quaternion octonion gauge theory of fundamental...

QUATERNION OCTONION GAUGETHEORY OF

FUNDAMENTAL INTERACTION(PARTICLE PHYSICS)

A THESIS

SUBMITTED TO

KUMAUN UNIVERSITY, NAINITAL

For

THE DEGREE OF

DOCTOR OF PHILOSOPHY (PHYSICS)

BY

GAURAV KARNATAK

UNDER THE SUPERVISION OF

Dr. P. S. BISHT

Department of Physics

Kumaun University

S. S. J. Campus

ALMORA-263601 [U. K.]

2012

CERTIFICATE

This is to certify that the entire work of the thesis entitled “Quaternion Octonion Gauge

Theory of Fundamental Interaction” has been carried out my supervision. This study em-

bodies the original contributions of the candidate to the best of my knowledge. Any part

there of has not been submitted earlier in any other thesis in any University. I approve the

submission of this thesis for the award of the degree of DOCTOR OF PHILOSOPHY (Ph.

D) in Physics. It is also certified that the candidate has put more than two hundred days

attendance, in last three and half years, in the Department of Physics, Kumaun University,

Soban Singh Jeena Campus, Almora for the completions of his work presented in this the-

sis.

Date: 29 /12 / 2012

Forwarded by

Head and Converner Dr. P. S. Bisht

Department of Physics Department of Physics

Kumaun University, Kumaun University,

S. S. J. Campus S. S. J. Campus

Almora Almora

i

ACKNOWLEDGEMENT

I would like to express my sincere gratitude to my supervisor Dr. P. S. Bisht for suggest-

ing problems, valuable guidance, Perpetual inspiration and encouragement throughout the

period of my research. Discussions with him have always inspired me to improve the qual-

ity of the manuscripts. My deepest thanks to him for his constant motivation and also for

making a conducive work environment throughout the thesis work, without his guidance

and support this work wouldn’t have been possible.

I am highly grateful to Prof. O. P. S. Negi for his continious interest, constant support and

guidance, perpetual inspiration and encouragement throughout the progress of this study.

His deep insight on the subject and vast experience has helped me a lot in learning the

subject. Time to time discussions of research problems with him motivated me to do more

and more work.

I would like to express my warmest thanks to Physics Department, Kumaun University

Soban Singh Jeena Campus, Almora where I succeeded to complete this work. I am grate-

ful to Prof. M. C. Durgapal, Head of Physics Department and teachers Prof. K. L. Shah,

Prof. B. C. Joshi and Smt. Pratibha Fuloria for their support and encouragement. I further

extend my thanks to other non - teaching and technical staff of Physics Department, Soban

Singh Jeena Campus, Almora, Mr. R. S. Rayal, Mr. B. S. Negi, Mr. J. C. Upadhayay, Mr.

Heera Singh Kharayat, Mr. Pramod Nailwal, Mr. Joga Ram, Mr. Rajendra Singh Rana, Mr.

Bheem Singh and Mr. Dan Singh. Freindly time to time co - operation from researchers Dr.

Pushpa, Pawan Kumar Joshi, Bhupesh Chandra Chanyal, Bhupendra Singh Chauhan and

Vishal Sharma of Physics Department, S. S. J. Campus Almora is greatly acknowledged.

Also my past and present friends Dr. Manoj Bisht, Dr. Sparsh Bhatt, Ganesh Nayal, Basant

ii

Pandey are greatly acknowledged for their help in various ways.

I am thankful to my father Sri P. C. Karnatak and my mother Smt. Janki Karnatak for their

moral support and continuous encouragement received through out my Ph. D. work. I am

also thankful to Prof. Jagat Singh Bisht and their family for his encouragement and inspir-

ing words to do the good work. My siblings Smt. Kiran Joshi, Reeta Karnatak, Poonam

Karnatak, Anjali Joshi and Deep Karnatak are specially acknowledged for their support and

care. Last but not least, I would like to thanks to Dr. Ila Sah, Sri C. N. Sah, Mr. Dinesh

Chandra Tewari and my nephew brother Dr. Ramesh Chandra Lohumi for encouraging me

to do the research work.

Date: 29/12 /2012

Place: Almora

(Gaurav Karnatak)

iii

PREFACE

The present thesis entitled “Quaternionion Octonion Gauge Theory of Fundamental In-

teraction” (Particle Physics) comprises the investigations carried by the auther over the pe-

riod of three and half years under the supervision of Dr. P. S. Bisht, Department of Physics,

Kumaun university, Soban Singh Jeena Campus, Almora. The present thesis embodies the

investigations towards the study of various field associated with dyons (particle carrying

simultaneously electric and magnetic charges) and gravito-dyons (particle carrying simul-

taneously gravitational or gravi-electric and Heavisidian or gravi-magnetic, the consistent

analyticity of quaternion and octonion, generalization of Schwinger-Zwanziger dyon, Va-

lidity of Ehrenfest theorem and electromagnetic tensor for dyons and the Physics beyond

the standard model where the monopoles and dyons play an important role.

The whole work is divided into Six chapters.

Chapter 1 is based on the survey, historical developments and findings related to monopoles,

dyons, quaternions, octonions and split octonions. The mathematical analyticity pertaining

to achieve the objectives carried out in the preceding chapters of the present thesis has also

been discussed. This chapter also contains the brief summary of the entire work done in

the thesis.

