This operator


acts on the symmetric coset S2 becoming the Laplace-Beltrami operator on the coset. Itseigenvalues are geometrically related to translations on S2 in the same manner as the eigen-values of the usual Laplace operator are related to translations on the plane. There are defi-nite eigenvalues, simultaneous with the Casimir operator, only along any arbitrary singlesu(2) direction, which we have taken as the direction of the even generator +E. The generatorE may be decomposed in terms of the even part and the complementary odd component E.We can not decompose E into components with definite expectation values. The splittinginto even and odd parts represents, respectively, the splitting of the group action into itsvertical even action on the fiber and a complementary odd translation on the base S2.

Consider that the exponential functions ek.r form a representation of the translation groupon the plane. The magnitude of the translation k is determined by the eigenvalue of the Laplaceoperator ,

k x k x k xe e k elD = = 2 , (14.4.1)where the absolute value of k is

( )mnm nk k kd=

1 2 . (14.4.2)

The odd subspace of su(2)Q, spanned by the two compact odd electromagnetic generatorsin E, is isomorphic to the odd subspace of the quaternion algebra, spanned by its orthonormalsubset qa. Associated to this orthonormal subset we have the Dirac operator q on a curvedbidimensional space. This Dirac operator represents the rotation operator L2 on the vector functionson the sphere which also corresponds to the laplacian bidimensional component. We obtain for thisaction, if we separate the wave function j into the SU(2)S eigenvector f and the SU(2)Q eigenvec-tor y, and use the fact that the Levi-Civita connection is symmetric,

E E iE- = 1 2

q 1

q 2

q 3

E E+ = 3

EQ

Figure 4.Electromagnetic

polar angle in su(2).


( ) ( ) [ ]

a a b a b a ba a b a b a b

aba b

q q q q q q q

g L

y y y y

y y y

= = +

= - = -D =

2

2 . (14.4.3)

This equation shows that the squared curved Dirac operator is the Laplace-Beltrami orsquared angular momentum Casimir operator on the sphere. Therefore, we may define the oddelectromagnetic generator E as the quaternion differential operator

a 2aE q C . (14.4.4)

The E direction in the odd tangent plane is indeterminable because there are no odd eigenvectorscommon with +E and E2. Nevertheless the absolute value of this quaternion must be the square rootof the absolute value of the Casimir quaternion. The absolute values of +E and E define a polarangle Q in the su(2) algebra as indicated in figure 4.

The angle Q is a property of the algebra representations, independent of the normalization asmay be verified by substituting E by NE. In section 7.4 we used the natural normalization of thegeometric algebra. The generators have twice the magnitude of the standard spin generators. Thisnormalization introduces factor of 2 in the respective commutation relations structure constantsand determines that the generator eigenvalues, characterized by the charge quantum numbers c, n,are twice the standard eigenvalues characterized by the spin quantum numbers j, m. Neverthe-less, it appears convenient to use the two different normalizations for the SU(2)Q and SU(2)Sisomorphic subgroups in accordance with the physical interpretation of the integer charge andhalf-integer spin quanta. With the standard normalization, in the base of the common eigenvec-tors of +E and C2, we have,

( ), ,C j m j j j m- = +2 1 , (14.4.5)

( ), , ,aaE j m q j m j j j m- = = +1 , (14.4.6)

, ,E j m m j m+ = . (14.4.7)

The electromagnetic generator has an indefinite azimuthal direction but a quantized polardirection determined by the possible translation values. Therefore we obtain, since the abso-lute values of the quaternions are the respective eigenvalues,

( ) ( )tan tanj c

m n

E C j j c c

m nE EQ Q

-

+ +

+ += = º = º

12 2 1 2

. (14.4.8)

The internal directions of the potential A and the current J in su(2)Q must be in the possibledirections of the electromagnetic generator E. In the near zone defined by 1rm- the Acomponents must be proportional to the possible even and odd translations. The total A vectormust lie in a cone, which we call the electrocone, defined by a quantized polar angle Qn

c relativeto an axis along the even direction and an arbitrary azimuthal angle.

The fundamental state ,c n that represents a charge quantum is ,1 1 and corresponds to


the ,1 12 2 electromagnetic rotation state. The corresponding Q½

½ angle is

( ) ( )tan tanE

Ep Q

-

+

+= = = º

12

12

1 12 2

12

13 3 . (14.4.9)

The su(2) connection excitations are functions over the bidimensional sphere SU(2)/U(1)generated by the -E odd generators. The complex charged raise and lower generators -E,which are different from the odd generator E, may be defined in terms de the real generators

E E iE- = 1 2 (14.4.10)and obey the relations

( ), ,E E c n n E c n+ - - = 2 . (14.4.11)

Therefore, these charged excitations, defined as representations of the SU(2)Q group, require asubstratum with a preferred direction along the even quaternion q3 in the su(2)Q sector. Onlythe complex solution, eq. (12.5.10), gives an adequate substratum. We shall call this substra-tum the odd complex substratum.

The odd complex substratum solutions in the su(2) sector reduce to

( )ˆˆ ˆ ˆ ˆcos sin a

adx A dx Aqm mm mw w k f k k k f- - - -= = =0

0 1 2 3 (14.4.12)

and should correspond to the two -E odd sphere generators. This odd su(2) subspace gener-ated by the complex solution is physically interpreted as an SU(2) electromagnetic odd sub-stratum, a vacuum, with potential -w which determines ranges which may be interpreted asmasses for the excitations of the electromagnetic connection in its surroundings. The range-m-1 is determined by the -w fundamental bidimensional component in the substratum equato-rial plane.

The even and odd components of the su(2) excitation in the E direction have to obey therelations,

A A A+ -+ =2 2 2

, (14.4.13)

Due to the quantization of the su(2) connection, in the near zone there only are two possibledefinite absolute values, associated to a fundamental representation, which are related by

sinA E

A EQ

- -

= = 1 2 . (14.4.14)

A fundamental su(2)Q connection excitation should be the spin 1 boson (SU(2)S represen-tation) which also is a fundamental SU(2)Q representation where the three component genera-tors keep the quantized relations corresponding to electromagnetic rotation eigenvalues ½, ½,similar to the frame excitation representation (proton). Of course, they differ regarding thespin SU(2)S because the former is a spin 1 representation and the latter is a spin ½ representa-tion. We shall call this excitation by the name complete fundamental excitation. When theangle Q is written without indices it should be understood that it refers to this representation.A fluctuation of the odd substratum


( ) ( ) qd w d w- - - = 0 (14.4.15)

does not provide a complete su(2)Q excitation.The su(2)Q currents are similarly quantized and define the formal odd charge. These cur-

rents and charges are related by

sinj E e

j E eQ

- - -

= = =1 2 . (14.4.16)

The g factor which expresses the geometric charge quantum in units of the formal odd charge(or weak charge) is

cscg Q= 1 2 . (14.4.17)

The only possible ranges associated with an su(2)Q excitation should be proportional to theonly possible values of the quantized connection

sinA

A gm

Qm

- -

= = =1

. (14.4.18)

The possible values of the range are

singm

g

Qwm

a a- = = 1 22 2

(14.4.19)

and

gmm

a=

2 2 . (14.4.20)

On the other hand, the excitation ranges should be provided by the excitation field equa-tion (14.3.7) as the absolute value of some substratum connection. The odd complex substra-tum admits a related substratum with an additional even component +w connection whichshould provide the value of the energy difference. This even part +w increases the modulus ofthe total connection. The possible value of +w should correspond to the allowed value of thetotal substratum connection absolute value m. We shall call this substratum the completecomplex substratum.

The value m2 represents the square of the bound energy of a complete (with its three com-ponents) su(2)Q fundamental excitation around the complete complex substratum with equa-tion,

( ) ( )A x A x xm pd ¢ - = - -2 2 4 . (14.4.21)

Since the orientations of the potential A and the substratum connection w are quantized,their vectorial decompositions into their even and odd parts are fixed by the representation ofthe connection. Therefore this m2 energy term may be split using the characteristic electroconeangle Q,


( ) ( ) ( )cos sin cos sinm m Q Q m Q m Q m m+ -= + = + º +2 22 2 2 2 2 2 . (14.4.22)

The energy parameter +m corresponds to the even component +w associated with theU(1) group generated by the k5 even electromagnetic generator.

This energy m produced by the excitation components may not be decomposedwithout dissociating the su(2) excitation due to the orientation quantization of itscomponents. If the complete excitation is disintegrated in its partial components, thecorresponding even equation is separated from the odd sector in K as indicated in eq.(14.3.7) and has an abelian connection. Therefore, no mass term appears in the evenequation, which is physically consistent with the zero mass of the photon. The en-ergy +m associated to +w in the k5 direction is available as free energy. For a shortduration the disintegration takes energy from the substratum. On the other hand, theenergy -m corresponds to the odd complex substratum and when the complete excita-tion is disintegrated, the energy appears as the mass term in the coupled equations(14.3.7) associated to the pair of odd generators -E

( ) ( ) ( ) ( )A x A L x xm m dw pd- + - ¢ - - + = - -2 2 21 4 . (14.4.23)

We may neglect the coupling term L1, as was done previously in eq. (14.3.7)), sothat the equation may approximately be written

( ) ( )A x A x xm pd- - - ¢ - = - -2 2 4 . (14.4.24)

14.4.1. Mass Values on Free Space.The fundamental fermionic solutions which represent the stable fermions have a

mass (energy), given by eq (12.3.27) in the defining representation. The only way ofcalibrating the geometrical mg which also appears in eq. (14.4.19) for w is throughthe only two physical masses proportional to mg by the integration required for theinduced representations as described in section 13.2. The proper current source onfree space for free -A su(2) connection excitations associated to high energy pro-tons corresponds to a full SL(2,) matter representation. We should calibrate mg

with the proton mass mp and obtain

p gm m= 4 . (14.4.25)

tr gm gmmw w w a m*= = =

1 2 24

. (14.4.26)

Considering these relations, the Q electrocone angle also determines the ratio ofthe two energies or masses associated to the fundamental excitation,

sinA

A

m

mQ- = . (14.4.27)

These relations determine that the SU(2) connection nuclear excitation energiesor masses are proportional to the proton mass mp,


. Gev. . Gev.g pA Z

m mm mm

a a= = = = » =

2 290 9177 91 188

2(14.4.28)

sin. Gev. . Gev.p

WA

mm m

Qm

a-

- = = = » =78 7370 80 422 (14.4.29)

These values indicate a relation with the weak intermediate bosons [6]. They also indicate arelation of this electroconic angle with Weinberg’s angle: Weinberg’s angle would be the comple-ment of the angle Q. They admit relative “dressed” rest mass corrections of order a, in particularthe charged bosons mass needs electromagnetic corrections. For example if we use the cor-rected value for Q, obtained from its relation to the experimental value of Weinberg’s angle,

sin

. Gev . GevpWA

mm m

Q

a- = = » =79 719 80 42

2. (14.4.30)

Let us define a monoexcitation as an excitation which is not a complete or odd SU(2)Q

representation and is associated to a single generator. A collision may excite a resonance atthe A fundamental connection excitation energy m

A determined by the complete complex

substratum. The fundamental connection excitation A which is an SU(2) representation mayalso be dissociated or disintegrated in its three components (+A, -A), when the energy issufficient, as two -A free monoexcitations, each one with mass (energy) A

m - determinedby the odd complex substratum, and a +A third free U(1) monoexcitation, with zero bare restmass determined by the even complex substratum component +w which is abelian.

This simply determines the existence of four spin-1 excitations or boson particles asso-ciated to the SU(2) connection and their theoretical mass values:

1.The resonance at energy m of the fundamental potential A, which has to decay neu-trally in its components (+A, -A) and may identified with the Z particle.

2.The pair of free monoexcitations -A, charged 1 with equal masses -m , which mayidentified with the W particles.

3.The free excitation +A, neutral and massless which may identified with the photon.In this manner we may give a geometric interpretation to the W, Z intermediate bosons and toWeinberg’s angle which is then equivalent to the complement of the Q electrocone angle.

14.4.2. Connection Excitations on a Lattice.The equation (14.3.7) for the odd SU(2) excitation displays additional terms which repre-

sent a geometric current source automatically produced by the same field excitation. Thiscurrent, with charged connection excitation components, may dominate the total effectivecurrent. In these cases the original wave equation (14.3.7), decoupled and time independent,determines a Helmholtz equation [7] for a coherent enhanced self-sustained collective po-tential wave which we may write as

( ) ( ) ( )( )A x A x j A xw pa d- = -2 2 24 . (14.4.31)

As an application, we may consider a magnetic su(2) excitations around a lattice withperiodic potentials in some media. Any wave in a periodic medium should be described as aBloch wave [8]. The properties of this waves are essentially based on the Floquet theory [9]


of differential equations. If we have a many excitation system composed by different particlespecies in the medium, we should consider the Bloch wave equations for Helmholtz's poten-tial and Pauli's particle equation for the different species. The wave functions behave as Blochwaves consisting of a plane wave envelope function and a periodic Bloch function unk(r) withthe same periodicity of the external potential,

( ) ( )ik r

nk nkr e u ry =

. (14.4.32)

The index n characterizes the energy bands of the solutions. A Bloch wave vector k repre-sents the conserved momentum, modulo addition of reciprocal lattice vectors. The wave groupvelocity is also conserved. Wave associated particles can propagate without scattering throughthe medium almost like free particles. There are possible transfers of energy and momentumamong the different waves.

For the collective A field the domain of the Bloch functions unk are lattice cells wheredisturbances arise. These disturbances should be produced by first order a terms in the cur-rent, eq. (14.4.31).

Equation (14.4.19) for the range m, using the mass mp gives mW as shown in the previoussection. For a collective state of excitations, an -A su(2) connection excitation state mayalso have a current source which corresponds to Sp(2,) matter representations with a com-mon k0 generator and the other fundamental mass me. The same expression (14.4.19), usingthe electron mass me, gives a lower energy mass for the connection excitation,

sin . . MeveeA

mm m

Qm

a-

- = = = =83 9171 42 88142

. (14.4.33)

Therefore, in this case, the complex massive collective SU(2) connection excitations canbe in two states of different range or mass which we indicate in general by mG. These excita-tions obey Bose-Einstein statistics and under certain conditions may condense into a collec-tive coherent state corresponding to its lowest energy state. The mass given by eq. (14.4.33)should be related to energy terms in the A potential Bloch functions unk. Using this mass mA

instead of the free space mass mW we also obtain a quantum relation required by the coherentglobal complete su(2) algebra between its odd and even components,

tan Am rA e AQ -- +=2 222 . (14.4.34)

14.5. Equations for Massless Fields.Since the effective connection mass arises from the cubic term, a curved substratum is a

necessary condition for mass. This is not a sufficient condition. If the curved substratumcorresponds to an abelian subalgebra there is no cubic term. Therefore, mass is associated to aconstant curved nonabelian substratum. Nevertheless, small energy correction terms due tointeraction field energies may be interpreted as small relative “dressed” physical rest masses forabelian massless excitations. Since the abelian Cartan subalgebra is tridimensional, we have amaximum of three massless long range excitation fields. In accordance with the discussion insection 7.2 each massless field is characterized by the helicity quantum number.

One of the three massless fields is associated to any one of the compact generators associ-ated to the electromagnetic SU(2). Therefore this massless excitation obeys Maxwell’s elec-tromagnetic equations and behaves like a photon, in flat space-time. The second massless fieldis associated to any one of the compact generators associated to the rotation SU(2). Therefore


this massless excitation obeys equations similar to Maxwell’s, but the action of the connec-tion is a rotation on spacelike vectors (spinning action) and should be interpreted as part ofgravitation. The third massless field is associated to any one of the noncompact generatorsassociated to relativistic boosts. The linear equations satisfied by this excitations are pseudomaxwellian equations and its action corresponds to Lorentz boosts (accelerations) and shouldalso be interpreted as part of gravitation.

We shall look for wavelike equations for the second massless excitation. A connectionform with a single algebraic component along the generator k2k3 is an even element of thealgebra which induces a connection in TM with a single algebraic component along the gen-erator of rotations in the 2-3 plane by means of the differential of the group homomorphismfrom SL(2,) to SO(3,1). We shall make the calculation in terms of orthonormal tetrad com-ponents. The metric components are constants, but the tetrads components are coordinate func-tions The connection and curvature forms may be written as

ˆˆ ˆˆ ˆ ˆE Aa a m

mb bw q= 1 , (14.5.1)

( )ˆ ˆ ˆ ˆˆ ˆ ˆ ˆd E dA A da a r r

r rb bw q q= +1 , (14.5.2)

( )ˆ ˆ ˆˆ ˆ ˆ [ ]E F E A dx dxa a a m n

m nb b bW ¶= = 1 1 . (14.5.3)

The field equations for the connection depend on the base space metric. For free space theyare

ˆ ˆˆ ˆd E d Fa ab b

W* * * *= =1 0 , (14.5.4)

which may be written, in terms of the codifferential or the metric and the Levi-Civita connec-tion,

( )L Ld g dxnl mn l md w w= - = 0 , (14.5.5)

formally looking like Maxwell equations. A similar equation should be obtained for the thirdmassless field. In general we have to solve the equations simultaneously with those relatingthe tetrad to the connection.

In accordance with previous use, we indicate by G the classical coefficients of the connec-tion in TM in general coordinates. In other words, the G are related to the components of theconnection forms w by a change to a coordinate frame. At each point m in the base manifold Mthe transformation to arbitrary coordinates are linear associative transformations with unit andinverse. The transformations at a fixed point may be considered matrices of GL(4,). Hence,the coefficients of the connection in arbitrary coordinates, at each point of a neighborhood Uof m, may be obtained from the connection 1-forms in an orthonormal frame, by a GL(4,)connection transformation at each point of U, in other words by a local section in a GL(4,)principal bundle over M. Since the SO(3,1) principal bundle may be identified with a subbundleof the former bundle, we may write the transformation of the connection w by an elementgÎGL(4),

w w ¶- -¢ = +1 1g g g g . (14.5.6)

Taking for g the matrix corresponding to the components of a coframe q in the arbitrary coor-dinate system and for u its inverse, we obtain the connection coefficients, in terms of the


connection forms, using matrix notation. We can multiply on the left by q to define themixed components

ˆˆ ˆ ˆˆ

a a b aln n l nbl

G w q ¶ qº + . (14.5.7)

The antisymmetric part of this relation gives, of course, the expression for the torsion,

[ ] [ ]ˆˆ ˆ ˆ ˆ

ˆ ] [ ][a a b a a

n l nln lnb lS w q ¶ q Gº + = , (14.5.8)

ˆˆ ˆ ˆˆda a a bb

S q w q= + . (14.5.9)

The compatibility of the connection with the metric determines a relation among the connec-tion form, the orthonormal frame and the Levi-Civita connection of the base space (Christoffelsymbols),

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )a g a g a ggl a m n gn a l m gm a n lw q q w q q w q q+ - = 0 , (14.5.10)

{ } ( )g g g gr a a amr ln am an lm al nmlnG S S= + + . (14.5.11)

A substratum solution which allows massless excitations, may be obtained by solvingsimultaneously equations (14.5.5, 14.5.9, 14.5.11).

14.5.1. Restrictions to Possible Solutions.We shall leave the general case for future consideration. Here we consider the case when

Einstein’s equations for the connection are also satisfied. Then the torsion is zero, the commonvacuum solutions of the field equation and the stress energy equation are Einstein spaces [10,11]. The curvature tensor of any abelian solution takes the form

( )ˆ ˆ ˆˆ ˆ ˆˆ ˆ [ ]ˆ ˆR E F E A dx dxa a a m n

mn m nbmn b b¶= = . (14.5.12)

As an example we assume plane waves, in other words, that the transverse hypersurface isa plane. The transverse metric is flat and we can use the trivial coordinates on the plane 2-3 tosimplify the equations. They imply that F is zero, except for one component,

F A A¶ ¶= -01 0 1 1 0 , (14.5.13)

which indicates that the connection is trivial for a transverse field. For Einstein spaces withzero cosmological constant there are no transverse solutions of this type.

14.6. Summary.We have given equations for the geometric connection excitations around a fixed substra-

tum geometry. In a particular case we showed that the excitation equations take the form of aYukawa equation. The corresponding solutions behave as short range fields, with a rangegiven by a constant associated to the substratum connection solution. This constant, which ingeneral depends on the particular representation, may also be interpreted as the mass of theparticle associated to the excitation.

In particular we studied the excitations which form an SU(2)Q representation. It was shown


References

1 G. González-Martín, USB preprint, 96a (1996).2 G. González-Martín, ArXiv physics/0009066, USB Report SB/F/274-99 (1999).3 G. González-Martín, ArXiv 0712.1538, USB Report SB/F349-07 (2007).4 H. Yukawa, in Foundations of Nuclear Physics, R. T. Beyer, ed. (Dover Publications,

New York), p. 139 (1949).5 G. González-Martín, I. Taboada, J. González, ArXiv physics/0405126, USB Reporte SB/F/

305.3-02, (2003).6 R. E. Marshak, Conceptual Foundations of Modern Particle Physics, (World Scientific,

Singapore) ch. 6 (1993).7 Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics, 1st edition

(McGraw-Hill,New York), Vol.1, Chap.5.8 F. Bloch, Z.Physik 52, 555 (1928).9 G. Floquet, Ann. École Norm. Sup. 12, 47 (1887).10 E. Fairchild, Phys. Rev. D14, 384 (1976).11 G. González-Martín, Gen. Rel. and Grav. 22, 481 (1990). See chapters 3 and 5.

that the generators associated to these excitations should obey the group commutation ruleswhich determine that the internal direction of a connection fluctuation A is along one of thepossible quantized directions. In consequence, the total vector A should lie on a cone definedby the quantized angle Q relative to a polar axis along the even direction with an arbitraryazimuthal angle.

The only possible excitation ranges of the fundamental su(2)Q representation should beproportional to the only two possible absolute values of its quantized connection, which arerelated by this polar angle. The corresponding proton source excites connection masses onfree space which are approximately equal to the experimental masses of the W, Z bosons,with possible corrections of order a. Weinberg’s angle is related to the polar angle Q. In thepresence of condensed matter, electron sources may excite lower connection mass excita-tions around a lattice with periodic potentials.

The possibility of having massless long range excitations which are classical fields arounda constant substratum is limited to the maximal abelian Cartan subalgebra which has dimension3. One of the massless excitations corresponds to the photon. The other two massless excita-tion would be associated to gravitation. The restriction to Einstein equations in vacuum forcesone particular solution to be zero.

15. WEAK NUCLEAR INTERACTION.

15.1. Introduction.It is possible to represent the weak nuclear interaction as a neutrino interaction within

these geometric ideas, in the same manner as gravitation and electromagnetism are repre-sented by the unified theory. The role that Clifford algebras play in the geometrical struc-ture of the theory provides a link to the nonclassical interactions theories. An even frameexcitation obeys the neutrino equation. The holonomy groups of the connection may beused geometrically to classify the interactions contained in the theory. The subgroupchain SL(2,) É Sp(2,) É SL(2,) characterizes a chain of subinteractions with re-ducing sectors of nonclassical interactions and has a symmetry SU(2)U(1) equal tothat of weak interactions, as shown in section 3.7.

Furthermore, the field equations of the theory admit a set of constant solutions for theconnection field in terms of a geometric fundamental unit of length [1]. We have seen inprevious chapters that we can associate the result of a physical measurement to one ofthese distances to obtain a calibration of the unit of length so that the corresponding scalecan be used to obtain other physical constants. The ratio of the proton and electron masseswas calculated, with great precision using these ideas [2]. Later we calculated the massesof the W an Z bosons [3] as indicated in the previous chapter. These results allow us toexpect that the geometric theory may really represent the weak interactions.

The standard model has had many successes in describing weak and strong interac-tions that lead to a general acceptance of the model. Nevertheless this model may beconsidered an effective theory of some other more general theory. History has shown usthat in many cases, progress in physics is attained by the evolution or replacement ofmodels which provide partial fits to experimental data by more general ones. The studyof geometrical elements to represent physical particles is a trend in this direction. There-fore, in the quest of unification, the lack of an a priori relation between geometry and thestandard model should not deter us from investigating the possible physical interpreta-tions of the odd sector of the connection which precisely holds the key to its relationwith nonclassical interactions. In particular, the odd sector may serve as a possible linkto the standard model. Even if this were to be impossible, as a matter of fact, there areother models [4] some of whose features may eventually offer a complementary approachto particles and their interactions and may be related to our theory.

As next task we consider here low energy aspects of weak interactions. In this chapterwe shall mainly concern ourselves with Sp(2,) and SL(2,) connections. A section e isrelated by charts (coordinates) to elements of the group, SL(2,), which are matrices thatform a frame of general SL(2,) column spinors, in the defining representation. The con-nection is an sl(4,) 1-form which acts naturally on the frame e (sections).

The field equation, which relates the derivatives of the curvature to a current source Jis

D k J JW pa* * *= = 4 , (15.1.1)

ˆˆJ e u em a mak= , (15.1.2)

161Weak Nuclear Interaction

in terms of the frame e, an orthonormal set of the algebra k, the correlation in spinorspaces and a space-time tetrad u. The coupling constant is 4pa, where a is the finestructure constant. To avoid confusion, in this chapter we use the symbol f for a generalsection reserving e for electron sections.

The field equation implies a conservation law for the geometric current, which deter-mines a generalized Dirac equation in terms of the local frames. This equation, for theeven an odd parts of a frame f reduces, under certain restrictions [5, 6] to

f f mfm mm mk ¶ k G+ - - -= = , (15.1.3)

f f mfm mm mk ¶ k G- - + += = , (15.1.4)

implying that a frame for a massive corpuscle must have odd and even parts. For an evenframe,

=f m- = 0 0 . (15.1.5)

Therefore, for an even frame we have, multiplying by k0,

fmms ¶ + = 0 , (15.1.6)

which is the equation normally associated with a neutrino field. A fluctuation of f+ on thefixed substratum obeys also the last equation.

We also have suggested that particles may be represented by excitations on a geomet-ric substratum. In particular, the electron and neutrino, at fixed states, correspond to ma-trices with only one nonzero column which form an algebraic representation of the group.

15.2. Geometric Weak Interaction.From previous discussion, [7] we consider that an electroweak interaction may be re-

lated to the action of the Sp(2,) holonomy group. The total interaction field should cor-respond to a connection G, a representation of Sp(2,). The total matter current should beassociated to an Sp(2,) frame f representing both the electron field e and the neutrinofield n.

At a point, the total frame f of the interacting e,n is related to an element of the groupSp(2,), a subspace of the geometric algebra R3,1=R(4). The frame f may be decomposedinto fields associated to the particles e,n by means of the addition operation within thealgebra. These e, n fields are not necessarily frames because addition does not preservethe group subspace Sp(2,) and geometrically, e and n are not sections of a principal fiberbundle but rather of an associated bundle with the Clifford algebra as fiber.

The source current J in the theory is

J f f f fk k= = , (15.2.1)

where f is the frame section associated to the total field of the electron and the neutrinoand k represents the orthonormal set. For the Sp(2,) group the correlation reduces toconjugation.

Due to the properties that a neutral particle field should have (there are no electromag-netic or massive effects), we consider that the effect produced by n should be small rela-tive to the effect of e. Hence we may assume that, in the composite system, n is a perturba-


tion of the order of the fine structure constant a, the only physical constant in the theory,

f e an= + . (15.2.2)

Then the current becomes

( ) ( ) ( )J e e e e e ean k an k a kn nk a nkn= + + = + + + 2 . (15.2.3)

The intermediate terms may be considered as a perturbation of order a to a substra-tum electronic matter frame. The perturbation current may be written, by splitting e intoits even and odd parts and noticing that n has only even part,

( ) ( )J e ek a h xk kn n k h k x+ +é ù- = + + +ê úë û

0 0 . (15.2.4)

As usual in particle theory, we neglect gravitation, which is taken as an even SL(2,)connection. If we look for effects not imputable to gravitation, it is logical to center ourattention on the odd part of the perturbation current as a candidate for the interactioncurrent,

( )j m m ma a hk n nk h- = + , (15.2.5)

which has the structure of the weak current. It should be noted that the neutrino n auto-matically associates itself, by Clifford addition, with the even part h of the electron. Thiscorresponds to the Weinberg-Salam association of the left handed components as a dou-blet, with the same Lorentz transformation properties.

If we apply perturbation theory to the field equations, we expand the connection G interms of the coupling constant a. We have

J J J Ja a= + + +20 1 2 , (15.2.6)

G G aG a G= + + +20 1 2 . (15.2.7)

The substratum equation and the first varied equation, which is second order in a,have the following structure,

( ) EED JW pa* = 4 , (15.2.8)

( )*D Jd W pad= 4 . (15.2.9)

In the static limit, the dJ excitations reduce to the su(2)Q+su(2)S subalgebra. The excitationsare SU(2)Q representations and therefore their components are subject to the cuantization oftheir electromagnetic orientations, in accordance with section 14.4. We should express theterm J1 as a function of its odd component. In this manner, we let the first order terms be, asa function of the representation electrocone angle Q,

sinJ j j Q-= =1 , (15.2.10)

WG =1 , (15.2.11)

and we obtain for the variation, the linear equation


( )d dW LW ja pa* + = 24 , (15.2.12)

where L is a linear first order differential operator determined by the substratum. Thisequation may be solved in principle using its Green’s function . The solution in terms ofcomponents with respect to a base E in the algebra is

( ) ( )ji i j

j xx xW dx nm mnpa ¢¢= ¢ -ò4 . (15.2.13)

It is well known that the second variation of a lagrangian serves as lagrangian for thefirst varied Euler equations. Therefore, the GJ term in the covariant derivative present inthe lagrangian [8] provides an interaction coupling term which may be taken as part of thelagrangian for the process in discussion. When the GJ term is taken in energy units, con-sidering that the lagrangian has an overall multiplier, it should lead to the interactionenergy for the process. The second variation (or differential) in a Taylor expansion of theenergy U, corresponds to the hessian of U,

( ) ( ), i i ji j i

U UU x U x x x

x x x¶ ¶

d d d d¶ ¶ ¶

= + + +210 0

2 . (15.2.14)

The GJ term gives the interaction lagrangian for the substratum, in energy units,

( )E tr E E E E EJ J j Am m mm m mG Gé ù= - + » -ê úë û

1 14 2 , (15.2.15)

where jA is clearly the electric energy. For the perturbations, the interaction lagrangian isgiven by 1/2 the second variation or dJ.dG,

( )tr W j j Wm mm ma- +2 1 1

4 2 = . (15.2.16)

It is clear that this interaction is carried by G or W. Nevertheless, we wish to obtain acurrent-current interaction to compare with other theories at low energies. Substitution ofeq. (15.2.13) in the last equation gives the corresponding action which, for clarity, weindicate by

( ) ( ) ( )tr j i l l ii j lx x E E E E j x j xdxdx m nmnpa é ù¢ ¢¢ - += - ê úë ûò3 1

42 , (15.2.17)

representing a current-current interaction hamiltonian, with a coupling constant derivedfrom the fine structure constant. This new expression may be interpreted as a weak inter-action Fermi lagrangian. The associated coupling constant is of the same order as thestandard weak interaction coupling constant, up to terms in the Green’s functions .

15.3. Relation with Fermi’s Theory.The action, in terms of elements of the algebra and the trace, corresponds to the scalar

product. For the case of the isotropic homogeneous constant substratum solution [9] theGreens function is a multiple of the unit matrix with respect to the algebra components.We assume here that for a class of solutions the Green’s function have this property. Thenfor this class of solutions, the last equation may be written as


( ) ( ) ( )tr x x j x jdxdx xm nmnpa ¢¢= - ¢-ò3 1

44 , (15.3.1)

where the j are matrices.The current defines an equivalent even current j- , the k0 component, by inserting k

0k0

and commuting, as follows,

( ) ( )† †j m m m m mhk k k n nk k k h k h s n n s h- = + = +0 0 0 0 0 , (15.3.2)

† + h.c.j h sn-- = . (15.3.3)

Each 2´2 block component of this 4 dimensional real current matrix is an even matrixwhich may be represented as a complex number, by the known isomorphism between thiscomplex algebra and a subalgebra of 2´2 matrices. Thus, we may use, instead of thegeneralized spinors forming the frame f, the standard spinor pairs, h1, h2 and n1, n2 whichform the matrices corresponding to the even part, respectively, of the electron and theneutrino. The first term of j- , which corresponds to a nonhermitian term, is

† †ˆ†B

A † †

m mm

m m

h s n h s nh s n

h s n h s n

é ùê ú= ê úë û

1 21 1

1 22 2

. (15.3.4)

In this manner, if we use the standard quantum mechanics notation in Weyl’s represen-tation, each component is of the form

( ) †e nj m m mY g g Y h s n- = + =51 1

2 , (15.3.5)

which may be recognized as the standard nonhermitian Fermi weak interaction current forthe electron neutrino system. We shall indicate this current by jF and we may write, for the0-0 matrix component j0

( )†F Fj j jm m m- = +1

0 2 . (15.3.6)

We may evaluate the trace of the currents in the expression for ,

( ) ( ) ( ) ( )† †tr tr tr trj j j j j j j jk k k k- - - - - - - -- - - - - -= = =0 0 0 0 . (15.3.7)

As discussed previously, within this theory, particles would be represented by excita-tions of frames. These fluctuations are matrices which correspond to a single represen-tation of the subgroup in question. This means that for each pair of spinors of a spinorframe, only one is active for a particular fluctuation matrix. In other words, for a fluctua-tion, only one of the components of the matrix in eq. (15.3.4) is nonzero and we mayomit the indices on the spinors,

( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )( )

† ††

† †

h.c. h.c.tr

h.c. h.c.

x x x xj j

x x x x

m nm n

m n

h s n h s n

h s n h s n

- -æ öé ùé ù ¢ ¢+ + ÷ç ê úê ú ÷ç = =÷ç ê úê ú ÷÷çè øë û ë û

¢ ¢+ ´ +

0 00 0

0 0 0 0

(15.3.8)


where the right side is in complex number notation instead of 2´2 even matrices. Thisterm combines with its hermitian conjugate. It should be noticed that in going from thereal 4´4 to the 2´2 complex matrix realizations the ¼ factor in front of the tracechanges to 1/2 and due to the form of the excitation matrices introduces a factor of 1/2.

The resultant expression for is

( ) ( ) ( )†0 0sin

x x j x j xdxdx m nmn

pa

Q- -¢ ¢¢= - -ò

3

2

2 . (15.3.9)

We now shall restrict the discussion to excitations around the constant complex sub-stratum solution indicated before, eq. (12.5.10). Then the Green’s function is determinedfrom the fluctuation equation,

( ) ( ) ˆˆ ˆ ˆ ˆg gd d m m Ja a r a a

rd d d ddw dw dw d pad* * + + =

2 22 2 2 2 2 4 , (15.3.10)

where mg has the constant value given by eq. (12.5.11), in the 44 representation. Wemay decouple the equations by assuming a zero value for the cross coupling termdwa

a, and introduce the radial coordinate r as was done in the mentioned section.The result is a Yukawa equation, where we have defined the constant coefficient insidethe parenthesis as the mass parameter m in the corresponding representation of inter-est. In general, the Green’s function has a Dirac time d function which allows inte-gration in t’. The spatial part of the Green’s function should provide an equivalentrange for the interaction. Since m is a constant, the equation reduces, for a pointsource, to the radial equation. The Green’s function is

x x re ex x r

m m

p p

¢- - ¢-- -= =

¢ ¢-1 1

4 4 . (15.3.11)

If we further assume that the currents vary slightly in the small region of integrationso that j(x’) approximately equals j(x), we obtain an approximation for the hamiltoniancontained in the geometric theory,

( ) ( )

( ) ( )

†w

†0 0

.sin

r ddx j x j x dr r e

dx j x j x

p

m Wa

pam Q

¥

¢-- -

- -

=¢¢ ¢= ·-

-·

ò ò ò

ò

42

0 00 0

2 2

12

2

. (15.3.12)

The value of the Q angle, which indicates the relation of the odd current with thetotal current, is determined by the component orientation quantization in the corre-sponding current representation. Since the current J is quadratic in terms of the framee, its excitation corresponds to a pair of su(2)Q fundamental excitations de and thecurrent excitation j is the representation with electromagnetic spin index i=1 in equa-tion (14.4.8). We have then the corresponding electrocone angle which we shall indi-cate by Q1 and


( )

( )sin sin

j j i i

j i ij jQ Q

- -

- +

+= = = = =

+ ++

11 1 2 2

1 231 1

. (15.3.13)

The expression of the currents in the integral may be rewritten in the following manner,

( )† † † †F F F F F Fj j j j j j j jm m m m

m m m m- - = + +1

0 0 4 2 . (15.3.14)

We can evaluate this scalar in any coordinate system. We may choose a system where jF isalong the timelike direction, that is, it has only a JF component. We may write then

( )† † † †

†

cos

cos

i iF F F F

F F

j j j j j j e e j j

j j

m j jm

mm

j

j

- - - - -= = + + =

=

0 0 2 2 0 210 0 0 00 0 04

2

2

, (15.3.15)

in terms of a phase j. The angle j measures the degree of nonhermiticity of jF. If j is zerojF is hermitic. Since j is not measured in standard experiments, we may expect its influenceon is through its average value <cos2j> and leave the expression in the form,

†cossinw F Fdxj j

pa jm Q

- < >= ò

2

2 21

2 , (15.3.16)

which determines the lagrangian and the current jF assumed in Fermi’s theory.The value of the weak interaction constant GF is determined from results of experi-

ments like the muon decay. As is it always done, in other weak interaction theories, wemay fit the value of the constant in front of the integral to its experimental value by fixingvalues of m and j,

cossin

FGpa jm Q

-- < >=

2

2 21

22

. (15.3.17)

On the other hand, we now have at our disposal the possibility of theoretically calcu-lating the value of GF from the knowledge of the constant substratum solution. The valueof the mass m is theoretically obtained in a geometric unit because of the used representa-tion,

- ggmm

a= 12 2

. (15.3.18)

This geometric unit g may be calibrated, in the SL(4,) representation induced bySL(2,), from the value of the proton mass mp in terms of a unit of mass. This is desirablebecause the expression for the proton and electron mass ratio agrees with the experimen-tal value [10]. The mass parameter for the proton, expressed in the geometric unit in thedefining representation, is

- gp gm m= 14 , (15.3.19)

and we may calibrate the fundamental unit of length g by the experimental value of the


proton mass. Because of similarities of the Fermi weak current with the electromagneticcurrent, which is hermitic, we may fix the phase related factor equal to 1. The theoreticalvalue of GF, determined from the value of the proton mass, without any other experimen-tal value, is

-2

-2

. GeVsin

. GeV

F

p

Gm

pa pam Q

a

-

-

= = = ´æ ö æ ö÷ ÷ç ç÷ ÷ç ç ÷ç÷ç è øè ø

» ´

522 2 2

1

5

2 2 2 2 1 176668 102

2 31 16639 10 . (15.3.20)

This value is subject to corrections of order a due to the approximations made.In the induced representation the calculated value of the geometric unit is

g . cm . f-= ´ =141 1 657012 10 0 1657012 . (15.3.21)

The general perturbation for the interaction of the electron and the neutrino fieldsincludes eqs. (15.3.5, 15.3.16, 15.3.20) which essentially are the current and lagrangianassumed in Fermi’s theory [11, 12] of weak interactions [13, 14] of leptons. Fermi’s theoryis contained, as a low energy limit within the unified theory of connections and frames, orphysical geometry. If the full theory is accepted there are certainly new effects and impli-cations, which should be determined without the simplifications made to display the rela-tion of our theory to low energy weak interactions.

In particular, we should not expect that electroweak theory is related to another cou-pling “constant”, Because of the nonlinearity of the theory, it is not correct to assume thatif we subtract from a full solution, a partial electromagnetic solution, we get another solu-tion. The same thing applies to strong forces. Nuclear interactions were historically intro-duced to account for physical phenomena not explained by electromagnetic and gravita-tional fields. From our point of view, we may say that nuclear effects theoretically corre-spond to the residue of subtracting solutions of linear equations from some solutions tononlinear equations and are residual in this sense.

15.4. Summary.It was shown that the general perturbation technique for the interaction of the electron

and neutrino fields leads to eqs. (15.3.5, 15.3.16, 15.3.20) which are the current, lagrangianand constant assumed in Fermi’s theory of weak interactions of leptons. Fermi’s theory isa low energy limit of the geometric unified theory. It is also clear that if the full theory isan adequate model there are certainly new effects to be considered.

The geometric fundamental unit of length, in the induced representation, was cali-brated in terms of the experimental value for the proton mass. From this calibration, theweak coupling constant G was calculated, obtaining a value which is close enough to thepresently accepted value, considering the simplifications made. Of course, we must con-sider in the future high energy applications which may shed some light into the relation ofthis geometry with the standard model.


References

1 G. González-Martín, USB preprint, 96a (1996). See chapter 12. 2 G. González-Martín, ArXiv physics/0009066, USB Report SB/F/274-99 (1999); See chapter

13. 3 G. González-Martín, ArXiv 0712.1538, USB Report SB/F349-07 (2007). 4 W. T. Grandy, Found. of Phys. 23, 439 (1993). 5 See section 3.5.1 6 G. González-Martín, Phys. Rev. D35, 1225 (1987). See chapter 3. 7 G. González-Martín, arXiv physics/0009045. USB Report SB/F/271-99 (1999). 8 See equation 3.2.2. 9 See chapter 12.10 See chapter 13.11 E, Fermi, Z. Physik 88, 161 (1934).12 E. Fermi, N. Cimento, 11, 1 (1934).13 R. Feynman and M. Gell-Mann, Phys. Rev, 109, 193 (1958).14 E. C. G. Sudarshan and R. E. Marshak, Phys. Rev. 109, 1860 (1958).

16. STRONG NUCLEAR INTERACTION.

16.1. Introduction.The odd geometric potential represented by the odd part of the connection corre-

sponding to the SU(2)Q generators determines a strong nuclear interaction. In particular,this geometric coupling provides an attractive magnetic potential with a 1/r4 strong ra-dial dependence which is important in nuclear processes. The interest in electromag-netic effects on the nuclear structure is old [1]. Nevertheless, here we present new ef-fects due to this geometric interaction. For certain applications it is not necessary to usethe complete theory and it is sufficient to take the triple electromagnetic coupling of theSU(2)Q subgroup.