Chapter 2 describes a self consistent and manifestly covariant theory for the dynamics of

four charges (masses) (namely electric, magnetic, gravitational, Heavisidian) in terms of

quaternion variables has been developed in simple, compact and consistent manner. Start-

ing with an invariant Lagrangian density and its quaternionic representation, we have ob-

tained the consistent field equation for the dynamics of four charges. It has been shown

that the present reformulation reproduces the dynamics of individual charges (masses) in

the absence of other charge (masses) as well as the generalized theory of dyons (gravito-

dyons) in the absence of gravito-dyons (dyons).

The manifestly covariant, dual symmetric and gauge invariant two potential theory of gen-

eralized electromagnetic fields of dyons and gravito-dyons has been developed consistently

from U (1)× U (1) gauge symmetry in Chapter 3. Corresponding field equations and equa-

iv

tion of motion are derived from Lagrangian formulation adopted for U (1) × U (1) gauge

symmetry for the justification of two four potentials of dyons and gravito-dyons.

Postulating the existence of magnetic monopole in electromagnetism and Heavisidian monopoles

in gravitational interactions, a unified theory of gravi-electromagnetism has been developed

on generalizing the Schwinger-Zwanziger formulation of dyon to quaternion in simple and

consistent manner in Chapter 4. Starting with the four Lorentz like forces on different

charges, we have generalized the Schwinger - Zwanziger quantization parameters in order

to obtain the angular momentum for unified fields of dyons and gravito - dyons (i.e. Gravi

- electromagnetism). The octonion covariant derivative has been discussed as the gauge

covariant derivative of generalized fields of dyons. The generalized four-potential, current

and fields of gravito-dyons in terms of split octonion variable, the U (1) abelian and SU (2)

non - Abelian gauge structure of dyons and gravito-dyons are also described consistently.

It is shown that the generalized four-current is not conserved but only the Noetherian

four-current is considered to be conserved one. It is concluded that the presence analysis

reproduces the theory of electric (gravitational) charge (mass) in the absence of magnetic

(Heavisidian) charge (mass) on the dyons (gravito-dyons) or vice versa.

The validity of Ehrenfest theorem with its classical correspondence has been justified for

the manifestly covariant equations of dyons in Chapter 5. We have also developed accord-

ingly the Lagrangian formulation for electromagnetic fields in a minimum coupled source

giving rise to the conserved current of dyons. Applying the Gupta subsidiary condition we

have extended the validity of the Ehrenfest theorem for U (1)× U (1) abelian gauge theory

of dyon. It is shown that the expectation value of the quantum equation of motion repro-

duces the classical equation of motion, which has been generalized the Ehrenfest theorem

in quantum field theory. Finally, we have discussed the energy momentum tensor and con-

servation laws for generalized fields of dyons.

Starting with the quaternion formulation of SU (2) × U (1) gauge theory of dyons and

gravito-dyons in Chapter 6, it is shown that the formulation characterizes the abelian and

non-Abelian gauge structure of dyons and gravito-dyons in terms of pure real and imagi-

nary units of quaternion. It is shown that the three quaternion units explain the structure of

Yang - Mill’s field while the seven octonion units provide the consistent structure of SU (3)C

v

gauge symmetry of quantum chromo dynamics. Grand Unified theories are discussed ac-

cordingly in terms of quaternions and octonions in terms of quaternion basis elements and

Pauli matrices as well as the Octonions and Gell - Mann matrices. Connection between the

unitary groups of GUT’s and the normed division algebra has been established to redescribe

the SU (5) gauge group. The SU (5) gauge group and its subgroup SU (3)C×SU (2)L×U (1)

has been analyzed in terms of quaternion and octonion basis elements. It is concluded that

the division algebra approach to the the theory of unification of fundamental interactions

as the case of GUT’s leads to the consequences towards the new understanding of these

theories which incorporate the existence of magnetic monopole and dyon.

vi

Contents

CERTIFICATE i

ACKNOWLEDGEMENT ii

PREFACE iv

1 GENERAL INTRODUCTION 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Summary of the present work . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 Electromagnetic Duality . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.2 Assymmetry of Maxwell’s equation of Duality non-invariance . . . . 16

1.3.3 Duality invariance and need of monopole . . . . . . . . . . . . . . . 18

1.3.4 Dirac Monopole and their properties . . . . . . . . . . . . . . . . . . 26

1.3.5 GUT’s and pt Hooft - Polyakov Monopoles . . . . . . . . . . . . . . . . 32

1.3.6 Why dyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.3.6.1 Wrong connection between spin and statistics . . . . . . . . 39

1.3.6.2 Witten Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.3.6.3 Recent conflict with existence with pure monopoles . . . . 40

1.3.6.4 Non-abelian dyons . . . . . . . . . . . . . . . . . . . . . . . 40

1.3.7 Schwinger- Zwanziger Dyons . . . . . . . . . . . . . . . . . . . . . . 41

1.3.8 Julia-Zee Dyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.3.9 Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.3.10 Octonion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

vii

CONTENTS CONTENTS

1.3.11 Split - Octonion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2 GENERALIZED GRAVI - ELECTROMAGNETISM 72

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.2 Linearization of Gravitational fields . . . . . . . . . . . . . . . . . . . . . . . 77

2.3 Analogy between electromagnetic and linear gravitational fields . . . . . . . 80

2.4 Postulation of Heavisidian (gravi - magnetic) monopoles . . . . . . . . . . . 84

2.5 Generalized fields of gravito - dyons . . . . . . . . . . . . . . . . . . . . . . 88

2.6 Quaternion charge and unified fields of dyons and gravito - dyons . . . . . 91

2.7 Lagrangian Formulation of dyons and gravito - dyons . . . . . . . . . . . . . 93

2.8 Equation of motion of a unified charge in quaternion form . . . . . . . . . . 101

2.9 Euler’s Lagrangian equation of motion of unified charges . . . . . . . . . . . 102

2.10 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3 ABELIAN GAUGE AND TWO POTENTIAL THEORY OF DYONS AND GRAVITO-

DYONS 117

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.2 Gauge symmetry of Dual Electromagnetism . . . . . . . . . . . . . . . . . . 121

3.3 Gauge symmetry of dual gravito-dynamics . . . . . . . . . . . . . . . . . . . 123

3.4 Dual symmetric covariant formulation of dyons . . . . . . . . . . . . . . . . 126

3.5 Dual symmetric covariant formulation of gravito-dyons . . . . . . . . . . . . 128

3.6 U (1)× U (1) gauge formulation of dyons . . . . . . . . . . . . . . . . . . . . 130

3.7 U (1)× U (1) gauge formulation of gravito-dyons . . . . . . . . . . . . . . . 136


4 GENERALIZATION OF SCHWINGER ZWANZIGER DYON TO QUATERNION 152

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.2 Schwinger - Zwanziger dyons in Electromagnetic fields . . . . . . . . . . . . 156

4.3 Schwinger - Zwanziger dyons in gravito - Heavisidian fields . . . . . . . . . 159

4.4 Generalization of Schwinger - Zwanziger dyons to Quaternions . . . . . . . 161

4.5 Quaternion Formulation of gravito - Heavisidian fields . . . . . . . . . . . . 163

4.6 Octonion gauge theory of dyons . . . . . . . . . . . . . . . . . . . . . . . . . 167

viii

CONTENTS CONTENTS

4.7 Octonion and unified fields of gravito-dyons . . . . . . . . . . . . . . . . . . 171


5 VALIDITY OF EHRENFEST’S THEOREM AND ENERGY MOMENTUM TENSOR

FOR DYONS 182

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5.2 Basics of Ehrenfest Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.3 Validity of Ehrenfest’s Theorem for Charged particles . . . . . . . . . . . . . 188

5.4 Validity of Ehrenfest Theorem for dual charged particles . . . . . . . . . . . 190

5.5 Validity of Ehrenfest’s Theorem for dyons . . . . . . . . . . . . . . . . . . . 192

5.6 Abelian gauge theory and Validity of Ehrenfest Theorem . . . . . . . . . . . 195

5.7 Energy Momentum Tensor of dyons . . . . . . . . . . . . . . . . . . . . . . . 198


6 QUATERNION-OCTONION GAUGE FORMULATION AND UNIFICATION OF FUN-

DAMENTAL INTERACTION 212

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

6.2 Quaternion formulation of SU (2)× U (1) gauge theory of dyons . . . . . . . 216

6.3 Quaternion formulation of SU (2)× U (1) gauge theory of gravito-dyons . . 221

6.4 Octonionic formulation of QCD . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.4.1 Gellmann λ matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.4.2 Relation between Octonion and Gellmann Matrices . . . . . . . . . . 225

6.4.3 Octonionic Reformulation of QCD . . . . . . . . . . . . . . . . . . . . 228

6.5 Quaternion-Octonion Formulation of Grand Unified Theory . . . . . . . . . 231


LIST OF PUBLICATION 248

CONFERENCES AND PAPER PRESENTED THERE IN 250

ix

Chapter 1

GENERAL INTRODUCTION

Chapter 1 General Introduction

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ABSTRACT

This chapter is based on the survey, historical developments and findings related to monopoles,

dyons, quaternions, octonions, split octonions. The mathematical analiticity pertaining to

achieve the objectives carried out in the preceding chapters of the present thesis has also been

discussed. This chapter also contains the brief summary of the entire work done in the thesis.

2


1.1 Introduction

Electromagnetism is the branch of science concerned with the forces that occur between

electrically charged particles. In electromagnetic theory these forces are explained using

electromagnetic fields. Electromagnetic force is one of the four fundamental interactions in

nature, the other three being the strong interaction, the weak interaction and gravitation.