16.2. Motion of an Excitation in a Nonrela-tivistic Approximation.

The equations of motion were discussed in sections 3.3, 3.4 and 12.4. They are linear-ized equations around the substratum solution with a connection fluctuation represent-ing self interactions. For a nonrelativistic approximation we may neglect the low veloc-ity terms of order v/c, which correspond to the boost sector of the algebra, in otherwords the boost and the hermitian parts of and ,

†h h= - , (16.2.1)

†x x= - , (16.2.2)

and we obtain

( ) ( )† † †i ic h x h x j= + = - - = -1 12 2 , (16.2.3)

( ) ( )† † †i ij h x h x c= - = - + = -1 12 2 . (16.2.4)

As usually done in relativistic quantum mechanics [2] in a nonrelativistic approxima-tion, we let

imtej j - , (16.2.5)

imtec c - , (16.2.6)

obtaining slowly varying functions of time with equations

( ) ( )

m mm m m

mm

i A i A A Aj s c j s c

j¡ s c¡

+ + - - + - + = - - +

- -

0 0 0

0 , (16.2.7)


( ) ( )

m mm m m

mm

i A i A A A

m

c s j c s j

c¡ s j¡ c

+ + - - + - + = + +

+ + -

0 0 0

0 2 . (16.2.8)

Neglecting the equations become after recognizing the space component Am asan odd magnetic vector potential,

( ) ( )mm m mi A A i A Aj s c+ - + - + + - + - =0 0 0 0 , (16.2.9)

0 0 02 mm m mi A i A A m A , (16.2.10)

where c is a small component, of order v/c relative to the large component j. We ne-glect the small terms which are, as usual, the c terms in equation (16.2.10) unless mul-tiplied by m, and substitute the resultant expression for in equation (16.2.9). The resultis

( )( ) ( )

m n

m m m n m m

i A A

i A A i A A

m

j

s s j

+ -

+ - + -

+ + -

+ - + +=

0 0 0

02

. (16.2.11)

Since we have the well known relation

( ). . . .a b ab i a bs s s= + ´ , (16.2.12)

substitution in equation (16.2.11 ) gives, denoting the 3-vector magnetic potential by A,

( ) ( ) ( )

( )( ) ( ) ( )

[

]

i A A i A Ai A A

m

A A A i A A

m m m

j

s s sj

+ - + -+ -

+ - - - +

+ - + + = - + +

´ + ´ ´- + -

0 0 02

2

(16.2.13)

which is a generalized Pauli’s equation [3] depending on the even and odd magnetic vec-tor potentials +A, A. The vector A decays exponentially in the characteristic Yukawaform as seen in section 14.4. Distances Wm r 1 define a subnuclear zone where the expo-

nential approximates to 1. Distances Am r 1 define a far zone where the exponential isnegligible. In the far zone we obtain Pauli’s equation,

( )i A Ai V

m ms

y yé ù+ ⋅´ê ú = - -ê úê úë û

2

0 2 2 . (16.2.14)

171Strong Nuclear Interaction

16.3. Magnetic Moments.According to physical geometry [4] a G-solution should include the three SU(2)Q genera-

tors. The equation of motion (16.2.13) displays the remarkable geometrical structure of trip-lets. We may associate the effect of a combination of three SU(2)Q connection components,one for each possible P in G, as three classically equivalent electromagnetic potentials A’s. Ithas been shown [5], in this geometry, that all long range fields correspond to fields associ-ated to the fiber bundle obtained by contracting the structure group SL(4,) to its even sub-group SL1(2,) which in turn correspond to classical fields. Therefore the long range com-ponent of the sl(4,)-connection coincides with the long range component of an sl1(2,)-connection corresponding to an even subalgebra sl(2,)u(1) of sl(4,) related to gravita-tional and electromagnetic fields. In fact any direction in the tridimensional geometric elec-tromagnetic su(2)Q subalgebra may be identified as a valid direction corresponding to thisremaining long range classical electromagnetic u(1). There is no preferred direction in su(2)Q.If we observe a long range electromagnetic field, we may always align the classical field Awith any of the 3 geometric electromagnetic k generators in su(2)Q, or a linear combination,by performing an SU(2)Q transformation. Nevertheless the two extra A’s should make addi-tional contributions to the magnetic energy of the short range G-system, as shown in equa-tion (16.2.13), and therefore to the corresponding magnetic moment [6]. In order to studythis energy we shall restrict ourselves to the subnuclear zone. See section 14.4 for moredetails about the electromagnetic SU(2)Q subgroup which is similar to the spin subgroupSU(2)S.

The internal direction of the potential A is along the possible directions of the electro-magnetic generator E in su(2)Q. The A components are proportional to the possible even andodd translations. In the subnuclear zone the total A vector lies in the electrocone defined bythe quantized polar angle Q [5] relative to the even direction axis and an arbitrary azimuthalangle,

( )tan i

n

E E i i

nE EQ

-

+ +

+= = º

12 2 1

. (16.3.1)

The action of the 4-potentials -Am or Am over a common proper vector in the algebramay be simply expressed in terms of the action of +Am,

( ) ( )( )( )( ) ( )( ) ( ) tan ,

r

r in

A A A n e i

e i n n r A A

mm m m m

mm m m

y y l y

l y Q m y X y

-

-

+ - -

- + +

= + = +

¢= + º + º +1 1 1 .(16.3.2)

The 4-potential Am has an orientation in the Clifford algebra with the angle Q’ respect tothe standard electromagnetic potential, which reduces in the subnuclear zone Wm r 1to the electroconic angle Q

ni required by the quantization of the SU(2)Q generator in the

representation (n,i), as indicated in section 14.4. The angle Q½

½ of the fundamental repre-sentation is approximately equal to the complement of Weinberg’s angle and it may sug-gest its geometric interpretation. When the angle Q is written without indices it shouldbe understood that it refers to the fundamental representation.

The magnetic energy coupled to the spin in equation (16.2.13) in the subnuclear zone is


( )( ) ( )

( )( ) ( ) ( ) ( ) tan in

B BU A A n n

m m m n

i iA A

n m m

s s s lj l j j

s sj Q j

+ -

+ +

æ ö⋅ ⋅ ÷ç= - ⋅ ´ + = - + = - + ÷ç ÷çè ø

æ ö+ ÷ç ÷ç ÷= - + ⋅ ´ = - + ⋅ ´ç ÷ç ÷ç ÷çè ø

1 2

12 2 2

11 1

2 2.

(16.3.3)We have introduced the electroconic angle Q representing the direction of the total gen-erator A for a G-system, with respect to the even direction in the su(2) algebra.

The fundamental state representing a proton is the SU(2)Q state with charge +1, cor-responding to electromagnetic rotation eigenvalues ½, ½. In terms of the even magneticfield the energy becomes

( )( ) ( )BU B

m ms

j s j+

+æ ö ++ ÷ç ÷ç ÷= - + = -ç ÷ç ÷ç ÷çè ø

1 21 1

2 2

12

1 311

2 2 . (16.3.4)

The value of Q is p/3. Statistically this direction corresponds to the average value of theprojected component, of a random classical direction, along the chosen even direction.

The first term in the parenthesis is related to the P-system which has only a U(1) electro-magnetic subgroup and the complications due to SU(2) are not present. The complementaryodd subspace corresponding to S2 does no exist. The orientation of the complete electro-magnetic connection A can always be taken along the definite even direction defined by thephysical u(1) algebra. The angle Q may be taken equal to zero. It corresponds to a P-system,associated to the electron, with only a k0 electromagnetic component. In this case the energyreduces to

( )U A Bm m

s s+ +- -= ´ =

1 12 2

. (16.3.5)

If it were possible to make a transformation that aligns the internal direction Q along theeven direction everywhere, we really would be dealing with a P-system because we actuallywould have restricted the connection to a P subgroup. A P-system provides a preferred direc-tion in the subgroup SU(2)Q of G, the only electromagnetic generator k0 of the associated Pgroup. If we use a P-test-particle (an electron) to interact with an external magnetic field B,we actually align the long range component A with this preferred direction. In this manner wemay explain the long-range physical experiments using an abelian electromagnetic field equa-tion with a remote source and a Dirac equation. If we use a G-test-particle (a proton) tointeract with an external field, the most we can do is to align the classical long-range compo-nent A with a random direction in su(2). Part of the total internal field iB is only observable ina short range region. To each B direction related by an SU(2)Q transformation, there corre-sponds a “partner” spin direction, defining associated scalar products s.B. The additionalscalar products, relative to a P-system, arise from the two additional geometric fields, and/or, equivalently, from the two additional copies of Pauli matrices knk in the universal Clif-ford algebra, determined by the additional spin-magnetic directions k, which originally werealong k1k2k3 and k5. The additional spin operators, introduced by the short-range “nonclassi-


cal” electromagnetic potentials, represent extra internal current loops that generate an “anoma-lous” increase of the intrinsic magnetic energy.

We have calculated the magnetic energy of a free G-system, with zero external field B, interms of the observable even internal field +B whose internal direction coincides with theeventual internal direction of an external field B. In this gedenken experiment, the magneticmoment is defined as the partial derivative of the magnetic energy, produced by the totalinternal generalized magnetic field iB, with respect to the even component +B, in whose inter-nal direction would align the internal direction of the external B field,

a

a

UB

m +

¶º -

¶ . (16.3.6)

The standard physical methods of magnetic moment measurement are nuclear paramag-netic resonance [7 , 8 ], molecular beams and optical spectroscopy [9]. In a real physicalexperiment, when the sample is placed in the external field, there is a change in the magneticenergy of either a G-system or a P-system. For both systems, our test particle will respond toan external field, sensing a variation of the electromagnetic field linked to the test particle, inthe internal direction of the even +B field which is the only observable component at longrange. The change in the magnetic energy, after restoring in the equation the fundamentalphysical constants e, , c which are all equal to 1 in our geometric units, is

( )tanU e S B S BU B g

B m ci i ii

D D Q m+

æ ö¶ ÷ç ÷= = - + = -ç ÷ç ÷ç¶ è ø2 1

2

(16.3.7)

where the variation seen by the test-particle is equal to the external field B, the angle Qi iszero for the electron or p/3 for the proton and mi is the respective mass. This expressiondefines the magneton (atomic or nuclear) mi

i

and the anomalous Landé gyromagnetic factor

gii.

The magnetic moments of the proton and electron result,

( )tana a

a e Sg

m ci i ii

sm Q m

æ ö÷ç ÷= + =ç ÷ç ÷çè ø2 1

2 2

, (16.3.8)

( )a a

ap p N

p

e Sg

m cs

m mæ ö÷ç ÷ç= + =÷ç ÷÷çè ø

2 1 32 2

, (16.3.9)

a aae e B

e

e Sg

m cs

m mæ ö÷ç ÷= - =ç ÷ç ÷çè ø

22 2

, (16.3.10)

determining the anomalous gyromagnetic factors of the proton, 2(2.732) and electron, -2.In QED the calculated values of the zero order magnetic moment for the electron and the

proton, given by its external Coulomb field scattering diagrams, are singular. In each case, theinteraction hamiltonian for the external Coulomb field differs by the respective additionalanomalous coefficient g, determined by equation (16.3.10). The effect of this difference isto adjoin the respective coefficient g to the external vertex in the diagram. Thus after


renormalization the zero order magnetic moment values for the proton and electron are pro-portional by the respective zero order factors g0. The radiative corrections for the electronhave been calculated by Schwinger [10, 11 , 12]. For both, electron and proton, the first ordercorrections, determined exclusively by the vertex part of the Coulomb scattering diagram,are proportional to the corresponding zero order terms. For the electron the vertex correc-tion diagram, formed by an internal photon line between the two internal fermion lines, givesSchwinger’s a/2p correction factor. For the proton the triple electromagnetic U(1) structurepresent in the SU(2) sector of the geometric interaction operator JG, determines three stan-dard j.A coupling terms. The triple structure is also present in the noncompact sector of thealgebra. This indicates that a full proton excitation description requires three boost momentaki. Consequently a full description of the external Coulomb field scattering diagram requiresa pair of internal fermion boost momentum triplets, instead of simply the pair of electronmomenta. We shall come back to this question in section 17.4. There are additional U(1)radiative processes between the six internal fermion boost lines corresponding to the twotriplets. Therefore, there are multiple additional internal vertex diagrams, obtained by per-mutation of the internal photon line among the six necessary internal fermion lines, whichcontribute to the first order correction. The multiplicity of the vertex corrections of the 6lines taken 2 at a time is

! !!( )! ! !

nM

p n p= = =

-6 15

2 4 . (16.3.11)

Each correction is equal to Schwinger’s value, a/2p, due the SU(2)Q group equivalence.

Up to first order, the proton magnetic moment is

. ( . ) .gg ap

æ ö÷ç= + = + =÷ç ÷çè ø01 151 2 7321 1 0 01742 2 7796

2 2 2 . (16.3.12)

Let us calculate the magnetic moment of a combination of GPL chain excitations.The total electromagnetic potential is the sum of the SU(2)Q A potential and a differentU(1) AU potential. Let j be a (½) eigenfunction of A and AU. The magnetic moment corre-sponds to [13]

( )( )

( )( )( ) ( )( )

U

U UU

U A A Am

i iBn n A A

m m n n

sj

s sl j j

+ -

+

= - ⋅ ´ + +

æ ö+ ÷ç- ⋅ - ÷ç ÷= + + = + ⋅ +ç ÷ç ÷+ç ÷çè ø

1 2

2

11

2 2.(16.3.13)

Including the radiative correction, with the 28 vertex multiplicity corresponding to 4P-excitations in the resultant su(2)+U(1) algebra, the magnetic moment of the combina-tion is

( ) ( ).a a a

an N N

p

e S Sg

m ca s

m m mp

æ öæ ö ÷ç÷ç ÷ç= - + + = = -÷ ÷ç ç÷ç ÷è ø ÷çè ø

282 1 3 2 1 2 1 9672 2 2

.

(16.3.14)


These results have a 0.5% discrepancy with the experimental values of the protonmagnetic moment (2.7928) and the neutron magnetic moment (1.9130). The neutron maybe considered a combination of the proton, the electron and the antineutrino.

16.4. The Modified Pauli Equation.If the generalized Pauli equation (16.2.13) in the subnuclear zone is useful in calcu-

lating the magnetic moments of the nucleons we may expect that this equation is in-volved in the nuclear interaction. In particular we are interested in applying it to thestudy the properties of the light stable nuclides [14]. Using eqs. (16.3.2) and (16.2.13) forthe quantum states of the potential A we obtain

( ) ( ) ( )

( )( )[

]

i A A m V

A A A mE

X X X

s X X X y y

+ + +

+ + +

- + ⋅ + - - - +

- ⋅ + ´ + ´ - ´ =

2 2 22 1 2 1

1 2 2 . (16.4.1)

A nonlinear substratum solution implies the self interaction between its excitation fieldand motion. It is normal to define the self potential of a charge as positive. With theadequate definition of the classic potential A, in accordance with section 12.4, we ob-tain,

( )A i Q Am m mG ++º = - , (16.4.2)

( ) ( )( ) ( )

( )( )

ˆ[ tan tan

tan tan

ˆ tan tan ]

m r m r

m r m r

m r m r

iA rm e e A

m e V e A

e A m e r A mE

G G

G G

G G

G

G

Q Q

Q Q s

s Q Q y y

- -

- -

- -

- - ⋅ - + - -

+ + - ⋅ ´

+ ⋅ + ´ - ´ =

2 2 22 1

2 1 2

1 2 . (16.4.3)

The relation of this fundamental excitation with its substratum determines its electro-magnetic potentials which depend on the bosonic mass mW. The excitation decomposes lin-early into local fundamental excitations, with relative signs of charge. These signs corre-spond to the action of the operator A, which now is considered an external field, on thewavefunction Y. Both are eigenvectors of the SU(2)Q physical charge operator with signscorresponding to their representations.

The SU(2)Q connection is short range [14]. At atomic distances in the far zone

Am r 1 Q is zero. For the atom, the wavefunction Y corresponds to the negatively chargedelectron excitation and the scalar potential A0 corresponds to the positively charged nucleusexcitation. We get the standard Pauli equation (16.2.14) with the classic U(1) electro-magnetic potential A when we introduce the charge -e in the usual units.

At small distances in the subnuclear zone Wm r 1 the tangent of the fundamental quan-tum electroconic angle Q, which is related to Weinberg’s angle, is 3½ and we obtain

( ) ( )( )( )

ˆ[

ˆ ]

iA rm A m V

A m r A A mE

G

Gs s y y

- - ⋅ - + - + +

+ ⋅ + ´ - ´ - ⋅ ´ =

2 23 2 2 2 1 3

1 3 3 2 3 2 . (16.4.4)


The presence of the odd A sector changes the sign of the potential term A2, in accor-dance with equations (16.2.13) and (12.4.13), so that this strong potential term is attrac-tive. In terms of the magnetic moment m and the charge q of a source we obtain an equa-tion which shows that the dominant effects are due to the magnetic terms which includean attractive nuclear r-4 potential,

( ) ( )

( )

[

ˆ]

rr ri

r r r

m qr r rm mE

rr rG

mm ms

s m s my y

´´ ´- - ⋅ - + + ⋅ ´

+⋅ ´ ´ ⋅ ´ ´- - + =

22

3 6 3

3 3

2 2 1 3

2 1 33 2 3 2

. (16.4.5)

This equation determines the motion of a combined y matter excitation field. Thegyromagnetic factor and the spin are internal geometric properties of an excitation whichare determined by the geometric electromagnetic operators present in the internal mag-netic energy term of its internal motion equation. Therefore, it is convenient to recog-nize a total geometric factor mg m= in the magnetic moment m which includes the gy-romagnetic factor and the spin, but excludes the mass which represents a secondary iner-tial reaction in the charge motion. Depending on the particular combined system consid-ered this mass may correspond to the system reduced mass or to a total mass.

16.5. The Proton-Electron-Proton Fields forthe Deuteron.

Consider that eq. (16.4.5) applies to a system of G and P excitations, in particular a com-bination of one electron field and two proton fields moving about the system center of mass.The total field is dominated by the electron magnetic field because its higher magnetic mo-ment m. The resultant SU(2) field is characterized by the system reduced mass which is theelectron mass m. As shown in chapter 7 there are flux quanta associated to the protons p andthe electron e, one for each particle, that should link among themselves. This makes conve-nient to use the flux lines or magnetic strings to characterize the links between particle mag-netic moments. The dominant potential is a strong attraction in the equatorial radial directionwhich may also be considered as the attraction among magnetic flux lines linked to fluxquanta. The three indicated quantum flux strings cannot be linked so that they fully attractthemselves as a whole without breaking the Pauli exclusion principle of the two p. To fullyattract themselves the only possibility is the necessary presence of a fourth string. This stringmust be supplied by the only neutral fundamental excitation in the theory, an L excitation orneutrino with a linked flux quantum.

The resultant system ( ), , ,p e pn corresponds to the deuteron decay products which we defineas the deuteroid. The only possible way to attractively link the flux lines determines that the sub-

system ( ),e en ¢º has the e and spins in the same direction. Hence e’ has spin 1, the charge,mass and the magnetic moment of the electron. The operators Ae of the electron e and Ap of

the protons p in a stable deuteroid ( ), ,p e p¢ constitute a single coherent total field operator A,


which depends on the reduced mass, and determines its fundamental SU(2) excitation. Inother words, the resultant A has the Clifford algebra orientation required by the fundamentalSU(2)Q quantum representation of the generator. This determines the geometric angle Q of Awith respect to the even electromagnetic potential +A. The resultant field affects the deu-teroid components and in particular the protons. The dominant internal magnetostatic energy

is the one that holds the deuteroid components ( ), ,p e p¢ together and determines its energyproper value E. We may say that this is a coherent symmetric quasistatic disposition of thetwo protons around the excited electron in a proton-electron-proton formation whichreduces eq. (16.4.5) to a stable combined wavefunction y around a magnetic center.

Because of the symmetry we may use cylindrical coordinates with the z axis along thedominant electron magnetic moment m. It is convenient to approximate the potentials tovery small r by only keeping the dominant r-4 potential term. The energy eigenvalue is

expressed as a mass fraction E me= 2 . In this manner we obtain [14] the equation

mz

me y

rr

é ùê úê úê ú- - - =ê úæ öê ú÷ç ÷+ê úç ÷ç ÷çê úè øë û

22 2 2

324

2

22 01

. ( 16.5.1)

This equation is not separable [15] due to the spatial polar angle q dependence containedin the cross products

Nevertheless, since orbital and spin angular momenta tend to align with the magneticB field we should expect that the wave function concentrates around the equatorial planeand most of the energy is in the equatorial region. A calculation of the negative gradientof the dominant potential term confirms this consideration,

( ) ( ) ( ) z

z zu u

zzz zr

m r mr rr rr r r

æ ö é ùæ ö÷ç ÷- -ç÷ ê úç ÷ç÷- = + -ç ÷ê ú÷ ç ÷ç ÷ ç +÷+ç ÷ê úç÷+ +÷ç è øè ø ë û

2 2 2 2

3 3 2 22 22 2 4 2 2

2 2 4 6 6

1.

(16.5.2)

The negative gradient has direction toward the equatorial plane z = 0 for all values of z andtoward the origin in an equatorial region defined by z r<2 22 . This equatorial region is lim-ited by two cones near the equator, characterized by an equatorial angle or latitude qe

( ) ( ), e e e zp q q p q q r- £ £ + »2 2 2 . (16.5.3)

In the limit of very small r the problem reduces to a bidimensional problem on the equatorialplane. Therefore, near this limit we should simplify by setting sin q » 1 , z » 0 ., obtaining aseparable approximate equation,

mm

e yr

é ùê ú + + =ê úë û

22 2 2

4

22 0 , (16.5.4)


mz

me y

r r r r j r

é ù¶ ¶ ¶ ¶ê ú+ + + + + =ê ú¶ ¶ ¶ ¶ë û

2 2 2 22 2

2 2 2 2 4

1 1 22 0 . (16.5.5)

Since the z dependence is originally through the spatial polar angle q, it is convenientto introduce in its place, in the equatorial zone, an angular latitude coordinate z,

tan z zz

r r= » . (16.5.6)

In terms of this coordinate in the equatorial region the equation becomes

mm

r r e r yr r j z r

é ù¶ ¶ ¶ ¶ê ú+ + + + + =ê ú¶ ¶ ¶ ¶ë û

2 2 2 22 2 2 2

2 2 2 2

22 0 (16.5.7)

because

z z z zz z z

z z z r z

æ ö æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶÷ç ÷ç» » »÷ ÷ç ç÷ ÷çç ÷ è ø¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø

22 2 2

2 2 2 2

1 . (16.5.8)

We assume a separable solution in the equatorial region, of the form

( ) ( ) ( )R Zy r L j z= . (16.5.9)

The separated equation for Z is

ZZa

z¶

=¶

22

2 (16.5.10)

where a is a real approximate separation constant.The cylindrical symmetry implies the conservation of the angular momentum azimuthal

component. The Pauli spinor y, or large component of the Dirac spinor, is a commoneigenspinor of the generators of rotations L

z, spin S

z and total rotation J

z. The eigenfunctions

may be constructed using the exponential functions,

( ), ,ie nj

L j n n+æ ö÷ç ÷= + = ç ÷ç ÷çè ø

12 0

. (16.5.11)

( ), , ie njL j n n-æ ö÷ç= - = ÷ç ÷ç ÷çè ø

12

0 . (16.5.12)

We require that the spatial part of the eigenfunction be single valued, which determines thatn is an integer. With this understanding, the separated equation for the angular function L±

is

, ,n n nj

¶ = -

¶

221 1

2 22 . (16.5.13)


Values of n mean that the deuteroid has orbital angular momentum around its center ofmass. From a mechanical point of view, this implies a possible rotation of the lighter elec-tron e around the center of mass of the two protons p. A nonzero n may be inconsistent withthe quasistatic mechanism, indicating an instability, and we should rule out all values n ¹ 0 .The stable model might only be possible for a static system.

In accordance with the indication at the end of section 16.4, the magnetic moment isdetermined by a geometric factor g that depends on the fields and charges in motion. Themagnetic moment is inversely proportional to the mass, which tends to oppose the mo-tion. It is convenient to make a change of variables to a radial dimensionless complexvariable z rationalized to the mass,

z zm mm g

re e

= =2 2 22 , (16.5.14)

where the electron g is ½. We obtain the separated radial equation which has irregular singu-lar points at zero and infinity [15],

( )z R zR z Rz

ge n aé ùæ ö÷çê ú¢¢ ¢+ + + - - =÷ç ÷çê úè øë û

2 2 2 22

12 0 . (16.5.15)

If we let uz e= we obtain

( ) ( )u uR e e Rge n a-é ù¢¢ + + - - =ê úë û2 2 2 22 0 , (16.5.16)

which is the modified Mathieu equation [15, 16, 17] with the parameters

q ge= 2 , (16.5.17)

a n a= -2 2 . (16.5.18)In the last equation the parameter a, which does not correspond to the r-4 dominant term,should be considered an effective parameter because, in addition to a2, it may include contri-butions from its r-3 associated term in equation (16.5.1). For example, if the radial coordi-nate r fluctuates around a value, it may contribute to give an effective value of unit order to a.

This approximate result regarding the application of the Mathieu equation for thedeuteroid under study might also be obtained using spherical coordinates.

16.6. Binding Energy of the Deuteron.According to Floquet’s theorem [18], a solution of eq. (16.5.16) includes an overall

factor, sue where s is a complex constant. If s is an integer we get the Mathieu functions oforder s. A Mathieu function [15] is one of the few special functions which are not aspecial case of the hypergeometric function and therefore the determination of its propervalues differs. The Mathieu functions can be expressed as a Fourier series. The seriesexpansion coefficients are proportional to the characteristic root values ar of a set ofcontinued fraction equations obtained from the three-term recursion relations amongthe expansion coefficients. These characteristic roots are obtained as a power series inq.

A solution of the Mathieu equation can also be expressed in terms of a series of


Bessel functions with the same expansion coefficients of the Fourier series. This Besselseries is convenient to find the asymptotic behavior of the solution. By replacing theBessel functions with the Neumann functions we obtain a solution of the second kind.Solutions of the third and fourth kind are obtained by combining the Bessel and Neumannfunctions to form Hankel functions. We want a regular solution at zero that vanishes atinfinity. We require that s = 0 to avoid a singular or oscillatory asymptotic behavior. Theappropriate negative exponential behavior at infinity is provided by the radial Mathieu func-

tion of zero order, of the third kind, or Mathieu-Hankel function, indicated by ( ),He q u0 .It is known that the characteristic roots and the corresponding series expansion coef-

ficients for the Mathieu functions have branch cut singularities on the imaginary q axis[17, 20] and the complex plane a of the corresponding Riemann surfaces. Therefore, theexpansion coefficients are multiple-valued. In the standard radial equation eigenvalue prob-lem, the energy is determined by eliminating the singularities of the hypergeometric seriesby requiring its coefficients to vanish after a certain order. In our case, the energy should bedetermined by eliminating the branch cut singularity of the Mathieu series by choosing ap-propriate coefficients. We should require that the solution coefficients of the R functionbe single valued as we required for the L function. To eliminate the possibility of multiple-valued coefficients and obtain a regular solution we must disregard all points q on all branchesof the Riemann surface except the points q0 common to all branches. We require that q be thetwo common points on the Riemann surface branches corresponding to the first two charac-teristic roots a

0 and a

2 which determine the rest of the expansion coefficients of the even

Mathieu functions, ( ),He q u0 , of period p.The value of the constant q0 has been calculated [19, 20, 21] with many significant digits.

The parameter q determines a single degenerate negative energy eigenvalue corresponding tobound states of the solution, as follows,

q iqge e= = = 02 , (16.6.1)

and we have a bound deuteroid solution with the value of binding energy per state

( ). . Mev.e e eE m q m me= = - = - » -22 20 0 1 4687686 1 10237 (16.6.2)

If both n = 12 states are occupied we have a model for the deuteron. The total en-

ergy is

. Mev. . Mev. dE U= - » - = -2 20474 2 2246 (16.6.3)

where Ud is the deuteron binding energy. It is the necessary energy to destroy the symmetric

quasistatic arrangement. It is required to disintegrate the deuteroid system into a proton andan electron-proton excitation or neutron, as indicated in the next section. This negative bind-ing energy is due to the SU(2)Q strong electromagnetic interaction. Therefore, the physicalreaction of a proton with a neutron, which produces a deuteron, must simultaneously releasethis quantity U

d of electromagnetic energy in radiation form. This determines the reaction

p n d g+ + . (16.6.4)

Since the component protons are in opposite angular momentum states, the deuteronspin is equal to the e’ system spin. If we find the correct equation for the spin 1 e’ systemit would display a magnetic energy term similar to the ones found previously where the


coefficient of the B field is

B e Be e eg Sm m m m¢ ¢ ¢= = = (16.6.5)

in terms of the Bohr magneton and a spin 1 matrix S. This equation rules out the spin 0 statefor e’ and for the deuteron. The deuteron can only be in the ±1 spin states. It also shows thate’ does not have an anomalous gyromagnetic factor.

16.7. The Electron-Proton Model for the Neu-tron.

If the stable 2-proton deuteroid suddenly looses one proton we may assume thatthe potential produced by the electron remains constant for the duration of a shorttransition, while the system becomes unstable. The symmetric quasistatic disposi-tion of the two protons around the electron in the proton-electron-proton forma-tion is destroyed, leading to a strongly-dynamic asymmetric disposition in an elec-tron-proton system. The fundamental SU(2) representation of the system potentialis replaced by separate electron U(1) and proton SU(2) representations respectively.The gyromagnetic factor g, as indicated previously, is determined by the SU(2) in-teraction. This combination now implies that the sum of the respective values de-termines the neutron anomalous gyromagnetic factor. During this transition the mag-netic parameters, in particular the anomalous gyromagnetic factor g, must adjustthemselves to the new situation in the unstable 1-proton system,

( ) pgm mr r

+ =3 32 1 3 . (16.7.1)

The other parameter that adjusts automatically is the energy eigenvalue E. The onlypotential terms in equation (16.4.5) that should undergo changes are those depending onthe magnetic parameters g and E. Thus, we consider the expression

( )W m mE gm m

er r

= + + = +2 23 32 2 1 3 2 (16.7.2)

which is the dominant expression containing the variable parameters g and E, while keepingr and the other potential terms constant in equation (16.4.5). It is clear that the total value ofW also remains constant while E and g change. Therefore,

dW mdE dgmr

= + =32 0 . (16.7.3)

The last equation relates the change of the energy eigenvalue to the change of the gyromag-netic factor while the su(2) magnetic fields are rearranged,

dEK

dg mmr

= - º3 , (16.7.4)

where K is taken as constant. We assume that the integration constant is equal to zero, whichmeans that E and g proportionally evolve toward the unstable state and W is zero after thecollision and during the evolution toward the unstable state. The energy eigenvalue is propor-


tional to the gyromagnetic factor and to the intrinsic magnetic moment potential term whichis a fraction of total effective potential

n

n p

EEEg g g

= = 0 . (16.7.5)

The ratio of the two nucleon energy eigenvalues is then approximately equal to the ratio ofthe gyromagnetic factors. We know from section 16.3 that both anomalous proton and neu-tron magnetic moments are determined and theoretically calculated from two different su(2)magnetic field arrangements in nucleon excitation systems with mass mp. The proton-elec-tron system undergoes su(2) magnetic field changes toward an unstable neutron system thateventually decays. The energy of the latter system is then theoretically determined by thesegyromagnetic factor theoretical values

( ). . . Mev. . Mev..n n p eE m m m

-= - = » - - =

1 967 1 10237 0 780 0 7822 780

(16.7.6)This system has a mass-excess equal to En which is given-up as energy when it decays into itsconstituents proton and electron.

The Pauli equation (16.4.5) for the internal motion relative to the center of mass,with reduced mass m, under the internal magnetic field, serves to calculate gyromag-netic effects and binding energies. For the neutron we have

( )( ) nn

p

g BmE

gs

y yé ù⋅ê ú- - - + =ê úê úë û

2 2 1 3 22

. (16.7.7)

The Pauli equation also serves to study the motion of the neutron system center of massunder an external magnetic field, now with the resultant total mass,

n p e nM M m E¢ = + + . (16.7.8)

In this case the equation, with the same geometric gyromagnetic factor in the internal mag-netic energy term, is

n n

i e Bi

M M c ts y

yé ùæ öæ ö ⋅ ¶÷ ÷ç çê ú÷ ÷- + + =ç ç÷ ÷ê úç ç÷ ÷ç¢ ¢÷ç ¶è øè øê úë û

2 32 12 2 2 2 . (16.7.9)

The system ( ), ,p e n is a spin ½ system because the proton spin is opposite to the

( ),e en ¢º spin. Thus, the neutron and proton forming the deuteron have their spins in thesame direction. The fact that the deuteron magnetic moment is the approximate sum ofthe proton and neutron moments may be explained because the presence of an externalmagnetic field forces one proton to react by itself while the rest of the system, p, e’reacts adjusting its fields as a neutron.


16.8. The many Deuteron Model.The presence of orbital angular momentum, l ¹ 0 , breaks the stable quasistatic mecha-

nism of the model. The equation determines only one stable bond excitation correspond-ing to the spin ±½ states of frame field excitations bound by the strong magnetic interaction.This deuteron “pep bond” is a (p,e’,p) fundamental excitation and supplies another cou-pling mechanism which allows the combination of more than 2 protons. The alpha par-ticle, or excitation a, may be considered as a 2-deuteron excitation [22].

The magnetic interaction range defines the subnuclear zone W pr m r1 , very smallin relation to the proton radius rp. Mathieu’s equation determines a single pep bond energyin a proton pair. In order to feel the strong attraction the proton centers must be inside thesubnuclear zone. Therefore the protons are essentially superposed. In the a model there are4 protons and 2 electrons which are also superposed and must share the magnetic field. Thestationary state, of minimum energy, is the symmetrized quantum superposition, as inthe hydrogen molecular ion. The electrons and protons are shared by all possible pep bondsin the system. The possible pep bonds obey the Pauli exclusion principle and are only thosesharing a single electron. All p participate in as many pep bonds as proton pairings.

One difference in the a model is that the reduced mass m of the system is approximatelyhalf the electron mass me due to the presence of two electrons in the system. The requiredcoordinate transformation to obtain Mathieu’s equation is then different from eq.(16.5.14),

e

z z zm m m mm g g

re e e

= = =2 2 2 222

, (16.8.1)

which changes q proportionally and determines twice the binding energy Ud’ per pair,

q e= 2 , (16.8.2)

( )d e dE E m q m q m Ue¢ ¢= = = =- =-22 20 02 2 2 2 4 2 . (16.8.3)

The energy d dE E¢ = 2 is the unique Mathieu equation eigenvalue even if e' is shared and isthe same for each pep magnetic bond. This fact together with the symmetry simplifies theenergy calculation of the n-body problem. There are symmetric cluster states whose ener-gies are determined by their pep bonds. The symmetric a cluster associated to 4 protons has6 pep bonds and the energy of these bonds minus the electronic mass difference due to esharing,

, ,

, ,

p e p e

i i ep e p em m m

¢ ¢

¢¢ ¢= -å å4 2 4 6 4 , (16.8.4)

( ) ,, . Mev.e d ep ep eE E m E m E Ua a¢¢

¢= - = - = - º » -4 64 2 4 6 4 28 501 . (16.8.5)

Similarly we have the same considerations for 3 protons. The symmetric cluster asso-ciated to 3 protons has 3 pep bonds with a unique Mathieu energy Ed,

( ) ,, . Mev. He Hee d ep ep eE E m E m E U¢¢ = - = - = - º » -3 33 33 2 3 2 7 636 . (16.8.6)


The exclusion principle prohibits 2e in a pep bond. An extra e may link to a p which is not in pepbond, which is impossible for the 4p in the a system, forming a neutron,

( )e n pE m m mD = - - , (16.8.7)

( ), . Mev.He H Hp eE E E E U¢ = + D = - º » -3 3 33 2 8 416 . (16.8.8)

These equations determine a mass relation identity for the A=3 isobars. Since our

notation uses the correspondence ( ) ( ), ,p e p n¢ =3 2 we may write

p n p n eH He He

eH He

m m m E E m m m E

m m m

+

+ +

= + + + D = + + +

º +

3 3 3

3 3 2

2 2(16.8.9)

which indicates a small b decay possibility of 3H+ into 3He+ from total energy consider-ations in spite of the unfavorable binding energy difference. It also implies that the cor-responding atoms 3He and 3H have essentially equal masses, up to the negligible Cou-lomb electron interaction.

We have obtained the binding energies for 4He and the isobars 3H, 3He within 1%. Weare lead by the model to consider that the pep bonds or deuterons act as the essential com-ponents of more complex clusters in electromagnetic interaction with additional nucle-ons. This is consistent with the protonic and neutronic numbers of the nuclides. Eitherproton in a pep bond when associated with e' may be considered a neutron. Therefore theenergy bond U

d determines a “nuclear” force among nucleons which is independent of a

neutron association.

16.8.1. Nuclear Barriers and Reactions.We should realize that the electromagnetic field plays a fundamental role in the geo-

metric theory. The deuteron is understood as a geometric field excitation, not as com-posed by particles. The dominant nuclear potential energy barrier for the deuteron “pepbond” is approximately given by the SU(2) magnetic A2 connection field term in equation(16.4.3). We may express the effective potential in a useful approximate manner by neglect-ing some terms in the equation on the equatorial plane, only keeping the dominant magneticterm and the asymptotic long distance electric term,

meU E

m m m

G r

y y yr r r r

-é ùæ ö é ù- ¶ - - ¶÷çê ú ê ú÷+ - - = + =ç ÷ê úç ê ú÷ç¶ ¶è ø ë ûë û

22 2

2 3 4 21 3 1 1 1

2 8 2 . (16.8.10)

The equatorial U barrier height critically depends on the values of the field masses mG and m.The location of the maximum in terms of the variable rmG is determined by the equation

( )mm e G rG r = -23 2 3 . (16.8.11)

It is possible to have particle fields which tunnel across the pep barrier producingnuclear reactions. If the pep system is on a periodic medium, like a lattice in condensedmatter, equation (14.4.33) gives the mass m

A of the SU(2) boson field corresponding to

the electron field mass m as source. Equation (16.8.11) for mA determines that the equa-

torial pep barrier maximum occurs at r equal 0.0215 fm, less than the proton radius, and


has a value of 3430 Gev. It is smaller, by a factor of order 10-13, than the equatorial barriermaximum on free space which is given by the SU(2) boson field high mass m

W and occurs

at much smaller value of r. This large reduction of the pep potential on preferred latticesurfaces greatly increases the probability of field tunneling through the barrier producinglow energy fusion reactions rather than the standard high energy reaction obtained on freespace. Since the “pep bond” is potentially present between any pair of proton fields in thetotal system composed of deuterons immersed in a lattice of heavy nuclei, there is a theo-retical probability that “pep systems” tunnel through the barriers among themselves andthe nuclei, producing alpha particles and nuclei transmutations.

There may be difficulties in achieving the required conditions. In metals like palla-dium, which has the natural property of absorbing large amounts of deuterium in its lat-tice, the conditions of having deuterons and free electrons available are partially satis-fied [23]. We indicate that it is also necessary to establish the Helmholtz eq. (14.4.31)for the connection field or boson field of mass m

A which acts as a catalyzer of the reac-

tion. The electron field and its magnetic SU(2) field should concentrate on the regionbetween two proton fields while simultaneously serving as source to the boson fieldequation. Further conditions are required to develop clusters of reacting material. [24]

16.9. Summary.The equation of motion, in a nonrelativistic approximation, is a modified Pauli equation

which allows us to perform approximate calculations. Using this equation we determinedexpressions for the anomalous magnetic moments of the electron, the proton and the neu-tron. The numerical results obtained for the magnetic moments are surprisingly close to theexperimental values. These values admit correction factors due to the approximations made.

The magnetic SU(2) connection action in the equation provides a model for the deuteronas composed by 2 protons and 1 electron. The “strong” SU(2)Q electromagnetism, withoutthe help of any other force, generates sufficient attractive short range potentials to providethe deuteron binding energy. This pep bond also determines the binding energy of the neutronan the light nuclides 4He, 3H, 3He. The consideration of neglected interaction terms may besufficient, without other forces, to explain the 1% differences in the nuclide energies.

The binding energy of the deuteron and other light nuclides may be essentially mag-netostatic and the nuclear fission and fusion reactions in a nuclide system would be mag-netic multipolar transitions. If a pep system interacts with an external system, in addi-tion to the radiative channel there should be other possible channels in the S dispersionmatrix. The total potential may have energy transfers in the non radiation near zone. Qua-sistatic channels may open between the nuclide system and the other system through thefield in this electromagnetic zone.

If the pep system is on a periodic medium, like a lattice in condensed matter, themagnetic potential barrier may be reduced. In this manner magnetic energy would betransferred with low radiation, as it usually happens between near electromagnetic andmatter systems. The mathematical expression for the nuclear potential generated by the domi-nant magnetic pep bond was shown in section 16.8.1. The maximum of the equatorial nuclearpotential barrier on a condensed matter lattice was calculated. We found, in fact, a largetheoretical reduction of this barrier maximum from its value on free space.

These results should not be overlooked because the implications they might have in theunderstanding of the process of nuclear fusion of light nuclei. The SU(2) electromagneticfield may be an active element of a fusion reaction. In particular it may provide possibleprocesses for the development of clean fusion energy without radiation problems.


References

1 J. F. Carlson and J. R. Oppenheimer, Phys. Rev. 41, 763 (1932).2 J. D. Bjorken, S. D. Drell, Relativistic Quantum Mechanics (Mc Graw-Hill, New York,

ch. 1, (1964).3 L. D. Landau, E. M. Lifshitz, Mécanique Quantique, Théorie non Relativiste (Ed. Mir,

Moscow), 2nd. Ed. p. 496 (1965).4 See sections 3. 6 and 4.2.5 See sections 2.3 and 14.46 G. González-Martín, I. Taboada, J. González, ArXiv physics/0405126, USB Reporte

SB/F/305.3-02, (2003).7 E. M. Purcell, H. C. Torrey, R. V. Pound, Phys. Rev. 69, p.37 (1946).8 F. Bloch, W. W. Hansen, M. Packard, Phys. Rev., 70, p. 474 (1946).9 N. F. Ramsey. Nuclear Moments, (Wiley, New York), (1953).10 J. Schwinger, Phys. Rev. 73, 416 (1948).11 J. Schwinger, Phys. Rev. 76, 790 (1949).12 J. M. Jauch, F. Rohrlich, The Theory of Photons and Electrons, Springer-Verlag, New

York), Second Ed., p342 (1976).13 G. González-Martín, ArXiv physics/0009066, USB Report SB/F/274-99 (1999). See

chapter 13.14 G. González-Martín, ArXiv 0712.1531, USB Report SB/F/350.2-07 (2007).15 Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics, 1st edition

(McGraw-Hill,New York), Vol.1, Chap.5, p.655, 673.16 E. Mathieu, J. Math. Pures Appl 13, 137 (1868).17 Gertrude Blanche,” Chapter 20 Mathieu Functions” in Milton Abramowitz and Irene A.

Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964).18 G. Floquet, Ann. École Norm. Sup. 12, 47 (1887).19 H. P. Mullholland and S. Goldstein, Phil. Mag. 8, 834 (1929).20 Josef Meixner, Friedrich W. Schäfke, and Gerhard Wolf, in Mathieu Functions and

Spheroidal Functions and Their Mathematical Foundations, edited by A. Dold and B.Eckman (Springer-Verlag), Berlin, Vol. 1, Chap.2, p.85.

21 C. J. Bouwkamp, Kon. Nederl. Akad. Wetensch. Proc. 51, 891 (1948).22 G. González-Martín, ArXiv 0805.0363, USB Report SB/F/361-08 (2008).23 Y. Iwamura, M. Sakano, Jpn. J. Appl. Phys. 41, 4642 (2002).24 Y. Arata, Y-C. Zhang, Proc. Jpn. Acad. 78(Ser, B), 57 (2002).

Implications of the existence of this “strong” electromagnetic SU(2) attraction should playa fundamental role in the photonuclear interactions, nuclear fusion, the structure of nuclides,the development of nuclear models, gamma ray spectroscopy and other related fields.

There may be electromagnetic radiation interchanges through the corresponding quan-tum or gamma photon. Strong localized magnetic fields related to directed gamma radiationin a polarized nuclear plasma may cause resonance effects which stimulate an electromag-netic fusion reaction and gamma emission.

One remarkable feature of the electromagnetic SU(2) is its triple geometric struc-ture. There should be additional experimental evidence in favor of this structure. In par-ticular this structure should reveal itself as resonance triplets associated to su(2) subex-citations inside nucleons. Experimental evidence for these phenomena should appear asquarks. Rather than fundamental blocks of matter, quarks may be considered excited in-ternal states that actually support the geometric electromagnetic SU(2) interaction.

17. THE GEOMETRIC STRUCTURE OFPARTICLES AND INTERACTIONS.

17.1. Introduction.The group of the geometric space-time structure of relativity is fundamental to the field

theories of elementary particles, which are representations of this group. Our geometric relativisticunified theory of gravitation and electromagnetism also determines the particle structure andtheir interaction [1, 2 ], on the contrary of general relativity which has not played such a funda-mental role. Some of the numerical values obtained in previous chapters related to particleproperties are indicated in the following paragraphs.