In 20th century the quantum theory of magnetic charge given by physicist P. A. M. Dirac in

1931 [1]. Dirac showed that if any magnetic monopoles exist in the universe, then all elec-

tric charge in the universe must be quantized [2]. The electric charge is, in fact, quantized,

which is consistent with the existence of monopoles [2]. Therefore, it remains an open

question whether or not monopoles exist. Further advances in theoretical particle physics,

particularly developments in grand unified theories and quantum gravity, have led to more

compelling arguments that monopoles do exist. These theories are not necessarily incon-

sistent with the experimental evidence. In some theoretical models, magnetic monopoles

are unlikely to be observed, because they are too massive to be created in particle acceler-

ators, and also too rare in the Universe to enter a particle detector with much probability

[3]. Maxwell’s equations of electromagnetism relate the electric and magnetic fields to

each other and to the motions of electric charges. The standard equations provide for elec-

tric charges, but no magnetic charges. Except for this difference, the Maxwell’s equations

are symmetric under the interchange of the electric and magnetic fields [4, 5]. One of

the defining advances in quantum theory was Dirac’s [1] work on developing a relativistic

quantum electromagnetism. Before his formulation, the presence of electric charge was

simply inserted into the equations of quantum mechanics, but in 1931 Dirac [1] showed

that a discrete charge naturally falls out of quantum mechanics. Polchinski [6], a string-

theorist, described the existence of monopoles as one of the safest bets that one can make

about physics not yet seen [3].

This ingenious suggestion in connection with the existence of monopole give rise to con-

siderable literature on the subject [7]-[21] to predict the mass, size, spin, and quantum

properties of monopoles. The confirmation of experimental evidence of these particles are

theoretical reasons were put forward for the existence of these particles and the case of

3


their existence of these particles was accepted as strong as that for other undiscovered par-

ticle. But in view of lack of experimental evidence the literature partially turned to negative

casting doubts on the existence of monopoles in the attempts to construct a classical theory

of electrodynamics in presence of both electric and magnetic charges. Rosenbaum [22], ar-

guing against the existence of monopole, proved that it is impossible to formulate an action

principle for a classical electrodynamical field of such charges unless an extra restriction,

contradicting Lorentz force law, is imposed on the path of magnetic charge.

Schwinger [23]-[29] formulated and analyzed what is now known as the Schwinger model,

quantum electrodynamics in one space and one time dimension, the first example of a con-

fining theory. Schwinger [23]-[29] suggest an electroweak gauge theory, an SU (2) gauge

group spontaneously broken to electromagnetic U (1) at long distances. Schwinger [23]-

[29] attempted to formulate a theory of quantum electrodynamics with point magnetic

monopoles, a program which met with limited success because monopoles are strongly in-

teracting when the quantum of charge is small. Schwinger considered source theory as a

substitute for field theory, although it is only a different point of view, a version of effective

field theory. It treats quantum fields as long-distance phenomena, and does not require a

well defined continuum limit. Source theory was considered overly formal and lacking in

distinctness from quantum field theory. Peres [20] pointed out the control vertical nature

[26] of these singular lines [30]-[34] and derived the charge quantization condition in

purely group theoretical manner without using string variables. Attempts were also made

to develop the theories related to the possibility of formulating an action integral in pres-

ence of electric and magnetic charges.

Schwinger [23]-[29] gave relativistically invariant quantum field theory of spin-12

magnetic

charges and its extension described by Zwanziger [35]-[38] to the particles carrying elec-

tric and magnetic charges though provided a natural generalization of electrodynamics,

suffered due to the fact that in the former the basic variables were not canonically conju-

gate while the later admitted the duality of number of variables to maintain locality while

the theory discussed by Biagojevic et. al. [39], suffers with the violation of Lorentz invari-

ance along with the attempt made by Brand’t et.al. [40]-[42] to maintain it led to the use of

controversial and unphysical string variables. On the other hand, the Lagrangian theory of

4


monopole developed by Ezawa and Tze [43] in terms of non-Abelian gauge symmetry lacks

in action principle in the presence of both electrically and magnetically charged fields.

In theoretical Physics, the pt Hooft - Polyakov [44, 45] monopole is a topological soliton

similar to the Dirac monopole but without any singularities. It arises in the case of a Yang

- Mill’s theory with a gauge group, coupled to a Higgs field which spontaneously breaks

it down to a smaller group via the Higgs mechanism. It was first found independently bypt Hooft - Polyakov [44, 45]. Unlike the Dirac monopole, the pt Hooft - Polyakov [44, 45]

monopole is a smooth solution with a finite total energy. The monopole problem refers to

the cosmological implications of Grand unification theories (GUT’s). Since monopoles are

generically produced in GUT during the cooling of the universe, and since they are expected

to be quite massive, their existence threatens to over close it. This is considered a problem

within the standard Big Bang theory. Cosmic inflation remedies the situation by diluting

any primordial abundance of magnetic monopoles [46].

During the past few years, after the report of Price et. al. [47], there has been a growing

interest in the development of a self consistent gauge field theory to establish the theo-

retical existence of these particles and to explain their group theoretical properties and

symmetries. In the mean time it became clear that the monopole can be understood better

in non-Abelian gauge theories in which it appears as classical solutions of the system which

have a topological meaning .

Julia and Zee [48] extended the idea of pt Hooft - Polyakov [44, 45] constructed the quan-

tum mechanical excitations of the fundamental monopole includes dyons which are the

particles carrying both electric and magnetic charges. They have arisen automatically from

the semi-classical quantization of the global charge rotation degrees of freedom of the

monopole. In an attempt to explain CP - violation in terms of non-zero θ angle of vac-

cum, Witten [49] has showed that monopoles are necessarily dyons with fractional electric

charge in general. Wilczek et. al. [50] have extended this model to disappearing dyons

and dyonium i.e. bound states of dyon-existence of these dyons can remove the objection

raised by Pantaleon [51] about the simultaneous recent experimental results of Fairbank

et. al. [52] about the reported evidence of fractional electric charge and that of Cabrera

[4] about an event interpreted as monopoles. Now it is widely recognized [53] that SU (5)

5


grand unified model [54] is a gauge theory that contains monopole and dyon solutions and

consequently monopoles and dyons have become intrinsic part of all current grand unified

theories and the question of their existence has gathered enormous potential importance in

connection of Vaccum [55, 56], their role in catalyzing proton decay [57, 58, 59] and also

in the unification’s of gravitation with electromagnetic fields [60, 61].

Quaternions were introduced by Hamilton [62, 63, 64] and octonions by Cayley [65] and

Graves [66]. These number systems were the first examples of hyper complex numbers or

algebras and they have had a significant impact of Mathematics and Physics. Because of

their beautiful and unique properties, they have attracted many to study the laws of nature

over the field of these numbers. Various authors [67]-[70] used quaternion in the context

of special relativity, general relativity [71]-[80], quantum mechanics [81]-[87], superlumi-

nal Lorentz transformation [88, 89] and gauge field theories [90]-[92] etc. Quaternionic

gauge theory studied by Finkelstein et. al. [82]-[84] and has been extended by Adler

[93] to construct non local U (2) gauge theory of composite quarks and leptons while on

the other hand Morita [94, 95] used quaternion to reformulate left-right Weinberg-Salam

theory with SU (2)L × SU (2)R × U (1) gauge structure. Furthermore, quaternion quantum

mechanics developed by Soucek [96] has been shown to the theory of tachyons. Group

theoretical study of quaternions has been extensively developed by some authors [97]-[99]

showing its extension to the algebra over real and complex numbers systems. Fields asso-

ciated with monopoles has been written by Rajput et. al. [100]-[106] in terms of simpler,

unique and consistent quaternion formulation.

Octonions form the widest normed algebra after the algebras of real numbers, complex

numbers, and quaternions. Since their discovery, almost three decades before Maxwell’s

equations, there have been various attempts to find appropriate uses for octonions in

Physics. Octonions have a natural local SU(3) gauge group, which is the color force gauge

group. Since the quaternions are a sub algebra of the octonions, the octonions naturally

unify electromagnetism, the weak force, and the color force, producing the SU(3) × SU(2)

× U(1) standard Model. The application to Physics pursued by Jordan, Neumann and

Wigner [107] observables are represented by octonionic Hermitian matrices [108]-[110]

in which system is quantized through associators rather than commutators. Foot and Joshi

6


[111] studied the Lorentz groups in 3, 4, 6 and 10 dimensions in terms of real, complex,

quaternions and octonions and also relate string theory with Jordan algebras [109]. Octo-

nion Gauge theory investigated by Lassig and Joshi [112] with a magnetic charge as well as

studied a connection between Nambu mechanics and non-associative algebras [113, 114].

The octonionic algebra has been in fact linked with a number of interesting subjects, SU(3)

color symmetry and quark confinement [115]-[117], exceptional GUT groups [118], Dirac

equation in the octonionic algebra [119], Representation of Clifford algebras [120], Hyper

complex quark fields and quantum chromo dynamics [121], Quantum mechanics [122],

space-time symmetries etc. Domokos and Koveski-Domokos [123] and Baez [124] used

octonions to construct algebra of dynamical variables satisfying the triality rule for quarks

with an automorphism group isomorphic to SU(3)C .

In theoretical Physics, quantum chromo dynamics (QCD) [125]-[131] is a theory of the

strong interaction (color force), a fundamental force describing the interactions between

quarks and gluons which make up hadrons (such as the proton, neutron or pion). It is the

study of the SU(3) Yang–Mill’s theory of color-charged fermions (the quarks). QCD is a

quantum field theory of a special kind called a non-abelian gauge theory, consisting of a

color field mediated by a set of exchange particles (the gluons). The theory is an important

part of the Standard Model of particle Physics [125]-[131]. A huge body of experimental

evidence for QCD has been gathered over the years. QCD enjoys two peculiar properties

(1) Confinement :- Which means that the force between quarks does not diminish as they

are separated. Because of this, it would take an infinite amount of energy to separate two

quarks, they are forever bound into hadrons such as the proton and the neutron. Although

analytically unproven, confinement is widely believed to be true because it explains the

consistent failure of free quark searches, and it is easy to demonstrate in lattice QCD.

(2) Asymptotic freedom :- Which means that in very high-energy reactions, quarks and

gluons interact very weakly.

Quarks are massive spin12

fermions which carry a color charge whose gauging is the con-

tent of QCD. Quarks are represented by Dirac fields in the fundamental representation 3 of

the gauge group SU(3). They also carry electric charge (either -13

or 23) and participate in

weak interactions as part of weak isospin doublets. They carry global quantum numbers

7


including the baryon number, which is 13

for each quark, hyper charge and one of the flavor

quantum numbers. Gluons are spin1 bosons which also carry color charges, since they lie

in the adjoint representation 8 of SU(3). They have no electric charge, do not participate

in the weak interactions, and have no flavor. They lie in the singlet representation 1 of all

these symmetry groups. Every quark has its own antiquark. The charge of each antiquark

is exactly the opposite of the corresponding quark. According to the rules of quantum field

theory, and the associated Feynman diagrams, the above theory gives rise to three basic

interactions, a quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon,

and two gluons may directly interact [125]-[131]. This contrasts with QED, in which only

the first kind of interaction occurs, since photons have no charge. Diagrams involving Fad-

deev–Popov ghosts must be considered to except in the unitary gauge.