The possible magnetic flux quantum of particles, eq. (7.5.3), is

he e

pf = =0 2 .

The mass ratios of the proton, electron and W, Z bosons, eqs. (13.3.39), (14.4.28), (14.4.29),are

( )( )

. .pRG

H R e

mV Kmm V C m

p= = = » =56 1836 1181 1836 153 ,

. Gev. . Gev.g pA Z

m mm mm

a a= = = = » =

2 290 9177 91 188

2

sinsin . Gev. . Gev.p

A WA

mm m m

QQ

a- = = = » =78 7370 80 42

2The proton and neutron magnetic moments, eqs. (16.3.12), (16.3.14), are

( ) ( ).p p N Np

e S Sg

m ca s

m m mp

æ öæ ö ÷ç÷ç ÷ç= + + = =÷ ÷ç ç÷ç ÷è ø ÷çè ø

152 1 3 1 2 2 77962 2 2

,

( ) ( ).n n N Np

e S Sg

m ca s

m m mp

æ öæ ö ÷ç÷ç ÷ç= - + + = = -÷ ÷ç ç÷ç ÷è ø ÷çè ø

282 1 3 2 1 2 1 9672 2 2

.

The deuteron and alpha particle binding energies and the neutron mass excess, eqs.(16.6.3), (16.8.5), (16.7.6) are

. Mev. . Mev. d e e dE m q m Ue= = - = - » - = -2 202 2 2 20474 2 2246 ,


( ) ( ), , . Mev. .e d ep e p eE E m U m Ua¢ ¢

¢= - =- - =- »- =-4 2 4 6 4 6 4 28 5009 28 28 ,

. Mev. . Mev.d nn n p e

p

E gE m m m

g= = » - - =0 780 0 782

2

These values and other previously indicated results are not “ad hoc” values but rathermathematical consequences of a geometric model of relativity. In other words, they aregeometrically determined by first principles, which is not true in other popular physicalparticle models where no equivalent numbers are produced. On the contrary, many of thesemodels are full of arbitrary parameters.

Therefore, in this chapter we shall discuss this question using geometric and groupfeatures and ideas, avoiding unnecessary aspects that may present obstacles in under-standing the physical implications of the geometry for physical particles. Previously, it isconvenient to review the principal ideas as an introduction to the general discussion.

The study of groups that act on the geometric structures of a physical theory maydetermine essential physical features without actually solving the equations of the theory.This particular approach, a geometric group realization as fiber bundle sections, is new inthe treatment of elementary particles and their interactions.

It is well known that the holonomy groups of a connection in a fiber bundle may beused to classify its possible connections. It is interesting to use this method to gain insightin the types of physical interactions represented by the connection in the unified geomet-ric theory.

In QFT certain particles are given masses by a Higgs mechanism [3 ] which relies oncertain symmetries which a vacuum solution may possess. This appears to assign to thevacuum a not completely passive role. We may try to assign to the vacuum an even moreactive role. This is accomplished by recognizing that the particle vacuum is a geometricspace with physical meaning, related to a unified nonlinear geometric theory of physics.We consider an approximation, to the geometric nonlinear theory, where the microscopicphysical objects are realized as linear geometric excitations around a nonlinear substratumgeometric space. This is consistent with the QFT interpretation of particles as vacuumexcitations. We interpret the excitation as particles and the substratum as the particlevacuum.

With this definition, a particle is acted upon by the substratum and is never really freeexcept in absolute empty space (zero substratum curvature). The substratum space carriesuniversal inertial properties. A free particle is an idealization. In a fundamental level, ifwe accept physics in the substratum we are assuming, in part, a (Parmenides) holisticprinciple which should be consistent with the ideas of Mach [4 ] and Einstein [5 ] thatassign fundamental importance of faraway matter in determining the inertial properties oflocal matter. On the contrary, if we assume only particles we are assuming a (Democritus)atomistic principle. This implies there is no physics in the substratum.

Restriction to holonomy subgroups has also implications for the equations of motionof matter, since within the theory particles are represented by frame excitations (sectionsof a bundle). We may expect that geometric association of frames to holonomy subgroupsmay naturally classify particles and associate them to interactions. Hopes to accomplishthis objectives are supported on results of previous work.

Constants associated to a certain connection appear as constant parameters in the exci-tation equations and play a fundamental role [6 ]. As in general relativity, the covariant

189Geometric Structure of Particles and Interactions

equations may be referred to frames (coordinates) which must be related to observersdetermined by the physical experiment in question. Otherwise theoretical results remainindeterminate. The freedom to choose a reference frame, arbitrary by a group transfor-mation, generates a class of equivalent solutions represented by one particular referenceframe er.

Any excitation must be associated to a definite substratum. An arbitrary observation ofan excitation property depends on both the excitation and the substratum, but the physicalobserver must be the same for both excitation and substratum. We may use the freedom toselect the reference frame, to refer the excitation to the physical frame defined by its ownsubstratum, which satisfies the nonlinear equation (1.4.2).

Then this trivial substratum is referred to itself, as in chapter 12, and the substratummatter frame eb, referred to er becomes the group identity I. Actually this generalizescomoving coordinates (coordinates adapted to dust matter geodesics) [7 ]. We adopt coor-dinates adapted to local substratum matter frames because the only not arbitrary frame isthe matter frame itself. Free matter shows no self acceleration nor self action. In its ownreference frame these effects actually disappear. Only constant self energy terms, deter-mined by the nonlinearity of the substratum, make sense and should be the origin of theconstant mass parameter.

At a small distance l, characteristic of free excitations, the connection and frame ofthe substratum appear symmetric. Mathematically we should say that the substratum is alocally symmetric space [8 ] or hyperbolic manifold . We recognize the necessary condi-tion that locally the substratum be a bundle that locally admits a maximal set of Killingvectors of the space-time symmetry, with zero Lie derivative of the connection [9 ]. Thismeans that there are space-time Killing coordinates such that the connection is constantnonzero in the region of particle interest. (A flat connection is too strong assumption).

It is clear from the definition of excitation that a free particle is a representation of thestructure group of the theory and consequently an algebraic element. A representation(and therefore a particle) is characterized by the eigenvalues of the Casimir operators. Thestates of a representation (particle) are characterized by the eigenvalues of a Cartan sub-space basis operators. This provides a set of algebraic quantum numbers to the excitation.Of course, we must somehow choose the respective representations associated to theseparticles. It has been indicated that the physical particles are representations of the holonomygroup of the connection, a subgroup of SL(2,), induced from the subgroup SL(2,) andrealized as functions on the coset spaces. In fact, it has been shown that new electromag-netic consequences of this theory leads to quantization of electric charge [10] and mag-netic flux, providing a plausible explanation for the fractional quantum Hall effect [11 ].The particle properties should be determined by the geometric structure.

One important issue is: How do we calculate mass in a consistent manner? Mass arisesfrom constants with inverse length dimension corresponding to a symmetric substratumsolution of the nonlinear field equation. These constants appear in the linear excitationequations (13.1.1) for a fermionic frame fluctuation de and (14.3.7) for a bosonic connec-tion fluctuation dw, in the defining representation, where m is the fermionic mass param-eter given by eq. (13.2.13) and w determines the bosonic mass parameter given by eq.(14.2.6). Both parameters are determined by the Cartan-Killing metric in terms of the con-stants of the substratum solution.

The geometric excitations on a substratum may be expanded as a perturbation aroundthe substratum in terms of a small parameter e characterizing the excitation and indicatingthat the zeroth order energy term or bare mass is given entirely by the substratum currentand connection, with small corrections depending on the excitation interaction energies.


These energy corrections may be interpreted as small relative “dressed” physical rest massesfor null mass excitations. The smallest dressed rest mass would correspond to the least inter-acting excitation. As indicated in previous work [12 ], these corrections correspond to ageometric quantum field theory.

In this manner, the particle bare mass parameter is not determined from the linearparticle equation itself, but rather from the holistic inertial nonlinear substratum solu-tion. The existence of a constant connection solution for these asymptotic fields (freeparticle substratum) provides a fundamental distance, due to nonlinearity, and gives amechanism to calculate mass ratios in terms of volumes of respective symmetric cosetspaces. The extreme case of absolute empty space which implies zero substratum curva-ture self interaction, is only mathematically possible in this theory for a universe with-out matter, because of the nonlinearity of the substratum field equations.

Nevertheless, the association of a mass to one of these geometric distances remainsarbitrary because this association actually calibrates a geometric scale of distance in termsof a physical mass scale. Since mass is inverse length, we may associate a standard physi-cal mass to the standard geometric length. After the association we may attempt to calcu-late ratios of masses. For example, if we select the mass of the proton excitation to cali-brate the geometric unit, we may calculate the mass of the electron excitation, as shown inequation (13.4.1) [13]. The result of this calculation, using the quotients of holonomygroups and the constant substratum solution, is the mass ratio which essentially agreeswith the experimental value. Similarly we can calculate the mass of the intermediate bosonsin terms of the proton mass as indicated by equations (14.4.28) and (14.4.29).

The equation of motion may be written, in a non relativistic approximation, as a modi-fied Pauli equation which allows us to calculate the proton and neutron magnetic mo-ments [14], the binding energies of the deuteron, the alpha particle, the A=4 isobars andthe neutron mass excess [15].

The numerical results obtained are surprisingly close to the experimental value. In thischapter we discuss how to determine the existence of other particles and their propertiesusing the algebraic and topological properties of the geometric model and in particular howto calculate the geometric mass ratios of these possible particles.

17.2. Geometric Classification of the Con-nection.

In order to classify the interactions we look for dynamical holonomy groups H of theassociated connection. It is clear that H must be a subgroup of the structure group of thetheory, SL(2,). For physical reasons we want that the interactions be associated to adynamic evolution of the matter sources. The dynamical effects are produced in the theoryby the action of the group, in particular accelerations should be produced by generatorsequivalent to the Lorentz boosts as seen by an observer. Therefore, we require that theboost generators, k0ka, be present in a connection identifiable with a dynamical interac-tion as seen by the observer associated to the Minkowski subspace generated vectoriallyby ka. Because of the nature of the source current,

ˆˆJ e u em a mak= , (17.2.1)

which corresponds to the adjoint action of the group on the algebra, it should be clear thatall generators of the form k[akb] of SL(2,) should be present in the dynamical holonomygroup of a physical interaction connection.


The even subgroup, generated by k[akb], is L and the previous discussion means that

L H GÍ Í . (17.2.2)

Furthermore, H must be simple. The different possibilities may be obtained from theknowledge of the subgroups of G.

The possible simple subgroups are as follows:1. The 10 dimensional subgroup P generated by ka, k[bkg]. This group is isomor-

phic to the groups generated by, k[akbkg], k[akb] and by k[akbkc], k[akb], k5 and infact to any subgroup generated by a linear combination of these three genera-tors;

2. The 6 dimensional subgroup L, corresponding to the even generators of the al-gebra, k[akb]. This group is isomorphic to the subgroups generated by ka, k[akb],and by k0k[akb], k[akb] and in fact to any subgroup generated by a linear combina-tion of these three generators;

2. The group G itself.The P subgroup is Sp(2,), as may be verified by explicitly showing that the genera-

tors satisfy the simplectic requirement [16 ]. This group is known to be homomorphic toSO(2,2), a De Sitter group. The L subgroup is Sp(2,), isomorphic to SL(2,). In addi-tion there are only two simple compact subgroups of G, nondynamical, generated by k[akb]

and k0, k5, k1k2k3, apart from the unidimensional subgroups.Then we have only three possible dynamical holonomy groups: L, P, or G. For each

case we have an equivalence class of connections and a possible physical interaction withinthe theory. As shown in section 3.7, the LPG chain internal symmetry is SU(2)U(1)and coincides with the symmetry of the weak interactions. Other holonomy groups are notdynamical, in the sense that they do not produce a geometrical accelerating action onmatter, as determined by an observer boost. This is the case of holonomy subgroups in thenon simple chain L U(1)L SU(2)L G which may represent electromagnetismbut do not provide, by their direct action, a geometric dynamics (force) on charged matter.The dynamics requires a separate Lorentz force postulate [15] .

17.3. Excitations Corresponding to Sub-groups.

If we restrict the group to either the P or L subgroups, the corresponding frame matri-ces (subframes) are elements of the subgroup. The total frame f, which is an element a ofthe algebra, decomposes into even and odd parts in accordance with the general Cliffordalgebra decomposition,

a a ak+ -= + 0 . (17.3.1)

From previous results, the equations of motion for the even and odd parts of f are undercertain restrictions [16] ,

f f mfm mm mk ¶ k G+ - - -= = , (17.3.2)

f f mfm mm mk ¶ k G- - + += = , (17.3.3)

implying that a frame for a massive particle must have odd and even parts. In our case, if


we set f- equal to zero we obtain also that m is zero. Therefore, for an L-subframe wehave multiplying by k0 ,

fmms ¶ + = 0 , (17.3.4)

the equation of the neutrino as discussed before.Furthermore, if we calculate the left handed and right handed components, we obtain,

omitting the indices,

( ) ( )L

xy g y g

h h

æ ö æ ö÷ ÷ç ç= + = + ÷ = ÷ç ç÷ ÷ç ç÷ ÷ç çè ø è ø5 51 1

2 2

01 1 , (17.3.5)

( ) ( )R

x xy g y g

h

æ ö æ ö÷ ÷ç ç= - = - ÷= ÷ç ç÷ ÷ç ç÷ ÷ç çè ø è ø5 51 1

2 21 10

. (17.3.6)

We have that the left handed component is equivalent to the h field which in turn isdefined in terms of the even f+ field. Similarly, we see that the right handed component isequivalent to the x field and consequently to the odd f- field. Therefore, an even framecorresponds to a left handed particle, as should be for a neutrino. The Lorentz frame exci-tation has neutrino like properties. It is necessary to point out that, since the mass in thisequation is zero, the isotropy subgroup, which preserves the momentum k, is SO(2,1)instead of SO(3) corresponding to massive particles. The induced representations of theneutrino excitation are sections over the null 3-hypercone valued in representations ofSO(2,1) and are characterized by the value of helicity.

A general Sp(2,) excitation has a k0 generator, which is an electromagnetic genera-tor. We expect that this excitation should represent a particle with one quantum of chargeand one quantum of spin. The component of the current along this generator coincideswith the standard electric current [17] in quantum mechanics. In addition, it was shownthat the general perturbation technique for the interaction of the electron and neutrinofields lead, essentially, to the current and lagrangian assumed in Fermi’s theory of weakinteractions [18] of leptons. This lead to the conjecture that a Sp(2,) excitation mayrepresent an electron. With this in mind, the value of the ratio of the proton to electronmasses was derived from the definition of mass in terms of energy, using the properties ofthe structure group and its subgroups and respective cosets in the theory as indicated in aprevious section.

It is possible to define the eigenvalue of the projection operator to the odd sector in thegeometric algebra as a quantum number which we shall call rarity. Furthermore, transfor-mations by the subgroup SL(2,) leave invariant the even and odd subspaces of a frame.Since these eigenspaces correspond to the left handed and right handed parts, or equiva-lent to the helicity eigenvalues, it is clear that, within a Sp(2,) frame, eL may be assigneda nL partner and may be considered a doublet, under another group, while eR has no nRpartner and may be considered a singlet. This indicates a possible link with the standardmodel of weak interactions.

17.4. Algebraic Structure of Particles.In general, the equation of motion for matter,


ˆˆe u em a m

m m ak k + =2 0 , (17.4.1)

applies to the three classes of dynamical frames, according to the three dynamical ho-lonomy groups. The three linearized equations, together with the nonlinear field equa-tion should have solutions related to a substratum state solution. Then we may associatean excitation to each class of frames around the substratum solution. These excitationsare elements of the Lie algebra of the structure group of the theory.

The maximally commuting subspace of the Lie algebra sl(4,), generated by the cho-sen regular element, [19 ] is a tridimensional Cartan subalgebra, which is spanned by thegenerators:

X k k=1 1 2 , (17.4.2)

X k k k k=2 0 1 2 3 , (17.4.3)

X k k=3 0 3 . (17.4.4)

It is clear that X1, and X2 are compact generators and therefore have imaginary eigen-values. Because of the way they were constructed, they should be associated, respectively,to the z-component of angular momentum and the electric charge. Both of them may bediagonalized simultaneously in terms of their imaginary eigenvalues.

The 4 members of the fundamental representation form a tetrahedron in the tridimen-sional Cartan space as described in chapter 7. They represent the combination of the twospin states and the two charge states of the associated particle.

charge spin flux negative charge with spin up

negative charge with spin down positive charge with spin up

positive charge with spin down

f

f

f

f

-

-

+

+

- + -- - ++ +++ --

1 1 11 1 11 111 11

The fundamental representation f of SL(2,), indicated by f+, f+

, f-, f-

groupstogether two excitation states of positive charge with two excitation states of negativecharge. The presence of opposite charges in a representation forces us to make a clarifica-tion. To avoid confusion we should restrict the term charge conjugation to indicate theoriginal Dirac operation to relate states of opposite charge.

There is also another fundamental representation f dual to f and of the same dimen-sions, with all signs reversed, which is inequivalent to the original one and represented bythe inverted tetrahedron in the Cartan subspace. One of the two representations is arbi-trarily assigned to represent the physical excitation. The mathematics of the representa-tion algebra indicates that the state may also be described as composed of 3 members ofits dual representation. In turn, the dual representation may also be similarly described ascomposed of 3 members of its dual, which is the original representation. This mutualdecomposition is reminiscent of the idea of “nuclear democracy” proposed in the 1960’s[20 ] but restricted to dual representations. To avoid confusion we restrict the use of theterm duality to relate these inequivalent representations.


In standard particle language the antiparticle is the dual particle which implies that italso is the charge conjugate. Nevertheless, this assignment of a physical particle-antipar-ticle pair to the fundamental representation and its inequivalent dual is not a mathemati-cal implication of standard particle theory, only a physical assumption. Since our funda-mental representation includes opposite charges it is not appropriate to consider thedual representation as antiparticles. We may, as well, simply say that the antiparticle isdetermined by the algebra conjugation. The dual particle is just a necessary dual math-ematical structure.

We have then for a particular state p of the fundamental representation,

( ) ( )p f f f f q q q + + + - + + -º = º , (17.4.5)

in terms of the dual states q. Similarly for a particular q state,

( ) ( )q q q q p p p + + + - + + -= = . (17.4.6)

It does not follow that the q necessarily are states of a different physical excitation, onlythat the q form a dual triplet mathematical representation of the p representing the sameexcitation. This allows a different physical interpretation for these mathematical construc-tions. It should be noted that all p, q have electric charge equal to the geometric unit,electron charge e. Since these excitations have particle properties, there is a dual math-ematical representation of the physical excitation (particle). We may raise the followingquestion? What happens if we physically identify p with the proton, which mathematicallymay be expressed as 3 q, interpreted as quarks? In our theory there is no need to assignfractional charges to quarks. In accordance with the “restricted nuclear democracy” wemay assume that the quark states, in turn, may be mathematically expressed as 3 protons p,which may justify the large experimental mass of these states.

The fundamental representation of Sp(2,) which may be displayed as a square in atwo dimensional Cartan space is e-

, e-, e+

, e+. Its dual Sp(2,) representation, obtained

by reversing all signs is mathematically the same e+, e+

e-, e-

. Similarly the SL(2,)representation n, n which may be displayed in a one dimensional Cartan space is math-ematically its own SL(2,) dual. Therefore, the only one of the three excitations with amathematically inequivalent dual structure is p. Since the Sp(2,) and SL(2,) subgroupsmay be imbedded in SL(2,), the corresponding Cartan spaces of Sp(2,) and SL(2,)may also be imbedded in the Cartan subspace of SL(2,). The plane subspace Q=-1,including dual states, has charges of one definite sign and is a representation of Sp(2,).Another plane, Q=1, containing opposite charges is another representation of Sp(2,)which completes the SL(2,) Cartan subspace.

The space SL(4,)/SU(2)SU(2) is a nonadimensional riemannian space of thenoncompact type. There are 9 boost generators Bm

a. The rotation SU(2), indicated by S,contained in the maximal compact SO(4) acts on the m index and the electromagnetic SU(2),indicated by Q, contained in SO(4) acts on the a index,

a n m mb a n bQ B S B ¢= . (17.4.7)

Similarly the subspace Sp(4,)/SU(2)U(1) is a hexadimensional riemannian space of thenoncompact type. The complementary subspace within SL(4,)/SU(2)SU(2) is 3-dimen-sional. There is a triple infinity of these subspaces within the total space, reflecting thetriple infinity of groups P in G, depending on the choice of an electromagnetic generatoramong the three possible ones in SU(2).


In Sp(4,)/SU(2)U(1), with a fixed electromagnetic generator, we have a vector inthe odd sector, representing a momentum k associated to an excitation e. Since we havethis situation in SL(4,)/SU(2)SU(2) for each generator in SU(2), we have, in effect, 3momenta, ki, that may characterize an excitation p. We must consider excitations charac-terized by 3 momenta, ki, which may be interpreted as three subexcitations.

Three P-subexcitations are necessary to characterize the protonic excitation p, but notthose of e nor those of n. Mathematically we have a system characterized by 3 trimomenta, ki,that may be scattered into another system of 3 trimomenta, kj’. A scattering analysis of excita-tions must include all momenta in some d functions that appear in the scattering results. Ex-perimentally this is a collision with a system of 3 pointlike scattering centers and is inter-preted as a collision with partons inside the proton. In general it is clear that a pointlike protonexcitation is not predicted by our theory. In particular, e scattering from p can be replaced bya sum of incoherent scatterings from P-excitations. Then group invariance implies that thecollision cross section can be expressed as two functions of the energy and momentum trans-fer and, for deep scattering, Bjorken’s scaling law is obtained.

17.5. Physical Interpretation in Terms ofParticles and Interactions.

We have found properties of the geometric excitations which are particlelike. We takethe position that this is no coincidence but indicates a geometrical structure for physics.The source current J depends on a frame. To each holonomy group we may associate aclass of frames thus giving three classes of matter.

As already discussed in section 4, the corresponding L-frame represents a zero baremass, neutral, spin ½, left handed pointlike geometric excitation, which obeys eq. (17.3.4).It has the particle properties of the neutrino.

For the P-connection, the corresponding P-frame represents a massive, negative charge-1, spin ½, geometric pointlike excitation with bare magnetic g-factor -2, which obeys theDirac equation. It has the properties of the electron.

For the G-connection, the corresponding G-frame represents a massive, charge 1, spin½, 3-point geometric excitation with bare magnetic g-factor 2(2.780), which obeys theDirac equation with a bare mass of 1836.12 times the bare mass of the previous excitation.It has the properties of the proton.

In this manner, we have associated to each of the three holonomy groups, one of theonly three known stable particles.

For the L-connection it is not difficult to recognize that the interaction is gravitation,from the discussion in previous chapters and the work of Carmelli [21 ]. Similarly, it alsowas shown in previous chapters that electromagnetism (without dynamics) is associatedto one of the SU(2) generators and that the physical Fermi weak interaction is related tothe odd sector of a P-connection.

We propose here that the P/L generators may be interpreted as an electroweak interac-tion and the G/P as strong nuclear interaction. Then the three dynamic holonomy classesof connections may correspond to three classes of interactions as follows:

1. The L-connection describes gravitational interaction;2. The P-connection describes coupled gravitational and electroweak interactions;3. The G-connection describes coupled gravitational, electroweak and strong in-

teractions.The L-frames obey equations that may be obtained from the general equations of mo-


tion when the frame e has only the even part e+. From the field equation it is seen that a P-frame generates a P-connection and that a G-frame generates a G-connection.

From this classification it follows, in agreement with the physical interactions that:1. All (matter) frames self interact gravitationally;2. L-frames self interact only gravitationally (uncharged matter);3. G-frames (hadrons) are the only frames which self interact strongly (hadronic

matter);4. G-frames self interact through all three interactions;5. G-frames (hadronic matter) and P-frames (leptonic matter) self interact

electroweakly and gravitationally;6. P-frames self interact gravitationally an electroweakly but not strongly (lep-

tonic matter).

17.6. Topological Structure of Particles.The framework of this theory is compatible with a phenomenological classification of

particles in a manner similar to what is normally done with the standard symmetry groups.First we should notice that the theory suggests naturally three stable ground particles

(n, e, p). In fact, if we consider the possibility of different levels of excited states, eachparticle may generate a class of unstable particles or resonances.

In particular, since the equations for each of the three particle classes are the same,differing only in the subgroup that applies, it may be expected that there is some relationamong corresponding levels of excitations for each class, forming families.

The ratio of the mass of the hadron in the ground level family (proton) to the mass ofthe lepton in the ground level family (electron) has been calculated from the ratio of vol-umes of coset spaces based on the existence of a trivial nonzero constant solution for thesubstratum connection. This trivial solution is the representative of a class of equivalentsolutions generated by the action of the structure group.

Now consider only topological properties, independent of the connection, of the spaceof complete solutions (substratum plus excitation solutions). Scattering processes of exci-tations around a given substratum should naturally display these properties. An incomingscattering solution is a jet bundle local section, over a world tube in the space-time basemanifold, which describes the evolution of the solution in terms of a timelike parametert from past infinity to some finite time t. Similarly, an outgoing solution is a local sec-tion from time t to future infinity. The local sections in the bundle represent classes ofsolutions relative to local observers. Scattering solutions at infinity are asymptoticallyfree excitation solutions around a substratum. The substrata (incoming and outgoing) areequivalent to each other and to the constant trivial solution if we choose observer framesadapted to the substrata.

The equations are of hyperbolic type and we should provide initial conditions, at pastinfinity t=-¥, for the solutions over an initial tridimensional spatial hypersurface -.We require that any incoming solution over the past infinity hypersurface - reduce to afree excitation around the substratum solution over the bidimensional subspace -(¥) atspatial infinity r=¥. Since the incoming solution substrata are equivalent to the trivialsubstratum solution at spatial infinity-(¥), we may treat this spatial infinity as a singlepoint, thus realizing a single point compactification of -, so the initial hypersurface- is homeomorphic to S3. All incoming solutions on - are classified by the functionsover S3. The same requirements may be applied to the outgoing remote future solutionsand, in fact, to any solution along an intermediate tridimensional hypersurface , a section


of the world tube. Thus, the final hypersurface at future infinity + is also homeomor-phic to S3. The incoming and outgoing substratum local sections over - and + must bepasted together in some common region around the present t, by the transition functionsof the bundle. The scattering interaction is represented by the group action of the transi-tion functions at t=0. All generators of the group produce a transformation to a differ-ent, but equivalent under the group, expression for the solution. If the holonomy groupof the solution is not the whole group, there is a reference frame that reduces the struc-ture group to the particular holonomy subgroup. But in general for arbitrary observers,there are solutions formally generated by SL(2,)=SL(4,). Since this transition re-gion, the “equator” ´R, has the topology of S3´R, the transition functions j define amapping, at the t=0 hypersurface,

: ( , )S SLj 3 2 , (17.6.1)

which is classified by the third homotopy group [22] of the structure group SL(2,) orthe respective holonomy subgroup. There are some solutions not deformable to the trivialsolution by a homeomorphism because j represents the twisting of local pieces of thebundle when glued together.

For the homotopy group of SL(2,) we get as shown in appendix E,

( ) ( ) ( )( )( )( ) ( )

( , )

( )

SL SU SU

SU SU Z Z

p p

p p

= Ä

= Ä = Ä

3 3

3 3

2 2 2

2 2

. (17.6.2)

Similarly, we have for the homotopy groups of the other two holonomy groups,

( ) ( )( ) ( )( , ) ( )Sp R SU SU Zp p p= Ä = =3 3 32 2 2 , (17.6.3)

( ) ( )( , ) ( )SL SU Zp p= =3 32 2 . (17.6.4)

These scattering solutions are characterized by topological quantum integer numbersn, for the three groups, and n’ only for group G, called winding or wrapping numbers. Inall cases the scattering solutions are characterized by one topological winding number nand in particular the hadronic scattering solutions have an additional topological windingnumber n’. This result implies that there are solutions wn, en that are not homotopicallyequivalent to w0, e0.

All p, e, n excitation solutions with a given n may be associated among themselvesbecause of the isomorphism of the homotopy groups Z. This is an equivalence relation.Two p, e, n excitation with the same n are in the same topological class determined by thesubstratum. Each topological class, characterized by the topological quantum number ndefines a physical class, a family of particles with the same winding number, which isrespected by transitions the same as the algebraic quantum numbers s, q, f.

For example, the association of a solution of each type to any value of n may be part ofa general scheme of relations labeled by n as follows:

, , ,

e l

e

l e l l

h p h p hm

m

m tn n n n n n

= = == = == =

0 1 2

0 1 2

0 1 2

, (17.6.5)


among the excitation levels of the electron (proper leptons), the excitation levels of theneutrino (other neutrinos) and the excitation levels of the proton (hadrons).

Since in our geometric theory the only constant that enters is a, the ratio of the massesof the excited states and the fundamental state may be related to this constant. It is pos-sible that the geometric theory may provide an explanation for the interesting approximaterelation of the masses of the leptons and the electron,

l

l em m na

æ ö÷ç= + ÷ç ÷ç ÷è øå 4

0

312

. (17.6.6)

This equation was proposed by Barut [23] with n interpreted as the number of pairs.Although this interpretation is also characterized by the group Z, it is not clear whetherboth interpretations are compatible.

In any case, the possibility exists that there be excited states which may be interpretedas a particle field composed of p, e and n fields. As a matter of fact Barut [24 ] has sug-gested that the muon should be considered as an electromagnetic excitation of the elec-tron. Although the details should be different, because Barut’s approach is only elec-tromagnetic in nature and ours is a unified interaction, we may conjecture also that themuon is an excited electron state with a leptonic winding number n=1. Similarly the twould be an excitation with winding number n=2

17.7. Geometric Excitation Masses.If the leptons are topological excitations it should be possible to calculate their mass

ratios [25] using the method given in chapter 13. The integrand JG in eq. (13.2.13), whichcorresponds to a fundamental representation, is a local substratum constant equal for allvalues of n. The number of possible states of a fundamental representation only dependson the integration volume. Physically, when integrating over the CR subspace of De Sitterspace C, corresponding to inequivalent observer states by an L transformation, we havecounted the number of L equivalence classes of states (points in the mass hyperboloid arein the same relativistic equivalence class as the rest state of the representation) in localform (only counting n=0 local states). Taking in consideration the global characteristicsof the topological excitation which would represent heavy leptons, the count should belarger, over all L equivalence classes of excitation states with wrapping n. What is thenumber of states for each value of n?

Since we are working with a manifold with atlas, the integration over the symmetricspace K is realized en the atlas charts. On the principal fiber bundle (E,M,G), the transi-tion function preserves the projection and acts, as a group element, over the fiber which isthe bundle structure group G [26 ]. For two neighborhoods U, V in the base manifold M,which corresponds to space-time, we have the following maps: homeomorphisms h fromthe bundle to the model space, transition functions j among the charts and partial identityi over the points mM,

( ) ( )

( ) ( )

U

V

h

U UV V

h

U E U M G

i

V E V M G

j

¾¾ ´

¾¾ ´

. (17.7.1)


The transition function j acts as an element of G over the incoming scattering solu-tion, defined on a hypersurface homeomorphic to S3, to produce the outgoing scatteringsolution. Physically j represents a collision interaction. The active representation is theinteraction of two excitations observed by a single observer. The passive representationis the observation of a single excitation by different observers. The nontrivial transitionfunctions of class [jn], where the index n represents the set of wrapping numbers (n, n’),are classified by the mappings

( ),S G S SL 3 3 4 . (17.7.2)

Consider G as a principal fiber bundle (G,K,G+). The induced representations, as-signed to points mM, are sections of an associated bundle to G which has Lorentzgroup representations [L] as fiber and is denoted by (D,K,[L],L) as shown in chapter13. Under a change of charts, representing a change of observers, the transition func-tion acts on the bundle D on the section JG over K, which we shall indicate by f(m).

The transition functions belong to the different classes determined by the thirdhomotopy group of G. The class [jn] may be expressed, using the homotopic productin terms of the class [j1]

[ ] [ ] [ ] [ ] [ ] [ ] [ ]nn nj j j j j j j= =0 1 1 1 0 11 2

. (17.7.3)

This class [jn] may be considered generated by the product of n independent ele-ments of class [j1] and one element of class [j0]. Observe that the homotopic map-ping

( ): , ,h S G K G+3 (17.7.4)

determines essentially a wrapping of subspaces s(m)G, wrappings homeomorphicto spheres S3, inside (G,K,G+). Each of the generating classes j is globally associatedto a generating wrapped subspace s in (G,K,G+), additional and necessary, linked tothe original trivial subspace by a transition function. In this manner the class [jn] isassociated to the trivial wrapping [s0] and to (n, n’) additional wrappings [s1], de-fined by the expression for the coordinates u of point mM

( ) ( )( )( ) ( ) ( )( )( ), ,UiV iu m s u m v m s v mj = . (17.7.5)

The excitation is not completely described by a single subspace that would corre-spond to a single observer s(m) in the passive representation. For global reasons wemust accept the presence of as many images of generating subspaces in the nontrivialchart as there are wrappings. Nevertheless some of the wrappings may be equivalentunder the subgroup of interest. For the trivial subbundle G there is only one indepen-dent wrapping, the trivial (n=0) wrapping s0, because the subspaces sn are equivalentunder a G transformation. For other subbundles, a tridimensional su(2) subalgebraacts on the complete group G fiber bundle, generating transformations of P and Lsubbundles which are not equivalent under the corresponding subgroup. It is pos-sible to generate no more than three wrappings si(m) in geometrically independentPi or Li subbundles in G. In other words any other wrapping may be obtained by somecombination of three transformations in Pi or Li. For this reason the number of physi-cally independent wrappings si in P or L bundles is n2 and determines three, andonly three, families of excitations.


17.7.1. Leptonic Masses.The actions of the n+1 subspaces si(m)Pi (0in) on a Lorentz representation over

C, the section f of fiber bundle (D,C,[L],L),

( ):f m C D , (17.7.6)

map the fiber p-1 over c onto fibers p-1 over sic by linear transformations li [27 ],

( ) ( )( ) ( ) ( )( ):i is m c m s m c mp p- -1 1 , (17.7.7)

( ) ( )( )( ) ( )( ) ( ) ( )( ), ,i i i is m f m c m l s m f m s m c m= , (15.7.8)

which determine a set of n+1 images of f, independent wrapped sections fi in the non-trivial chart, one for each independent wrapping,

( ) :if m C D , (17.7.9)

( ) ( ) ( )i i i i if s c l s f s cº . (17.7.10)

The Lorentz representations fi(sic) may be independent of any other fa(sac). Therefore, thephysically possible states of a wrapped excitation of class [jn] correspond 1 to 1 to all Lequivalence classes of states determined in each one of these n+1 section images fi. Usingthe passive representation, on each image there are momentum coordinates ki

m relative tocorresponding local observers i, states physically independent among themselves. Theintegration should be done over all independent variables k0

m,k1m,...kn

m , that is over a set ofCR equivalent spaces contained in a product of n+1 De Sitter spaces C.

As we indicated before for the substrate solutions, the integrand JG, corresponding toa fundamental representation, is a local constant equal for all values of n. The bare massesare determined by the volumes of integration. Let us denote the integration space whichsupports fi(m) by C*, equal to the product of n+1 copies of the original CR space, corre-sponding to the n+1 observed wrappings,

( )* n

RC C+

=1

. (17.7.11)

For the trivial case (n=0) the calculation is strictly local, it is not necessary to considerwrapped charts and the integration over CR may be done locally on any chart of the trivialclass. The subspace s0 is determined by a local observer.

For wrapped states (n0) we are forced to use the nontrivial transition functions [jn],with nonzero wrapping n0, which relates a wrapped chart of class [n] with the trivialchart. Therefore it is necessary to consider simultaneously (globally) these pairs of chartclasses associated to the incoming and outgoing scattering solutions respectively. Physi-cally it is necessary to consider all L-equivalence classes of possible states in accor-dance with observers related to nontrivial charts obtained from the trivial chart. There-fore, for the class [n0] it is necessary to integrate over all possible charts with “glo-bally wrapped” sections, produced by the action of transition functions. In this case thetransition functions may be any element of the group P. We restrict the integration tothose non trivial charts which are inequivalent under a relativity transformation L. Thenumber of possible charts corresponds to the volume of the P/L subspace CR of L-equiva-


lence classes.The linear topological excitations are generated by the linear infinitesimal action of

the differential j* of the transition function j which represents the t=0 interaction. Whenrestricting the transition functions to be elements of the subgroup P,

( )UhV V V P¾¾ = ´ , (17.7.12)

we should only consider the effect of the differential mapping j*. The odd subspace inthe P algebra, as differential mapping, generates the coset C. The only generator inequiva-lent to an L “boost” is the compact u(1) generator. The differential action produces asmaller space of L-equivalence classes. If we denote the compact subgroup U(1)SU(2)of P by H we obtain the noncompact coset BC

PB

H= . (17.7.13)

The group P may be expressed as a principal fiber bundle (P,B,H) over B. If we injectL in P, the image of the rotation subgroup in L should be the SU(2) subgroup in H but theimage of the “boost” sector in L is not uniquely defined in B. The group L acts on Ppreserving its image in P. The u(1) subalgebra of H acts on B, as a translation within P,mapping the C subspace onto another subspace C’ in B which is P-equivalent to C. Never-theless C´ is not L-equivalent to C. Therefore, the possible L-inequivalent states corre-spond to the translation action by the U(1) compact subgroup not related [26] with therotation SU(2). The number of L-equivalence classes of these possible charts, over whichwe should integrate, is determined by the volume of this electromagnetic U(1) group. Thetotal wrapped space CT for an n-excitation (n0) is the space formed by all possible trans-lated, not relativistically equivalent, C* spaces

( ) *TC U C= ´1 . (17.7.14)

The number of states (different k values) is, therefore, proportional to the volume ofthis total CT space, which may be calculated,

( ) ( )( ) ( )( ) nTRV C V U V C n

+= ´ ¹

11 0 . (17.7.15)

The bare mass of the trivial excitation (n=0) which is the electron, as previously indi-cated, is proportional to the volume of CR [28 ]

( )RV Cp

=16

3 . (17.7.16)

The bare masses corresponding to the n-excitations (n0) are proportional to the vol-umes

( ) n

TV C np

p+æ ö÷ç= < £÷ç ÷çè ø

1164 0 23

, (17.7.17)

that may be expressed in terms of the electron bare mass m0,


n

nmn

mp

pæ ö÷ç= < £÷ç ÷çè ø0

16 4 0 23

. (17.7.18)

For excitations with wrappings 1 and 2 we have

( ) . .R

m mV C

m mt

m

p= = = » =2

1

16 16 75516 16 8183

, (17.7.19)

( )

. .

. Mev

m m

m Om

pp

a

æ ö÷ç= = ´÷ç ÷çè ø

= » +

1 016 4 0 5109989 210 5516

3107 5916 , (17.7.20)

( )

. .

. Mev

m m

m Ot

pp

a

æ ö÷ç= = ´÷ç ÷çè ø

= » +

2

2 016 4 0 5109989 3527 825

31802 7 , (17.7.21)

which correspond to the bare masses of the m and t leptons, due to the electromagneticU(1) interaction. Due to the electron U(1) electromagnetic interaction in the topologi-cal excitation, we should apply higher order energy corrections to mi. For the subbundleL we can make a similar calculation. The bare masses of the neutrinos for the 3 familiesare zero because the volume of the L/L coset space is zero, as indicated in section 13.4.We shall refer to these topological excitations, respectively, as TnP-excitations and TnL-excitations.

17.7.2. Mesonic Masses.The chart transformations under general G transition functions translate, within G, the

CR region of integration. The only possible additional L-inequivalent states correspond tothe action of compact group sectors. In order to find the different possibilities we considerthe two related chains of symmetric spaces in G, not related to L

( ) ( )

( ) ( )

R RG K C I

SU SU U U I

É É É

È È È

É É É2 2 1 1 . (17.7.22)

The compact symmetric subspaces CR and KR, respectively, contain one and two U(1)electromagnetic subgroups of the three equivalent U(1) subgroups contained in the elec-tromagnetic SU(2). We note that one U(1) is common to both subspaces. We are inter-ested in extending the translation within G, beyond the U(1) transition function, by adjoin-ing compact sectors. The physical interpretation is that these excitations are subjectedto additional nonlinear interactions which increase the energy and therefore their masses.In the same manner as an observable relativistic motion increases the rest mass to a


kinetic mass, an observable relativistic interaction increases the free mass to a dynamicmass. This relativistic effect, included in the mass definition, is realized by the action ofthe transition functions at t=0.

For n0 topological excitations the transition functions may be extended from U(1) to CR.The physical interpretation is that these topological excitations are subjected to the additionalelectroweak nonlinear interactions of a leptonic P-system. We say that the excitations are“masked” by the nonlinear electroweak interactions. The geometric mass expressions for theclass [n0] masked excitations should be multiplied by the volume of the corresponding com-plete subspace of L-equivalence classes V(CR) rather than by its subspace V(U(1)). The weakinteraction energy acquired by the n-excitations corresponds to the product of the values fromequations (17.7.20) and (17.7.21) by the ratio of the volumes,

( )( )( )

.RV Cm mm mV U

p

m

¢= = » =1

1

4 1 32095731

. (17.7.23)

If we use the experimental values for the lepton masses, we obtain the masked geo-metric masses,

( ). Mev m m Op a¢ = » +1 140 8778 , (17.7.24)

( ) Mev fm m O a¢ = » +2 2369 . (17.7.25)

We shall refer to these CR masked (electroweakly interacting) topological excitations asTnPC-excitations. The TnPC-excitations may be considered components of mesons. In par-ticular, a masked muon m’ or T1PC -excitation, joined to a low energy T1L-excitation, hasthe geometric mass and other properties of the pion p.

For n0 topological excitations the transition functions may be further extended be-yond CR to include the sector of KR corresponding to both U(1) subgroups in KR which isan SU(2)/U(1) compact sector which may be identified with the sphere S2. The physicalinterpretation is that these excitations are subjected to the additional strong S2 electromag-netic interaction of a hadronic G-system. The parametrization of group spaces and theirsymmetric cosets is, to a certain extent, arbitrary. They map different points in the linearLie algebra to the same group operation. Since both equivalent U(1) subgroups in S2 havethe same significance because of the symmetry of S2, they both should contribute equallyto the invariant volume of S2 and it is convenient to choose a parametrization which ex-plicitly displays this fact. A parametrization which accomplishes this is

( ) ( ) ( )( ) exp exp exp ( )SU J J J Ua a a S S+ + - - + -= =3 32 1 , (17.7.26)

( )( ) ( ) ( )( ) ( ) ( )V SU V V U V Vp S S p S+ - += = =2 22 16 1 4 . (17.7.27)

We adjoin the group subspace SKR, defined by the expression in equation (17.7.26), tosubspace CR. Because of the extended integration in S, the n-excitations acquire a stronginteraction energy corresponding to the ratio of the volumes

( )( )

( ) ( ) . .R K

R

V Cm mV

m V C mp

SS p

¢¢= = = = » =

¢121

1

4 3 5449 3 5371 . (17.7.28)


We obtain the geometric masses for n=1 and n=2,

( ) ( ) ( ). Mev m m V m m Op p KS p a¢¢= = = = +12

1 4 494 76 . (17.7.29)

( ) ( ) Mev K

mm m V m m

mt

m

pS p

æ öæ ö ÷ç÷ç ÷¢¢ ¢ ¢ ç= = » =÷ ÷ç ç÷ç ÷è ø ÷çè ø

12

2 2 116 4 8303

3. (17.7.30)

These excitations are not contained in the P sector defined by the n=0 wrapping, butrather in a combination of inequivalent P sectors inside G. Therefore, strictly, they donot have a proper constituent topological P-excitation. We shall refer to these S masked(strongly interacting) topological excitations as TnPS-excitations. In particular, a maskedmuon m’’ or T1PS-excitation, joined to a low energy T1L-excitation by the strong interac-tion, has the geometric mass and other properties of the kaon K.