In order to avoid the use of string singularities of monopoles and keeping in view, the

formalism necessary to describe them has been clumsy and not manifestly covariant. Ra-

jput et.al. [132]-[134] have undertaken the study of second quantization and interaction

of generalized electromagnetic fields associated with spin-1 and spin-1/2 particles carrying

electric and magnetic charges. Throughout the work generalized fields, generalized charge,

generalized current and generalized potential associated with these doubly charge particles

have been taken as complex quantities with their electric and magnetic constituents real

and imaginary parts. Undertaking the study of rotationally symmetric and gauge invari-

ant angular momentum operators of dyons, it has already been shown that the presence

of magnetic charge on dyons directly leads to a residual angular momentum and chirality

dependent multiplicity in eigen values of third component of angular momentum operator

[135]. The quaternionic formulation of generalized field equations, generalized potential,

generalized current and Lorentz force equation of dyons has been investigated [136, 137]

in a unique, consistent and simple manner.

Bi quaternion formulation of generalized field equation of dyons has been shown [138,

139] in simple, unique, self consistent and manifestly covariant one. Instead of of real

quaternions, complex quaternions (bi quaternions) method of description of generalized

electromagnetic fields of dyons has been adopted and the corresponding physical quanti-

ties respectively. The generalized equations of dyons are described in simple and compact

8


bi quaternion forms [140, 141]. The Dirac - Maxwell equations, equation of motion, po-

tential fields and Lagrangian density associated with generalized filed of dyons [142] and

gravito dyons [143] for their manifestly covariant theory and self contained structure of

dyons has been reformulated. Non - Abelian gauge theory of dyons and gravito-dyons has

been developed in terms of quaternions [144] and octonions [145, 146] gauge group and it

is shown that the gauge structure characterize Abelian U(1) and non - Abelian SU(2) gauge

structures. Application of quaternions and to Super symmetry quantum mechanics are ex-

plored [147] and Quaternion - Octonion gauge analyticity are described consistently [148]

while the behavior of tachyons in Super symmetry has also been analyzed [149, 150].

1.2 Summary of the present work

Keeping in view the importance of the above work and problems raised thereby, the

present thesis embodies the investigations towards the study of various field associated

with dyons (particle carrying simultaneously electric and magnetic charges) and gravito -

dyons (particle carrying simultaneously gravitational or gravi-electric and Heavisidian or

gravi-magnetic), the consistent analyticity of quaternion and octonion, generalization of

Schwinger-Zwanziger dyon, Validity of Ehrenfest theorem and electromagnetic tensor for

dyons and the physics beyond the standard model where the monopoles and dyons play an

important role.

The chapter 1 of the thesis gives the history of monopoles, dyons, quaternions, octonions,

split - octonion and work carried out by different authors in this subject.

In chapter 2, a self-consistent and manifestly covariant theory for the dynamics of simul-

taneously existing four charges (masses) (namely electric, magnetic, gravitational, Heavi-

sidian) in simple, compact and consistent manner [151]. Starting with the structural sym-

metry between linear gravity and electromagnetism, the chirality quantization condition

for both dyons and gravito-dyons, has been reproduced. Accordingly, the quaternionic

form of unified charge and unified fields of dyons and gravito-dyons, which reproduces

the dynamics of Maxwell’s field equations associated with electric, magnetic, gravitational

and Heavisidian charges, has been developed in a consistent way. Considering an invari-

ant Lagrangian density and its quaternionic representation, the consistent field equation

9


and equation of motion for the dynamics of four charges are obtained. It is shown that

the Euler Lagrangian equations provides the field equations associated with each charge

(namely electric, magnetic, gravitational and Heavisidian) after taking care of usual method

of variation with respect to potential. These corresponding equations may immediately

be generalized to the unified field equations of a particle simultaneously containing four

charges namely electric, magnetic, gravitational and Heavisidian charges (masses). It has

been shown that the present reformulation reproduces the dynamics of individual charges

(masses) in the absence of other charge (masses) as well as the generalized theory of dyons

(gravito-dyons) in the absence gravito-dyons (dyons) and vice versa.

Keeping in view the recent potential importance of monopoles (dyons) and the applications

of electromagnetic duality, in chapter 3 we have made an attempt to revisit the analogous

consistent formulation of dual electrodynamics subjected by the magnetic monopole only.