The physical interpretation of these masked leptons suggests that the correspondinggeometric excitations may be combined to form lepton pairs which may be consideredparticles. In accordance with this interpretation, the combinations under nuclear interac-tions are only possible if there is, at least, one masked lepton. Take the n=1 topologicalleptons and construct a doublet l of a cSU(2) combinatorial group associating a maskedmuon and a stable lepton. We should indicate that the conjugation in the geometric alge-bra, which is equivalent to the dual operation in sp(4,), is not the dual operation incsu(2). Both l and its conjugate are csu(2) fundamental representations 2. The productrepresentation is

Ä = Å2 2 3 1 . (17.7.31)The first possibility is that m is a weakly masked m’, part of the SU(2) leptonic system

doublet (m’, n) characterized by the charge Q and the muonic number Lm. We get theamplitude for the p representation:

( ) ( )( ), , , ,

, ,

xm n m n p p p

nm m m nn m n m m nn

- +é ùé ù é ù¢ ¢Ä º Åê úë û ë û ë ûé ù¢ ¢ ¢ ¢ ¢ ¢= - Å +ê úë û

0

1 12 2

. (17.7.32)

The only other possibility is the more complex combination where the muon is a stronglymasked m’’. In addition to the coupling to n, since m’’ has a strong S2 electromagnetic interac-tion, m’’ also couples strongly to the electron e, forming two related SU(2) hadronic systems.We substitute n by e as the leptonic doublet partner. We get the K representation amplitude:

( ) ( )( ), , , ,

, ,

L Lxm n m n K K K

nm m m nn m n m m nn

- +é ùé ù é ù¢¢ ¢¢ ¢Ä º Åê úë û ë û ë ûé ù¢¢ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢= - Å +ê úë û

1 12 2

, (17.7.33)

( ) ( )( ), , , ,

, ,

s Se e x

e ee e ee

m m K K K

m m m m m m

é ùé ù é ù¢¢ ¢¢ ¢Ä º Åê úë û ë û ë ûé ù¢¢ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢= - Å +ê úë û

0 0

1 12 2

. (17.7.34)

The masses of these p and K geometric particles equal the masses of ground states


belonging to the product representation of these representations. Their geometricalmasses, essentially determined by the mass of its principal component or masked heavylepton as indicated in equations (17.7.24) and (17.7.29), approximately correspond tothe masses of all physical pions and kaons. We may also define

( ) ( )L Sx x x eem m nn h¢ ¢ ¢ ¢¢ ¢¢ ¢= + = + + º1 122

2 . (17.7.35)

( )x x h¢+ º12 . (17.7.36)

These definitions correspond to physical particles whose geometrical masses, essen-tially determined by the masked heavy lepton mass, are approximately,

( ) Mev . Mevxm O mha¢ ¢» + » =990 957 8 (17.7.37)

and using the experimental value of mh’,

( ) Mev . Mevxm O mha» + » =549 547 3 . (17.7.38)

These results suggest a higher approximate symmetry for the pK combination, whichmay be contained in the pseudoscalar meson representation as indicated in figure 5. Inother words, the scalar mesons may be considered lepton-antilepton pairs, as suggestedby Barut [29].

In addition to the proper topological P-excitations just discussed, we may considersubexcitations inside G. For [n=0] P-subexcitations, considered in G , there is a set of C sub-spaces that correspond to the infinite ways of choosing PG, not to the unique C defined for anelectron PP. These additional C spaces are not P-equivalent and must be included in theintegration. For a triple electromagnetic interaction there are as many C spaces as U(1) sub-groups inside SU(2), in other words, as points on the sphere SU(2)/U(1). The volume of integra-tion should be multiplied by 4p. The same thing happens with L-excitations. In other words,class [n=0] leptonic subexcitation masses m0, present in a hadronic G-system, are masked by thenonlinear strong interactions. We shall refer to these S2 masked (strongly interacting)subexcitations as GS-subexcitations. The GS-subexcitations geometric mass values correspond-ing to the fundamental leptonic excitations are 4p times the previous leptonic geometric massvalues,

( ) ( ), , , , , . , , . , , . , Gevem e m tn m n t n¢¢¢ = 0 00641 0 1 77 0 29 9 0 . (17.7.39)

The masked excitations are characterized by the same quantum numbers that character-ize the leptonic excitation but the value of the masked geometric mass includes the energyincrease due to the other interactions. If the masked excitations were to be ejected (in-jected) from (to) a G-system they would loose (gain) the energy due to the extra interac-tions, and they would exit (enter) with the bare geometric mass value, as standard freeleptons. There are no observable free particles with the masked masses. Experimentallythe masked mass would never be detected by long-range methods. These masked leptonicexcitations behave as quarks.

In particular, the GS-subexcitations correspond, one to one, to the leptons. These GS-subexcitations determine a hexadimensional space. Consequently any other possible geo-metric GS-subexcitation can be expressed as a linear superposition of these fundamentalmasked geometric subexcitations. Alternately we may interpret some of these subexcitationsas quark states. The structure of masked leptonic subexcitations is geometrically equiva-


lent to the physical quark structure. For example, a superposition of 2 low velocity maskedelectrons e plus 1 masked low energy neutrino ne has a 2/3 charge and a total 4.2 Mevinvariant mass and may be interpreted as a u quark. Similarly a superposition of 2 maskedlow velocity muons m plus 1 masked low energy neutrino nm has 2/3 charge and 1.2 Gevtotal mass. It should be noted that, if free quarks are unobservable, all experimental in-formation about their masses comes from bound quark states (meson resonances) andmust depend somehow on theoretical arguments about these states. There is no quarkconfinement problem.

17.8. Barut’s Model.As shown in section 17.4, a proton contains leptonic subexcitations. We may con-

sider that there are GS-subexcitations inside the hadrons. In this manner we may identifysix flavors of geometric masked leptonic excitations inside all hadrons providing anequivalent quark flavor structure. There is also another triple structure associated to thedual representations, also shown in that section, which could in principle provide an-other quark structure. Nevertheless, we have said duality is just a necessary mathemati-cal equivalent structure. Additionally, there is no spin-statistics problem with these geo-metric components and there is no need for another degree of freedom (color). On theother hand, Barut has suggested [29, 30, 31, 32, 33] a process to construct particles interms of the stable particles and the muon. Therefore, it appears better to constitutegeometric excitations which represent particles starting from the geometric fundamen-tal excitations G (the proton), P (the electron) and L (the neutrino), together with thestrongly interacting masked leptonic TnPS-excitations. In this section the symbols m, m’,m’’ and t represent masked unstable leptons. On hadronic time scales a weakly maskedmuon is considered stable and may be represented here by its stable components ee .

In order to adapt our geometric model to Barut’s model we establish that the quantumnumbers of an excitation correspond to the quantum numbers of its stable components. Inthis manner, following Barut’s ideas, we define the charge of an excitation as the netnumber of charge quanta of the component excitations,

p

pe

p

0 p

0 p

n pe

0 p e

pe

0 p

0 p

Barionic

octet J P 12 .

Figure 5.


p eQ N N= - . (17.8.1)

Similarly, the barionic (atomic) number of an excitation is the number of constituentG-excitations, representatives of protons p,

pA Nº , (17.8.2)

and the leptonic number of an excitation is the number of constituent P-excitations and L-excitations, in n classes according to their topological Tn excited states, representatives ofelectrons e and neutrinos n,

eL N N nº + . (17.8.3)

In a hadronic system of excitations strangeness is the number of component muons m,masked by the strong interaction, capable of a T1PS-excitation or m’’,

S Nm- º , (17.8.4)

and, similarly, beauty is the number of strongly masked t leptonic components capable ofT2PS -excitations,

B N tº , (17.8.5)

Charm and truth are, respectively, the number of component T1L-excitations and T2L-exci-tations, representatives of neutrinos nm and nt,

( )C Nn mº , (17.8.6)

( )T Nn tº . (17.8.7)

e e 16

0K e K

K 0K e

0 e e

12 2

12

2 e e

0 12

Pseudo scalar mesonic

octet J P 0 .

Figure 6.


The isotopic spin depends on the number of stable constituent G-excitations, P-excita-tions and L-excitations representatives of p, e and n in the following manner,

p eI N N N nº - +32 . (17.8.8)

From these definitions we may derive the Gell-Mann-Nakano-Nishijima formula,

( )Q I A S= + +13 2 , (17.8.9)

and define strong hypercharge

Y A Sº + . (17.8.10)In addition to these Barut definitions, it is also possible to define weak hypercharge for a P-excitation in terms of the leptonic number and the rarity r [34] of the representative alge-braic element,

( )y L rº - + . (17.8.11)

The hadronic states contain protons. In figure 4 we show the barionic octet JP=1/2+ andin figure 6 the barionic decuplet JP=3/2+. The meson states are bound states of two leptonsll- as indicated in the mesonic octect JP=0- in figure 5.

Furthermore, the geometric theory allows a discussion of the approximate quantuminteraction between two dressed leptonic excitation states forming a system (quarkonium?).We speculate that the nonrelativistic effective potential should be similar to the one inQCD because there are similarities of the mathematics of the theories. If this were thecase, (this has to be shown) the nonrelativistic effective potential would be [35 ]

( ) aV r kr

r= - + (17.8.12)

p e

p

pe e 0 pe p pe

0 p e

pe p e

0 p

0 p

Figure 7.

Barionic

decuplet J P 32 .


and then we may fit the experimental Y excited levels accordingly, as is done in QCD.The full frame excitation has protonlike properties. Within the geometric theory the

quark excitations are not the fundamental building blocks of matter. They are only subex-citations which are useful and necessary in describing a series of hadronic excitations.

17.9. Relation with Particle Theory.The geometric triple structure determines various structures compatible with the quark

description. On top of this geometry, as is done in the standard model over the geometry ofspecial relativity, it is possible to add an approximate structure to help in the phenomeno-logical understanding of the physical particles. It is known that heavy nuclei may be par-tially understood by using groups SU(N), U(N), O(N) to associate protons and neutrons ina dynamic symmetric or supersymmetric manner [36 ]. This is essentially the use of grouptheory to study the combinations of the two building blocks, protons and neutrons, as-sumed to form nuclei. In the same manner we can use the geometric groups to describe thecombinations of the three fundamental geometrical building blocks introduced by the physi-cal geometry, protons, electrons and neutrinos assumed to form other particles or clusters.

We are interested in combining G, P and L excitations. We attach L excitations to Pexcitations and then to G excitations. As indicated in section 3.7, this combination may notbe done uniquely, rather it depends on the identification of a subgroup H with a subspacein the group fiber bundle space G. Any noncompact generator is equivalent to a boost orexternal symmetry, by the adjoint action of a compact generator. Thus, the compact genera-tors generate an internal symmetry of the combination of excitations.

In particular, the identification of L within P is not unique. The group L may be expressedas the principal bundle (L,3B,S), where the fiber S is the SU(2)R associated to rotations and3B is the tridimensional boost symmetric space,

( , )( )R

SLB

SU=3 2

2

. (17.9.1)

The group P may be expressed as the principal bundle (P,6B,SU(1)) where 6B is the hexa-dimensional double boost symmetric space,

( , )( ) ( )R Q

SpB

SU U=

Ä6 4

2 1

. (17.9.2)

The choice of the boost sector of L (even generators), inside P, depends on the action ofthe U(1) group in the fiber of P, generated by k0, a rotation within the double boost sectorin P.

[ ],k k k k=0 1 0 12 . (17.9.3)

Mathematically this corresponds to the adjoint action of the only compact generator in theodd quotient P/L.

Similarly, the identification of P inside G is not unique either. The group G may beexpressed as the bundle (G,9B,SQ) where Q is the SU(2)Q associated to charge and 9B isthe nonadimensional triple boost symmetric space,


( , )( ) ( )R Q

SLB

SU SU=

Ä9 4

2 2

. (17.9.4)

The choice of the boost sectors of L and P sector inside G depends on the action of theSU(2) in the fiber of G, generated by the compact electromagnetic generators. Math-ematically this corresponds to the action of all compact generators in the quotient G/L.The total internal symmetry of first identifying LP and then (LP)G is the product ofthe two groups. Therefore, there is an internal symmetry action of a group equal toSU(2)U(1), but different to the subgroups of G, on the identification of the subgroupsin the chain GPL.

There is an induced symmetry on the combination of L and P excitations on G excita-tions corresponding to this SU(2)U(1) symmetry. The action of this combinatorial groupmay be interpreted as the determination of the possible combinations of L and P excitationsto give flavor to states of G and P excitations and thus related to the weak interactions.

This SU(2)F group relates electron equivalent P-excitation states or neutrino equivalentL-excitation states. Clearly at sufficiently high energies the mass of any excitation kine-matically appears very small and its effects on results are small deviations from those of azero mass excitation which always corresponds to an excitation of the even subgroup orL-excitation. For this reason, at high energies, the even part (left handed part) of a P-excitation may be related to an even L-excitation by an SU(2) transformation. Both cor-responding physical leptonic excitations, the even part e+ or e0 and n, may be consideredmembers of a doublet, labeled by rarity 0 or weak hypercharge -1, while the odd part e- ore1 is a singlet labeled by rarity 1 or weak hypercharge -2. This weak interaction associa-tion of leptons into hypercharged states has the approximate symmetry SU(2)F. In addi-tion, there is an undetermined orientation of L1 inside G which depends of the action ofSU(2)U(1). Under this approach, the standard physical electromagnetic potential A,the connection of U+(1) in SL1(2,), has an orientation angle within the chiral SU(2)U(1)which should be related to the polar angle Q in eq. 14.4.8.

Since our description may be made in terms of the proton representation or, alter-nately, in terms of its dual quark representation as indicated in section 17.7.4, we couldobtain complementary dual realizations of the relations among different experimental re-sults. These are really different perceptions or pictures of the same physical reality ofmatter. The proton or G-excitation has a quark structure: it behaves as if formed by threepoints. Using the quark representation to build other particles, we may consider that in Gthe 3 quark states span a 2 dimensional Cartan subspace. There is a combinatorial unitarysymmetry, SU(3), related to this subspace of a type A3 Cartan space, which may be inter-preted as the color of the combinations of states of G and thus related to strong interac-tions. States of G-excitations display an SU(3)C symmetry.

In this manner, the physical geometry determines a combinatorial internal symmetryfor excitations characterized by the combinatorial symmetry group SC,

( ) ( ) ( )C C FS SU SU U= Ä Ä3 2 1 . (17.9.5)

Since high energy experiments are excitation scattering processes their relevant physicsis related to transitions which should keep these excitation symmetries in accordancewith the combinatorial SC group. At high energies we may expect the appearance of highmass excitation resonances with these symmetries.

If the probability of combination is propagated infinitesimally by the connection, wehave the elements of a high energy quasi standard model. It is possible, in an approximate


way, to “gauge” these groups to obtain an approximate dynamic theory. This is essentiallywhat the standard model does from certain general features of experimental and theo-retical results. In particular the SU(3) transformation among a triplet of components, orthe quark-parton model [37, 38], and the SU(2)L transformation of the chiral parts of adoublet, or the electroweak theory [39, 40], are the starting points of the model [41].Since these features are present in the physical geometry, as shown in previous sections,we have the real possibility of building an approximate “quasi standard model” on top ofthis geometry. The groups actually relate discrete asymptotic in and out states in a scat-tering theory of excitations.

Nevertheless, there is another alternative to the approach indicated in the previousparagraphs. It may be better to continue the development of a fundamental physical geom-etry model letting the mathematical structure (geometry) guide us, as Dirac once sug-gested, without imposing other preconceived models. We started from a geometric rep-resentation of the fundamental physical concepts of matter and interaction by a fiberbundle (E,M,G) over spacetime. There are subgroups GPL where L is a Lorentz spinorgroup and P a curved Poincaré group. The compact SU(2)R subgroup in L is interpreted asphysical rotations. The other two compact subgroups in P and G are U(1) and SU(2)Q andare interpreted as internal electromagnetic rotations. The complementary noncompact sec-tors are interpreted as generalized (internal) boosts. Physically we require that quantumnumbers of product representations in the chain GPL obey an addition rule. From thephysical point of view we recognize that all chain subgroups are related. We combine L-excitations, P-excitations and G-excitations using mathematical and physical consider-ations. As indicated in sections 17.7 and 16.3, combinations of excitations appear todetermine masses, magnetic moments, quantum numbers, etc. of physical particles.

A physical geometry should share certain features and results with the standard modelwhile there would be differences in some other features. Due to the vastness of highenergy physics experimental results, it is not clear, at this time, how these differencescompare with experiments, in particular because the geometrical ideas and the groupSL(4,) [42] may introduce a rearrangement of experimental results. Actually, the ap-proach should be a program to extend Physical Geometry to the higher energy excita-tions in terms of the stable fundamental excitations. It is true that the standard model hasled to success in finding particle symmetries, but this does not imply there is no betterway of reordering the high energy results.

17.10. Summary.The geometric triple structure determines various physical triple structures in the classifica-

tion of particles. The geometric unified theory represents interactions that may be classified intothree classes by the dynamical holonomy groups of the possible physical connections. Thethree classes correspond to gravitation, electroweak and strong physical interactions.

Solutions to the three corresponding frame excitation equations representing matter,show algebraic and topological quantum numbers. The algebraic numbers correspond toelectric charge, spin and magnetic flux. The topological numbers correspond to wrapping(winding) numbers of higher levels of excitation, defining three 2-member leptonic families.

Using these numbers, calculated mass ratios and other properties, we are able to iden-tify three classes of fundamental fermion matter excitations and a fundamental bosonfield excitation corresponding to the stable physical particles: neutrino, electron, pro-ton and photon. The calculated masses and magnetic moments agree with the experimen-tal values. Two of these fundamental stable excitations have a bare rest mass equal to zero: the


neutrino and the photon. Small energy correction terms due to interaction field energies maybe interpreted as small relative “dressed” physical rest masses for these null mass excitations.The smallest dressed rest mass would correspond to the least interacting particle.

The combination of the three fundamental fermion excitations has a calculated magneticmoment in agreement with the experimental value for the neutron magnetic moment. Themass values and magnetic moment values of the 6 topological excitations agree withthose of the leptons. The pion, kaon and other mesons may be considered as systems oftwo masked leptons. The mass of the pseudo scalar mesons may be understood as themass of the ground states of the product representation of two masked fundamental lep-ton representations.

The geometry determines the mass spectrum of geometric excitation ground states,which for low masses, essentially agrees with the physical particle mass spectrum. It ispossible to classify particles in terms of the stable fermion particles: the proton, the elec-tron and the neutrino (p, e, n), together with the masked leptonic excitations.

The proton shows a triple structure in terms of subexcitations which geometricallycorrespond to these leptonic excitations. We identify six types of geometric dressed lep-tonic subexcitations inside all hadrons providing an equivalent quark flavor structure.

The interactions felt by these excitations conform to the general classification schemeof particles into neutrinos, leptons and hadrons. The combinations of the three particlesdisplay an SU(3)SU(2)U(1) symmetry.

References

1 G. González-Martín, ArXiv 0712.1538, (2007). 2 G. González-Martín, ArXiv physics/0405097, (2004). 3 J. C. Taylor, Gauge Theories of Weak Interactions (Cambridge Univ. Press, Cambridge),

ch 6 (1976). 4 E. Mach, The Science of Mechanics, 5th English ed. (Open Court, LaSalle), ch.1 (1947). 5 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton),

p.55 (1956). 6 G. González-Martín, ArXiv physics/0009052, (2000).. 7 W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco)

p. 715 (1973). 8 E. Cartan, Assoc. Avanc. Sc. Lyon, p. 53 (1926). 9 A. Trautman, Geometrical Aspects of Gauge Configurations, preprint IFT/4/81, Acta

Phys. Austriaca, Supl. (1981).10 G. González-Martín, Gen. Rel. Grav. 23, 827 (1991). See chapter 7.11 G. González-Martín, ArXiv cond-mat/0009181, (2000). See chapter 11.12 G. González-Martín, Gen. Rel. Grav. 24, 501 (1992). See chapter 6.13 G. González-Martín, ArXiv physics/0009066, (2000). See chapter 13.14 G. González-Martín, ArXiv physics/0405126, (2004). See chapter 16.15 G. González-Martín, ArXiv physics/0712,1531, (2007). See chapter 16.16 R. Gilmore, Lie Groups, Lie Algebras and Some of their Applications (John Wiley

and Sons, New York), p.188 (1974).17See section 2.6.1.18 G. González-Martín, ArXiv physics/0009045, (2000), See chapter 14.19 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

p. 137 (1962).


20 G. F. Chew, S. C. Frautschi, Phys. Rev. Lett. 7, 394 (1961).21 M. Carmelli, Ann: Phys. 71, 603 (1972).22 R. E. Marshak, Conceptual Foundations of Modern Particle Physics, (World

ScientificSingapore) ch. 10 (1993).23 A. O. Barut, Phys. Rev. Lett. B42, 1251 (1979).24 A. O. Barut, Phys. Lett. B73, 310 (1978).25 G. González-Martín, ArXiv physics/0405097, (2004).26 See appendix E.27 R. Hermann, Lie Groups for Physicists (W. A. Benjamin, New York) p. 53 (1966).28 See chapter 1229 A. O. Barut, Surv. High Energy Phys. 1, 113 (1980).30 A. O. Barut, in Lecture notes in Physics, 94, (Springer, Berlin) (1979).31 A. O. Barut, Quantum Theory and Structure of Space-time, 5, L. Castell et al, eds.

(Hauser, Munich) (1983).32 A. O. Barut, Physics Reports, 172, 1 (1989).33 W. T. Grandy, Found. of Phys. 23, 439 (1993).34 See appendix A, section A2.35 E. Eichen, K. Gottfried, T. Kinoshita, K. Lane, T. Yan, Phys. Rev. D21, 203 (1980).36 A. Arima, F. Iachello, Phys. Rev. Lett., 25, 1069 (1974).37 M. Gell-Mann, Phys. Lett. 8, 214 (1964).38 G. Zweig, CERN Report 8182/Th. 401 (unpublished) (1964).39 S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).40 A. Salam, in Elementary Particle Theory, ed. N. Swartholm (Almquist and Wissell,

Stockholm) (1968).41 S. Weinberg, The Quantum Theory of Fields, (Cambridge Univ. Press, Cambridge)

V. 2, 384 (1996).42 Y. Ne’eman, Dj. Sijacki, Phys. Lett., 157B, 267.

18. THE NEUTRINO EQUATION ANDOSCILLATIONS.

18.1. Introduction. It has been known for a long time that light deflection experiments confirm that the null

cone is curved by gravitation as predicted by Einstein’s General Relativity. The curvatureaffects particle trajectories as mass does. It may be interpreted using the energy-massequivalence principle that particles acquire a curvature mass-energy. This means that weakgravitational effects should not be neglected “a priori” when studying very fast particlesmoving on trajectories on the curved null cone or nearby mass hyperboloids. In this chap-ter we consider the motion of a neutrino wave in the Earth Schwarzschild geometry anddiscuss its differences with a light wave instead of simply assuming geodesic motion on aflat null cone.

In order to discuss the motion of a neutrino in the proper physical context we shouldfirst make some theoretical-historical considerations. Cartan introduced geometric spinors[1, 2, 3] in 1913. Weyl [4] soon developed a physical theory for a 2-component (Weyl)spinor. Pauli used the spin matrices to formulate the theory of non relativistic spin [5] in1927. Dirac developed the standard 4-component relativistic spinor theory for massivefermions in 1928. It is now well known that if the particle has zero mass the Dirac equationsdecouple into 2-component Weyl zero mass n spinor equations [6] which may be written interms of the Pauli matrices s and the Planck constant as eq. (3.5.3)

Since the introduction of a (neutrino) particle [7] in beta decay by Pauli in 1930 the mainquestion debated was whether its mass was zero or very small. Traditionally the Weylequation was used to describe the neutrino. Some experiments suggest that the neutrino isnot a massless particle and has a real rest mass. Today the bound for the electron neutrinomass is estimated to be of the order of a few eV and questions arise about the adequateequation to use. Such small masses may also be questionable because they would be of thesame order of the curvature effects described here. It is also well known that the experimen-tal mass of particles is generally explained by field energy corrections added to an assumedfinite “bare” particle mass to obtain a theoretical “dressed” mass in the process of massrenormalization [8, 9] for Dirac’s equation. If the curvature effects of the gravitationalpotential were equivalent to the effects of a small finite dressed mass-energy we similarlymay explain the small neutrino mass starting from a zero “bare” neutrino mass. Or equiva-lently, the neutrino oscillations [10, 11] could also be explained by the dressed mass-energy contribution to a zero bare mass neutrino. Both points of view may be consistentwith observed neutrino oscillations. Certainly these questions are still open for furtherexperiments and theoretical ideas.

When there is a gravitational field and spacetime is curved we expect the neutrinospinor field n to obey the geometric covariant equation determined from the integrabilityconditions of the full sl(4,) field equation restricted to the gravitational spinor sl(2,)connection 1-form w as generally indicated in chapters 3 and 4. The influence of gravita-tional curvature on the neutrino motion would be consistent with the old Bohr idea [12, 13]that the neutrino may be related to gravitation.

215Neutrino Equation and Oscillations

18.2. The Action of the Even Subgroup L.The use of differential and spinor analysis displays the fundamental principles in-

volved in this problem. We apply the spinor moving frame formulation under the SL(4,)structure group of the geometry as presented in chapters 1, 2 and in particular sections3.1 to 3.4. The local physical action is through the spinor connection.

The interactions are determined by the sl(4,) connection generators which form abasis in the R3,1 Clifford algebra. When the interaction reduces to only a gravitational fieldthe group reduces to its even SL(4,) subgroup or SL(2,). The fundamental representa-tion of SL(2,) is a zero-mass 2-spinor particle as shown in section 13.4, which has theproperties of a neutrino. Within the theory this appears as the simplest geometric structureassociated to spacetime.

The even sl(4,) connection generators on the manifold M include antisymmetric ten-sors of type (1, 1) which obey J2=-1 determining a complex structure on a real 4-vectorspace as discussed in section 13.3.2. The corresponding complex vector space has 2 com-plex dimensions. This complex structure J determines an integral complex structure onsubspaces of the spinor fiber spaces VMm at points m on M which form the fiber space ofthe spinor vector bundle VM introduced in section 3.1. These spaces are isomorphic to thecomplex tangent vector spaces TMm

C. The restriction to the even sl(4,) generators deter-mines a unique complex structure on the manifold M. The notation indicates that TMC andthe original fiber bundle VM are related bundle representations of spacetime with pointslocally identified by the physical spacetime events. The curved complex manifold may bea valid representation of spacetime, capable of representing a physical relativistic localexpression in terms of time and space variables related to complex 2-spinors.

There should be a local bundle map,

: CmTM TMm , (18.2.1)

which allows a physical interpretation on M and the definition of complex coordinatesassociated to the real coordinates on M. The 2 to 1 mapping from 2-spinors to 4-vectorsdetermines a local relation between the Lie algebras of the linear transformations on theprincipal bundles SL(2,) or the even subgroup SL+(2,) over M and SO(3,1) over M.

On the null cone the isotropy group is SO(2,1) instead o SO(3) as may be seen fromsection 7.2. The null tangent plane may be perceived as decomposed into a visual nullplane w and a complementary complex plane z. We may write [14] for coordinates x, y, s,l=it on M, the pulled-back complex coordinates w, z

( )Re Imz z z s i l wm m* *= + º + º1 1 1 , (18.2.2)

( )Re Imz z z x i y zm m* *= + º + º2 2 2 , (18.2.3)

where s is chosen along the null neutrino trajectory.The plane spanned by the complex dw and its conjugate is tangent to a Lorentzian

surface adapted to the neutrino motion. The volume element on this complex [15] spinorsubspace spanned by the differential forms is, omitting the complex coordinate indices,

( )C dw dw i ds dlS m m S* *= = =2 2 . (18.2.4)

This means that the volume (area) element on this spinor complex tangent subspace de-termines twice the volume (area) element on the corresponding 2-dimensional real vec-tor tangent space. The dual complex vector base [15] on this complex subspace is formed


by the vectors

( ) ( )i

w s lm m* *

æ ö÷¶ ¶ ¶ç ÷ç= - ÷ç ÷ç¶ ÷¶ ¶ ÷çè ø

12

, (18.2.5)

( ) ( )i

w s lm m* *

æ ö÷¶ ¶ ¶ç ÷ç= + ÷ç ÷ç¶ ÷¶ ¶ ÷çè ø

12

, (18.2.6)

which are mapped to TM as null vectors tangent to the null cone bicharacteristics,

iw s l s

m* --

æ ö¶ ¶ ¶ ¶÷ç= - º º ¶÷ç ÷çè ø¶ ¶ ¶ ¶12

, (18.2.7)

iw s l s

m* ++

æ ö¶ ¶ ¶ ¶÷ç= + º º ¶÷ç ÷çè ø¶ ¶ ¶ ¶12

. (18.2.8)

The complex conjugation on the complex tangent w plane reflects the l imaginary timecomponent. The substitution l=it determines that w represents a null coordinate. Theeffect of conjugation on the null tangent plane s-t reflects the time component of the nullcoordinate. This determines the Lorentzian line element on the null surface. We have adaptedthe complex w subspace to the null direction of the neutrino motion.

18.3. An even L-Spinor Neutrino Equation.Consider now the equation of motion for a spinor on spacetime. The spinor excitation

motion may be described as a particular case of the general equation obtained from eq.(1.4.8). This is accomplished by a connection valued on the sl(4,) algebra and its cor-responding curvature as discussed in section 12.4. We consider the particular eqs. 12.4.5and 12.4.9. These equations decouple for the even and odd spinor parts h, x when m iszero and the only field is the gravitational connection G, as in the case of flat space. Theresultant equations correspond to covariant parallel transportation of the spinor fields.

We should distinguish the geometric spinor structure from its position complex coordi-nates on the complex w surface in spacetime. The resultant direction of motion is a nullvector k which should be given in relation to the orthonormal set km of the Clifford algebra.This determines the null Lorentzian subspace implicit in the spinor structure. The complexderivatives should also be given in relation to the orthonormal set of the Clifford alge-bra. To preserve both the complex spinor and gravitational structures we project the complexcovariant derivatives onto an orthonormal set adapted to the SO(2,1) isotropy on the nullcone. In order to do this, we should define a set kA in the Clifford algebra [16] adapted tonull vectors and complex coordinates in terms of the fundamental orthonormal set,

( ) ( ), z x y z x yi ik k k k k k= + = -1 12 2

, (18.3.1)


( ) ( ) ( ) , w s l s t s wik k k k k k k k k k+ -= + = + = + º =01 1 12 2 2 .

(18.3.2)The desired scalar neutrino spinor equation is the projection of the spinor derivative on

the spinor wave vector or equivalently on the w tangent plane represented by kw. Thisprojection determines a coordinate invariant scalar curved operator,

AAk h = 0 . (18.3.3)

Since there is no motion along the complex z direction the equation of motion may besimply written

A w wA w wk h k h k h = + = 0 . (18.3.4)

If we take the second derivative of eq. (18.3.3) the quadratic operator is of the form

A BA Bk k h = 0 . (18.3.5)

The kA anticommuting properties and the curvature symmetries determine that there are nocontributions from mixed wz components. This operator introduces the even 4 by 4 sl(4,)representation k0km for the Pauli sm matrices. Decomposing the second derivative in itssymmetric and antisymmetric parts we obtain using the commutation relations,

{ }, ,A B A B CA B A B w w ww

sk k h k k h h k k W hé ù + = ¶ ¶ + =ê úë û01 1

2 2 0 . (18.3.6)

18.4. A Lorentzian-section Curvature Scalar.The first term in eq. 18.3.6 is a differential operator and the second term is an algebraic

operator. Both act on the spinor wavefunction h which is a function over a 1-dimensionalnull characteristic of the null cone. We use the geometrically significant fact that Lorentz-ian surfaces are characterized by pairs of null directions parametrized by a space affineparameter s. The spinor observation implies a projected curvature form CW on the spinorwave null directions in eq. (18.3.6). These directions are determined by t2 = s2 on thenull cone. Points on a null cone are characterized by points on spheres of radius s=trelated by Lorentz dilation transformations. The unit sphere S2 [2] is a cross section ofthe null cone at t=1 and corresponds to the celestial sphere. A null direction is charac-terized by a point on the unit sphere S2.

The measured spinor wavefunction must be expressed using the same coordinates asthe gravitational field. There is a 2-dimensional flat space spanned by the timelike andspacelike unit vectors. This plane, which contains the 4-velocity, is tangent to a Lorentziansurface. There is a well defined curvature tensor on this surface. We let the physical prob-lem determine which is the significant Lorentzian curvature scalar. Equation (18.3.6) deter-mines a single curvature scalar component by contraction of the curvature tensor with thewavevectors k . We should note that the Gaussian curvature may be generalized on manifoldsprovided with moving frames and special affine transformation groups on surfaces [17, 18].

In Schwarzchild coordinates the sl(2,) curvature forms CW are defined by the wellknown curvature tensor or the so(3,1) forms W indicated in section 5.3. Both sets of formsare equal with vector generators and differential forms replaced by the spinor generatorsand complex volume forms S [14]. This replacement implies two factors of 1/2 in CW. Theyare due to the double-valued homomorphism between the SO(3,1) real vector transforma-


tions and the SL(2,) spinor transformations [2, 3, 19] and the relation of real and complexvolume elements [15], as indicated in appendix A6 and eq. (18.2.4).

Equation (18.3.6) determines a single curvature scalar component by contraction of thecurvature tensor with the physical null wavevector of the spinor wavefunction h(ks). For aspinor motion on a direction tangent to any great circle on a Schwarzchild sphere (horizon-tal directions) with angular longitude coordinate f it should be clear that s=rf. The otherdirections are obtained by polar angle q rotations of s toward the Schwarzschild r radialcoordinate direction (vertical direction). Since the Schwarzschild coordinates are not iso-tropic the curvature varies with the field direction as indicated in eqs. (5.3.12-5.3.13). Thesl(2,) curvature 2-forms CW on the spinor subspace corresponding to null paths along s inradial and horizontal trajectories, in terms of the Newtonian potential j, are

Cr r

E dr dt dw dwr r r

j j s S jsW W

æ ö ÷ç= º =÷ç ÷÷çè ø

1 1 10

2 2 22 2

2 2 2, (18.4.1)

Ch h

E d dt dw dwr r r

j f j s S jsW W

æ ö ÷ç= º =÷ç ÷÷çè ø

3 3 30

2 2 22 2 4. (18.4.2)

The sign is chosen so the gravitational boost is attractive, along the negative radial direc-tion. We should project these forms on the wave vector k using the adapted coordinates was indicated in the curvature term CW in eq. (18.3.6).

Consider the laplacian differential wave operator in eq. (18.3.6). It should also be ex-pressed in rotational coordinates as the curvature forms. For a horizontal direction thespacelike component of the wave operator corresponds to the angular momentum operatorL which, without loss of generality, may be chosen along any great circle parametrized bythe angular coordinate f,

( ) ( )sinsin sin

Lr

r rr r rq

q q q q f fæ ö¶ ¶ ¶ ¶ ¶ ¶÷ç + + = ÷ç ÷÷ç ¶ ¶ ¶ ¶ ¶ ¶è ø

2 2 22

2 2 2 2 2 21 1 1 1

.

(18.4.3)

The angular momentum operator appears with an extra r-2 coordinate factor. The circular arc rfis the rotation parameter s of L2 on the circle at r. The action of SL(2,) is through boosts alonghorizontal and radial directions. The isotropy subgroup on the null subspace is SO(2,1) asindicated in sections 7.2. The null path SO(2,1) boost operator generates a hyperbolic arc.Thus we recognize the extra factor r-2 in eqs. (18.4.1) and (18.4.2) as a Jacobian transforma-tion factor between curved Schwarzchild and orthonormal Clifford algebra coordinates. Itshould be present in the km spherical matrices because they transform as vector densities.The 4-dimensional complex wave operator eq. (18.3.6) may be written in terms of a nullboost operator P2 on the null w subspace and a curvature scalar K,

w w sww

s

r r rn P n Kn

s W n¶ ¶

+ = - + =2

2 2 2 0 , (18.4.4)

where the dot indicates a Pauli matrix product. We explicitly define a Lorentzian-sectioncurvature scalar K from the W components.

The curvature scalar K may be considered a generalization of the sectional curvature


for lorentzian metrics [14, 17, 20]. The wave vector k defines a polar angle q with respect to theradial gravitational field direction. This angle determines the Lorentzian section of interest,spanned by ds and dt in the null-cone subspace. We express K in terms of a curvature relatedform dw spanned by dt, dr and a horizontal dh to obtain an equation for any k direction,

( ) ( )ww w w w ws s sr

j jK k k W k k s w w w wº = ¶ ¶ = ¶ ¶2 0 0

4 4, , . (18.4.5)

The 1-form dw has a radial component which is twice the ds radial component, due to theanisotropy factor of 2 in Schwarzchild’s coordinates, eqs. (5.3.12-5.3.13). We express K interms of the polar angle q and the components of dw,

( ) ( )

( )cos sin cos sin

cos sin cos sin , r h t r h t

dr dh dt dr dh dtj

K q q q q

q q q q

= + + + -

¶ + ¶ - ¶ ¶ + ¶ + ¶

2 24

2 2, (18.4.6)

( )( )

cos coscos ,

sin sin sin ,

r t r t

h t h t

dr dt

dh dt

q qK j q

j q q q

¶ - ¶ ¶ + ¶= -

¶ - ¶ ¶ + ¶-

2 2

2 2 2 . (18.4.7)

We obtain K for any k direction of the Schwarzchild coordinates,

( )cos sinj

K q q-

= +2 224

. (18.4.8)

The desired neutrino equation (18.3.6) in a gravitational field determines a Laplace-Beltrami operator on the complex surface in the complex manifold bundle,

( ) ( ) ( )ssr

tP n n n K n

æ ö¶ ¶ ÷ç= - = =÷ç ÷ç ÷¶ ¶è ø

2 22 1

4 , (18.4.9)

in terms of the function K. The same equation may be obtained using Dirac’s matrices ofthe R13 Clifford Algebra. This is possible because the even parts of the correspondingalgebras are isomorphic.

Equation (18.4.9) is consistent with a flat Laplacian on a null surface in the TM bundle.The units are undefined in this expression. The equation shows that curvature producesthe same effects of mass. The dimensionless source K should be related to the funda-mental geometric energy provided by the geometry which also defines the units of timeand distance for the motion of this L-excitation.

18.5. Neutrino Mass-Energy. In the even sl(4,) spinorial model, the neutrino moves under the action of the con-

nection as the covariant eq. (18.3.3) on a curved spinor bundle over the complex space-time,


AAk n = 0 . (18.5.1)

The bending of the neutrino trajectory is locally generated by the transformed spinor-section curvature, eq. (18.4.1), which determines the source K of the neutrino radial equa-tion and may be experimentally detected as a neutrino mass-energy.

The gravitational potential j and the curvature scalar K define a neutrino mass-en-ergy scale E0 at the constant Earth radius r=R . Using mass ratios relative to the funda-mental geometric energy M, defined in chapter 12, the operator in eq. (18.4.9) may bemapped and interpreted as a Klein-Gordon energy-momentum operator P2. Using eqs. (18.2.7,18.2.8) in terms of the t, s coordinates with equal to unity we obtain,

CsP m

t sn n n n

æ öæ ö æ ö¶ ¶ ÷ç ÷ ÷ç ç ÷çD = - =÷ ÷ ÷ç çç ÷ ÷ç ç ÷è ø è ø ÷ç ¶ ¶è ø

2 22 1

4 . (18.5.2)

If we restore the Planck constant in the equation, the neutrino energy operator is

( )( )t t

hE i i

V Un = ¶ = ¶1

2 1 . (18.5.3)

Planck’s energy relation is satisfied for the natural 2-spinor frequency which is half thevector or 4-spinor frequency because the spinor linear group SU(2) volume is twice thevolume of the orthogonal group SO(3).

The mass-energy calculated from the Lorentzian-section curvature scalar K may bedefined from the radial line curvature k,

r

jk K

-º =2

2 , (18.5.4)

.gR

c

jk --

= = = ´ 51 319 102 2

. (18.5.5)

A particle dressed mass under a connection field is expected to include a small correc-tion proportional to the fundamental inverse length or mass M and the field line curvature,as indicated at the end of section 13.4. As discussed in section 17.7.1 the leptons corre-spond to topological excitation waves [21] characterized by winding (or rather wrapping)topological numbers n which may determine leptonic flavor levels. We calculated the lep-ton bare mass ratios [22, 23] in terms of algebraic volumes of group symmetric subspacesand M. We may include the gravitational curvature correction k in the expression for theneutrino mass eq. (13.4.2) and obtain the resultant neutrino dressed mass-energy in termsof the electron mass,

eRR

L Pm V m V

LLn k k kæ öæ ö æ ö÷ç ÷ ÷ç ç÷= + = =ç ÷ ÷ç ç÷÷ ÷çç ç ÷ç è øè øè ø

M M . (18.5.6)

We obtain the energy factors from the spinor-section curvature ratios using the vol-umes of the same previously used symmetric spaces related to the S(M) bundle. In particu-lar the geometric factor indicated by eq. (17.7.16) is the factor of proportionality of the


electron bare rest mass with respect to M. This factor gives the physical energy in terms ofthe electron mass,

( )r e e

R

m mE

V C

j jk

p- -

= = =03

2 16 2M . (18.5.7)

On the Earth surface this effect is higher than expected because it depends on thesquare root of the dimensionless gravitational potential ratio j. The horizontal spinor-section curvature is a half of the radial spinor-section curvature. The numerical mass-energy values on the Earth surface range from a minimum on a horizontal trajectory to amaximum on a radial vertical trajectory as indicated by eq, (18.4.4),

( )cosh remE E

jq

p£- + £

2 22 2 20 010

312 2

, (18.5.8)

( ). eV r e

R

mE m

V Cn

j-= =0 0 569

2 . (18.5.9)

The minimum value of mass-energy corresponds to a horizontal neutrino trajectory onthe Earth surface,

( ). eV h e

R

mE E m

V C n

j-º = =0 0 0 402

2 . (18.5.10)

We take the characteristic mass-energy scale E0 equal to this minimum value. The mass-energy increases with the polar angle relative to the vertical direction.

If this gravitational effect is neglected there may be significant errors in the calculationof small rest masses at very-high-energy processes. The errors would be of the order of theenergy E0 related to the linear curvature k. Some of these theoretical curvature effects werefirst reported [24] within the context of neutrino velocity determinations.

18.6. Topological Squared Energy Effects.The group of the connection also produces a gravitational action on neutrinos of differ-

ent type. The same lepton model used in the previous section represents the muon andtauon neutrinos as topological excitations of the electron neutrino. This implies a variationof the geometric mass-energy range for these excited neutrinos.

The action of the geometric sl(2,) connection) on the covariant neutrino wave equa-tion is similar to the action of rotation generators on gravitational precession [25] experi-ments. The SL(2,) group acts on its spin representations. In particular we are consider-ing representations induced by its SU(2) subgroup, designated as H, which are spinorfunctions on the SL(2,)/SU(2) coset. This 3-dimensional coset space may be related tothe null cone. The SL(2,) neutrino representations have zero rest mass and are geo-metrically related to the massive leptons [21] through the homotopy groups of SL(2,),Sp(4,) and SU(2) which are isomorphic to the integers Z as indicated in section 17.7.1.