Gauge formulation has been adopted accordingly to derive the dual Maxwell’s equation,

equation of motion and Bianchi identity for dual electric charge (i.e. magnetic monopole)

from the minimum action principle [152]. Accordingly, we have discussed the dual symmet-

ric and manifestly covariant formulation of generalized fields of dyons in order to obtain the

generalized Dirac-Maxwell’s (GDM) field equations and Lorentz force equation of motion of

dyons in terms of two four potentials. Two potential theory of magnetic monopoles (dyons)

has been justified from U (1) × U (1) gauge symmetry. Consequently, the gauge symmetric

and dual invariant manifestly covariant theory has been discussed consistently from the

U e (1) × U g (1) gauge symmetry. It has been emphasized that the two U (1) gauge groups

act in different manner whereas the first U e (1) act on the Dirac spinors while the other

group U g (1) act on Dirac iso-spinors. We have also developed accordingly the consistent

Lagrangian formulation for the justification of two gauge potentials of dyons. Accordingly,

the analogous gauge formulation has been adopted in order to derive the dual Maxwell’s

equation, equation of motion and Bianchi identity for dual electric charge (i.e. magnetic

monopole) and dual gravitational charge (i.e. Heavisidian monopole) from the minimum

action principle. Likewise, we have discussed the dual symmetric and manifestly covariant

formulation of generalized fields of dyons and gravito-dyons in order to obtain the gen-

eralized Dirac-Maxwell’s (GDM) field equations and Lorentz force equation of motion of

10


dyons and gravito-dyons in terms of two four potentials. Two potential theory of mag-

netic monopoles (dyons) and gravi-Heavisidian (gravito-dyons) have been justified from

U (1)×U (1) gauge symmetry. Consequently, the gauge symmetric and dual invariant man-

ifestly covariant theory has been reformulated consistently from the U (1) × U (1) gauge

symmetry for the dyons and gravito-dyons as a separate case. It is shown that Um (1) acts

on the Dirac spinors while the second unitary abelian group Uh (1) acts on Dirac iso-spinors

due to presence of Heavisidian monopole. Furthermore, the consistent Lagrangian formu-

lation has been developed in order to obtain consistent field equations for the justification

of two gauge potentials of dyons and gravito-dyons.

In chapter 4, postulation of the existence of magnetic monopole in electromagnetism and

Heavisidian monopoles in gravitational interactions, a unified theory of Gravi - electromag-

netism has been developed on generalizing the Schwinger - Zwanziger formulation of dyon

to quaternion in simple and consistent manner [153]. This chapter comprises six sections.

In section 4.2 we have undertaken the study of Schwinger - Zwanziger dyon in electro-

magnetic field. Starting with the force experienced by an electric charge, the generalized

Dirac Maxwell’s equations are obtained and shown to be manifestly covariant and dual

invariant. It is shown that the angular momentum of dyons lead to the chirality quanti-

zation condition as the extension of Dirac quantization condition. Similar techniques are

applied in section (4.3) where we have undertaken consistently study of dual invariant and

manifestly covariant theory of Schwinger - Zwanziger dyon in Gravito-Heavisidian fields.

Starting with the four Lorentz like forces on different charges, in section 4.4 we have gen-

eralized the Schwinger - Zwanziger quantization parameters as quaternion valued in order

to obtain the angular momentum for unified fields of dyons and gravito-dyons. Taking

the unified charge as quaternion, in section 4.5 we have reformulated manifestly covariant

and consistent theory for the dynamics of four charges namely electric, magnetic, gravita-

tional and Heavisidian associated with gravi-electromagnetism. It has been shown that the

combined theory describes the interaction of particles in terms of four coupling parameters

which we name as the different chirality parameters associated to electric, magnetic, gravi-

tational, and Heavisidian charges. On applying the various quaternion conjugations, it has

also been emphasized that the combined theory of gravitation and electromagnetism repro-

11


duces the dynamics of generalized electromagnetic (gravito - Heavisidian) fields of dyons

(gravito-dyons) in the absence generalized gravito Heavisidian (electromagnetic) fields of

gravito-dyons (dyons) or vice versa. Section 4.6 describes the octonionic gauge theory of

dyons and it has been shown that the octonion gauge formalism may hope for the better

understanding of these two type of forces of dyons in non-Abelian and abelian limits. Oc-

tonion formalism has been shown to be simple, compact, consistent and unique one. It

reproduces the theories of electromagnetism and gravitation separately in the absence of

each other. The O- derivative, reduces to partial derivative in the absence of split octonion.

Split octonions are described in terms of U (1)×SU (2) gauge group simultaneously to give

rise the abelian (point like) and non - Abelian (extended structure) of dyons. This gauge

group plays the role of U (1)×SU (2) Salam Weinberg theory of electro-weak interaction in

the absence of Heavisidian gravity and taking the magnetic charge on dyons vanishing. It

has been shown that the enlarged gauge group U (1)× SU (2) explains the built in duality

to reproduce abelian and non - Abelian gauge structure of dyons. Similarly in section 4.7,

the same techniques of octonion gauge formalism are applied to the case of gravito-dyons

with the same consideration.

In chapter 5, the validity of Ehrenfest theorem for charged particle, dual charged particle

in the case of separate electric and magnetic charge and dyons along with the energy mo-

mentum tensor for generalized fields of dyons is described [154]. Starting from the basic

definition of Ehrenfest theorem in section 5.2 and in section 5.3 we have shown the va-

lidity of Ehrenfest theorem in the case of a Dirac particle moving in the electromagnetic

field carrying electric charge and it is shown that the Ehrenfest theorem is valid for a Dirac

particle moving in an electromagnetic field. Section 5.4 describes the validity of Ehrenfest

theorem of magnetic charge (i.e. monopole) and the generalization of Ehrenfest theorem

for magnetic monopole has been obtained. In section 5.5 starting from the Hamiltonian of