The third homotopy group of a space is defined as the classes of mappings from the 3-spheres to the space. The number of produced images is called the winding or wrapping


number. The images in H allow alternative actions of SU(2) and additional neutrino waveequation states on the images, which we call channels.

The number of energy-degenerate group elements in each wound space level is the Hgroup volume V. The squared mass-energy, which is proportional to the curvature K, shouldbe a characteristic parameter of any neutrino wave channel. Since all states are equivalentunder the H group they equally share the curvature energy at any given point and directionon spacetime. We may consider that there are neutrino energy currents along channels orparallel paths in spacetime. Each parallel path corresponds to a neutrino of differenttype. The available energy-mass corresponding to any n-wound neutrino channel is theproduct of the number of states Vn by the energy-mass E.

The volume of the spin subgroup V(H) represents a geometric inertial opposition orresistance to the energy currents in the parallel channels. This is due to the H group action,directly on the L/H coset, which corresponds to an action of the group H on its inverse H-1 anddetermines that the volume appears as the divisor. These actions, the energy and the trajec-tories for wave states with n=1,2 winding numbers vary discretely from their n=0 values.These variations are due to the varied number of H group elements under the action of theZ homotopy group and should be of order V -1 between adjacent n levels.

The variation of the volume in the neutrino family, together with the presence of non-zero curvature, produces a variation of the neutrino energy flow per channel state associatedto the mass-energy source term. We may take K and the volume V0 as constants but there arevolume variations for the other channels. The available potential energy for all states in eachof the n active channels determines that the energy varies inversely proportional to the vol-ume as a function of n as n varies discretely from zero,

( ) ( )V E V E n V nK = =22 2 2 2 20 00M . (18.6.1)

The SU(2) group H acts through the connection as a derivative. We may assume that theaction of the homotopy group Z produces a variation of the energy from its n=0 value by aderivation with respect to the geometric SU(2) group volume V which determines the number ofpossible elements. We define a map from the homotopy group Z to a discrete subset of approxi-mate difference operators Dn

V in the set of operators with a derivative as operation,

( )( )

, ,n

nV n

V VZ n

VD

æ ö¶ ÷ç ÷ç º ÷ç ÷ç ¶ ÷÷çè ø

2 211 2 3 , (18.6.2)

( ) ( ) ( ) ( )( ), , , !nn nV V V V V n VD D D - - -= - - - +21 2 1 22 1 6 1 1 . (18.6.3)

The resultant difference operators determine discrete squared energy differences DE2

for waves in each channel n,

( )V

EE E

V VK

D D--

= =2

1 2 2 00 10

22 , (18.6.4)

( )V

EE E

V VK

D D= =2

2 2 2 00 212

66 . (18.6.5)


A wave current in a level n should imply currents in all lower levels. Due to the propor-tionality of the equations, the energy in the channels up to winding number 2 and theirdifferences satisfy the relations

( )( )E E E E

V SU

æ ö÷ç ÷ç= + = - ÷ç ÷ç ÷çè ø

2 2 2 21 0 10 0

212

, (18.6.6)

( )( ) ( )( )E E E E

V SU V SU

æ ö÷ç= + = - + ÷ç ÷çè ø2 2 2 22 1 21 0 2

2 612 2

, (18.6.7)

EE E E ED D

p= - = »

22 2 2 2001 0 1 022

216

. (18.6.8)

In a gravitational field these gravitational energy excitations must include waves of the3 possible n values For any pair of n states the neutrino wave transitions should includeterms of the form

( ) ( )cos sini k x i k xT e e k x k xm m

m m m mm m

D - D= + = D = D22 4 2 (18.6.9)

which indicate transition oscillations among the neutrino states. The states determined byeqs. (18.6.6, 18.6.7) are energy eigenstates with wavefunctions nn. They may be taken as theneutrino mass states of the standard theory [26] of neutrino oscillations.

Observed very small experimental neutrino masses and neutrino oscillations [26]may be caused by this gravitational effect. Instead of constant neutrino masses we reallywould have the null cone gravitational curvature and small mass-energy differences de-termined by variable neutrino winding numbers under the curvature effect.

The potential neutrino energies En may be found from eqs. (18.5.10, 18.6.6, 18.6.7).

In particular, the neutrino “dressed” mass in experiments on the Earth surface would de-pend on its spacelike direction because the Schwarzschild metric Lorentzian-section cur-vature is not isotropic as indicated in section 18.4. We express the numerical energyresults in terms of the trajectory polar angle q with respect vertical radial direction usingthe standard Earth gravitational values which may be taken as constants. We have, respec-tively, for Earth neutrinos associated with the short or long energy-ranges,

( )( )2 2

cos. eV . eV

EE

qD q

p- -

+´ £ = £ ´

2 203 2 3

01 2

2 12 05 10 4 10 10

16 , (18.6.10)

( )( )

( )2 2

cos. eV . eV

EE

qD q

p- -

+´ £ = £ ´

2 205 2 5

21 22

6 13 89 10 7 79 10

16 . (18.6.11)

In general we can say that the neutrino mass-energy relations would be variable, depend-ing on the gravitational potential in each experiment. A neutrino is created by a nuclear reac-tion in a definite state. From this initial state neutrino waves are driven and completely deter-mined by the curvature lens effect until they are destroyed by a final nuclear collision. If thecurvature scalar is zero, neutrino waves in all channels move equally free on a flat null cone.


Neutrinos in outer space, where interactions may be neglected, would have zero line curvaturescalars and would appear as massless particles travelling as photons on flat null geodesics.

Atmospheric neutrino experiments [26] indicate a 2.4 10-3 eV2 result, 17% higher than the2.04 10-3 eV2 minimum energy value of the short-range-oscillation. An average trajectory incli-nation may produce a variation of the spinor-section curvature and sufficient to account forthe difference.

The reactor and solar neutrino experiments indicate a combined result [27], 2% lower thanthe maximum value of the long-range-oscillation. The solar neutrino wave mass-energy mayhave a diurnal anisotropy due to the Earth rotation. There is a variation of the spinor-sectioncurvature which may sufficiently increase the observed values beyond the average value.

18.7. Summary.We have obtained the following results: 1- The even sl(4,) spinorial neutrino model

determines a neutrino mass-energy proportional to the line curvature. 2- The energy-massvariations for wound neutrinos are proportional to the gravitational curvature and determinefundamental squared energy differences among the neutrino states.

All neutrino curvature effects on the Earth surface vary with the trajectory direction dueto the anisotropic presence of the potential term j in the Schwarzschild spacetime metric

The experimental results of neutrino mass-energy measurements and oscillations appearto be consistent with these theoretical results and their physical interpretation.

At the Earth surface the dimensionless radius of curvature of null cone trajectories is ofthe same order (10-5) of the square root of the potential j(r). When averaging experimentalmass-energy values it is necessary to consider the dependence with respect to the trajectoryinclination. If there are additional weak fields on spacetime there may be a higher effectivepotential j and the numerical results may increase. In general for stronger fields a unifiedgeometry would produce stronger particle effects.

The spinor and spacetime complex structures determine the momentum operator for theneutrino spacetime. It satisfies Planck’s energy relation for the natural neutrino frequency,proportional to the spin. The neutrino waves may be driven and consistently determined bythe curvature scalar until they are destroyed by a final nuclear collision. Bohr might havebeen correct when he presented the idea that neutrinos may be related to gravitation [12].

There has been a search for a graviton associated to the metric interaction field. We seethat the metric may be given by an orthonormal vector frame determined by a spinor frame.If we look at the gravitational curvature action we find the spin-½ neutrino which may beconsidered an excitation of a spinor frame field and therefore related to the metric. Insteadof a metric wave we have a neutrino wave with energy proportional to the frequency, whichmay be considered the real graviton. This is consistent with the fact that gravitational andneutrino fields are the only ones not blocked by the Earth crust. The nonlinearity of the uni-fied theory does not provide a clear cut distinction between interaction fields and matterfields. We realize that there may be objections to this interpretation. Further theoreticalanalysis and experiments may be required to understand the physical effects of the nonlinearspinor geometry associated to particles.

References

1 E. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane,Bul. Soc. Math. France 41, 53 (1913).


2 R. Penrose & W. Rindler, Spinors and space-time, (Cambridge University Press. Cambridge) V 1 (1984).

3 E. Cartan, The theoty of Spinors, (Hermann, Paris) (1966). 4 H. Weyl, Gruppentheorie und Quantenmechanik, (Hirzel, Leipzig) (1928). 5 W. Pauli, Zur Quantenmechanik des magnetischen Elektrons, Zeitschrift fur Physik, 43, p.

601. (1927). 6 J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, (McGraw-Hill, New York),

p. 257 7 W. Pauli, letter to Lisa Meitner et al (1930). 8 J. Schwinger, Editor, Selected Papers on Quantum Electrodynamics (Dover, N. Y.) (1958) 9 S. Weinberg, The Quantum Theory of Fields. (Cambridge University Press, N. Y.) (1995),

Vol. I, 43910 B. Pontecorvo, Neutrino Experiments and the Problem of Conservation of Leptonic

Charge, Zh. Ekspe. Teor. Fiz. 53, 1717 (1967); Sov. Phys. JETP 26, 984 (1968).11 Y. Fukuda et al., (Kamiokande Collaboration), Measurements of the Solar Neutrino Flux

from Super Kamiokande’s First 300 Days, Phys. Rev. Lett. 81, 1158 (1998).12 W. Pauli, Wissenschaftlicher Briefwechsel, vol 2, 1930-1939, K. von Meyen, ed. (Springer

Verlag, Berlin/Heidelberg/New York/Tokio) (1985), Bohr to Pauli, 15 March 1934, p. 308.13 D. I. Blokhintsev and F. M. Galperin, Gipoteza neutrino i zakon sokhraneniya energii, Pod

znamenem marxisma, # 6, p. 147.14 G. Gonzalez-Martin, arXiv: 1212.2277v2 (12/2012)15 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

(1962), p. 291.16 I. Porteus,Topological Geometry, (Van Nostrand Reinhold, London), ch 13 (1969).17 M. Spivak, Differential Geometry, 3rd. ed. (Publish or Perish, Houston), Vol. 4, p 64

(1999).18 M. Spivak, Differential Geometry, 3rd. ed. (Publish or Perish, Houston), Vol. 3, ps. 71, 128

(1999)19 R. Gilmore, Lie Groups, Lie Algebras and some of their Aplications (John Wiley and Sons,

New York) (1974).20 B. O’Neill, Semi-Riemannian Geometry, (Academic Press, San Diego) (1983).21 G. R. González-Martín, Lepton and Meson Masses, arXiv: physics /0405097 (2003).22 G. González-Martín, A Geometric Definition of Mass, Gen. Rel. and Grav. 26, 1177 (1994).23 G. R. González-Martín, Importance of symmetric spaces in the determination of masses. Rev

Mex de Fis., 49 Sup. 3, 118 (2003).24 G. R. González-Martín, arXiv: 1110.3287v3 (11/2011).25 G. R. González-Martín, Relativistic Gyroscopic Motion and Aberration, Phys. Rev. D14, 399

(1976).26 Review of Particle Physics, C. Amsler, et al., Physics Letters B667, 1 (2008).27 S. Abe et al, (KamLAND Collaboration), Precision Measurement of Neutrino Oscillation

Parameters with KamLAND, Phys. Rev. Let. 100, 20 (2008); arXive:0801.4589.

19. THE ALPHA CONSTANT.

19.1. Introduction.It is known that the fine structure constant a is essentially equal to an algebraic expres-

sion in terms of p and entire numbers which appears to be related to the quotient of thevolume of certain groups [1, 2]. This expression may also be determined by the invariantmeasure corresponding to the structure group of the unified geometric theory [3, 4]. Usingthe geometric fact that space-time is a hyperbolic manifold modeled on the symmetric spaceK, the invariant measure on this symmetric space obtained from associated groups is trans-ported to space-time. This relation is valid because the fiber of the tangent space to space-time is the image of a minkowskian subspace of the geometric algebra, as seen in chapter 2.

19.2. A Geometric Measure.The current *J is a 3-form on M valued in the Clifford algebra A. It is constructed

starting from a vector field on the symmetric space K. This space is G/G+ where G isthe simple group whose action produces the automorphisms of A and G+ is the evensubgroup, relative to the orthonormal base of the algebra. The vector field is theimage, under the Clifford injection k of a vector field on space-time M. This injectionallows us to define *J as the pullback form of a 3-form on K. The integration of this 3-form on a tridimensional boundary of a region R in K is equivalent to the integration ofthe pullback 3-form on a tridimensional boundary of the image of the region R in thespace time M. The forms are defined by the existence of a geometric invariant measureon G/G+. The constant coefficient of this invariant measure may be calculated in theparticular case where the fiber bundle is flat and the field equation reduces to thelinear equation equivalent to electromagnetism. This relation defines a geometric in-terpretation for the coupling constant of the geometric unified theory: “The couplingconstant is the constant coefficient of *J introduced by the invariant measure on thesymmetric space G/G+”.

19.2.1. Symmetric Space K.As indicated before, the group G is SL(2,) and the even subgroup G+ is SL1(2,).

The symmetric space K is a noncompact real form of the complex symmetric spacecorresponding to the complex extension of the noncompact SU(2,2) and its quotients.The corresponding series of symmetric spaces coincides with the series character-ized by the group SO(4,2) as shown in appendix B. In particular we can identify thequotients with the same character, +4, in order to write the series of spaces in thefollowing form,

( , ) ( , ) ( )( ) ( ) ( , ) ( ) ( ) ( )SO SL SO

R KSO SO SL SO SO SO

º » @ @ »´ ´ ´4 2 4 6

4 2 2 2 4 2

.

(19.2.1)These quotients include the two symmetric spaces of interest: the noncompact

riemannian hermitian R and the noncompact pseudoriemannian nonhermitian K sym-metric spaces. Since some of these groups and quotients are noncompact we shall use

227The Alpha Constant

the normalized invariant measure mN calculated from a known measure, as usually donewhen working with noncompact groups. For compact groups the integral of the invari-ant measure over the group parameter space gives the group volume. In general, thenormalized measure gives only the functional structure of the volume element, inother words, the invariant measure up to a multiplicative constant.

Previously we have discussed the symmetric space K [5]. The center of G, whichis not discrete, contains a generating element k5 whose square is -1. We shall desig-nate by 2J the restriction of ad(k5) to the tangent space TKk. This space, has for basethe 8 matrices ka, kbk5, and it is the proper subspace corresponding to the eigenvalue -1 ofthe operator J2, or,

( ) , ,

J x y x y

x y

l l l ll l l l

l ll l

k k k k k k k k

k k k

é ùé ù+ = + =ê úê úë ûë û- -

2 15 5 5 54

5 . (19.2.2)

The endomorphism J defines an almost complex structure over K. In addition, us-ing the Cartan-Killing metric, in the Clifford representation we have shown that thecomplex structure preserves the pseudoriemannian (minkowskian) metric. Furthermorethe torsion S vanishes,

( ) [ ] [ ] [ ] [ ], , , , ,S a b a b J Ja b J a Jb Ja Jb= + + - = 0 . (19.2.3)

In this form, the conditions for J to be an integrable complex structure, invariant by G aremet and the space K is a nonhermitian complex symmetric space.

19.2.2. Realization of the Symmetric Space K as aUnit Polydisc.

The bilinear complex metric on K is invariant under SO(4,) and does not have a definitesignature. Using Weyl’s unitary trick on the complex minkowskian coordinates of the sym-metric space K, with components xl, yl, we find that its complex structure is related to thecomplex structure of R. The generators of the coset K are 2 compact and 6 noncompactinstead of the 8 noncompact generators of the coset R. Both cosets have the matricialstructure [6],

*

*

x y

x y

x yK

x y

x y

x y

é ùé ùé ùê úê úê úê úê úê úê úê úê úê úê úê úê úê úê ú= ê úê úê ú ê úê úë û ë ûê úé ùê úé ùê úê úê úê úê úê úë û ë ûë û

0 0

1 1

2 2

3 3

4 4

5 5

. (19.2.4)

where the lower right submatrix is


x x x yx y

y x y yx y

é ù é ù+ · ·ê ú ê ú=ê ú ê ú· + ·ë ûë û

14 4 2

5 5

11

. (19.2.5)

The conditions imposed by the associated groups SL(2,) and SO(4,2) over thecorresponding coordinates on these spaces, expressed by the scalar product in thissubmatrix, are related respectively by the minkowskian and euclidian metrics.

Define the six complex coordinates ta on R that relate this space to the complexspace C6, where R is immersed, that is,

a a at x iy a= + £ £0 5 . (19.2.6)

These coordinates may be expressed in terms of the four corresponding coordinatesua on K by recognizing the scalar product on eq. (19.2.5),

( ) t t u u Im n m nmn mn m nd h k k m«- = £ £0 3 , (19.2.7)

and if we introduce new coordinates t on K,m mt u= , (19.2.8)

t i u=0 0 , (19.2.9)we find the same conditions on the coordinates t on K that exist on the coordinates ton R.

The conditions over the coordinates t4 and t5 allow us to reduce to C5 the complexspace where the realization of K is immersed. If we introduce the four complex pro-jective coordinates zm, we obtain the realization,

tz

t it

mm m= £ £

-4 5 0 3 . (19.2.10)

If we indicate the transposed by z’ the conditions on these coordinates are those ofthe unit polydisc, D4ÌD5, defined by

( ) { }; , n nD K z C zz zz zz¢ ¢ ¢= Î + - > <21 2 0 1 . (19.2.11)

In this manner, the complex coordinates define a holomorphic diffeomorphism h ofK onto the interior of a bounded symmetric domain D. The bounded realization of thespace K is the unit polydisc D4(K) [6]. This realization D4(K) corresponds to thebounded realization of the space R, the unit polydisc D4(R), by a change in coordi-nates. These octadimensional spaces are part of a series of real forms of symmetricspaces which may be indicated by 8Sn,4-n and are determined by the compact groupwhich characterizes the series. In our case the group is SU(4) and the spaces areshown in the equations 4.9 y 4.10 in appendix B. The polydisc D4 provides realiza-tions of each member of this series according to the complex coordinates used.Although the interior of D4 is not compact we can apply the existing mathematicaltechniques of the classical bounded domains to study the spaces 8Sn,4-n. In particularwe can find normalized invariant measures for the spaces K and R.


19.2.3. Invariant Measure on the Polydisc.In order to construct these measures it is convenient to define certain domains [7 ]

related to Dn. Silov’s boundary, the generalization of the circle as the boundary of the 1dimensional complex disk, is established by the Fourier transformation on the symmetricspace Dn. It is the characteristic space of Dn, in other words it allows us to characterize theholomorphic functions on Dn by their values on this boundary. It is defined by

( ) { }; , , n i nQ K xe x R xxqx q p¢= = Î = £ £1 0 . (19.2.12)

Poisson’s kernel Pn(z,x) over Dn´ Qn is defined as the euclidian invariant measure on thecharacteristic space Qn. This kernel has the value

( )( )

( ) ( )( ),

n

n nn

zz zzP z

V Q z z

xx x

¢ ¢+ -=

¢´ - -

2 21 2 , (19.2.13)

determined by Hua [7]. The actual construction of the measure over Q4, due to Wyler, isindicated in the next section. The expression for the harmonic functions over Dn is

( ) ( ) ( ),n

n

Q

z P z dj x j x x= ò , (19.2.14)

which, for the case of the disc, reduces to a solution the Dirichlet problem using Poisson’sintegral formula which gives the harmonic function knowing its value on a circle bound-ary,

( ) ( ) ( ),z P z dp

j q j q q= ò2

0

. (19.2.15)

A kahlerian structure over Dn is defined by Bergman’s kernel which has the following propertyover the harmonic functions,

( ) ( ) ( ),n

n

D

z B z w w dwj j= ò . (19.2.16)

The Poisson and Bergman kernels define normalized forms over Qn and Dn respectivelyif the harmonic function is the unit function.

The invariant measure pulled back from the complex bounded domains toward eachone of these symmetric spaces in the series have a common numeric coefficient becausethey are the different real forms of the same complexified group and the use of normalizedmeasures. There are tetradimensional subspaces 4Sn,4-1 which correspond to symmetricspaces whose tangent spaces at one point are copies of the original orthogonal spaceRn,4-n. These spaces are the real forms determined by the compact subgroup USp(4) andits compact symmetric subspace S4 which characterizes the series, shown in the equa-tions 4.11 and 4.12 in appendix B. The polydisc D4 realizes each one of the 4Sn,4-n insubspaces according to the complex coordinates used in the group parametrization, re-


lated by the Weyl unitary trick. In order to pull back an invariant measure from the com-plex bounded domains it is necessary to embed topologically all the 4Sn,4-n in some Silovboundary Qn. It is clear from eq. 19.2.12 that the necessary topological embedding of S4,which characterizes the series, is not possible in Q4. It is necessary to embed in Q5 whichis topologically adequate. Furthermore since the realizations of all these spaces are de-fined in C5 the measure on D4 is obtained from the kahlerian measure on D5, in the samemanner as the measure on the sphere Sn is obtained from the euclidian measure en Rn+1.This determines the use of groups of higher dimensions. In order to evaluate this mea-sure, it is necessary to use the immersion, i:D4D5 defined on the intersection of the D5

and the plane z5=0. Since D5 is a homogeneous space under the action of the compactgroup SO(5)SO(2), using this group and SO(5,2) we may obtain the measure on thequotient R [6 ], which is equal the measure on K. There are injective mappings,

h iM K D Dk¾¾ ¾¾ ¾¾4 8 4 5 , (19.2.17)which correspond to Clifford’s mapping k and the holomorphic mapping h. The topologi-cal embedding of M in Q5 may be expressed by,

( )( )i h M Qbk ¾¾4 5 . (19.2.18)

The indicated mappings i and b allow us to induce a pullback form from the characteristicspace Q5, boundary of D5, to D4, that we shall indicate as follows,

( ) ( ) ( )4 4 4 5Q Q Qi D i h M i h M Qb m b m k m b k** * é ù é ù é ù= º Íê ú ê ú ê úë û ë û ë û . (19.2.19)

19.3. Wyler’s Measure on the K Space.In principle we may obtain the common coefficient of the measures of the symmetric

spaces in the series using any corresponding group in the series. The more transparentcalculation is using the group SO(5,2) as Wyler did. In what follows, we indicate thiscalculation of the value of the constant coefficient of the measure on Q5, Silov’s boundaryof D5. This measure is obtained by constructing Poisson’s measure invariant under gen-eral complex coordinate transformations on D5 by the group SO(5,2) of analytic mappingsof D5 onto itself.

The calculation is based on the following proposition: The isotropy subgroup at theorigin, SO(5)SO(2), acts transitively over Q5 and Poisson’s kernel Pn(z,x), harmonic onDn, represents an invariant measure of this action of SO(5)SO(2) over Q5. Poisson’skernel, applied to a volume element, defines a normalized invariant form over the charac-teristic space,

( )( )( ) ( ),

,5

055 5 5

QCQ

N

Qj z dQ P z d

V Q V Q

mx xm x x

é ùê úë ûé ù = = =ê úë û , (19.3.1)

in terms of jC0, which indicates the complex jacobian or determinant of the jacobian matrixJC0 of the transformation zG(z) of the group SO(5,2) which carries the point z into theorigin. The form mQ represents an invariant not normalized form over Q5 defined by theseequations. The kahlerian structure of the complex bounded domains (polydiscs) Dn is de-termined by the geometric structure defined by the Bergman kernel and the metric h it


defines [8]. This kernel is determined by any orthonormal base fn of the Hilbert subspaceformed by the holomorphic functions over the polydisc Dn. Since the solutions of theinvariant differential operators, obtained by their inverse operators or Green’s functions,are holomorphic harmonic functions over Dn we should take as invariant measure the onedefined by this kahlerian structure of Dn [9]. The group SO(5,2), of coordinate transfor-mations, acts on D5 and consequently on Q5. The Poisson kernel, although invariant underthe isotropy subgroup SO(5)SO(2) at the origin, is not invariant under the full groupSO(5,2). An invariant measure under the latter group G requires a volume density on the cosetG/GI,, to be incorporated as an additional factor in mN.

Hua established a relation between the Bergman kernel and the volume density overthe domain D5. The Bergman kernel may be written as

( ) ( )( )

( )det C

n

J zB

V DV D zz zz= =

¢ ¢´ + -

2

5 525

1

1 2 . (19.3.2)

The complex Bergman metric hC is defined by the invariant bilinear form

( ) ( )( )

C CdzJ z J z dzds

V D

¢ ¢=2

5 . (19.3.3)

In a natural manner the Bergman kernel provides a proportional volume density r for thedomains D5, as indicated by Hua,

( ) ( )det CV D B J zr = =25

5 . (19.3.4)

The real part of the kahlerian Bergman metric induces a real riemannian metric hR whichshould be used when restricting the complex coordinates to the real coordinates of Silov’sboundary Q5. We want to pass from the boundary Q5 to the interior of D5 when replacing thereal coordinates x by the complex coordinates z. We consider the effect on Q5 of the trans-formation zG(z), where G is the complete SO(5,2). A factor equal to the determinant of alocal real euclidian metric at the origin, which is a tensor density of weight -2, is trans-formed by the real jacobian jR related to the complex jacobian jC of the transformation,

( ) ( ) ( ) ( )det det det2 4 41R R R C C Rh j h j j h¢ ¢= = = . (19.3.5)

The jacobian jC, which includes transformations by the isotropy subgroup at the origin andits corresponding coset G/GI, should be expressed using a coset decomposition of thejacobian in two factors corresponding respectively to the subgroup and the coset. TheBergman kernel is proportional to the determinant of the Bergman metric hC [8, 9]. In order toeliminate this constant proportionality factor V(D5), which appears in the determinant of theBergman metric h, it is necessary to make a transformation that affects this determinant by theaction of the inverse jacobian. This allows us to evaluate the geometric coefficient jg intro-duced by this transformation in the invariant measure,

( )4 4 4 4 4 5 40 0 0C C C g C Cj j j j j V D j-¢= = = . (19.3.6)

The invariant measure under the full SO(5,2) group is obtained by the application of jC


to the measure in eq. (19.3.1). The jC0 jacobians combine within the invariant form m toobtain

( )( )( )

( )( )( )

,

145

5 5 50 05 5

gQ Qg C N C

V Djj j Q j z d Q

V Q V Qm z z mé ù é ù¢ ¢¢= =ê ú ê úë û ë û . (19.3.7)

To obtain the invariant measure, on D4, it is necessary to reduce the action of theisotropy group I(5,2) to the isotropy subgroup I(4,2). This allows us to define a normalizedinvariant measure on the polydisc D4,

( )( )

( )( )( )

( )( )

, ,, ,

145

5

4 2 4 25 2 5 2

Q Q QNg N g

V DV I V Ii i i i

V I V IV Qb m b m b m a b m* * * * * * * *

é ù é ùë û ë û¢= = =é ù é ùë û ë û

.(19.3.8)

The inverse of the measure of the isotropy groups quotient is

( ) ( )( ) ( )

( )( )

( )( )

V SO SO V SOV S

V SO SO V SOp p

G

é ù é ù´ë û ë û= = = =é ù é ù´ë û ë û

5 3 224

52

5 2 5 2 234 2 4

. (19.3.9)

Under this reduction the coefficient of Poisson’s kernel over Q5, the constant coeffi-cient of the normalized measure mNg in eq. (19.3.8), defines the coefficient of the measureover D4,

( )( )( )

( )( )

( )( ) ( )( )( ) ( )( )

,,g

V D V D V SOV I

V IV Q V Q V SOa

é ù ´ë û= =é ù ´ë û

1 14 45 5

5 5

44 25 2 5

. (19.3.10)

The indicated volumes are known. The volume of the polydisc is

( )!

nn

nV Dn

p-= 12

(19.3.11)

and the volume of Silov’s boundary is the inverse of the coefficient in the Poisson kernel,

( )( )

n

n

nV Q

pG

+

=2 1

2

2 . (19.3.12)

In particular we have,

( )!

V Dp

=´

55

42 5 , (19.3.13)

( )V Qp

=3 3

5 23

. (19.3.14)

Substitution of theses expressions in equation (19.3.10) gives Wyler’s coefficient of the


induced invariant measure,

!g Qp

ap

æ ö÷çé ù ÷= çê ú ÷ë û ç ÷ç ´è ø

142 5

46 5 4

32 2 5

. (19.3.15)

19.4. Value of the Geometric Coefficient.Equation (19.3.8) provides a geometric normalized invariant measure mNg on the sym-

metric space K and allows us to induce a similar pullback measure on space-time M

[ ] [ ] ( ) [ ]Q QNg Ng g gM h i M h i M Mm k b m k a b m k a m k* * * * * * é ùº = ºë û . (19.4.1)

The use of a normalized measure makes sense and is necessary on a noncompact sub-space of M. In order to calculate the coefficient of the invariant measure on this noncom-pact space we must restrict the measure mNg to this subspace by integrating over a comple-mentary compact subspace. The latter space should be the characteristic compact sub-space of the harmonic functions on M. There is a contribution from the integration on thisspace to the value of the geometric coefficient on the complementary noncompact sub-space. Since mNg is defined on the cotangent space at a point *TMm and only depends onthe properties of the symmetric fiber K at m, this contribution should be independent ofany solution of the field equation. Therefore, without loss of generality we may assumespherical symmetry and staticity conditions which allow a decomposition of space-timeM4 and its forms in two orthogonal subspaces, spatial compact spheres S2 and the comple-mentary noncompact space-time M2 with their forms ,

( ) ( ) ( )( )Ng gM M Sm k a m k m ké ù é ù é ù¢= ê ú ê ú ê úë û ë û ë û4 2 2 2 2

. (19.4.2)

The space M2 locally has the null-cone structure of a relativistic bidimesional Minkowskispace.

In order to find the complete coefficient we are free to restrict the problem to thespecial particular case of pure electromagnetism on a flat space-time M. All solutions of thisrestricted problem may be found as a sum of fundamental solutions which correspond tothe Green’s function for the electromagnetic field. The Green’s function determines thefield of a point source which always corresponds to a spherically symmetric static fieldrelative to an observer at rest with the source. The spherically symmetric harmonic po-tential solutions are determined, using Poisson’s integral formula, by its value on a bound-ary sphere. We see that the characteristic boundary space, where integration should beperformed, is the sphere S2 in R3. Geometrically, in accordance with our theory, the formmNg

should be integrated over the image of a sphere, determined by the Clifford mappingk, on the space K. Therefore, the characteristic space is k(S2). Since S2 is compact themeasure provided by the volume element m does not require normalization. The measuremNg

integrated over the characteristic space gives,

Ng g g g

S S Sk k

m a m m a m k m pa m*¢ ¢= = =ò ò ò2 2 2

2 2 2 22 8 . (19.4.3)

The volume of k(S2), indicated in the previous equation, is twice the volume of S2 due to


the 2-1 homomorphism between standard spinors and vectors determined by its homo-morphic groups SU(2) and SO(3).

After the integration we obtain a geometric measure restricted to the M2 subspace,

( ) ( ) ( )2 2 2 2 2 2 2 28Ng gM M M Mm k pa m k a m k a k m*é ù é ù é ù é ù= º = ê úê ú ê ú ê ú ë ûë û ë û ë û . (19.4.4)

Hence the complete geometric coefficient a is obtained substituting, in the last equa-tion, the value of Wyler’s coefficient calculated in the previous section,

! . e

p p pa a

p p

æ ö æ ö÷ç ÷ç÷= = = »÷ç ç÷ ÷ç ç÷ç è ø´è ø

1 13 2 5 4 4

6 5 4 3

2 3 9 12 2 5 16 120 137 03608245

, (19.4.5)

which is equal to the experimental physical value of the alpha constant ae.In a natural manner we may obtain a geometric distance element, a 1-form mg, using

the interior product of the normalized geometric 2-form 2mNg by the 4-velocity unit vectoru of the physical current J,

g NgM M u M u Mm k m ak m a m* *é ùé ù é ù é ùº = ºê ú ê ú ê úê úë û ë û ë ûë û21 1 2 2 2 1 1

. (19.4.6)

If we take 1m as a radial differential dr and use it to construct an euclidian volume elementd3x for R3 orthogonal to u. We have a geometric not normalized 3-volume form wg,

[ ]3 1 2 3 2 2g M d d x drr dw a m W a a W= = = . (19.4.7)

The physical current 3-form *J, dual to the current along u, defines a geometric current *Jg,

gJ Ja* *º . (19.4.8)

and the field equation may be written as follows,

gD J JW p pa* * *= º4 4 . (19.4.9)

In chapter 12 we showed that the substratum connection is proportional to the current1-form. Therefore, the geometric current 3-form that determines the substratum connectionalso defines a natural substratum 4-form,

s g g g gJ m J Jw * *¢ = . (19.4.10)

19.5. Summary.The geometric coupling constant a of the unified theory is the coefficient of the invari-

ant measure obtained from the related complex domains and may be calculated from thevolumes of certain symmetric spaces related to the structure group of the theory and itssubgroups. The numerical value obtained equals the physical value of the fine structurealpha constant. There is no additional arbitrary constant in the theory.

This invariant measure essentially determines that the material source of the field equa-tion is represented by a geometric current. If we assume the geometric current we canobtain back the invariant measure.


References

1 A. Wyler, Acad, Sci. Paris, Comtes Rendus, 269A, 743 (1969).2 A. Wyler, Acad, Sci. Paris, Comtes Rendus, 271A, 180 (1971).3 G. González-Martín, Gen. Rel. Grav. 26, 1177 (1994). See chapter 9.4 G. González-Martín, ArXiv physics/0009051, USB Report SB/F/277-00, (2000).5 See section 13.3.2.6 R. Gilmore, Lie Groups, Lie Algebras and some of their applications (John Wiley and

Sons, New York), ch. 9 (1974). 7 L. K. Hua, Harmonic Analysis in the Classical Domains (Science Press, Peking) (1958),

translated by l Ebner, A Korányi (American Mathematical Soc., Providence) (1963). 8 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

p. 298 (1962). 9 A. Wyler, The Complex Light Cone, Symmetric Space of the Conformal Group, Report,

(The Institute for Advance Study, Princeton) p. 15 (1972).

A. GEOMETRIC ALGEBRA.

A.1. Introduction.We present here certain necessary notions of Clifford algebras and spinors. The treat-

ment is not complete, but serves to establish concepts and the notation used in the book. Ageneral knowledge of algebra and orthonormal spaces is assumed. For more details seethe references [1, 2, 3, 4].

A.1.1. Clifford Algebras and Spinors.For any finite n dimensional real orthogonal space X=Rp,q of signature (p,q), there is a

real associative algebra, A=Rp,q, with unit I, containing isomorphic copies of R and X aslinear subspaces in such a way that, for all x in X,

x x x= -2 . (1.1)If the algebra is generated as a ring by the copies of R and X, or equivalently, as a real

algebra by {I} and X then A is said to be a geometric (Clifford) algebra of X. The Cliffordalgebras of maximum dimension 2n are unique up to isomorphism and called universalClifford algebras.

More strictly there is a linear injection k such that for all x

( )( ) ( ),x x x I g x x Ik = - = -2

. (1.2)

It may be proved that if ei is an orthonormal base in X, its image in A satisfies

( )i i ie e e= - =2 1 , (1.3)

( )( )i j i j j ie e e e e e= + =12 0 . (1.4)

The involution of A induced by the orthogonal involution -1X will be denoted by a andcalled the main involution. The antiinvolutions of A induced by the orthonormal involu-tions 1X and -1X will be denoted by a and a called respectively reversion and conjuga-tion.

A Clifford algebra may be considered to act on a spinor space. In fact, each universalClifford algebra Rp,q is the endomorphism algebra of a right Q-linear space V called thespinor space of the orthogonal space Rp,q

A right linear space over a noncommutative ring Q consists of an additive group and amap

X Q X´ , (1.5)

( ),x q xq x q¾¾ = , (1.6)

such that the usual distributivity and unity axioms hold and satisfy

237Geometric Algebra

( )x q q x qq¢ ¢= . (1.7)

Similarly a left linear space satisfies

( )q q x q q x¢ ¢= . (1.8)

The spinor space does not properly have a metric structure. A correlation x on a Q-linear space is a right to left Q-semilinear map to its dual space,

: LX Xx ¾¾ , (1.9)

( ) ( )x q q xfx x= , (1.10)

( )x x xx = . (1.11)

Any correlation on the spinor space induces an antiinvolution of the endomorphismalgebra, namely the corresponding adjoint antiinvolution. Consider the spaces X, Y, finitedimensional right Q-linear spaces. Given a Q-linear map,

:t X Y¾¾ , (1.12)we define the adjoint map, with respect to correlations x in X and h in Y, as the unique t*,

:t Y X* ¾¾ , (1.13)which for all x, y, satisfies

( ) ( )y tx t y xxh *= . (1.14)

A map on the spinor space does not necessarily preserves a correlation. A correlatedautomorphism t is a right Q-linear map such that for all a,bX,

( )ta tb a bh x= , (1.15)

which may be shown to be equivalent to the condition on t

t t* = 1 . (1.16)The following theorem may be proved. Let V be the spinor space for the orthogonal

space Rp,q, with Rp,q the endomorphism algebra. Then if p>0, (p,q)(1,0), the conjugationantiinvolution in Rp,q coincides with the adjoint antiinvolution of End(V) induced by aneutral semilinear correlation on V. This theorem indicates that there is a correlation in thespinor space associated to the conjugation in A.

The geometric algebra for the real field, as an orthonormal space R0,1 with a metric ofpositive definite signature, is the complex field , regarded as a real algebra. The quater-nion field , also regarded as a real algebra, is a geometric algebra both for R0,2 and R0,3.The absolute value of a quaternion product obeys the relation

ab abab baab bbaa a b= = = =2 2 2 , (1.17)

Appendix A PHYSICAL GEOMETRY238

ab a b= . (1.18)

The universal Clifford algebras may be constructed by induction. The step in the in-duction process is given by the following theorem. Let X be a -linear space, where is or 2 and is , or and let h={ei} be an orthonormal subset of End(X) of type(p,q), generating End(X) as a real algebra. Then an orthonormal subset h’ of type (p+1,q+1)generating End(X2) is given by the following matrices,

, ,i

i

e eeh

e ee

ì üé ù é ùé ù -ï ïï ïê ú ê úê ú¢ = í ýê ú ê úê úï ï-ë û ë û ë ûï ïî þ

0 0

0 0

0 000 00

, (1.19)

where e0 is the identity in . It is understood the use of the standard identification of theeven part + with , + with , e0 with 1 and exp(ip) with -1.

A.2. A Representation for the Algebra A.In particular the orthonormal subset for the algebra R3,1 is obtained from the orthonor-

mal subset for the algebra R2,0. The latter corresponds to the bidimensional orthogonalspace with signature (-1,-1). The two real Pauli matrices satisfy the properties of the or-thonormal subset for this algebra. The Clifford algebra for the orthonormal space of oppo-site signature (1,1) is the quaternion algebra , generated by its orthonormal subset, thequaternions j, k which may be represented by the imaginary matrices is1 and is3. Introducethe real anticommuting matrices,

ré ùê ú= ê úë û

0 11 0

, (2.1))

sé ùê ú= ê ú-ë û

1 00 1

. (2.2))

These anticommuting elements form an orthonormal subset which may also be designatedby,

{ },ir r s= , (2.3)

and generates two additional matrices

eé ùê ú= ê ú-ë û

0 11 0

, (2.4)

eé ùê ú= ê úë û

1 00 1

. (2.5)

which together form a base for the real 2´2 matrices,


[ ] [ ]er s e r r r r r r= 1 2 1 2 1 1 . (2.6)

The R(2) matrices form the noncommutative ring of 2-dimensional real matri-ces, which is the R2,0 Clifford algebra. These matrices are similar to the quater-nions but they are not isomorphic algebras. The R(2) ring is not a division ring asthe quaternion ring and their products are not equivalent. It is associated with theorthogonal space R2,0 instead of the R0,2 associated to the quaternions. Neverthe-less, their even parts are isomorphic and may be identified with the complex num-bers . Because of the similarities we shall designate this nondivision ring as thepseudoquaternions .

It follows that the algebra R3,1 is obtained from R2,0 by setting

, , ,e e

he e

r sr s

ì üé ù é ù é ù é ù-ï ïï ïê ú ê ú ê ú ê ú¢ = í ýê ú ê ú ê ú ê úï ï- -ë û ë û ë û ë ûï ïî þ

0 0 0 00 0 0 0

. (2.7)

This subset generates the algebra R3,1 that we shall designate as A. It is convenientto use another equivalent orthonormal subset ka defined using only the odd matri-ces ri.

, , ,r r s r

kr r s r

ì üé ù é ù é ù é ù-ï ïï ïê ú ê ú ê ú ê ú= í ýê ú ê ú ê ú ê úï ï-ë û ë û ë û ë ûï ïî þ

0 0 0 00 0 0 0

. (2.8)

From the geometric meaning of the algebra, the elements ka are associated totriads of opposite orientation. In principle, both subsets with opposite signs may beused as part of the orthonormal subset ka of the algebra. The arbitrary sign is deter-mined by a standard relation of products of Pauli matrices, in terms of Hodge dualitye ijk and Clifford (complex) duality i. We should adhere to the same convention in thechoice of sign for the ka . The standard orientation in space-time [u0,u1,u2,u3] inducesan orientation in the geometric algebra which should be used to define the Cliffordgeometric duality k0k1k2k3. This tetradimensional duality operation denoted by k5,and Hodge duality should relate the matrices representing the Pauli matrices withinthe algebra, preserving the standard relation. Mathematically,

ik k k k k s s s s k k k k k k= º = =0 1 2 3 5 0 1 2 3 0 1 0 2 0 3 , (2.9)

i is k k= 0 . (2.10)

The elements of the algebra may be expressed as 2´2 matrices over the ring. Explic-itly in this representation, the orthonormal subset of the algebra A is, up to equivalenceunder automorphisms,

rk

r

é ùê úê úé ùê úê ú= = ê úê ú- -ë û ê úê ú-ë û

0

0 0 0 10 0 0 1 0

0 0 1 0 01 0 0 0

, (2.11)


0 r

kr

é ù-ê úê úé ù- -ê úê ú= = ê úê úë û ê úê úë û

1

0 1 0 00 1 0 0 0

0 0 0 10 0 1 0

, (2.12

sk

s

é ùê úê úé ù -ê úê ú= = ê úê úë û ê úê ú-ë û

2

1 0 0 00 0 1 0 0

0 0 0 1 00 0 0 1

, (2.13

rk

r

é ùê úê úé ùê úê ú= = ê úê úë û ê úê úë û

3

0 0 0 10 0 0 1 0

0 0 1 0 01 0 0 0

. (2.14)

This orthonormal subset km generates the rest of the base,

I

Ik k s

é ùê úê ú é ùê ú ê ú= = =ê ú ê úë ûê úê úë û

0 1 1

0 0 1 00 0 0 1 01 0 0 0 00 1 0 0

, (2.15)

ek k s

e

é ù-ê úê ú é ù-ê ú ê ú= = =ê ú ê úë ûê úê ú-ë û

0 2 2

0 0 0 10 0 1 0 00 1 0 0 01 0 0 0

, (2.16)

I

Ik k s

é ùê úê ú é ùê ú ê ú= = =ê ú ê ú- -ë ûê úê ú-ë û

0 3 3

1 0 0 00 1 0 0 00 0 1 0 00 0 0 1

, (2.17)


ie

k k se

é ùê úê ú é ù-ê ú ê ú= = =ê ú ê ú- -ë ûê úê úë û

1 2 3

0 1 0 01 0 0 0 00 0 0 1 00 0 1 0

, (2.18)

ie

k k se

é ùê úê ú é ù-ê ú ê ú= = =ê ú ê úë ûê úê ú-ë û

2 3 1

0 0 0 10 0 1 0 00 1 0 0 01 0 0 0

, (2.19)

Ii

Ik k s

é ùê úê ú é ùê ú ê ú= = =ê ú ê ú- -ë ûê úê ú-ë û

3 1 2

0 0 1 00 0 0 1 01 0 0 0 00 1 0 0

, (2.20)

sk k k

s

é ùê úê ú é ù-ê ú ê ú= =ê ú ê ú- -ë ûê úê úë û

1 2 3

0 0 1 00 0 0 1 01 0 0 0 0

0 1 0 0

, (2.21)

sk k k

s

é ù-ê úê ú é ù-ê ú ê ú= =ê ú ê úë ûê úê ú-ë û

0 2 3

1 0 0 00 1 0 0 00 0 1 0 00 0 0 1

, (2.22)

rk k k

r

é ùê úê ú é ùê ú ê ú= =ê ú ê úë ûê úê úë û

0 3 1

0 1 0 01 0 0 0 00 0 0 1 00 0 1 0

, (2.23)


sk k k

s

é ùê úê ú é ù-ê ú ê ú= =ê ú ê úë ûê úê ú-ë û

0 1 2

0 0 1 00 0 0 1 01 0 0 0 00 1 0 0

, (2.24)

Ii

I

ek k k k

e

é ùê úê ú é ù é ù-ê ú ê ú ê ú= = =ê ú ê ú ê úë û ë ûê úê ú-ë û

0 1 2 3

0 1 0 01 0 0 0 0 00 0 0 1 0 00 0 1 0

. (2.25)

The orthonormal subset km of the algebra R3,1 obeys the relation

( ) I e Ia b ab abk k h h= - » -2 4 , (2.26 )

where the flat metric is taken with the timelike signature (1, -1, -1 -1). This algebra is notisomorphic to Dirac’s algebra R1,3, which obeys also the last equation but with the metrictaken with the spacelike signature (-1, 1, 1, 1) and is generated by Dirac’s g matrices.Nevertheless their even subalgebras are isomorphic.

For any element a of the algebra A, the conjugate antiinvolution is

† †a ak k= 0 0 (2.27)

which corresponds to Dirac’s conjugation. The antiinvolution † is the transpose of the real 4x 4 matrices. The main involution,

†a ak k= 5 5 , (2.28)

induces a direct sum decomposition of A into its even subalgebra A+ and the complemen-tary odd part A-. Associated to the eigenvalues of the main involution, we may introduce aquantum number called rarity r, defined by the eigenvalues of the odd projection operator,

( ) ˆrar a aa -=

2 . (2.29)

The algebra A acts on a linear spinor space V. Each element of A is a 2´2 matrix over thepseudoquaternion ring . Let V be a 2 dimensional module over , the spinor space of R3,1.The elements of V may be represented by 2-columns 4-rows matrices

r r

r rk

k r r

r r

é ùé ùê úê úê úê úé ù ë ûê úê ú ê úê ú é ùê úë û ê úê úê úê úë ûë û

11 21

12 221

2 13 23

14 24

. (2.30)


A correlation in the spinor space V is induced by conjugation in the algebra. For anyspinor v of V,

† †v v k= 0 . (2.31)

It is possible to give another representation to this geometric algebra A. Define thefields and of matrices isomorphic to the quaternion field, with respective bases,

{ }, , ,Ial k k k k k k= 1 2 2 3 3 1 , (2.32)

{ }, , ,q Ia k k k k k k k k= 0 1 2 3 0 1 2 3 . (2.33)

The fields and commute. We may define the tensor product which has for base

E qab a bl= Ä . (2.34)

The set satisfies the postulates to form a ring which we shall call the biquater-nions with product and addition defined by

q q qql l ll¢ ¢ ¢ ¢Ä ´ Ä º Ä , (2.35)

( ) ( )q q q ql l l l¢ ¢ ¢ ¢Ä + Ä º + Ä + . (2.36)

The orthonormal subset ka may be expressed in terms of the ring

{ }, , ,q q q qak l l l l= 1 0 2 1 2 2 2 3 . (2.37)

This set of tensor product of matrices is homomorphic to the set of 44 matrices generatedby the orthonormal subset with the standard matrix product. The homomorphism is 2 to 1due to the direct product in the definition. The base E spans the geometric Clifford algebraA.

A.3. Correlated Space of the Group G.The conjugation in V does not give an invariant product. Nevertheless, there is an

invariant product related to V preserved by the group of correlated automorphisms. Thegroup SL(4, ) is known to preserve the correlation [1] of the space (hbR)4, which hasdimensionality twice that of V. In the following sections the relation among these spacesand the usual Weyl and Dirac spinors is discussed. The invariant product and the corre-lated space S under correlated automorphisms SL(4, ) may be obtained as indicated byPorteus [1]. There is an alternate way to obtain it from the group and its inverse as fol-lows.

Associated to R3,1 we consider a spinor of the first kind v which transforms under

automorphisms as gSL(4,) and a spinor of the second kind w which transforms as -1gand form the space S with 8 dimensional real vectors of the form

; v

v v w ww

y -é ùê ú ¢ ¢= = =ê úë û

1g g . (3.1)


Now, associated to a spinor base (frame) e, introduce a correlation e induced from thealgebra conjugation e by the hyperbolic rule

[ ]w vy = , (3.2)

which may also be written in terms of 8´8 real matrices indicated by diagonal 2´2 ma-trices with 4´4 submatrices,

†I v

I w

ky

k

æ öé ù é ù é ù÷çê ú ê ú ê ú÷=ç ÷çê ú ê ú ê ú÷çè øë û ë ûë û

0

0

0 00 0

. (3.3)

It is clear that frame e and e 1 are of opposite kind and together form a base in S.Let us represent the automorphism of S, induced by the action of the frame element g,

defining a mapping r to diagonal 2´2 matrices with 4´4 submatrices of the form

:G Gr 2 , (3.4)

( ) Gr -

é ùê ú= = Îê úë û

2 21

g 0g g

0 g . (3.5)

The map r induces its derivative map at the identity. It should be clear that there is aninduced correlation in the algebra which for any a in the algebra sl(4,) gives the relation

:I A Ar* 2 , (3.6)

( ) ( ) ,I

aa a a A sl

ar*

é ùê ú= = Î ºê ú-ë û

2 04

0 , (3.7)

where 2A represents diagonal 2´2 matrices with 4´4 submatrices determined by an elementa of the algebra.

Using the orthonormal subset km we may define four matrices 2k which belong to 2A by

mm

m

kk

k

é ùê úº ê ú-ë û

2 00 (3.8)

and a matrix g,

I

Ig

é ùê ú= ê úë û

00

. (3.9)

Then we may define 2g as follows:

( ) ( )~ † †y y k g y= =2 2 2 20g g g , (3.10)

where


† † Gg k k g-

-é ùê ú= = = Îê úë û

12 2 2 2 2 1 2

0 0g 0

g g g0 g

. (3.11)

Since 2g is the inverse of 2g, we have that the correlation in S is preserved by the corre-

lated automorphism 2g of the space S.The operation ~ may be transported to 2A. If we indicate the elements of 2G and 2A by

its component elements 2g and 2a, we have

( ) ( )exp expa a-= = = -2 2 2 1 2g g , (3.12)

~a a

a aa a

é ù é ù-ê ú ê ú= = = -ê ú ê ú-ë û ë û

2 20 00 0

, (3.13)

which has an inverse.The Clifford product in A induces a product in 2A, indicated by ,

a bA

a b

é ù é ùê ú ê ú ê ú ê ú- -ë û ë û

20 00 0

, (3.14)

by means of the expression,

( ) ( ){ } ,I I Ia b a b a b Ar r r- -* * *º Î2 2 1 2 1 2 2 2 2 , (3.15)

which explicitly gives

a b ab

a b ba

é ù é ù é ùê ú ê ú ê ú=ê ú ê ú ê ú- - -ë û ë û ë û

0 0 00 0 0

. (3.16)

If we represent the matrices 2aÎ2A by their A component a, the product may be indi-cated by the Clifford product in A,

a b a b«2 2 . (3.17)In a strict manner we should have a notation which distinguishes between the ele-

ments associated to S and those associated to V. Nevertheless, it may be convenient toindicate corresponding elements associated to both spaces with the same symbol, hop-ing that it can be determined from the context whether the element is associated to S orto V. This convention is used in some places throughout the book. Practically this meansthat all calculations may be made using elements a Î A. At the end, the other componentin 2A may be obtained by conjugation:

aba b

ab

é ùê ú ê ú-ë û

0

0 , (3.18)

which explicitly agrees with


aba b

ba


2 2 00

. (3.19)

A.4. Relation of R3,1 with R2,0.It is of interest to discuss the relation among the corresponding involutions and

antiinvolutions of the algebras R3,1 and R2,0. The latter, as indicated before, is generated bythe elements r1 (equal to r) and r2 (equal to s) which obey the relations

( )ir =2 1 . (4.1)

r r r r e= - = -1 2 2 1 , (4.2)

e = -2 1 . (4.3)The transposition operation on these matrices gives the reversion antiinvolution of the

Clifford algebra, which we shall indicate by *. The main involution of the algebra indicatedby ^ may be expressed as

a ae e*= , (4.4)and the conjugation antiinvolution is a^*.

It is clear that we have the following relations:

ˆi ir r= - , (4.5)

e e= , (4.6)showing that the even subalgebra is spanned by 1 and e. This even subalgebra is isomor-phic with the complex numbers. The reversion antiinvolution reduces to complex conjuga-tion within the even subalgebra. This is the reason for selecting the symbol * to indicatereversion in R2,0. We also have the relations

i ir r* = , (4.7)

e e* = - . (4.8)If we indicate by the ring generated by these matrices it should be clear that the

group SL(4,) is homomorphic to the group SL(2,), where each element of the 2 x 2 matrixbelongs to . The operations on the algebra may be expressed using those of as fol-lows. The conjugation is

† †a a a t tk k k k* *= =0 0 0 0 , (4.9)

where t indicates the transpose of the 2 x 2 matrices and † the transpose of the 4 x 4matrices. It should be clear that, with this notation the operation † is equivalent to theoperations *t in the even subalgebra, as it should be to agree with the complex notation forhermitian conjugation. Similarly, the main involution is

†a a a tk k k k*= =5 5 5 5 , (4.10)


ˆ ˆˆ

ˆ ˆb c b c

ad f d f

e ee e

é ùé ù é ù é ù- ê úê ú ê ú ê ú= = ê úê ú ê ú ê ú-ë û ë û ë û ê úë û

0 00 0

(4.11)

and the reversion is

ˆ ˆa a t tk k* *= 0 0 . (4.12)

The group SL(2, ) acts on a 2-dimensional space over the ring , which is the associ-ated spinor space V. An element of V may be decomposed over the complex ring,

ii

ii

a b a ba a bv

c d c dc c d

re rr

re r

é ù é ù é ù é ù++ +ê ú ê ú ê ú ê ú= = = +ê ú ê ú ê ú ê ú++ + ë û ë û ë ûë û

1 2

1 2

. (4.13)

The R3,1 spinor space V is composed of two R2,0 spinor spaces. The even part of V isisomorphic to the usual complex spinor space associated to SL(2,), the even part ofSL(2,).

A.5. Relation between Spinors of GroupsG and L.

The group SL(4,) is homomorphic to the group SL(2,). The conjugation in the Cliffordalgebra is

† †a ak k= 0 0 , (5.1)

_ † † † †

† † a b a b d b

c d c d c a

r r r r r rr r r r r r

é ùé ù é ù é ù é ù -ê úê ú ê ú ê ú ê ú= = ê úê ú ê ú ê ú ê ú- - -ë û ë û ë û ë û ë û

0 00 0

, (5.2)

and in the spinor space, for xÎV,

†

† † a

b ab

rx r r

r

æ öé ù é ù÷ç é ùê ú ê ú÷= = -ç ÷ ê úç ë ûê ú ê ú÷ç -è øë û ë û

00

. (5.3)

If we restrict to SL(2,), the even subgroup that preserves the metric, the conjugationreduces to hyperbolic interchange of complex components since the ring reduces to itscomplex subring. We have then

† † † †

† † a b I a b I d b

c d I c d I c a

- é ùé ù é ù é ù é ù -ê úê ú ê ú ê ú ê ú= = ê úê ú ê ú ê ú ê ú- - -ë û ë û ë û ë û ë û

0 00 0 , (5.4)

†

† † I a

b aI b

xæ öé ù é ù÷ç é ùê ú ê ú÷= = -ç ÷ ê úç ë ûê ú ê ú÷ç -è øë û ë û

00

. (5.5)


The 2´2 matrices a, b, c, d, may be expressed as complex numbers,

b ax * *é ù= -ê úë û , (5.6)

or in matricial form

( )†x ex= . (5.7)

In this form the spinors x are seen to correspond to Weyl 2-spinors. The operation ofmatrix multiplication may be written with indices, using e to raise and lower them,

BAB Ae x xº , (5.8)

where care must be taken in lowering and raising indices because of the antisymmetry of e.We shall adopt the convention that the sign is positive when the line joining adjacentsummed indices is /. We also indicate indices of the inequivalent complex conjugatedspinors with a dot. In this manner we write

BA BAx e x= - , (5.9)

x x= 21 , (5.10)

x x= - 12 , (5.11)

x x* *= 21

, (5.12)

x x* *= - 12

. (5.13)

We also note the following relations:

ABABe e= , (5.14)

BCABe e e= - = 21 , (5.15)

A BA C BA C AB BC Cx x e e x e x d= = = . (5.16)

The even part of the orthonormal subset k corresponds to the Pauli matrices s,

X AA X

b ba as s d= 2

, (5.17)

X B B XA AY Y

mms s d d= 2

. (5.18)

Since s represents the even part of the mapping k from Minkowski space into the algebra A,it serves as a mapping between vectors and second rank SL(2,) spinors. We may inter-change spinor and vector indices using s, as follows,

ab AWBXc CY

T T« , (5.19)


WXC W X CA BABY Y

T Tg aba b gs s s=

, (5.20)

X AAX

va ax s h =

. (5.21)

If we have a 2-component spinor over the action of the even subgroup L of thegroup G, may be expressed in terms of the even and odd parts of the original spinor,which are 2-spinors over the complex field,

( )l l l l l ln h rx h rx h r x*= + = + = + , (5.22)

lh h¢ = , (5.23)

*lx x¢ = , (5.24)

and the even and odd parts transform as inequivalent conjugate representations of L. Wemay form a four component complex spinor,

xx

yhh

é ùê úê úê ú= ê úê úê úê úë û

1

2

1

2

, (5.25)

which may be recognized as a Dirac spinor.

A.6. Even Spinor Frames and VectorFrames.

An even spin frame f is a matrix in SL(2,), composed of two spinors that satisfy

ˆ ˆˆ ˆ

C DA B AB A B A BCD

e f f e x h h x= = - , (6.1 )

which defines a vector frame q in a canonical way using the homomorphism that sendsSL(2,) to SO(3,1),

ˆ †tram a mq s f s f= 1

2 . (6.2)

The metric associated to this tetrad is

Y B A X Z D C WA X C WB Y D Z

g a bmn ab m nh f s f s f s f s= 1

4 , (6.3)

AC X YA CXY

gmn m ne e s s=

. (6.4)

The metric also satisfies,


( )detdeti

gi

aa

q q q q s qq q q q

é ù+ -ê ú == ê ú+ -ê úë û

20 3 1

1 2 0 3 , (6.5)

in terms of the hermitian matrices s. An SO(3,1) pseudorotation leaves the metric in-variant. The new hermitian matrix induced by the rotated tetrad is related to the old by atransformation l of a group L composed of complex 2´2 matrices. If the metric remainsinvariant under L,

( )†detg g l laas q¢ = = , (6.6)

the following relation holds:†det detl l = 1 , (6.7)

and

det il e a= . (6.8)The group of transformations L which preserve the metric is SL1(2,), the special lineargroup in 2 dimensions over the complex field with complex determinant of unit modu-lus.

A.7. Derivative of the Orthonormal Set.Consider the bundle AM with fiber A over space-time M. Consider a base Ea in the

algebra. There is a way to define a connection on AM so the set Ea has null covariantderivative [5 ]. There is an internal base for which all matrices Ea are equal for all points.Since the holonomy group is the same for all points, one can, for a fixed point m0ÎM andevery point mÎM assign a path from m0 to m. The set U of matrices Ea, generated by theholonomy group is composed of analytic functions of the coordinates. Then, an internalbase transformation U-1(m) will result in a set of matrices independent of the point m.

There is a connection in AM defined from the given connection on VM, associated to E.The covariant derivative of the matrices Ea is

ba a a a b aE E E E Em m m m m¶ G G G = + - - , (7.1)

where we may write the terms involving the connection in VM as

[ ], ,d d ba d a d a bE E E c Em m mG G Gé ù = =ê úë û (7.2)

and obtain

( )d b ba a da a bE E c Em m m m¶ G G = + - . (7.3)

If we now define the connection in AM by the relation

b b da dacm mG Gº , (7.4)

we have

( ) aaE Emm

¶ = = 0 . (7.5)


In particular, for an orthonormal subset, composed of four of the matrices Ea anddesignated by the symbol ka we have

( ) ˆ ˆˆ , bbEm a mama

k G k Gé ù = = -ê úë û0 . (7.6)

References

1 I. Porteous, Topological Geometry, (Van Nostrand Reinhold, London), ch 13 (1969).2 E. Cartan, Theory of Spinors, (M.I.T. Press, Cambridge) (1966).3 R. Penrose, W. Rindler, Spinors and Space-Time (Cambridge University Press,4 R. Penrose, in Relativity, Groups and Topology, ed. by DeWitt and DeWitt (Gordon

and Breach Sc. Publ., New York), p. 565 (1963).5 Loos, J. Math. Phys. 8, 2114 (1967).

B. GROUPS AND SYMMETRIC SPACES.We present certain Lie groups and symmetric spaces related to the Clifford algebra.

For a general treatment see, for example, the books of Gilmore and Helgason indicated inthe references

B.1. Lie Groups.The Lie groups have a differentiable manifold structure in addition to their algebraic

group structure. They may be given a principal bundle structure, with a subgroup H asfiber and a symmetric space M as base. As shown in section 4, the group cosets are sym-metric spaces,

M G H= . (1.1)

If we write the left coset decomposition as

G MH= , (1.2)we clearly display the right action of H on G, and on itself as a vertical action on the fiberof the principal bundle (G,M,H).

B.1.1. The Differential of a Map.The Lie algebra of a group is the algebra of the differential generators of the group. To

see this geometrically, we first introduce the concept of differential of a map. Assume wehave two manifolds M, N and a map m,

:M Nm ¾¾ . (1.3)

Let tÎTM, then we have tÎTMm for some mÎM. Take a curve g with t as the tangent vectorat m. Since

:R Mg ¾¾ , (1.4)

( ):R Nm g ¾¾ (1.5)

is a curve in N. Denote its tangent at m(m) by m*t. Define the map

:TM TNm* ¾¾ . (1.6)

The definition of m* could conceivable depend on the choice of g. We shall show that m* isindependent of g using an equivalent definition. If tÎTM, it is sufficient to say how m*tÎTNoperates on the functions F(n) where n is m(m). If fÎF(n), we have fmÎF(m) and wedefine,

( )( ) ( ) ( ) , , t f t f f F N t TM t TNm m m* *= Î Î Î . (1.7)

This definition is independent of g and we see that it is equivalent to the first definition asfollows,

253Groups and Symmetric Spaces

( )( ) ( )( ) ( )( )tangent d df f f

d dm g m g m g

l l= =

0 0 , (1.8)

( ) ( )( ) ( )tangent t f f t fm g m m* = = . (1.9)

Hence the two definitions agree.It may be shown that m* is linear,

( )t s t sm m m* * *+ = + , (1.10)

( )at a tm m* *= (1.11)

and also, if we have the map

:N Pt , (1.12)that we have the chain rule,

( )t m t m* **= . (1.13)

We should note that if N=R, m* reduces to the differential.It should be clear that, for the projections p in TM and p in TN,

:TM Mp ¾¾ , (1.14)

:p TN N¾¾ , (1.15)

we have

t m p t m np m m*= = = . (1.16)

If we have a field of forms a on N, which is a section in *TN, we may define a sectionin *TM by composition. We define a*m on the point m in M by,

( )mmm a a m**º , (1.17)

and we have for any vector field n in M,

( )( ) ( )v vm a a m**= . (1.18)

Therefore we obtain a field of differential forms on M, called the pullback form a*m,which determines a section in *TM.

B.1.2. The Lie Algebra of a Group.The Lie algebra A of a Lie group G is the vector space TGI with a product inherited

from the group product. To define this product we introduce the right (left) invariantvector fields on the group manifold G as follows. Consider right multiplication by anelement aÎG,

:aR G G¾¾ (1.19)

Appendix B PHYSICAL GEOMETRY254

( ) ,aR b ba a b G= Î , (1.20)

which is a diffeomorphism with inverse Ra-1. The differential of this map,

:a b baR TG TG* , (1.21)

is an isomorphism of the tangent spaces. A vector field on G is a right invariant vector

field, denoted by X , if it satisfies

aR X X a G* = Î , (1.22)

for all aÎG. This means that

a b baR X X* = , (1.23)

which is true if and only if

a I aR X X* = , (1.24)

a b I ba I baR R X R X X* * *= = . (1.25)

Consequently given XIÎTGI there is a unique right invariant vector field X with the value

XI at the identity. It may be shown that X is C¥. In a similar manner we may define leftinvariant vector fields .

The invariant vector fields allow us to define an operation in TGI, using the Lie deriva-tive, by setting

[ ] ( ), ,X II

X Y Y X Yé ù= = ê úë û . (1.26)

The vector space TGI with this operation as product forms the Lie algebra of the group G.This product obeys

[ ] [ ], ,X Y Y X= - , (1.27)

[ ] [ ] [ ], , , , , ,X Y Z Y Z X Z X Yé ù é ù é ù+ +ë û ë û ë û . (1.28)

The structure constants are defined using a base Xa in the algebra,

, ki j i j kX X c Xé ù =ë û . (1.29)

The adjoint representation may be defined in terms of right and left multiplication.Consider

:L R G G- ¾¾1g g , (1.30)

( ) :L R TG TG-

*¾¾1

g g . (1.31)

We may define


( )AdI

L R-

*º 1

g g g , (1.32)

which clearly acts on the Lie algebra A,

Ad :A A¾¾g , (1.33)

and belongs to End(A).Consider now that

( )Ad : EndG A¾¾ , (1.34)

( )Ad : End EndTG T A A* ¾¾ = . (1.35)

We may define

ad Ad I*º , (1.36)

which clearly also acts on A,

ad: EndA A . (1.37)In terms of explicit expressions, we have if

, ae G a Al= Î Îg g , (1.38)

( ) [ ] ( )( )exp ,bL R e b a b Ol l l l- = + +1 2 3g g . (1.39)

Since the identity corresponds to l=0 we may write,

( ) ( ) [ ] ( ),L R b b a b Ol l l l-

*= + +1 2 3

g g , (1.40)

( ) [ ] ( )Ad ,b b a b Ol l= + + 2g . (1.41)

If we expand around g=I, we have

( ) ( ) ( )Ad Ad adI ab b bl= + +g , (1.42)

but since AdI is the identity, comparing the last two equations, we obtain

( ) [ ]ad ,a b a b= . (1.43)

It should be noted that

( ) ( )expAd exp adaal l= . (1.44)

B.2. Cartan Subspace.The structure constants satisfy the commutation rules and the Jacobi identity. There-

fore, they give a representation R of the algebra, called the regular representation,


( ) ,k ki k ij k i jj

X X c X X Xé ù= = ë ûR , (2.1)

which corresponds to the adjoint representation

( ) [ ]ad ,a b a b= . (2.2)

The Cartan-Killing metric is defined using the trace in this regular representation,

( ) ( ) ( )( ), tra b a b= R Rg . (2.3)

In any representation we can define a metric in the same manner. In general the metricsin different representations are neither equal nor proportional. Nevertheless, if the alge-bra is simple, the product of two vectors is a property of the algebra only. If the repre-sentations are related by addition of representations, we can mathematically relate themetrics in corresponding representations (for example see the Wigner-Eckart theorem).In this case, the internal product in any representation may be expressed in terms of anabstract product. In applications we use a normalized metric so that the norm of the unitoperator, in an associated Clifford algebra, is 1.

Consider the eigenvalue equation in the regular representation,

( ) [ ]ad ,a x a x xl= = . (2.4)

This equation leads to a secular equation which has solutions determined by the complexroots of the associated polynomial.

For any element H in A and each eigenvalue l of adH consider the subspace

( ) ( ){ }, : ad kA H X A H I Xl l= Î - = 0 . (2.5)

It may be proved that A is a direct sum over these subspaces,

( ),r

ii

A A H l=

= å0

. (2.6)

There is a zero eigenvalue subspace, called the Cartan subspace, spanned by generatorswhich commute among themselves. A Cartan subalgebra is not uniquely determined butdepends on a regular element of the complete algebra. There is an automorphism of thecomplexified algebra which links any pair of Cartan subalgebras. An element H is regularif

( ) ( )( )dim , min dim ,A H A X=0 0 . (2.7)

The n generators which span a Cartan space may be considered the components of avector H. The corresponding eigenvalues may be represented in the Cartan space by theweight vector w,

w wy y=H w . (2.8)

Similarly, the roots may be represented in the Cartan subspace as root vectors r(a). Infact, we can transfer all the information contained in the roots and the Cartan-Killingmetric to the smaller Cartan space. The commutation relations may be expressed in the


canonical form,

,i jH Hé ù =ë û 0 , (2.9)

[ ] ( ),E Ea aa=H , (2.10)

[ ] ( ),E Ea a a- = · H , (2.11)

E EN E

,

0 , (2.12)

where N is a normalization factor.The Cartan root spaces can be classified using the relations among the different pos-

sible roots which form geometrical figures in this space. In this manner, it may be shownthat there are four infinite series of these spaces, designated by the letters A, B, C y D.Each element of a series is determined by the dimension of the Cartan space. In additionthere are other exceptional classes, of limited dimensions, designated by E, F, G. There arethe following isomorphisms among some of these spaces,

1 1 1 2 2 3 3

2 1 1 4 4 5 5

A B C B C A DD A A A E D E

= = = =

= ´ = = . (2.13)

It may be proved the following theorem, due to Cartan: a noncompact Lie algebra A’can be decomposed as a direct sum,

'A A A+ -= Å , (2.14)

where the Cartan-Killing metric is negative definite when restricted to A+ and positivedefinite when restricted to A- and there exists a Z2- gradation,

,A A A+ + +é ù Ìë û , (2.15)

,A A A+ - -é ù Ìë û , (2.16)

[ ],A A A- - +Ì . (2.17)

In other words, A+ is compact and A- is noncompact. We say that A+ is the maximalcompact subalgebra of A’. There is a real compact algebra associated to A’, a real form ofthe complex extension AC of A’, defined by Weyl’s unitary trick,

A A iA+ -º Å . (2.18)

In general, the action of involutive automorphisms of the complex extension of thealgebra produces a decomposition which meets the Z2-gradation of the subspaces witheigenvalues +1 and -1,

TAT A- =1 , (2.19)


T 2 1 . (2.20)There are only three different automorphisms of this type,

*T = , (2.21)

,p

p qq

IT I

I

é ù+ê ú= ºê ú-ë û

00 , (2.22)

,p

p pp

IT J

I

é ù+ê ú= ºê ú-ë û

00 . (2.23)

The following theorem may be proved: Let A’ y B’ be two real semisimple Lie alge-bras, with respective Cartan decompositions. Then A’ y B’ are isomorphic if and only ifthe compact real forms A and B are isomorphic by an isomorphism which sends A+ to B+.

This shows that the semisimple algebras may be classified determining the pairs (A,A+)that is, a compact algebra and a maximal compact subalgebra. The algebras related to thisdecomposition are obtained, from the compact form, using Weyl’s unitary trick on the -1eigenvalue subspace of the indicated involutions. The real forms of the groups are ob-tained from the compact form G applying the same procedure. The classical groups corre-spond to the A, B, C y D series of Cartan spaces.

The An-1 space characterizes the complex group SL(n,) and all its real forms. Thecompact real form is SU(n). Under complex conjugation (type I) we construct the groupSL(4,) which is the normal real form of the complex group. Under the involution Ip,q(type III) we obtain the group SU(p,q). Under the combination of Jn,n and conjugation(type II) we obtain the group SU*(2n).

The Bn space characterizes the complex group SO(2n+1,). The compact real form isSO(2n+1,) which also is the normal real form. The conjugation is trivial and the involu-tion Jn,n does not exist. Therefore the only nontrivial automorphism is Ip,q (type I) whichdetermines the group SO(p,q) for q+p odd.

The Cn space characterizes the complex group Sp(2n,). The compact real form isUSp(2n). The conjugation and Jn,n produce the same result (type I), the group Sp(2n,)which is the normal real form. The automorphism Ip,q (type II) determines the groupUSp(2p,2q).

The Dn space characterizes the complex group SO(2n,). The compact real form isSO(2n,) which also is the normal real form. The conjugation is trivial. The automor-phism Ip,q (type I) determines the group SO(p,q) for p+q even. The automorphism Jn,n(type III) determines the group SO*(2n).

B.3. The Group G.Inside the Clifford algebra A there is a group defined as the subset of elements of A

with inverse. Since the algebra is the determinant geometric element, this group is gen-erated by the exponentiation of the algebra in terms of the orthonormal base. Both havenatural actions on a linear space V which is called the associated spinor space of A.

This group in A is formed by the linear transformations GL(2,), composed of all 2´2invertible matrices aÎA over. In other words it is the linear group in two dimensions over


the ring of pseudoquaternions (real 2´2 submatrices). The elements of GL(2,) maybe represented by 4´4 real matrices, denoted by R(4). We may define the determinant inGL(2,) by the corresponding determinant in R(4). This group produces linear endo-morphisms of the spinor space V. We are interested in the unimodular subgroup SL(2,),the group G of inner automorphisms of R3,1.

In particular the inner automorphisms of the algebra, produced by G, are of the form,

a a-¢ = 1g g . (3.1)

where g is an element of GL(2,) contained in the algebra A. Since the algebra is theadjoint representation, this action corresponds to the adjoint group of G, Ad(G), acting onA.

We are also interested in the automorphism produced by the covering groups. Sincethe covering group of a direct sum algebra is the product of the covering groups of theinvariant components of the algebra, we have

( ) ( ), ,GL R SL= Ä2 4 . (3.2)

The adjoint of the center of this group, acting on the algebra, corresponds to the iden-tity. Then, there are trivial automorphisms produced by the R+ subgroup. The nontrivialautomorphisms correspond to the covering group of SL(4,). The group SL(2,) is rep-resented by SL(4,) and has for Lie algebra sl(4,). Then we must have a homomorphism

( ) ( , ),i

SLSL

D=

42 , (3.3)

where Di is a discrete subgroup of the center.In algebraic topology there is the general homotopy sequence [1 ], relating different

homotopy groups pk,

( ) ( ) ( ) ( ) ( ))i pk k k k kB H G B HD Dp p p p p* * * *+ -¾¾ ¾¾ ¾¾ ¾¾1 1 (3.4)

for a fiber bundle G with fiber H over the base space B.The first homotopy group of SL(4,) may be determined as follows. We have the

general homotopic exact sequence,

( )( )

( ) ( )( )

( ), ,

,SL SL

SO SLSO SO

p p p pé ù é ùê ú ê úé ù é ù¾¾ ¾¾ ¾¾ë û ë ûê ú ê úê ú ê úë û ë û

2 1 1 1

4 44 4

4 4 . (3.5)

The base space B is the noncompact riemannian symmetric space generated by the noncom-pact sector complementary to the maximal compact subalgebra,

( )( )

( )( ) ( )

, ,SL SLB

SO SU SU= =

Ä4 4

4 2 2

. (3.6)

Due to the homotopic properties of B, which is contractile, its homotopy groups are theidentity and this sequence collapses to a short exact sequence,

( ) ( ),I SO SL Ip pé ù é ù ë û ë û1 14 4 , (3.7)


which implies that the corresponding homotopy groups are isomorphic,

( ) ( ),SL SO Zp pé ù é ù= =ë û ë û1 1 24 4 , (3.8)

and SL(4,) is doubly connected as the well known SO(4). The covering group of SO(4)is

( ) ( )( )SO SU SU= Ä4 2 2 . (3.9)

To get the covering group of G we need the sequence

( ) ( )I SU SU G Ip p é ùé ù Ä ë û ë û1 12 2 . (3.10)

This distinction can be accomplished in the enveloping Clifford algebra A, using the rep-resentation of the algebra provided by its biquaternion structure. In order to have roomfor an SU(2)SU(2) subgroup we have to break the equality

qe e Ipl p = 4 (3.11)

of matrix multiplication, for l=1, q=1. The operations provided by the ring deter-mine that the addition inverse (negative) of the identity) in , denoted by -I, is not equiva-lent to the respective inverse in , denoted by -I.

It is possible to define matrices over the ring . In particular consider GL(1,),the linear group in one dimension over the abstract noncommutative ring . Definethe subgroup of unit modulus biquaternions (two quaternions of unit moduli), SL(1,).This group is generated by the Lie algebra

( ), a a aba a a bsl I I q b qdq l df d lÄ = Ä + Ä + Ä1 , (3.12)

which is isomorphic to sl(4,). A group obtained by exponentiation of this algebra ishomomorphic to SL(4,)

If we use the tensor product , as in the algebra A, the discrete invariant center ofthe group is determined by

Il = 1 , (3.13)

qI = 1 . (3.14)

Therefore, the special linear group SL(1,) has the invariant discrete subgroup Dcomposed of the elements

{ } { } { } { }{ }, , , , , , ,I I I I I I I I- - - - . (3.15)

The invariant subgroups of D are

{ },I I I= , (3.16)

{ } { }{ }, , ,I I I I Z- = 2

, (3.17.


{ } { }{ }, , ,I I I I Z- = 2

, (3.18)

{ } { }{ }, , ,I I I I Z- - = 2

, (3.19)

which are distinct, although the last three are isomorphic.If we use the standard identification defining the operations -I and -I to be equivalent,

as in matrix multiplication, we clearly obtain the group SL(4,). The invariant discretesubgroup of SL(4,) may be written as

{ } { }{ } { }, , , ,DI I I I I I

Z-= = -4 4

2 . (3.20)

In other words, SL(4,) is doubly connected and we obtain dividing by the equiva-lence relation,

( , ) ( , )( , )SL SLSL

Z ZÄ

= =2 2

1 44 , (3.21)

from which we can identify the covering group of SL(4,), up to an isomorphism,

( , ) ( , )SL SL= Ä4 1 . (3.22)

The compact subalgebra is generated by

( ), ( ) ( )a aC a asl q su sudq l dfÄ = Å = Å1 2 2 (3.23)

and we obtain the maximal compact subgroup

( ) ( )H SU SU= Ä2 2 . (3.24)

The group G has the structure of a fiber bundle over B with fiber the compact subgroupH, corresponding to a left coset decomposition,

G BH= . (3.25)The nonsimple group U(1)SL(2,) is the even subgroup of SL(1,) and the ho-

momorphic SL1(2,), the 2´2 matrices of complex determinant of unit modulus, is theeven subgroup of SL(2,) or SL(4,).

B.4. Symmetric Spaces.The symmetric spaces are manifolds whose curvature tensor is invariant under all par-

allel transportations,

R = 0 . (4.1)Consider any geodesic passing through a point p=g(0), in a riemannian manifold M. Thegeodesic symmetry sp symmetrically maps points along the geodesic,

( ) ( ):ps g l g l¾¾ - . (4.2)


A riemannian manifold is a locally symmetric space if for each pM there is a normalneighborhood Np where the geodesic symmetry is an isometry.

It may be shown that the curvature invariance is equivalent to the condition that thegeodesic symmetry with respect to each point is a local isometry. Therefore, the sym-metric spaces have a transitive group of isometries G’ and may be represented by a quotientG’/H. [2 , 3 ]. The classification of these coset spaces was done by Cartan using thisrelation of the symmetric spaces to Lie group quotients.

The Cartan-Killing metric for a group quotient space, G’/H, is taken as the metric inthe subspace of the algebra of G’, complementary to the algebra of H. The exponentiationof this subspace is a globally symmetric space [4 ] because any point and its neighborhoodcan be translated to any other point by a group operation. In this way it is possible to showthat the metric is invariant.

Since both the group G’ and the subgroup H are related to compact groups by means ofinvolutive automorphisms, there are different quotients related among each other by twoinvolutions which we shall indicate as s and t. The possibilities for the involution t areexactly the same available for the involutions s, indicated in a previous section. Hence,the classification of the symmetric spaces is determined applying a pair of these involu-tions, taken from the indicated set, which commute and exhaust the possibilities. Thesimultaneous eigenvalues of this pair serve to describe A,

A A A A A++ +- -+ --= Å Å Å . (4.3)

The involutive automorphism t serves to select a compact subgroup, starting from acompact group G

( )expG A At+ ++ -+= Å (4.4)

and the complementary compact symmetric space with definite metric is

( )( )exp Gi A A

G t+- --

+

Å = . (4.5)

The other automorphism s serves to convert the compact subgroup to a noncompact sub-group

( ) ( )exp expG A A A iA Gs st t+ ++ -+ ++ -+ += Å ¾¾ Å = (4.6)

and to convert the symmetric space with definite metric in one with indefinite metric

( )( ) ( )( )exp expG Gi A A i A iA

G G

ss

st t

++ -- +- --+ +

= Å ¾¾ Å = . (4.7)

The symmetric space G/G+ has a negative definite metric derived from the Cartan-Killing metric. The dual space G*/G+, where G* is the maximal noncompact group formand G+ is the maximal compact subgroup, has an equal but positive definite metric, de-rived from the Cartan-Killing metric restricted to the complementary subspace of A*. Bothspaces are, therefore, riemannian spaces.

There is a theorem that says that the noncompact irreducible hermitian symmetric spacesare exactly the manifolds G/H where G is a connected noncompact simple group with


center {I} and H is a maximal compact subgroup of G with a nondiscrete center [5 ].There is a standard notation for the classification of riemannian spaces, indicating theCartan subspace (A,B…), the involution type (I,II,III) and the dimensions that character-ize the groups (2n,p,q).

There are other real forms of the complex group GC between the two extremes Gy G* and therefore there is a series of symmetric spaces which are real forms of thecomplex extension of the quotient,

CC

CG G

G G+ +

æ ö÷ç =÷ç ÷çè ø , (4.8)

which fall between the two extreme riemannian spaces. These intermediate spaceshave an indefinite metric and are considered pseudoriemannian spaces. The differ-ent real forms within the series corresponding to a complex symmetric space areclassified by their characters. The character of a real form is defined as the trace ofthe canonical form of the metric. This integer, corresponds to the difference in thenumber of compact and noncompact generators. The series may be characterized byits noncompact end group.

The series of quotient spaces related to the A3 Cartan space are of interest. Inparticular we choose the involutive automorphism t of type AIII(p=2,q=2) whichdetermines a heptadimensional compact subgroup G+. We obtain, in this manner, aseries of octadimensional spaces, characterized by the noncompact group SU(2,2),corresponding to the riemannian space G/G+ and its dual G*/G+,

( ) *( ) ( , )( ) ( ) ( ) ( , ) ( ) ( , ) ( , )

( ) ( , ) ( , ) ( ) ( ) ( ) ( )

SU SU SUSU SU U SL SO SL SO

SL ,R SUSL SO SU SU U

» » »Ä Ä Ä Ä

» »Ä Ä Ä

4 4 2 22 2 1 2 2 2 1 1

4 2 22 2 2 2 1

.(4.9)

Due to the isomorphism of the spaces A3 and D3 we have the isomorphic series, of spacescharacterized by the noncompact group SO(4,2), corresponding to the riemannian cosetspace G/G+ and its dual G*/G+ with involution t of the type BDI(p=4,q=2),

( ) ( , ) ( , )( ) ( ) ( , ) ( ) ( , ) ( , )

( , ) ( , ) ( , ) ( ) ( ) ( )

SO SO SOSO SO SO SO SO SO

SO SOSO SO SO SO

» »Ä Ä Ä

» »Ä Ä

6 5 1 4 24 2 3 1 2 3 1 1 1

3 3 4 23 1 2 4 2

. (4.10)

The characters of the real forms of both isomorphic series are -8, -4, 0, +4, +8.Other series of interest are the ones related to the C2 Cartan space. In particular, we

choose the involutive automorphism t of the type CII(p=2,q=2) which determines a hexa-dimensional compact subgroup. We obtain a series of tetradimensional spaces, charac-terized by the noncompact group USp(2,2), corresponding to the riemannian space G/G+ andits dual G*/G+,


( ) ( , ) ( , )( ) ( ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( ) ( )

USp USp SpUSp USp Sp Sp Sp

Sp USpSp USp USp

» » »Ä Ä

» »Ä

4 2 2 42 2 2 2 2

4 2 22 2 2

. (4.11)

Also, due to the isomorphism of the spaces B2 and C2, we have the isomorphic series ofspaces characterized by the noncompact group SO(4,1) corresponding to the riemannianspace G/G+ and its dual G*/G+ with involution t of the type BDI(p=4,q=1),

( ) ( , ) ( , ) ( , ) ( , )( ) ( , ) ( , ) ( , ) ( )

SO SO SO SO SOSO SO SO SO SO

» » » »5 4 1 3 2 3 2 4 14 3 1 2 2 3 1 4

. (4.12)

The characters of the real forms of both isomorphic series are -4, -2, 0, +2, +4.

References

1 G. W. Whitehead, Elements of Homotoy Theory (Springer Verlag, New York) (1978).2 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, NewYork)

p. 163 (1962).3 E. Cartan, Bull. Soc. Math. France 55, 114 (1927).4 R. Gilmore, Lie Groups, Lie Algebras and Some of their Applications (John Wiley

and Sons, New York), p. 350 (1974).5 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

p. 310 (1962).

C. CONNECTIONS ON FIBER BUNDLES.We present certain necessary notions of connections on fiber bundles. The treatment is

not complete, but serves to establish concepts and the notation used in the book. A generalknowledge of differential geometry and fiber bundles is assumed. For more details see thereferences [1, 2, 3, 4].

C.1. A Fundamental Field.Let A be the algebra of a Lie group G acting on a manifold M on the right. For every aÎA,

we have the curve in G,

( )expt ta¾¾ . (1.1)

For each point mÎM this gives rise to a curve in M,tat me¾¾ . (1.2)

Denote by ma the tangent vector to M at t=0. Thus, we have a vector field a on M. The

vector a can also be described as follows: for mÎM let the map

:m G Ms ¾¾ , (1.3)

( )m ms =g g , (1.4)

then a is

( )m m Ia as *= . (1.5)

It is clear that, if we denote by S the set of sections of a fiber bundle,

( ):A S TMs* is linear. It also may be proved that

[ ], ,a b a bé ù= ê úë û

. (1.6)

If we now take M to be a principal bundle E, we know that G acts on the right withoutfixed point. Therefore, we have a fundamental vector field a corresponding to any aÎA.For every eÎE the map

ea a¾¾ (1.7)

is an isomorphism since G acts freely. Since the maps Rg:EE takes the vertical spaces tothemselves, the set of all ea is precisely the set of vertical vectors at pe.

Appendix C PHYSICAL GEOMETRY266

C.2. The Ehresmann Connection.A connection in a principal bundle (E,M,G,p), with algebra A corresponding to the

structure group G, is an A valued 1-form w on E such that

:TE Aw ¾¾ , (2.1)

( )a aw = , (2.2)

( ) ( )Ad R t t t TEw w-* = Î1

g g . (2.3)

For every eÎE the map

: e eTE A e Ew ¾¾ Î (2.4)

is onto, so its kernel He is a subspace of TEe having the same dimensions as M. This He iscalled the horizontal subspace of TEe determined by the connection. Tangent vectors in Heare called horizontal vectors. It may be proved that H is a C¥ distribution which satisfies

e e eTE V H= Å , (2.5)

e eH R H*=g g . (2.6)

It may also be shown that H, satisfying these equations, determines the connection formw.

From the direct sum decomposition and the fact that the vertical subspace Ve is thekernel of

: e eTE TMpp* ¾¾ , (2.7)

it is clear that

: e eH TMpp* ¾¾ (2.8)

is an isomorphism for each e. Consequently, for each vector field X in TM there is a uniquevector field X* on E which is horizontal everywhere and projects back to X,

e eX Xpp ** = , (2.9)

which is called the lift of X. It may be proved that

R X X* ** =g , (2.10)

( )X Y X Y* * *+ = + , (2.11)

( ) :fX f X f M Rp* *= , (2.12)

[ ], hor ,X Y X Y* * *é ù= ê úë û . (2.13)

267Connections on Fiber Bundles

If c is a piecewise C1 curve in E, we say that c is horizontal if and only if all its tangentvectors are horizontal. At those points where c is not C1 we require that both c’+ and c’- behorizontal. If

[ ]: ,c M¾¾0 1 (2.14)

is a piecewise C1 curve in M. We define the lift of c to be the horizontal curve

[ ]: ,c E* ¾¾0 1 , (2.15)

which projects back to c,

c cp * = . (2.16)This curve is unique given the initial point c*(0), which projects to c(0).

We can now define parallel translation of fibers of the principal bundle E along anycurve c on M, as the vector space isomorphism,

( ) ( ):t c c tt p p- -¾¾1 10 , (2.17)

defined by

( ) ( )tc c tt * *=0 . (2.18)

If the principal bundle is a bundle of frames e, bases of a n dimensional vector space V,it is naturally associated to the vector bundle VM with the space V as fiber. A section e ofE induces an isomorphism,

: nee R VMp¾¾ , (2.19)

and we have

( ) ( ):t e ee e VM VMp p tt - ¾¾1 , (2.20)

which allows us to define parallel transplantation of vectors of V, sections of VM, from theparallel transplantation of e

( ) ( ) t m t m m mY e e Y Y VMt t -= Î1 . (2.21)

We can now define the covariant derivative,

( )( ) ( )( )lim X t

Y c t Y cY

t

t-

- =

1

0

0 , (2.22)

where c(0) is m and

( ) m mX c TM m c= Î = 0 . (2.23)

In particular, if the vector bundle VM is the tangent bundle TM, the principal bundle Eis a frame bundle associated to the tangent bundle TM. Then both vectors in the covariantdifferentiation obey X,YÎTMm and the derivative defines a Koszul connection on thebundle TM. If the structure group G can be reduced to the real orthogonal group O(p,q)there is a metric g, group invariant with vanishing covariant derivative, on the base space M.


C.3. Tensorial k-forms.Given a vector space V with a finite number of dimensions and a linear representation

(G) of the structure group in V, we may define tensors of the type (G) over the principalbundle E as the mapping

:t E V¾¾ (3.1)

such that

( ) -e et t G= Î1

g g g . (3.2)

Consider an atlas covering the base space with the charts Ui and a local section ei of thefiber bundle E in each chart. For every mÎUiÇUj we have an element gj

i of G such that

i ij i j je e G= Îg g (3.3)

and we may consider a local tensor on M,

i it e t*º , (3.4)

( ) ( )ij e mme t t* = , (3.5)

obtaining

( ) ( )( )- ij j im m

e t e t* *= 1 g . (3.6)

One may see that the si*t may be considered as the images, by the local homeomor-phisms, of a section in a fiber bundle over M with fiber V and structural group (G) whichmay be called the tensor bundle of type (G).

A q-form a over E, with values in V is called a tensorial q-form of type (G) if it satisfiesthe two following properties,

( ), , vq iX X X X T E TEa = Î Ì1 2 0 , (3.7)

( ) ( ) ( ), , , ,-q qR X R X R X X X Xa a* * * = 1

g 1 g 2 g 1 2g , (3.8)

where TvE is the vertical subbundle.A tensor of type (G) may be considered as a 0-form. If we have an atlas with local

sections ei we may define local q-forms on M with values in V,

( ) ( ) ( ), , , , , ,i q i q i i i qX X X e X X X e X e X e Xa a a** * *

¢ = =1 2 1 2 1 2 , (3.9)

which satisfy

( ) ( ) ( ), , , ,- ij q j i qX X X X X Xa a¢ ¢= 1

1 2 1 2g . (3.10)

Inversely, a set of local ai’ which cover M and satisfying the last equation, determinea tensorial q-form over E.

It should be clear that a tensorial q-form with values in V defines canonically a tensorover E, an element of the tensor bundle over with fiber VÄq where q is the space


of q-forms.Further, consider a q-form a on E with values in a vector space V. We define a (q+1)-

form valued in V, the covariant exterior derivative of a, designated by the expression

( ) ( ), , , ,q qD X X X d hX hX hXa a+ +=1 2 1 1 2 1 , (3.11)

where XiÎTEm and hX is the horizontal part of X.It is clear that if any of the vectors X1, X2, ...Xq is vertical,

( ), , qD X X Xa =1 2 0 . (3.12)

In addition we have

( ) ( )( ) ( ) ( )

( ) ( )

, , , ,

, , , ,

, ,

k

-

-k

D R X R X R X d hR X hR X

d R hX R hX d hX hX

D X X X

a a

a a

a

* * * + * *

* *

+

=

= =

=

g 1 g 2 g 1 g 1 g 2

1g 1 g 2 1 2

11 2 1

g

g

. (3.13)

We have, therefore, that if a is a tensorial q-form of type (G), then Da is a tensorial (q+1)-form of the same type.

C.4. Curvature and Torsion.In any principal bundle with a connection we may define the curvature tensorial 2-form

W by

DW wº . (4.1)If we have the bundle of frames of the tangent space TM, we can define an Rn valued 1-

form by projecting and taking components,

: ne eTE RQ ¾¾ , (4.2)

e eQ p-*= 1 , (4.3)

which we call the canonical or soldering form Q of the dual form coframe q of the tan-gent vector frame u.

For a section s we have the form on M

( ) ( ) ( ) ( )( ) ( )( )-= m m m ms ms Y sY s m sY s m Y Y TMQ Q p* -* * *= = Î1 1 . (4.4)

If we take for Y all the vectors of a tangent frame u and consider a local section

( )s m u= , (4.5)

we get

u u I- =1 , (4.6)which are the components of coframe q with respect to the frame u. Clearly s*Q coin-cides with q.

We can define the torsion form S in terms of this canonical form,


DS Qº . (4.7)If the torsion is zero and the frames are orthonormal the connection is riemannian orpseudoriemannian, depending on the structure group O(p,q).

If we define a basic vector field in a frame bundle, for any point xÎRn,

( )( )ˆ ex x*

= , (4.8)

the following relations may be proved,

( )ˆQ x x= , (4.9)

( ) ( ) ,R GL nx* -= Î1g g g , (4.10)

( ) ( )ˆ, ,a a a gl n Ax xé ù = Î ºê úë û , (4.11)

[ ], , e ea h H h H a AÎ Î Î . (4.12)

The transformation properties of the forms W, Q, S are given by

( ) ( ) ( ), Ad ,R X Y X YW W* -= 1g g , (4.13)

( )R XQ Q* -= 1g g , (4.14)

( ) ( ), ,R X Y X YS S* -= 1g g . (4.15)

It may be proved that these forms obey a set of structural equations, which are,

( ) ( ) ( ) ( ), , ,X Y d X Y X YW w w wé ù= + ë û , (4.16)

( ) ( ) ( ) ( ) ( ) ( )( ), ,X Y d X Y X Y Y XS Q w Q w Q= + - , (4.17)

which may be written in terms of ordinary forms. With respect to a standard basis e of Rn,

iieQ Q= , (4.18)

iieS S= , (4.19)

and a standard basis E of gl(n,)

j ii jEw w= , (4.20)

j ii jEW W= . (4.21)

Then we have


i i i kj j k jdW w w w= + , (4.22)

i i i kkdS Q w Q= + , (4.23)

and omitting the indices,

dW w w w= + , (4.24)

dS Q w Q= + . (4.25)

These forms satisfy certain identities, called the Bianchi’s identities. In any principalbundle, the first Bianchi’s identity may be shown to be

DW = 0 (4.26)

and if the bundle is a frame bundle of TM, the second Bianchi identity is

DS W Q= . (4.27)

C.5. Induced Connection, Curvature andTorsion Forms.

If we have local sections ei, in E, we may induce corresponding forms on the base space,connection ei*w, dual frame ei*Q, curvature ei*W and torsion ei*S.

For every tangent vector m ÎTM on point m there corresponds the vectors in TE suchthat

ie m TE* Î , (5.1)

ie m mp* * = . (5.2)

If we have a change of reference sections,

ij i je e= g , (5.3)

we get

i ij i j i je e e= +g g , (5.4)

where the tangent vector g ÎTG at gÎG, corresponds to an element a of the Lie algebra A ofgroup G such that

( )a d m-= 1g g . (5.5)

If we let w act on e we have

( ) ( ) ( )i ij i j i je e ew w w= +g g , (5.6)

but


( ) ( ) ( )Ad ie R e ew w w-*= = 1

g gg , (5.7)

( ) ( )( ) ( )ie d m d mw w - -= =1 1g g g g g , (5.8)

giving

( ) ( ) ( )Adj ie m e m d mw w- -* *= +1 1

g g g , (5.9)

( )Adj i ie e d e dw w w* - * - - * -= + = +1 1 1 1g g g g g g g , (5.10)

which is the transformation law for a local connection form on M under a change offrame.

The covariant derivative may also be used to define the local connection forms e*winduced from the local section e of the principal bundle E, in components with respect tothe e and a frame u of TM, as follows,

( )e e e w* = , (5.11)

which in components may be written as

( )BM MA B A

e e em mw* = . (5.12)

The action on the right is expressed by summation over frame indices. This equation givesthe rate of change of the frame e referred to the frame e itself, along curves in M withtangents the elements of the frame u.

The structure group may act on the principal bundle fiber, a vector base e of an associ-ated vector bundle, in two ways: first as an action on the different vectors of the basetransforming them into each other, the active transformation on the right, and second as anaction on the components of the base with respect to another base, the passive transforma-tion on the left. In the associated principal bundle, the structure group may be takenisomorphic to the set of bases. We may indicate the different members of a base by an indexwith a bar. In this manner, a bundle action of the group on the right or active transformationmay be written as

e e ba ab= g , (5.13)

while a group action on the left or passive transformation may be written as,

e em m ll= g . (5.14)

The connection form w defined in terms of the right action on the base (an activetransformation), has an expression in terms of a left action (a passive transformation) on avector v of the fiber V

( ) ( )ev ev e v e e v vw ¶* = + = + . (5.15 )

The covariant derivative of a vector field v in VM is, in components,


M M M NNv v e vm m m¶ w* = + , (5.16)

or for short,

v v e v¶ w* = + . (5.17)In particular, for the vector components of the frame e, referred to another arbitraryframe s, we may write,

e e s e e e¶ w ¶ G* = + = + . (5.18)where we denoted by G the set of components (connection coefficients) of s*w withrespect to the arbitrary frame s.

The covariant derivative of a tensorial q-form f may be expressed by

( ) ( ) ( )( )

( ) ( )

ˆ,

ˆ ˆ , ,

i

qi

q A i qi

i j

i j i j qi j

D A A A A A A

A A A A A A

f f

f

++

+ +=

++

<

= - +

é ù- ë û

å

å

11

1 2 1 1 11

1 1

1

1

. (5.19)

For a vector valued 0-form v the last expression reduces, in a coordinate basis,

M M M NNem m m¶ w* = +v v v . (5.20)

The action on the left is expressed by summation over component indices. Similarly for avector valued 1-form q we have

( ),D m n m n n mq ¶ ¶ q q= - . (5.21)

If the bundle VM is TM, we may associate holonomic coordinate frames ¶ sections, tothe coordinates of the atlas on the base manifold M. Associated to an arbitrary frame,section s of E, and its dual (inverse) coframe, there are, respectively matrices u, q, whichare elements of the group GL(4,). We may transform the connection form expressionin the coordinate frame, ¶*w, to the expression in the arbitrary frame, s*w , by its trans-formation law, eq. (5.10). We must take for g the group element u, whose action from theright maps ¶ to s,

( )s u duw q ¶ w q* *= + . (5.22)

The coordinate expression of the connection coefficients G of ¶*w may be obtained bythe inverse transformation,

( )u s udG ¶ w w q q* *º = + . (5.23)

We may write then, for a base u in TM,

u u ul l l nm a m a nm a¶ G = + , (5.24)

or for short,

u u u¶ G = + . (5.25)


If we have a form F of degree p with values in L(S,S), in other words matrix valued, wemay write the covariant derivative

,D d sF F w F*é ù= + ê úë û , (5.26)

where the bracket is defined for two matrices a and b of degree p and q respectively by

[ ] ( ), pqa b a b b a= - - 1 . (5.27)

By repeated application of these relations and using the definition of curvature, we obtain

,D sF W F*é ù= ê úë û2 . (5.28)

References

1 M. Spivak, Differential Geometry (Publish or Perish, Berkeley), ch. 8 (1970).2 A. Lichnerowicz, Théorie Globale des Connexions et des Groupes d’Holonomie, (Ed.

Cremonese, Roma), p. 62, 101 (1972).3 J. A. Schouten, Ricci-Calculus, 2nd ed. (Springer -Verlag, Berlin), (1954).4 I. Vaissman, Cohomology of Differential Forms, (Marcel Dekker, New York), (1973).

D. JET BUNDLES.

D.1. Jet Bundles.Consider a C bundle E. Let S(E) denote the space of all sections of E and Ck(E) the

subspaces of Ck sections. Let e Î E be given such that p(e) = m and U be an open neigh-borhood of m with local coordinates xm and such that there exists a trivialization

: U mE U Ej ¢ ´ (1.1)

and let W be an open neighborhood of em, in the fiber Em with local coordinates yi . Indicatepartial derivatives of order | a | by

... nnx x

aa

a a

¶¶

¶ ¶=

11

, (1.2)

where a = {a1,....an,} are n-tuples of nonnegative integers and |a|=a1+…+an.We define an equivalence relation on the sections at a point, by stating that (x, S1(x)) is

equivalent to (x’, S2 (x’)) if

x x¢ = , (1.3)

( )( ) ( )( )i i

m my s y sa a¶ ¶=1 2 (1.4)

and indicate the quotient of S(E) by this equivalence relation by Jk(E)m. For a section s letjks(m) be the class of sections with equal derivatives up to order k, at m. Uniquely we cantake the Jk(E)m as fibers of a C¥ vector bundle JkE over M so that jks Î C¥(JkE) for allsections s Î C¥(E), which we call the kth order jet bundle of bundle E [1, 2]. The linearmap

( ) ( ): kkj C E C J E¥ ¥ (1.5)

is called the k-jet prolongation map of a section.Two equivalent sections of JkE have the same partial derivatives at the specified point.

The J1E has for vertical space at a point, all classes of sections that have the same valueand the same first partial derivatives at the specified point.

There is a natural projection p from JkE to E defined by sending a point jksm Î JkE overm Î M to the point of E which has for fiber at m the order zero equivalence class of jksm.We have

( )k m mp j s s= . (1.6)

The vertical space JkEe is the set of all classes of sections, defined near m=p(e), suchthat s(m)=e, relative to the equivalence relation s1~s2 if and only if (¶ ks1)m~(¶ ks2)m. Theelement of JkEe defined by the section s is called the k-jet of s at m, indicated by jksm

The functions (xm,yi,zim ) defined on p-1[j-1(U´W)] by the rules

Appendix D PHYSICAL GEOMETRY276

( ) ( )mx j s x mm m=1 , (1.7)

( ) ( )( )i imy j s y s m=1 , (1.8)

( ) ( )( )i

im

yz j s s m

xm m

¶¶

=1 , (1.9)

where yi = yi(xm) are the equations of the section s relative to the coordinates xm, yi and thetrivialization j which we have chosen, form a system of local coordinates on J1E, calledcanonical coordinates on J1E.

It can be seen that this formalism provides a natural setting for the discussion of differ-ential equations. One can think of a differential equation system as defined by a set ofrelations of the form

( )..., , , ...i i i iAF x y z z zm

m mn mn x = 0 , (1.10)

where the z coordinates are linked to the partial derivatives of the independent variables yi

in terms of the dependent variables xm. Geometrically, the functions FA represent surfacesin the jet bundle JkE determining a subspace SÎJkE. A solution of the system may bethought of as a section s in JkE satisfying

kj s SÌ . (1.11)

It is possible to introduce a fundamental form associated to the jet bundle which maybe used to geometrically characterize variational problems. For this purpose we shall fol-low the work of García [3, 4, 5]. Let us transport the fiber of TEv to J1E constructing theinduced bundle p*TEv as follows. For a given point js(m) Î J1E we obtain a point of E,

( )( ) ( ) p j s m s m E= Î1 . (1.12)

Let p*TEv be the set {j1s(m), vs(m)} where vs(m)ÎTEv indicates a vector at s(m) satisfying

( )( ) ( )( ) ( )s mp j s m q v s m= =1 , (1.13)

where q : TEv E. The bundle p*TEv has the projection

: vq p TE J E*¢ 1 , (1.14)

( ) ( )( ) ( ), s mq j s m v j s m¢ =1 1 . (1.15)

Now we can define a differentiable 1-form on J1E, called structure form Q, valued onp*TEv. If U is an open neighborhood of e in J1E with canonical coordinates (xm,yi,zi

m) thenp*¶/¶yi is a base for the fiber of p*TEv and Q has the following expression:

: vTJ E p T EQ *1 , (1.16)

277Jet Bundles

( ) ( ) aa

V V p V TJ Ey¶

Q q¶

*= Î 1 , (1.17)

where the qi are the ordinary 1-forms

a a ady z dx aaq = - . (1.18)

It follows then that the 1-jet prolongation jls of a section s on E is the unique section onJ1E such that

p j s s=1 , (1.19)

j sQ = 0 . (1.20)

It is convenient to introduce the following definition. An infinitesimal contact trans-formation is generated by a vector field V, a section of TJkE, by means of the Lie deriva-tive, if it satisfies

V hQ Q= · , (1.21)

where h is a homomorphism of the fiber of p*TEv and · is the usual bilinear product.It may be proved that there are prolongations of vectors in analogy with prolongation

of sections. Let V be a section in TE. Then there exists a unique p-projectable infinitesi-mal contact transformation jV, a section of TJ1E, such that p*jV = V. We shall call this jVthe 1-jet prolongation of V. The map V jV is an injection of the Lie algebra of vectorfields on E into the algebra of vector fields on J1E.

D.2. Critical Sections And Jacobi Vectors.If we have an orientation h on the base manifold M which we shall call the volume

element of M, we can inject it by (p p)* in the algebra of differential forms on J1E. Wecan speak of the 4-form Lh in J1E where L is the lagrangian. We can define a functional onthe set of differentiable sections s of E by

( )js

LA s h= ò , (2.1)

where js is the 1-jet prolongation of s. The functional A is defined on the set S of thosesections such that the integral exists.

We shall call the differential of the functional A at a given section sÎS, denoted by dAs,the linear functional on the space {j1V} of the infinitesimal contact transformations withcompact support defined by

( ) ( )s jVjs

A jV Ld h= ò . (2.2)

We shall say that a section s is critical when

Appendix D PHYSICAL GEOMETRY278

( )sA jVd = 0 . (2.3)

We may define an (m-1)-form L, called the Legendre transformation form, by

: m vTJ E p TEL - * *Ä 1 1 , (2.4)

with the following local expression:

ii p dyL L *= · , (2.5)

where

!m

i i

J Ldx dx dx

m za b

mabm

¶L e

¶-= - 1

. (2.6)

We can also define the Poincaré-Cartan form P, associated to the given variationalproblem as the m-form

: mTJ E RP Ä 1 , (2.7)

LP Q L h= · - , (2.8)

where the exterior product is taken with respect to the bilinear product defined by thenotion of duality .

The expression for the Lie derivative in eq. (2.2) may be calculated. For every infini-tesimal canonical transformation jV we have

( ) ( ) ( ) ( )j V L jV D f d jVh Q L h P Q L¢= · + - + · , (2.9)

where L’ is a p*TEv* valued (m-1)-form on J1E and f is a unique section of p*TEv*. Thesymbol indicates the interior product of forms. We shall call the resultant m-form DL+fh,the Euler-Lagrange form associated to the given variational problem.

Noticing that Qjs = 0 and neglecting the exact form in eq. (2.9) we get from eq. (2.2)that a section s Î S is critical if and only if the Euler-Lagrange form is zero on the 1-jetextension of the section (Euler equation)

( )js

D fL h+ = 0 . (2.10)

Given a section s of E, let Xs be the space of all sections Vs of s*TvE. By identifying Mwith s(M) by the section s, every VsÌXs, defines a vertical vector field of E defined ons(M). Then there is a vector field V of J1E uniquely defined on j1s(M) such that

s spV V* = , (2.11)

( )( )sV js M

Q = 0 . (2.12)

From the definition of jet prolongation we may show that sV is the value taken on j1s(M)by the 1-jet prolongation j1V of some local extension V of Vs .

If s is critical we can define the hessian of L at s as the bilinear symmetric functional

279Jet Bundles

References

1 R. S. Palais, Foundations of Global Non Linear Analysis, (W. A. Benjamin, New York)(1968).

2 R. Hermann, The Geometry of Non Linear Differential Equations, BäcklundTransformations and Solitons (Math. Sci. Press, Brookline) (1976).

3 P. L. García, Symp. Math. 14, 219 (1974).4 P. L. García, J. Diff. Geom. 12, 209 (1977).5 P. L. García, Rep. on Math. Phys. 13, 337 (1978).

:s s sA Rd X X´ 2 , (2.13)

( ) ( ) ( ) ( ),s s s jVjV jVjs js

A V V L jV D fd h Q L h¢ ¢¢ = = +ò ò2 . (2.14)

The kernel of (d2L)s is the subspace of Xs defined by those vector fields VsÌXs, suchthat

( )sjV D fL h+ = 0 . (2.15)

We define the Jacobi vector fields on the critical section s as those vector fields whichsatisfy the last equation for the second variation (or linear varied field equation).

E. SOME PROPERTIES OF FIBERBUNDLES.

We group here certain definitions regarding manifolds, fiber bundles and their topol-ogy. For a complete treatment see the references [1, 2, 3, 4].

E.1. Manifolds.An atlas valued in of a topological pseudocategory or -atlas is a triple (Q(E),A,h)

consisting of the following:1-A pseudocategory Q(E) formed by the nonempty open subspaces Ui of a Hausdorff spaceE, as objects, and the partial identity maps, as morphisms,

ii j i jU U U U e EÇ ¹ Æ $ ¬¾ Î ; (1.1)

2- A covariant functor (U)

( ):Q E , (1.2)

that assigns, for each object (chart) of Q(E), an object of the topological pseudocategory;3- A functorial isomorphism which sends the covariant imbedding functor I, of Q(E) inEns, to transition functions

:h I j (1.3)

and determines the local mappings (homeomorphism), hU for each U, which obey thefollowing commutative diagram

( )

( )

U

V

h

U UV V

h

U U

i

V V

j

¾¾

¾¾

(1.4)

and the local transition functions jVU which are bijective mappings (homeomorphisms)

which may form a group,

( ) ( ):UV U Vh U V h U Vj Ç Ç , (1.5)

U UV V V Uh i hj -= 1 . (1.6)

An atlas is a covering atlas if the union of all U is E. Two atlases are compatible if theirsum is also an atlas valued in . An atlas is complete if it is a covering atlas and containsall compatible atlases.

A -manifold is the Hausdorff space E together with a complete -atlas. If the objectsof are Banach, Hilbert or Euclid spaces we obtain respectively Banach, Hilbert or Euclidmanifolds. If the objects of are n-dimensional symmetric spaces X with its group G of

281Some Properties of Fiber Bundles

similarities as transition functions we obtain an (X,G)-manifold [5 ]. In particular we havea hyperbolic manifold if X is a hyperbolic space H and G is its isometry group I(H).

E.2. Fiber Bundles.Define a fiber bundle in the following manner [1]. Consider

:p E M , (2.1)

an object of S(Top), the arrow category of topological spaces (category formed by allmorphisms of the category Top as objects and commutative diagrams as morphisms). Ifthe mapping p is surjective it is called a projection, E is the total space, M is the base spaceand the triple is a projected space. If m Î M we define the vertical space,

( ) verticalmp m E- º1 . (2.2)

The projected space is called a sheaf if the projection is a local homeomorphism. Eachpoint e of E has an open neighborhood homeomorphic to an open neighborhood of p(e).

Define pseudocategory (M,F) taking as objects the products U´F, where U are theopen subsets of M and F is a topological space, and as morphisms preserving the projec-tion over U,

:U F V Fa ´ ´ , (2.3)

pr id pr V U U Va = Ç ¹ Æ . (2.4)

A fiber bundle is the manifold formed by a projected space E over M with an atlasvalued in(M,F) compatible with the projection p. If the transition functions of the atlasform a group we obtain a fiber bundle with structure group G. If the fiber coincides withthe structure group we have a principal fiber bundle (E,M,G).

In a principal bundle (E,M,G,p) a local section s over an open set UÌM is a C¥ map

:s U E¾¾ , (2.5)

s Ip = , (2.6)with ps the identity in U. Then, if s is global, any point e may be written as s(m)g and(m,g)ÎM´G is a direct product, a trivialization of E. If a principal bundle has a globalsection the bundle is trivial.

Consider a principal bundle p:EM with structural group G and corresponding Liealgebra A. Denote by TGE the bundle of G-invariant vector fields on E. Denote by AdE theadjoint bundle of E, which is the fiber bundle associated with E by the adjoint representa-tion of G. It is the subbundle of TGE defined by G-invariant vector fields which are tangentto the fiber (vertical). It is a bundle of Lie algebras.

A connection on the principal bundle may be defined by an splitting of the short exactsequence of vector bundles [6 ],

GAdE T E TMp ¾¾ 0 0 , (2.7)

where the splitting

: GTM T E H Vw = Å (2.8)

Appendix E PHYSICAL GEOMETRY282

is a homomorphism defining the horizontal subspaces of the principal bundle. These hori-zontal subspaces define a connection form w.

A connection on E may be identified with a section of the bundle of connections p:WMdefined as follows. For a point mÎM, let Wm be the set of homomorphisms w: TM TEsuch that

Ip w = . (2.9)Define W = UMWm letting p be the natural projection of W onto M. It may be seen thateach point wm, of the vertical space Wm of the fiber bundle W corresponds to a vector spacecomplement of AdE in TGE. It is known [7] that the space of linear complements of avector subspace in a vector space has a natural affine structure. Therefore the fiber of thebundle W is an affine space with linear part L(TGE / AdE, AdE) » L(TM, AdE).

E.3. Homotopic Product.There are topological spaces characterized by an entire number associated to the homo-

topy groups, which we shall call winding, or more appropriately, wrapping numbers n forthe different homotopy groups. For a complete treatment of this subject see references[2, 3].

The neighborhood of curves C(Y,y0) in a manifold Y is the collection of all continuousmappings

:f I Y1 (3.1)

of the unit interval which satisfy

( ) ( )f y f= =00 1 . (3.2)

Let f and g be two mappings in C(Y,y0). The juxtaposition of f and g is the element ofC(Y,y0) given by

( )( ) ( ) ( )( ) ( )

f g x f x x

g x x

* = £ £

= £ £

12 0 212 12 . (3.3)

Similarly consider the mappings from the n-cube into Y

: nf I Y (3.4)

such that they send the boundary of In to point y0 of Y. The boundary bIn of In is defined asthe points that satisfy

( )n

i iix x

=- = 1

1 0 . (3.5)

That is, f satisfies

( )nf I yb = 0 . (3.6)

Let us denote the set of these mappings by Cn(Y,y0).Define a homotopy relation in Cn(Y,y0) saying that f and g are homotopic modulus y0 if

there is a continuous mapping

283Some Properties of Fiber Bundles

: nh I I Y´ 1 (3.7)

which satisfies

( ) ( ), nh x f x x I= Î0 , (3.8)

( ) ( ), nh x g x x I= Î1 , (3.9)

( ), nh I t y tb = £ £0 0 1 . (3.10)

This is an equivalence relation in Cn(Y,y0) and decomposes it into equivalence classesformed by the arcwise connected components of Cn(Y,y0).

The juxtaposition of two elements of Cn(Y,y0) is the element given by

( )( ) ( ) ( )( ) ( )

, ,

, ,

n

n

f g x f x x x x

g x x x x

* = £ £

= - £ £

1 2 1

1 2 1

12 0 212 1 12

. (3.11)

The homotopy group pn of Y on the point y0 is defined as the classes of Cn(Y,y0) with thegroup operation defined by juxtaposition,

[ ] [ ] [ ]f g f g= * . (3.12)

These definitions, based on the mappings from the n-cube into Y, may be expressed asmappings from spheres

: nh S Y . (3.13)

E.4. Third Homotopy Group.In particular we are interested in the third homotopy group of group spaces, taken as

fiber bundles over cosets. We have the general homotopy sequence [3],

( ) ( ) ( ) ( ))i pG GH GH HDp p p p* * *¾¾ ¾¾ ¾¾4 3 3 3 , (4.1)

To determine p3(G), we recognize that the exponential mapping from its maximal noncom-pact subalgebra is a diffeomorphism [4] onto the coset G/H where H is the maximalcompact subgroup. This coset is a noncompact riemannian space which is contractile andits third homotopy group is the identity. Therefore, we have an exact short sequence

{ } ( ) ( ) { }*iH Gp p¾¾ ¾¾ ¾¾3 30 0 , (4.2)

which implies the intermediate mapping is an isomorphism and

( ) ( )G Hp p=3 3 (4.3)

where H is the maximal compact subgroup.It is also known that there is an isomorphism between the homotopy groups of a group

and its covering group, except for the first homotopy group [8 ]. Then we obtain for the

Appendix E PHYSICAL GEOMETRY284

References

1 I. Vaisman, Cohomology of Differential Forms, (Marcel Dekker, New York), p. (1973).2 J. G. Hocking, G.S. Young, Topology (Addison-Wesley, Reading) p. 159 (1961).3 G. W. Whitehead, Elements of Homotoy Theory (Springer Verlag, New York) (1978).4 S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

p. 214 (1962).5 J. G. Ratcliffe, Foundations of Hyperbolic Manifolds (Springer-Verlag, New York) (1994). 6 P. L. Garcia, Rep. on Math. Phys. 13, 337 (1978). 7 I. Porteous, Topological Geometry, (Van Nostrand Reinhold, London), ch 13 (1969). 8 F. H. Croom, Basic Concepst in Algebraic Topology, (Springer Verlag, New York) (1978).

homotopy group of SL(4,),

( ) ( ) ( )( )( )( ) ( )

( , )

( )

SL SU SU

SU SU Z Z

p p

p p

= Ä

= Ä = Ä

3 3

3 3

4 2 2

2 2

. (4.4)

Similarly, we have for the homotopy groups of the other two possible holonomy groups,

( ) ( )( ) ( )( , ) ( )Sp R SU SU Zp p p= Ä = =3 3 34 2 2 , (4.5)

( ) ( )( , ) ( )SL SU Zp p= =3 32 2 . (4.6)

F. NEWTON’S GRAVITATION AND GEO-METRIC THEORIES.

F.1. Spacetime Limits.

F.1.1. Instantaneous Propagation.The development of the geometric physical ideas is based on relativity, gravitation and

electrodynamics. We have shown that the fiber bundle geometry determines a generalizedEinstein equation which is related to the space-time even curvature. This curvature is an Ehres-mann curvature determined by the Lorentz SO(3,1) group. We also know that in Newton’stheory, gravitational signals travel instantaneously and it is natural to expect that PhysicalGeometry, as any other Einstein type theory, would reduce to Newton’s theory in a limit ofinstantaneous propagation. This relation with Newton’s theory is simplified using the geo-metric representation of the latter which is the Newton-Cartan theory [1, 2, 3]. The mainmathematical difference is that Cartan’s formulation does not provide a metric but relieson a nonriemannian affine connection, a tensor of valence (2,0) and rank 3 (singular metric)and a scalar time function. Since we have taken the connection of the required space-timefiber bundle geometry as the fundamental representation of interactions in general, we are ina position to express this relation appropriately by essentially using the contraction of theSO(3,1) group to ISO(3) which is in accordance with the Newton-Cartan theory.

It is known that gravitational signals should propagate in wave fronts formed on thecharacteristic hypersurface of the system of differential equations. On these hypersur-faces, characterized by a null normal vector, the Cauchy initial value problem cannot besolved without restriction. At each point of this hypersurface there is also a tangent char-acteristic local cone of null vectors. The rays of the propagating disturbances orbicharacteristics of the system are its normal null geodesics. To obtain a limit, we shall lookfor a limit of the characteristic gravitational null cones of the corresponding theories.

We introduced a frame u of orthonormal tetrads as is common in gravitation [4, 5, 6]. Thisframe is related to the relativistic electromagnetic light-cone structure at each space-time point.

ˆˆˆˆg u ua b

mn m nabh= . (1.1)

We now split the metric covariantly into

ˆ ˆ ˆˆˆˆ

a bab

g u u u u hmn m n m n mn mne d t e= - º +0 0 2 2 (1.2)

which serves as a definition for singular tensors t of rank 1 and h of rank 3. Strictly, thestructure expressed by equation (1.2), is pseudoriemannian but we shall call it rieman-nian as customary.

We expect that h determines a tridimensional space connection

( )ab

amn m nb n mb b mn

hh h hG = ¶ + ¶ -¶

2(1.3)

Appendix F PHYSICAL GEOMETRY286

related to Newton’s geometry.In special relativity it is assumed that the maximum speed of signals given is equal

to the speed of light determined from Maxwell equations. For the moment we do notmake any assumptions about a possible relation between the speeds of gravitationalend electromagnetic signals. Thus, we allow for the possibility that the speed of propa-gation of disturbances in the gravitational metric field may differ from the speed oflight. The units of distance and time are assumed determined by processes indepen-dent of gravitation. The unit of time may be defined by some definite atomic frequency.The unit of length may be derived from the unit of time by the requirement that thespeed of light be the unit velocity. In this way all components of the metric tensor aredimensionless.

If we now take a null vector l, we know

ˆ ˆg l l u u l l h l l cm n m n m n

mn m n mn= + =0 0 2 0 , (1.4)

therefore we find

( ) ( )ˆ ˆ

h l l l

u l u l

m nmn

a aa a

e

-= =

2

2 220 0

1(1.5)

where we have defined the scalar l. The expression in the denominator is the pro-jection of the null vector on the timelike vector of the tetrad. We now have aninterpretation for the characteristic dimensionless scalar e related to the character-istic null cones,

ˆ lu lm

me º 0 . (1.6)

To make the usual assumption of equality of the speed of gravitational and elec-tromagnetic waves we just set e equal to 1 in our equations. The mathematicalassumption that any nonzero null vector be orthogonal to the timelike vector ofthe tetrad is equivalent to the physical assumption that gravitational signals propa-gate instantaneously. This geometric assumption will be used to establish the desiredlimit.

We shall investigate the geometry expressed by equations (1.2) and (1.3) in themathematical limit e0, l0, which we shall call the limit of instantaneous propa-gation [7]. In this limit the speed of gravitational signals tends to infinity. Thelocal gravitational null cone at a point, or locus of all null vectors, opens as thislimit is approached, reducing to a single local hyperplane, normal to the timedirection, separating the absolute future from the absolute past. We can call thishyperplane the local plane of simultaneity. When e equals one, we get the usualRiemann geometry of Einstein type theories. When e equals 0 we obtain the limitof instantaneity where gravitational wave fronts propagate with infinite speed alongthe limit of the bicharacteristics of the metric. Otherwise we have Riemann spaceswhere the gravitational equations hold but light does not necessarily travel alongthe bicharacteristic of the metric. This limit is equivalent to a stationary limit inwhich e would represent the ratio of a characteristic velocity over c and may beused as a low velocity approximation.

287Newton’s Gravitation and Geometric Theories

F.1.2. Local Limit.We assume that the singular dependence of the metric on the parameter e is the

“minimal dependence” required by the local agreement with special relativity.

ˆˆˆab

abh

d


1 00 . (1.7)

In a sense, what we are doing is introducing a geometrical structure in our manifold andinvestigating the limits as the scalar e goes to zero. To define the concept of limits ofspace-times we follow, in general, the work of Geroch [8] and extend it to the case ofaffine spaces [7]. By a geometrical structure we mean a general statement of the types offields under consideration, that is, the number of connections, the number and valencesof tensor fields, etc. By a realization of a geometrical structure we mean a connectedHausdorff manifold, equipped with fields of the type described by the given geometricalstructure.

Let M and M’ be manifolds with realizations f and f’, respectively, of a givengeometrical structure. By an isoaffinity we mean a diffeomorphism of M onto a subset ofM’ which takes f into f’, preserving the geometrical structure. Consider a one param-eter family of manifolds, that is, for each value of a parameter e >0 we have a tetradimen-sional manifold M(e) with some geometrical structure. We are interested in finding thelimits of this family as e goes to zero. Assume now that the manifolds M(e) may be puttogether to make the Hausdorff pentadimensional manifold M’. Each M(e) is to be asubmanifold of M’. The parameter e now represents a scalar field on M’ while thegeometric structure on M(e) defines a geometrical structure on M’.

The problem of finding limits of the family (M(e), f(e)) amounts to that of placing asuitable boundary on M’. We define a limit space of M(e) as a five-manifold M’ withboundary M’, equipped with a continuous geometric structure, a scalar field e and asmooth one to one mapping Y of M into the interior of M’ such that the followingconditions are satisfied:

1) Y is an isoaffinity, i.e. takes f into f’ and e into e’.2) M’ is the region e=0, required to be connected, Hausdorff and nonempty.

The first condition ensures that M’ really represents M with a boundary attached; thesecond condition ensures that the boundary represents a limit as e goes to zero.

It is always possible, given a limit space M’ of M to find some continuous C0 familyof frames which assumes a limit as e0. By a continuous Cl family of frames in M wemean a tetrad u(e) attached to a single point m(e) of M(e) for each e0 such that theu(e) are continuous vector fields of class Cl in the parameter e along the smooth curvein M’ defined by points m(e). Let us represent points in M(e) in a neighborhood ofm(e) in the normal coordinate system based on u(e). In terms of these coordinates,the components of the geometric objects in the M(e) approach a limit as e0 and thelimiting components are precisely the components of a geometric object in M’ in aneighborhood of m(0). Thus the family of frames u(e) uniquely defines a geometricalstructure in the limit space M’ at least in a sufficiently small neighborhood of u(0).Global properties of the limit will be discussed in the next section.

Then, we can say that a limit of the family of manifolds M(e) is characterized locallyby a continuous C0 family of frames. We have related the metric at anyone point by thedefining equations (1.1) and (1.7) to a frame at that point. This frame represents the restframe of an observer located at that point, and defines the local null cone which the


observer may determine by experiments. As we vary e opening the null cone, we ob-tain a one parameter family of Einstein type theories, each with a different value of e,until we reach the limit theory at e=0. Each theory is represented by a tetradimen-sional manifold (space-time) M(e0) and may be put together to form a pentadimen-sional manifold M’.

In each of the manifolds M(e0) we have an observer, which defines a rest frame atsome point. We have to define how we map points and observers from one manifold toanother with different values of e. That is, we are free to specify when one single pointand one single observer at M(e’) are to be considered the same as another point andobserver at M(e). In a more mathematical language, we define a curve in M’ whichcontains the set of single points in each M(e) which are considered equivalent pointsand we define a family of frames in M which contains the set of single frames (ob-servers) in each M(e) which are considered equivalent frames. We are free to definethis equivalence of points and frames only for a single point and frame, because thegeometrical structure that we shall assume is a rigid structure (in the sense of Geroch),and the equivalence of other points and frames is then determined by the geometricalstructure, as we shall see in the next section.

F.1.3. Global Limit.Geroch [8] uses the metric to prove that the limit space has a unique extension. In

our case the metric is singular in the limit and we have to generalize [7] the theoremsproved by Geroch. In order to do that we will rely on the affine connection rather thanon the metric. We know that in the tetradimensional formulation of Newton’s theory,the affine connection is not singular [7]. Our objective is to show that the newtonianconnection is in fact the limit of the Riemann connection, but for the moment we justassume that our manifold M’ is provided with a continuous affine connection which, asindicated by Geroch [8] should give us gives us the necessary rigid geometrical structure.Let us represent points in M(e0) in terms of the system of normal coordinates basedon u(e0). In terms of these coordinates the components of the connection in M(e0)have a limit as e0 and therefore define the affine connection in M’ in a neighbor-hood of the point m(0).

Let M and M’ be connected space-times provided with an affine structure and let (v1,v2... vn) be any collection v of n nonzero vectors at point in m in M. We construct abroken geodesic as follows. Let g1 be the geodesic, which passes through m and whosetangent vector at m is v1. Choose an affine parameter l such that, at m, it is zero andobeys

vaa l =1 1 . (1.8)

Let m1 denote the point on g1, unit affine distance from m. Parallel transport the n-1vectors v along g1, to m1. Now repeat this construction with these n-1 vectors at m1 andthus define a point m2. Repeat with n-2 vectors at m2 and thus define a point m3, etc.After the n step we obtain a point mn (We restrict ourselves to n-tuples of vectors v at mfor which, at each step in the construction, the appropriate geodesic can be extended aunit affine length). Since any arbitrary point m’ of M may be joined to m by a brokengeodesic, we may always choose n and the set v so that m’ is mn.

Let Y, Y* be two isoaffiniies from M to M’ which take a tetrad u at m into anothertetrad u’ at m’. Then Y and Y* have the same action on any vector v and both Y(m) andY*(m) are defined by broken geodesics in M’ with the same set of initial vectors. There-


fore, Y coincides with Y* for each point m of M and we have (Theorem 1) [8]: there isat most one isoaffinity of M into M’ which takes G to G’and u to u’. We say that wehave a rigid structure of order 1

Let M1 and M2 be two limit spaces of M. We say that M2, is an extension of M1 ifthere exists a smooth mapping of M1 into M2 which preserves the geometrical struc-ture and leaves invariant each point of M. The previous theorem implies that, if M1 isan extension of M2 and M2 is also an extension of M2 then M1 equals M2.

Construct the disjoint union of all extensions of M’ and denote it by N. We nowdefine an equivalence relation in N. If m1 belongs to M’1 and m2

belongs to M’2 we

write m1m2 if there exists a family of frames in M which in M1, has a limit at m1 and in M2has a limit at m2. We see that whenever m1m2 there exist neighborhoods of m1 and m2which are also identified. Thus the set of equivalence classes form, in a natural way, alimit space M+. By construction M+ is an extension of every extension of M’. If M* is anextension of M+, we must have M* equal M+ and therefore M+ has no proper extension.Then we have (theorem 2) [8]: every limit space M’ has a unique extension M+ such that(1) M+ has no proper extension and (2) M+ is an extension of every extension of M’.

We shall define a family M(e)of manifolds which represent the points, connectionsand frames (observers) in the original tetradimensional Riemann manifold. If the fam-ily of frames defined by one equivalence class of points and observers has a limit inM’, it singles out a particular boundary limit geometry among all the possible geom-etries in the limit of instantaneity.

The question of whether a particular family of frames, constructed in this way has alimit or not, cannot be settled unless we add more information. By requiring that thisobserver family of tetrads be continuous, we have introduced an important physicalassumption. The limit of instantaneity is singled out by requiring that any family offrames defined by equations (1.1) and (1.7) is in fact continuous C0. In other words,once we specify how the tangent space of a particular point m of M is to be mappedinto the tangent space of a particular point m’ of M’, the behavior of Y everywhere isdetermined by the action of Y on the tangent space. Therefore every differentiablefamily of frames either defines no limit space or else defines a limit space which ismaximal. In our case, we can say that the newtonian limit to be obtained in a smallneighborhood by the use of the family of tetrads u may be extended uniquely.

F.1.4. Postulates.In order to discuss newtonian space-time geometry as a limit of relativistic space-

time geometry we have made the following postulates:1. - There is a diffeomorphism Y of M into the interior of M’ which is an isoaffinity (i.

e. takes G into G’, u into u’, e into e’, etc.).2. - The boundary M’ is the region given by e equal to zero. We require that M’ be

connected, Hausdorff and nonempty. We also require that M’ be a space with the sametopology of the spaces M(e0).

3. - A family of tetrads u(e0) corresponding to a single observer at a single pointand related to the Riemann metric by equations (1.1) and (1.7) is continuous of class C0

along a curve g in M’ (assumption of instantaneity).Now, in order to insure the necessary rigidity, we add a fourth postulate:4. - The mapping, given in riemannian geometry, of the tetrad field u to the affine

connection G is continuous of class C0 along the curve g in M’.


F.2. Geometric Rigidity Condition.We know that in a Riemann space the connection is metric and the covariant de-

rivative of the metric tensor vanishes

ga mn = 0 . (2.1)

Our general postulate (4) implies that this equation has a limit in the boundary M’ ina neighborhood of the limit point of the curve g. We consider that this equation givesus a mapping between the tetrad field and the connection field, which we may write asfollows

ˆ:uam G1 . (2.2)

Since the mapping m is continuous (postulate 4) and its domain is continuous (postulate3) we conclude that its range, the connection, is also continuous of class C0 along thecurve g in M’. The affine connection at point m’ in the boundary M’ should be obtainedsmoothly from the Riemann connection as e0. This provides the necessary rigidity toour limit space M’ as described in a previous section.

A scalar time coordinate function 0t, defined in the boundary space M can be ex-tended from a point 0m’ in the boundary M to a half open neighborhood U’ of 0m’ in M’by assigning the same value of t to equivalent points in each M’(e). Thus we havedefined a function t on U’. We can define the orthogonal vector field t on U’

t tm m= ¶ . (2.3)

By using the freedom we have in selecting the direction of the tetrads in each M(e)without changing the metric [4, 5] we can choose u(e) so that its vector u0 aligns withthe direction of the vector field t at every point in U’.

ˆu tm mz=0 . (2.4)

In a previous section we have split the metric into

ˆ ˆ ˆˆˆˆ

a bab

g u u u u hmn m n m n mn mne d t e= - º +0 0 2 2 (2.5)

and its inverse into

ˆˆˆ ˆ ˆˆ

aba b

g u u u umn m n m ne d-= - 20 0

. (2.6)

If we express the tetrad field in the neighborhood of a point m in the curve g, in termsof the coordinates adapted to the tetrad at point m the components of the tetrad fieldshould be continuous in a neighborhood of the curve g. We have then that there existnonsingular normal coordinates, based in the family of frames u, where both t and h arenot singular as e0. In this limit we obtain a relation valid in U’ in M’. Using adaptedcoordinates indicated by , in the neighborhood U’

tm nd0 (2.7)

and we can write


mn

zt


2 00 0

. (2.8)

We also obtain, in U, because of the orthogonality of the tetrad,

ˆ

iu um

m =0 0 , (2.9)

mnhh

mné ùê úê ú-ë û

0 00

(2.10)

and its normalization,

ˆˆu u tm

m z=0 2 00

1 , (2.11)

mt

Am z-é ù

ê úê úë û

2

, (2.12)

m

n m n

A

A A Amn z

tz

-é ùê úê úë û

2

2 , (2.13)

where A is the projection of the vector tm on the hypersurface of constant time t. The3-vector A vanishes if tm is hypersurface orthogonal.

From the relation of the metric g and its inverse we obtain

mmn

n

A Ah

A hmn

z zz

é ù-ê úê ú-ë û

4 2 2

2 (2.14)

where h is a 3-metric in the t constant hypersurface and

nm mnA h A . (2.15)

It follows from equation (2.1) that

haa mn mnt e= - 2 . (2.16)

This equation is equivalent to the expression of the connection from the tetrads. Inother words, it does not impose a restriction in the order of the connection. The onlypossible contribution to a singularity of the connection comes from the term e-2 associ-ated to h in the metric. Taking in consideration the orthogonality of h and t we have forthe possible singularities

( )h hab ab

m nb n mb b mn b mnt t t te e

-¶ +¶ -¶ = ¶2 22 2

. (2.17)

From equations (2.8) and (2.16) we see that, for small e,


( )Oa z e¶ =2 2 , (2.18)

so that the derivatives are not singular as e0. The connection has a limit as e0,providing the necessary rigid structure to the pentadimensional space M. Since z(0) isconstant it may be set equal to 1 by redefining the function 0t in M. This implies that tmay be put in the form

( )t t t tmn m n m nt z e j= º +2 21 2 (2.19)

where j remains finite as e0. We also have

ta m =0 0 . (2.20)

One point in the original Riemann space is equivalent to one point in the boundaryspace when they are represented by broken geodesics constructed from vectors withthe same components with respect to our family of tetrads. We assumed in our postu-lates that the topology in the boundary M’ (newtonian space) is the same as the topol-ogy in the spaces M(e0) (Riemann spaces) so that separate points are equivalent toseparate points. This introduces a one to one correspondence between points in Ri-emann and Newton spaces. The equivalence classes of points in the pentadimensionalmanifold M, discussed in the previous section, can be extended to the limit space M’

F.3. Boundary Geometric Connection.Rather than attempt to associate the metric with the newtonian gravitation theory

we have chosen to take the connection as fundamental. The affine connection in theboundary M’ is the limit of the connection in the Riemann space. It is important tostate that this limit is nontrivial, that is, it does not lead to tetradimensional flat affinespace, showing there are gravitational effects in this limit.

We define the boundary affine connection or 0G as the limit, as e0 of the connec-tion in the Riemann space,

( )lim gg g g

aba

mn m nb n mb b mneG

= ¶ + ¶ - ¶0

0 2 . (3.1)

In this expression, we substitute the metric g by t and h and note that the t derivativesterm becomes in adapted coordinates

( ){ }t t t t tm nb n mb b mn b n m m n m n bt t t e j j e j¶ + ¶ - ¶ = ¶ + ¶ - ¶2 22 2 . (3.2)

Then, keeping in mind that t is orthogonal to h, we obtain for 0G

( )( ) ( )

( ) ( )lim

h t t t t t t t

h t th h h

ab a b am n b n m m n

aab a bmn e

m nb n mb b mn

e e j j e j j

G ee j

ì üï ï- + + ¶ + ¶ + ¶ï ïï ïï ïï ï= í ýæ ö÷ï ïç ÷+ + + ¶ + ¶ -¶ï ïç ÷ï ïç ÷çè øï ïï ïî þ

2 2 2

02

0 2

1 2

1 22 2

(3.3)


( )hh t t h h h

aba ab

mn m n b m nb n mb b mnG j= - ¶ + ¶ + ¶ -¶0

0 0 0 0 0 0

2(3.4)

which is true in any coordinate system.In detail we have, in adapted coordinates,

mnG0 0 0 , (3.5)

( )a ab abb b bh h A AG j¶ - ¶ + ¶0 0 0 0 0 0 2

00 0 2 , (3.6)

( )ab

am m b mb b m

hA h AG

-¶ - ¶ - ¶

00 0 0 0

0 02 , (3.7)

mG0 00 0 , (3.8)

( )ab

a amn mn m nb n mb b mn

hh h hG G ¶ + ¶ -¶

00 0 0 0 0

2 . (3.9)

These results for our boundary space are good in a sufficiently small neighborhoodof a point, but using the general results of the previous sections we know that the limitspace obtained can be extended uniquely. The connection in the boundary has thestructure of the newtonian connection. We can define

( )hh t t h h h

aba ab

mn m n b m nb n mb b mnG jº - ¶ + ¶ + ¶ -¶2

(3.10)

and split the connection as follows

( )( )

( )

t th h h t t

t t

a ba a

mn mn m nb n mb b mn b m n

am n

eG G e j j

e j

= + + ¶ + ¶ -¶ - ¶

- ¶

22

2

1 2 22

2

. (3.11)

It follows that the Riemann tensor is not singular as e0 in adapted coordinates.Nevertheless, the scalar R may be singular. The limit of the Ricci tensor is, in terms of 3-tensors on the tridimensional subspace,

m m n mm m m nR G G G G - ¶ -0 0 0 0 0 0

00 00 0 0 0 0 , (3.12)

n nm n m m nR G G -¶0 0 0 0

0 0 0 , (3.13)

mn mnR R0 0 . (3.14)


F.4. Newtonian Connection.Up to now we have concentrated in discussions of the limits of the tetrad and con-

nection. The boundary connection, defined above by our limiting procedure, will nowbe shown to be the same connection given for a tetradimensional formulation of Newton’stheory and therefore we shall call it the newtonian connection. If we have a theory ofgravitation with a field equation relating the geometry to a tensor representing matterwe may add another postulate to accomplish this objective. In general, the stress energytensor Tmn may be defined as the contravariant tensor defined by variations of the fieldlagrangian with respect to the metric gmn. We may consider the solution to the gravita-tional nonlinear field equation to be a nonlinear mapping between the stress energytensor field and the connection field. We should require this mapping to be differen-tiable. Therefore we will make the assumption that the nonlinear equations which arisein our analysis are expressions of differentiable nonlinear mappings.

Postulate 5.- The mapping, given by the gravitational field equation, of the affineconnection G and the tetrad field u to the stress-energy tensor T is continuous of classC0 along the curve g in M’.

For the geometric structure of Einstein type theories, including the physical geomet-ric theory, this postulate implies that the field equations are valid in the limit e0 (theyare hereditary). We consider that the field equations give us a mapping between thestress energy tensor and the tetrad, which we write as follows

( ): ,u Tm G 2 . (4.1)

Postulates 4 and 5 imply that the stress-energy tensor should also be continuousalong the curve g in M’. In the limit the relation between the scalars R and T is broken dueto singularity in the metric. Although R may be singular, we have then that T is nonsingu-lar as e0 in adapted coordinates. We obtain, from the field equations displaying theexplicit dependence on the metric, the following equations in the limit

lim limR g g T g g Tab abmn ma nb mn abe e

k

æ ö÷ç= - ÷ç ÷çè ø0 0

12

, (4.2)

( )limR T t t t t O T t t t tab abmn a b m n a b m ne

k e k

æ ö÷ç= + =÷ç ÷çè ø0 2 0 0

0

1 12 2

. (4.3)

In particular

( )lim lim limmn mn m n mnR R g g T g g T Oab aba b abe e e

k e

æ ö÷ç= = - = =÷ç ÷çè ø0 2

0 0 0

1 02

. (4.4)

This equation implies the vanishing of the tridimensional Ricci tensor on the time hyper-surfaces t in the boundary as e0. In three dimensions the Riemann tensor has only sixnonvanishing components as a consequence of its symmetry properties. Hence, thevanishing of this Ricci tensor implies the vanishing of the tridimensional Riemann ten-sor on the boundary. Therefore the tridimensional space in the boundary is flat and its

metric ( )h 0 can be put equal to d, as it should be to have agreement with Newton’stheory (the tetradimensional space still remains curved).


Equations (3.13) implies

( )om a m a a mR A A-

¶ ¶ -¶ =0 0 01 02

. (4.5)

This equation accepts solutions, in appropriate coordinates, with vanishing A. Theboundary singular tensors associated to the metric become, finally,

mnté ùê úê úë û

0 1 00 0

, (4.6)

mn

hmn d


0 0 00

. (4.7)

The vector tm is now orthogonal to hypersurfaces of simultaneity t as it should be forNewton’s theory. The hypersurfaces of simultaneity of the boundary space are reallythe hyperplanes of simultaneity of the Newton theory.

With these tensors, the boundary affine connection given by equation (3.4) reducesto the newtonian connection. This connection may be split into a tridimensional newto-nian inertial connection and a newtonian gravitational field tensor,

h t ta a abmn mn b m nG G j= + ¶0 0 0 (4.8)

and the Riemann tensor becomes

a aba a bR t t t tmn m n m nG d j= = ¶ ¶0 0 0 0

00 . (4.9)

The field equation reduces to Newton’s field equation in the tetradimensional for-mulation if we define the matter energy density r,

R t tmn m nkr=0 . (4.10)

Furthermore, this equation reduces to Poisson’s equation

aa G¶ ¶ j p r= 4 . (4.11)

The geodesic equation for the newtonian connection 0G gives us the newtonian equa-tions for a monopole. For an arbitrary stress-energy tensor T the Einstein equationsof motion are given by

T aba = 0 (4.12)

if there is sufficient knowledge about T (i.e. knowledge about the structure of thebody). This equation reduces exactly to the newtonian equations of motion for a fluid

T aba =0 0 0 . (4.13)

Therefore, the limit e0 reduces Einstein type theories to Newton’s theory if weaccept the general postulates indicated in sections 1.4 and 4.


References

1 E. Cartan, Ann. Ecole Norm. 40, 325 (1923).2 E. Cartan, Ann. Ecole Norm. 41, 1 (1924).3 K. Friedrichs, Math. Ann. 98, 566 (1927).4 F. Pirani, Acta Phys. Pol. 15, 389 (1956).5 F. Pirani, Bull. Acda. Pol. Sci. C1 III, 5, 143 (1957).6 C. Pellegrini, J. Plebanski, Mat. Fys. Skr. Dan. Vid. Selsk. 2, 4 (1963).7 G. González-Martín, Boston U. Ph. D. Dissertation, unpublished (1970).8 R. Geroch, Comm. Math. Phys. 13, 180 (1969).

297Index

INDEX

A

adjoint 237affinity 102algebra

Clifford 236Dirac 242

space-time 11enveloping 58geometric 236Grassmann 59Lie 253operator 56

alpha constantas a geometric coefficient

of invariant measure on coset 226geometric expression for 234

alpha particlebinding energy 183

angleelectroconic 151, 162, 165, 171, 175Weinberg's 155, 171, 175

angular momentum 85orbital levels 109quantum number 66

in Hall effect 108, 114annihilation operator 63antiinvolution

conjugation 236reversion 236

atlas 280automorphism

correlated 237of algebra A 10

B

barionicnumber

of excitation 207barions 208barrier

nuclear potential 184Barut's model

for particles 198base

correlated spinorsgeometric 244

orthonormalClifford 240

base spaceof manifold 281

beautyof an excitation 207

Bergman's kernel 231Bianchi identities 271

in physical geometry 5biquaternion 243Bjorken

scaling law 195bond

nuclear 185pep 184, 185

Bose-Einsteinstatistics 101

bosonic field 62boundary 282bracket

as anticommutator 62as commutator 62as derivation 61for connection Jacobi fields 98for matter Jacobi fields 97operation 60reduction to Lie bracket 61

bundleaffine 61fiber 281jet 275of connections 282

in physical geometry 61of frames 267principal 281

in physical geometry 15SM 86

C

canonicalcoordinates

on jet bundle 276coupling

gravitation to spin particle 7Cartan

canonical generators 67

PHYSICAL GEOMETRY298decomposition 67generators

by orthonormal subset 69geometric 71in particle classification 193

structure equations 270subalgebras

automorphism between 70regular element 70

subspace 67in particle classification 193

Cartan-Killing metric 16definition 256in different representations 123in gravitation 45in induced representation 123, 135in mass definition 94

for connection excitations 147in the substratum 116

Casimir operatorof electromagnetic SU(2) 66

in magnetic moments 149of rotation SU(2) 66

change of energyas intensive parameter 102

chargeelectric

opposite signs 76measurement of 86quantum 73

in Hall effect 108, 114index in Hall effect 111measurement 87

quantum number 66charm

of an excitation 207classical particle

multipole structure 37orthonormal tetrad 32

classical systemsprobabilistic behavior 106

classificationof connections 190

group G 191group L 191group P 191

of interactions 195of matter excitations 195

Clifford algebra 9

coframe 16color

of an excitation 210commutation relation

Cartan canonical form 256Heisenberg 95of Lie algebra 254quantum field theory 63

comoving reference frame 117complex

structure 140, 141neutrino spinor 215

conductivityfractions

FQHE 113conjugation 236

charge 73, 193connection

as horizontal subspace 266bundle of connections 282

as section of a bundle 61bundle of connections 282

as splitting 61of exact sequence 281

classification 190coefficients 273cosmic background 93Ehresmann 266even part 21in physical geometry 16in physical principle 5induced on base 271Koszul 267Levi-Civita 47metric compatible 36

pseudoriemannian 270riemannian 270

newtonian 294odd part 21on boundary 292on fiber bundle 266pulled-back 16space-time 36substratum 136total 93trivial 120

connection excitationsmass or range 156

connection field 62

299Indexexcitations

as photon 156equation for 148mass for 147

free 99massless 156

connection fieldsexcitations

in a lattice 155, 159conservation equation 6

of matter current 18constant

coupling 5identification 87

electron charge 87fine structure alpha 87gravitational 54

determined by substratum 123Planck's 91unit velocity 90weak interactions 166

contact transformationinfinitesimal 277

coordinatecomoving 117

correlation 237induced by conjugation 243

cosetC

in mass ratios 138volume of 138

complex structure 227in mass ratios 140

hyperboloid 139in structure group G 226

in mass ratios 134K 226

equivalence relation in 141in mass ratios 141matricial structure 227realization as polydisc 228

real formsseries of 140

symmetricin structure group 134

volume of 140Coulomb's law 88coupling

canonical

gravitation to spin particle 7nonabelian 25

covariance principle 1generalized 4

covariant derivative 267creation operator 63critical sections 57

in jet bundle 277current

canonical 82conservation 6electric 100interaction 162source 5

geometric 226curvature 16

definition 269energy effects 214, 224in physical geometry 5induced on base 271Maxwell 35Riemann 35Schwarzschild

exterior 52substratum 120

curvature parameter 121of symmetric space 54

D

De Sitter space 66derivative

covariant 267of Clifford base 250

covariant exteriorin terms of covariant derivatives 273of tensorial q-forms 269

Lie 58deuteron

binding energy 180model 176

Diracalgebra 11equation 95

2-component form 22general form 92

generalized equation 6explicit mass term 126

spinor 24

PHYSICAL GEOMETRY300driving force function 102dual formulation

of the geometric theory 25duality

Clifford 239Hodge 239of representations 74, 193

E

Ehrenfest’s theorem 95eigenvalue

mass 131Einstein

equation 44spaces 50tensor 48

Einstein equation 49Einstein-Maxwell theory 2electric charge

quantum 87quantum number 108, 114

electric current 100electric current density 27electrocone

angle 151, 162, 165, 171quantum subspace

in electromagnetic su(2) 151electrodynamics

classical 35Lorentz equation of motion 42quantum 99

electromagnetismgeneralized 42generator

quantization 149standard 27

generators 86identification 42

Maxwell's equations 35standard 12strong 185subgroup 42vector potential 125

in Pauli's equation 170magnetic moments 151, 171

electron 195as unit of charge 88charge

negative 76roles of 87

fieldfree 99

left handed componentneutrino partner for 192

magnetic momentradiative corrections 174

mass 144empty space

solution 50energy

bindingalpha particle 183deuteron 180fusion 186

dark 49, 124equipartition 133, 135magnetic

of excitation 171energy level

degenerateof orbiting electron 110

degenerate superfluxedpossible quanta 111

firstmagnetization flux quanta 110net flux quanta 109of orbiting electron 109

Landau 108energy momentum

equation 19and gravitation 44

energy momentum tensordue to torsion 47from coset fields 46geometric field 49matter current 46

energy operator 95entangled states

Schrödinger 104enveloping algebra

of vector fields 60equation

Cartanstructure 270

conservationdipole 39

Dirac 95

301Index2-component form 22and mass 90generalized 6

Einstein 49energy momentum 19

and gravitation 44for classical motion 40for trivial substratum 119geodesic motion 42Helmholtz 155Klein-Gordon 26Lorentz 42Mathieu

deuteron 179Maxwell 35multipole 39newtonian limit 53newtonian motion 295

in physical geometry 35Newton's field 295Pauli

at nuclear distances 175generalized equation 170

Poisson 295in physical geometry 53

Stehphenson-Yang 49Yang 35Yukawa 148

equation of motion 17and conservation equation 5with explicit mass 125

nonrelativistic approximation 169equipartition

energy 133, 135equivalence

principle 1generalized 4

relationin coset K 141under relativity boosts 136

relativistic relationapplication 143

Euler equation 278even connection

induced from connection 33evolution

active view 98passive view 98

excitation

as extensive parameter 102as particle 188connection

mass 146m-corpuscle 104point 82potential 102, 106

for Young's experiment 105representation 132topological 198

expectation value 83as eigenvalues 85of charge 86

experimental flux quanta 77extensive parameter 102external space 131

F

familieslepton 199particle 197

Fermicurrent

for weak interactions 164lagrangian

for weak interactions 163theory

lagrangian and current 166Fermi-Dirac

statistics 101fermionic field 62field

bosonic 63fermionic 63

field equation 5from variational principle 17substratum solution 116

field quantization 56flavor

of an excitation 207, 210flux

quantum 73, 176flux density

of excitations 101form

canonicalof coframe 269

connection 266

PHYSICAL GEOMETRY302curvature 269Euler-Lagrange 278fundamental on jet bundle 276Legendre transformation 278Poincaré-Cartan 278pullback 253soldering 269

in physical geometry 36structure

of jet bundle 276tensorial 268torsion 270

fractional Hall effect 77fractional quantization 77frame 15

as set of states 83classification 196composite

of interacting systems 161correlated 244eigenvector 83even part of 21matter 15

current 6odd part of 21pure states 85reference 15

in jet bundle 98spinor

even 249of double dimensions 244

subframeof interacting system 191

frame excitationsas group representations 65as particles 65

frame sectionsas wavefunctions 85

free connection field 99free matter frame 99free particle field 99function

Green'sfor connection excitations 148from fluctuation equation 165in weak interactions 163

transition 280functional

Dirac 82

hessian 278fundamental form

of jet bundle 276fusion

binding energy 186

G

Gell-Mann-Nakano-Nishijima formula 208generator

as quantum operator 85electromagnetic 86

quantization 149standard 27

rotation 84geometric

electrodynamics 99geometric algebra 246

algebra Aautomorphisms 26decomposition 21

Clifford productgrade of 59

exterior product 59in physical geometry 9interior product 59Lie product 59null subset 216orthonormal subset 16R2,0 246R3,1 242

geometric currentassociated to geometric measure

in coset K 226normalized 234

associated with excitations 81of a matter excitation

general form 83geometric excitation

annihilation operator 63creation operator 63equation for connection 148G-particle

compared with P-particle 76defines positive charge 76quantum numbers 73

L-particlequantum numbers 76

number operator 63

303IndexP-particle

negative charge 76quantum numbers 75, 76

point 82representation on symmetric space 132

geometric groupG 65subgroup L (Lorentz) 65subgroup of electromagnetism 66subgroup of rotations 66subgroup P 65

geometric measurementas functional 82result of 83

geometric spin 84geometric structure

triple 171, 195, 199geometric unit

of charge 88of mass 166

geometrical regime 101geometrical theory 6geometry

as germ of quantum physics 80substratum 115

global nonlinear analysis 57gradation

of product 60Z 257

gravitationconstant 54

determined by substratum 123effects on neutrino 214, 224Einstein

equation 44energy momentum tensor

geometric 49internal problem 50massless excitation 157

equations for 158newtonian limit 52

and the substratum 122spherically symmetric field 50torsion 47Yang's theory 35

groupDe Sitter 65G 65geometric 65

homotopy 283third 283

ISO(3,1) 66L 65Lie 252Lorentz 65

in physical geometry 9in relativity 1

O(p,q)and the metric 267and torsion 270

P 65Poincaré 66SL(2,) 65

and relativity 3homotopy group of 284topopological numbers 197

SL(2,)in physical geometry 4

SL(2,) 258covering group of 261homotopy group of 197in physical geometry 4physical significance 10subgroups 65

SL(4,)and topological numbers 197covering group of 261homotopy group of 284in physical geometry 4

SO(3,1) 1SO(3,2) 65SO(4) 65SO(4,) 141Sp(2,)

homotopy group 197Sp(4,)

homotopy group of 284topological numbers 197

structure 281in physical geometry 3

SU(2) 65and electric charge 86and neutrinos 222electromagnetic 42rotation 84

SU(3)xSU(2)xU(1) 210U(1) 3

group action

PHYSICAL GEOMETRY304on itself 131

H

Hall conductivityand FQHE 112quantum values 77

Hall effectcarrier model

magnetic vortex 111conductivity 112, 114

fractional values 112plateaus 113

half levelpopulation 111

Landau levelfilling condition 111

vortex modellinked flux 111

hamiltonianinteraction 100

Heisenberg commutation relation 95helicity 67

massless fields 156Helmholtz

equation 155hermiticity

degree of non 166hessian 278holonomy

group 160Sp(2,) 161

homomorphismbetween SO(3,1) and SL(2,) 34

homotopy 282group 283

third 221, 283product 199

homotopy sequence 259hyperbolic manifold 281

structure 12substratum 117

hyperbolic solution 121hyperboloid

parametrizationC space 139K space 142

unit 139hypercharge

strongof an excitation 208

weakof an excitation 208

I

identitiesBianchi 271

in physical geometry 5inclusion

mapping 9inclusion of L into G 33induced representation 132

as sections of vector bundle 65Cartan-Killing metric

in mass ratios 135in the substratum 123

induced from subgroup 65of G 66

injectionlinear 236

integrability condition 5intensive parameter 102interaction

classificationG/P generators as strong nuclear 195L-connection as gravitation 195P/L generators as electroweak 195

interaction connection solution 58interaction current 162

in standard quantum notation 164interaction energy

net 100interactions

electromagnetic 35generators 42

gravitational 44geometric 49in physical geometry 35

strong 196weak 160

interference experimentYoung's double 104

internal problem in gravitationspherical symmetry

equations for 50internal space 131invariant measure

305Indexnormalized 227

involutionmain 236

irreducible representationfundamental 73

isotopic spinof an excitation 208

isotropy groupaction on Silov's boundary 232

J

Jacobifields

canonical algebraic structure 59double algebraic structure 59vector 279

operatorevolving 98fixed 98interaction effects 100

vector fieldon connection solutions 61on frame solutions 58

jet bundle 275in physical geometry 57

jet prolongation 57of sections 275of vectors 277

K

Kaluza 3kaon 204Killing vector 117Klein-Gordon equation 26

L

lagrangianfor first varied equation 163

lawCoulomb 88

left handed componentof spinor 24

particle 192lepton

as topological excitation 198, 220, 221families 199masses 201. See mass values

general expression 201masked 203muon 202tau 202

leptonic numberof an excitation 207

Levi-Civita connection 47Lie

algebra 253derivative 58

generalized 62group 252product 254

limitnewtonian

and the substratum 122general potulates 289in physical geometry 52of space-times 287significance 285

limit boundaryconnection on 292Riemann tensor on 293rigid structure 289

linearization 57, 64first varied equation 162

long range fieldselectromagnetic

and magnetic moment 171Lorentz

equation of motion 5from current conservation 42

groupin physical geometry 9in relativity 1

Lorentzian sectioncurvature 217

scalar 219, 223

M

Mach 1, 129magnetic field

motion in constant field 108magnetic flux

bare fluxquantum number 109

enclosed by loop 109magnetization

PHYSICAL GEOMETRY306quanta 109

net fluxpossible quanta in level 110, 111quantum number 109

orbital quanta 109discrete change in 110

quantum 73, 176quantum number 108, 114

magnetic moment 171anomalous increase 173electron 173neutron 175proton 173, 174

manifold 280Hilbert 280

in physical geometry 63hyperbolic 281

and particles 189in physical geometry 12substratum 117

of sections 57of solutions 57

many excitations regime 101map

adjoint 237differential of 252inclusion 9jet prolongation 275non linear differentiable 57

markoffian system 102mass

as parameterin the substratum 120

as parameter in Dirac's equation 90and the substratum 125for different representations 94for particles 133

bare 133and the substratum 120corrections to 134corrections to mass ratios 144in Dirac's equation 125in general 90in induced representation 137of G-excitations 137of H-excitations 137

connection excitations 146definition 94excess

neutron 182gravitational

Scwarzschild 51in the induced representation 123

mass ratios 135of leptons 201

general expression 201phenomenological relation 198

parameter 91bosonic 189change in 92constant 94fermionic 189in QED 100

units of inverse length 91mass density

newtonian 53mass eigenvalue 131mass values

eta 205heavy leptons

muon/electron 202tau/electron 202

kaon/pion 204masked 203pion/muón 203stable particles

electron/proton 144neutrino/proton 144

weak bosons 155mass-energy 221

neutrino 220variation of 222

massless fields 156electromagnetic excitation 156gravitational excitation 157

Mathieuequation

for deuteron 179functions

hyperbolic radial 180matter

classification 196dark 49, 122, 124

matter current 16matter field 62matter frame 16

as set of states 83Clifford structure of 86

307Indexcurrent 6eigenvector 83free 99in physical geometry 15pure states 85solution 58

matter state 13maximal compact subalgebra 257Maxwell curvature 35measurement

current subject of 81mathematical representation 82physical 80

Meissner effect 78mesons 208metric 267

Cartan-Killing 256in gravitation 45in mass ratios 133in the substratum 116

in physical geometry 16space-time

defined by spinor frame 36gravitational 47in newtonian limit 52in relativity 1in terms of newtonian tensors 285in variational principle 17Schwarzschild 50

symmetric bilinear complex 227in K space 141

modelBarut's 206quark-parton 211standard 211Weinberg-Salam 211

moduleover ring A 60

momentmagnetic

in physical geometry 171momentum

angular 85canonical 109conjugate 82kinetic 109operator 95

and geometric Casimir operator 132geometric point of view 20

in physical geometry 131tetrad 20vector

quantum mechanics point of view 20momentum space

generalized 131multiplicative relation

among quantum numbers 72multipole decomposition 37multipole equations 39muon

massas topological excitation 202

N

neutrinoas Lorentz frame excitation 192, 221as perturbation of composite frame 161association with even part of electron 192

in weak interactions 162channel

for each type 222energy

currents 222for each type 223variation 222

equation 23, 214, 217in weak interactions 161mass 144, 214mass-energy 220

Earth values 223oscillations 214, 223properties 195spinor structure 215SU(2) action 222

neutron 175mass excess 182

Newtongravitation 52mass density 53

newtonianconnection 294curvature tensor 295equation of motion 295field equation 295limit

and the substratum 122general postulates 289

PHYSICAL GEOMETRY308in physical geometry 52of space-times 287significance 285

nonabelian couplingdifferences with quantum mechanics 25

nuclearbond 185

nuclear barrierpep 184

O

observable 81observer

complete 10standard space-time 11

operationbracket 60

operatorCasimir 132

electromagnetic SU(2) 149LaPlace-Beltrami 132momentum 131quantum

bosonic 63fermionic 63

orientationentanglement 9standard 239

orthonormalframe 16subset 16

in physical geometry 6

P

parallel translationof fiber 267

parametrizationin terms of arcs

C space 139K space 142

particle3-point

as proton 195alpha

binding energy 183as excitation 188families 197field

free 99left handed components 192number operator 63right handed components 192weak boson 155

particlesBarut's model 206geometric classification

families 197G-frame as hadronic matter 196L-frame as neutral matter 196P-frame as leptonic matter 196

leptons 202masses 190, 198, 202, 203, 204mesons 204properties

charge 193flux 193magnetic moments 171masses 189. See mass valuesmoments. See magnetic momentspin 193wrapping 197

stableelectron 195neutrino 195proton 195

Pauli’s equationgeneralized 170

at nuclear distances 175pep

bond 184, 185nuclear barrier 184

perturbationof substratum 162

photonas connection excitation 156mass 155, 156

physical geometryand electromagnetic charge 65and gravitation 44and measurements 80and quantum field theory 56

pictureHeisenberg 98interaction 100Schrödinger 98

pion 203Planck

309Indexconstant 91

Poincarégroup 66

Poissonequation 295

in physical geometry 53kernel 229

polydiscunit 228

volume of 232potential 94

and substratum curvature 121dynamic

due to matter 93for the substratum 120of substratum 93

in mass definition 136strong

magnetic 169principal bundle 281

in physical geometry 15principle

atomistic 188holistic 188inertia 1, 129of covariance 1

generalized 4of equivalence 1

generalized 4of relativity

generalized 10of special relativity 9Schwinger's action 62uncertainty

Heisenberg's 95variational 16

probabilistc behaviorclassical system 106

probabilistic behaviorquantum mechanics 106

probabilistic interpretation 103in Young's interference experiment 106

productClifford

of double dimensions 245convolution 94

in mass ratios 135homotopy 283Lie 254

in QED 97projection

of fiber bundle 281proton 195

chargedefined 76

magnetic momentcorrections 174

mass 144quark structure 194

3-point particle 195pseudoquaternion 239pullback form 253

Q

quantization relation 98quantum

fundamentalangular momentum 73electric charge 73magnetic flux 73

quantum electrodynamics 97standard techniques 100

quantum field theory 56variational principle 62

quantum mechanicscomplex structure of 86of free particles 23probabilistic interpretation 106

quantum numbersalgebraic 193topological 197

quantum of flux 77quantum operator 58

change in 62quark

as dual representation of G 194as geometric excitation 195, 206as leptonic excitation 205, 206flavor 206

quarkoniumconjecture 208

quaternion 238, 243pseudo 239

quotient spacesAIII Series 263CII Series 263

PHYSICAL GEOMETRY310

R

radiationfield

free 99range

connection excitations 156long

field 156magnetic field 171

rarityof an excitation 208operator

as odd projector 242reaction

self field 63reference frame 15

comoving 117in jet bundle 98

regimegeometrical holophysical 101microscopic excitation 101

regular elementas spin and charge generators 71of algebra 256

relationcommutation or anticommutation 63quantization 98

relativity of interactions 13representation

adjoint 254dual 193dual fundamental of group G

as quarks 194fundamental 193fundamental of G

as proton 194fundamental of L 76

as neutrino 194, 221fundamental of P 75

as electron 194regular 255

residual setof unexplained phenomena 29

resistivityfractions

FQHE 113reversion 236Ricci

tensor 48Riemann

curvature form 35tensor 47

Riemann tensoron limit boundary 293

right handed componentof spinor 24

particle 192rigid structure

of limit boundary 289ring

biquaternion 243pseudoquaternions 246

root spaceA1 74, 76A3 67, 73C2 74, 75classification 257collapse from A3 to C2 76collapse from C2 to A1 76

root vectorcanonical 68geometric, for G 72geometric, for P subgroup 75

rotation and spinors 9rotation generator 84

S

scaling lawBjorken 195

scattering solutionincoming 196outgoing 196topological classification

by functions on S 196Schawrzschild

exterior solutioncurvature 52

interior solution 51Schrödinger

picture 98section

critical 277self interaction

nonlinear equation 30self reaction 5semisimple algebra

311Indexclass A 258class B 258class C 258class D 258classification 258

Silov’s boundary 229volume of 232

SL(2,) group 12soldering form 36solution

constantnon zero 117

empty space 50substratum 94

source current 5as canonical currrent

electromagnetic 86space

affine 61base

of fiber bundle 281characteristic 229De Sitter 66Einstein 50external 131internal 131orthogonal 236Rp,q 236symmetric 261. See cosettotal

of fiber bundle 281space time

metric 36special relativity

principle 9generalization 10

spingeometric 84isotopic

of an excitation 208quantum 73

spinor 236Dirac 249

in physical geometry 24Weyl 248

standard modelrelation with 211

statesmatter 13

statistical interpretation 101statistics

Bose-Einstein 101Fermi-Dirac 101

strangenessof an excitation 207

strong interactionas G/L symmetric space 195quark-parton model 211SU(3) symmetry (QCD) 210

structural equationsCartan 270

structurecomplex 140hyperbolic 12triple geometric 171, 186, 195, 199

structure constants 116structure form

of jet bundle 276structure group

of bundle 281in physical geometry 3

subalgebraCartan 256

subframerelated to subgroup 191

subsetorthonormal 238

in physical geometry 6subspace

Cartan 256horizontal

in tangent bundle 266vertical

in tangent bundle 266substratum 115

as carrier of inertial properties 188as particle vacuum 188connection 136curvature 120form orthogonalization

Schmidt 118in mass definition 94mass parameter 122potential 93solution 117

substratum sectionevolving 98fixed 98

PHYSICAL GEOMETRY312free motion 100moving 98

symmetric space 261. See cosetclassification 262in newtonian limit 54

symmetrycombinatorial

SU(3)xSU(2)xU(1) 210geodesic 261

T

tangent spaceto solution space 57

taumass

as leptonic excitation 202tensor

Einstein 48Ricci 48Stephenson 47Weyl 48

tensorial formpotential 93

test particleorthonormal tetrad 32

tetrahedronfundamental representation

projection to square 75tetrahedron representation

fundamental states 73in particle classification 193

theoryalready unified 2Einstein-Maxwell 2Fermi

of weak interactions 163geometrical 6

QCD 210QED 97QFD 210

Kaluza 3Newton 52of antisymmetric field 3Weinberg-Salam 210Weyl's unified 3Yang 35

topologicalclassification of scattering solutions 196

quantum numbers 197excitation 197

quantum numbers 198torsion 47

gravitational 47in physical geometry 36on base 271

torsion form 270total space

of fiber bundle 281transformation

active 272passive 272

transition functions 280transpose 246triple structure

geometric 171, 195, 199trivial connection 120

as substratum 129, 147curvature 120

trivial substratum 117equation for 119

truthof an excitation 207

U

ultra relativity 10small effects 13

unit of chargegaussian 88natural 88

V

vaccumexcitation 101

vacuumas substratum 188gravitational 124

variationgenerator of 81of matter section 58of observer section 58

variation generatoras quantum operator 85

variational principle 16vector

Jacobi 98Killing 117

313Indexroot 256weight 256

vector fieldJacobi 279jet prolongation

in jet bundles 277lift of 266principal bundle fundamental 265right invariant 254

vector potentialelectromagnetic 125in Pauli's equation 170magnetic moments 151, 171

volume of symmetric spacein terms of boost integral

of C space 140of K space 143

of C space 138of K space 140up to equivalence relation 143

W

wavem-particle 104

weak interaction 160constant

from constant substratum solution 166Fermi current 164Fermi lagrangian 163QFD 210Weinberg-Salam theory 210

weight vectorcanonical 68geometric 72geometric, for G 256geometric, for P subgroup 74

Weinberg-Salam theory 211Weinberg’s angle 155, 171, 175Weyl 3

tensor 48winding number 282

for excitations 197neutrinos 222, 223

wrappingof subspaces 199

wrapping number 282Wyler’s measure

coefficient of 232on Silov's boundary 230

Y

Yang’s equationgravitation 35

Younginterference experiment 104

Yukawaequation 148

A Bibliography on Geometry and RelativityM. Atiyah, Topology Seminar Notes, (Harvard University, Cambridge) p. 23 (1962).G. Blanche, Chapter 20 “Mathieu Functions” in Milton Abramowitz, Irene A.Stegun, eds.,

Handbook of Mathematical Functions (Dover, New York) (1964).C. J. Bouwkamp, Kon. Nederl. Akad. Wetensch. Proc. 51, 891 (1948).E. Cartan, Assoc. Avanc. Sc. Lyon, p. 53 (1926).E. Cartan, The Theory of Spinors (Dover, New York) (1981).B. Doubrovine, S. Novikov, A, Fomenko, Géométrie Contemporaine, Méthodes et

Applications (Ed. Mir, Moscow), tranlated by V. Kotliar, Vol. 2, p. 62 (1982).A. Einstein, The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton) (1956);

Ann. Physik, 49, 769 (1916); Preuss. Ak. der Wissenschaft, Phys. Math. Klasse,Sitzungsberichte, p.18 (1925)

A. Einstein, L. Infeld, B. Hoffmann, Ann. Math. 39, 65 (1938).A. Einstein, M. Grossmann, Zeit. Math. Phys. 62, 225 (1913).A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935).E. Fairchild, Phys. Rev. D14, 384 (1976).G. Floquet, Ann. École Norm. Sup. 12, 47 (1887).P. L. Garcia, Rep. on Math. Phys. 13, 337 (1978).R. Gilmore, Lie Groups, Lie Algebras and some of their Aplications (John Wiley and Sons,

New York) (1974).S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York)

(1962).R. Hermann, Lie Groups for Physicists (W. A. Benjamin, New York) (1966); preprint HUTP-

77/AO12 (1977).L. Infeld, B. L. van der Waerden, Sitzber. Preuss. Akad. Wiss. Physik Math. K1, 380 (1933).J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, New York), Second Ed.

(1975).A. Lichnerowicz, Théorie Globale des Connexions et des Groupes d’Holonomie, (Ed.

Cremonese, Roma) (1972); Comptes Rendus 252, 3742 (1961); Comtes Rendus 253, 40(1961).

E. Mathieu, J. Math. Pures Appl. 13, 137 (1868).J. Meixner, F. W. Schäfke, G. Wolf, in Mathieu Functions and Spheroidal Func tions and

Their Mathematical Foundations, edited by A. Dold and B. Eckman (Springer-Verlag),Berlin, Vol. 1, Chap.2, p.85.

W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco)(1973).

C. Moller, K. Dan. Vidensk Selsk. Mat. Fys. Medd. 39, 1 (1978).P. M. Morse and H. Feshbach, Methods of Theoretical Physics, 1st edition (McGraw-Hill,

New York) (1953). .H. P. Mullholland and S. Goldstein, Phil. Mag. 8, 834 (1929).I. Newton, in Sir IsaacNewton’s Mathematical Principles of Natural Phylosophy and his

System of the World, edited by F. Cajori (Univ. of California Press, Berkeley and LosAngeles) (1934).

B. O’Neill, Semi-Riemannian Geometry, (Academic Press, New York) (1983).

315

R. S. Palais, Foundations of Global Non linear Analysis, (W. A. Benjamin, New York)(1968).

R. Pavelle, Phys. Rev. Lett. 34, 1114 (1975); Phys. Rev. Lett. 37, 961 (1976).I. Porteous, Topological Geometry, (Van Nostrand Reinhold, London) (1969).R. Penrose & W. Rindler, Spinors nd space-time Vol. I (University Press, Canbridge)

(1086).J. G. Ratcliffe, Foundations of Hyperbolic Manifolds (Springer-Verlag, New York) (1994).E. Schrödinger, Space-time Structure, (University Press, Cambridge) (1963).M. Spivak, Vols. I to V, Differential Geometry (Publish or Perish, Berkeley) (1970).A. Trautmann, F. A, E. Pirani, H. Bondi, Brandeis 1964 Lectures on General Relativity(Prentice-Hall, London) (1065)W. Tulczyjew, Acta Phys. Pol. 18, 393 (1959).H. Weyl, The Theory of Groups and Quantum Mechanics, (Dover, New York) (1931).E. P. Wigner, Ann. Math. 40, 149 (1939).A. Wyler, Acad, Sci. Paris, Comtes Rendus 269A, 743 (1969); Comtes Rendus 271A, 180

(1971).C. N. Yang, Phys. Rev. Lett. 33, 445 (1974).

physical geometry - prof.usb.veprof.usb.ve/ggonzalm/invstg/pblc/phsclgmtr.pdf · first attempts...

Documents