Dirac fields in presence of dyon (particle carrying simultaneously the electric and magnetic

charge) we have discussed the validity of Ehrenfest theorem in a static case for electric and

magnetic charge. It is also shown that the equation of motion of dyons may be visualized

as the generalization of Ehrenfest theorem for dyons moving in generalized electromag-

netic fields. In section 5.6 we have developed accordingly the Lagrangian formulation for

12


the electromagnetic fields in a minimum coupled source which justify the conserved Dirac

current for dyons. Applying the Gupta subsidiary condition, we have also reproduced the

classical equation of motion and the validity of Ehrenfest theorem to abelian quantum field

theory has been checked and verified. It is shown that the expectation value of the quantum

equation of motion reproduces the classical equation of motion which is the generalized

form of the Ehrenfest theorem in quantum field theory. Section 5.7 describes the energy

momentum tensor of generalized fields of dyons and energy momentum conservation laws

are discussed consistently for dyons. Here we have also discussed the momentum operator,

Hamiltonian and Poynting vector for generalized electromagnetic fields in a manifest and

consistent way.

Keeping all these facts in mind, in chapter 6 we have made an attempt to discuss the

quaternion formulation of SU (2) × U (1) gauge theories of dyons and gravito-dyons fol-

lowed by the SU (3) gauge structure and grand unified theories in terms of octonion. It has

been emphasized that the three quaternion units explain the structure of Yang - Mill’s field

while the seven octonion units provide the consistent structure of SU (3)c gauge symme-

try of quantum chromo dynamics (QCD) as these are well connected with the well known

SU (3) Gellmann matrices [155]. The symmetry breaking mechanism of non-Abelian gauge

theories in terms of quaternion and octonion opens the window towards the discovery of

two type of gauge bosons associated with electric and magnetic charges. Here, we have

also developed the quaternion formulation of SU (2) × U (1) gauge theory of dyons and

gravito-dyons. It has been shown that the structure of Yang - Mill’s field is explained by

three quaternion units while the seven octonion units provide the consistent structure of

SU(3)C gauge symmetry of quantum chromo dynamics (QCD) with their inter connectivity

with the well-known SU(3) Gellmann λ matrices. In this case the gauge fields describe

the potential and currents associated with the generalized fields of dyons particles carrying

simultaneously the electric and magnetic charges. Grand Unified theories are discussed

in terms of quaternions and octonions by using the relation between quaternion basis el-

ements with Pauli matrices and Octonions with Gell Mann matrices. Connection between

the unitary groups of GUT’s and the normed division algebra has been established to re-

describe the SU (5) gauge group. We have thus described the SU (5) gauge group and its

13


subgroup SU (3)C × SU (2)L × U (1) by using quaternion and octonion basis elements. As

such, the connection between U (1) gauge group and complex number, SU (2) gauge group

and quaternions and SU (3) and octonions is established. It is concluded that the division

algebra approach to the theory of unification of fundamental interactions as the case of

GUT’s leads to the consequences towards the new understanding of these theories which

incorporate the existence of magnetic monopole and dyon. As such it is concluded that the

division algebra approach to the theory of unification of fundamental interactions as the

case of GUT’s leads to the consequences towards the new understanding of these theories

which incorporate the existence of magnetic monopole and dyon. Three different imaginar-

ies associated octonion formulation may be identified with three different colors (red, blue

and green) while the Gell Mann Nishijima are described in terms of simple and compact

notations of octonion basis elements.

1.3 Mathematical Methods

1.3.1 Electromagnetic Duality

The concept of duality has received prominent attention in modern gauge theories. It

provides useful tools to construct solutions to the field equations, namely, those which are

self-dual or anti-self-dual, or allows to study regimes of the theory which prevent the use

of perturbation expansions. These features seem particularly important for recent devel-

opments in string theory and its derivatives, such as branes, M-theory [156]. Duality in

classical electromagnetic theory was discovered by Heaviside [157] a century ago for the

Maxwell equations in vacuum.

−→∇ ×

−→E =− ∂

−→H

∂t;

−→∇ ×

−→H =

∂−→E

∂t; (1.1)

where−→E and

−→H are respectively the electric and magnetic field strength, exchange among

themselves under the replacements,

14


−→E −→

−→H ;

−→H −→ −

−→E . (1.2)

This symmetry of the system, duality (Heaviside duality) is the electric field to be consid-

ered equivalent to the magnetic field. It seems that Larmor [158] generalized Heaviside

duality to a continuous transformation

−→E =−→E cos θ +

−→H sin θ;

−→H =−

−→E sin θ +

−→H cos θ; (1.3)

where 0 ≤ θ ≤ π2. This emphasizes that there is complete ambiguity or equivalently, con-

tinuous freedom, in the choice of electric and magnetic fields for the radiation fields. With

the advent of non-abelian gauge theories for elementary particle physics, a lot of work in

Physics and Mathematics has been performed to clarify the meaning and applicability of

duality, as exposed above, or in its modern non-abelian versions. Let the recall that the

original fields in the Maxwell equations have electric charges or currents and time varying

fields as their sources. Even in vacuum, electric fields are considered the ones accelerating

electric charges parallel to their direction, whereas magnetic fields provide transverse accel-

eration for electric charges. Magnetic materials are related to elementary magnetic dipoles

but single isolated magnetic charges have never been observed. Dirac [1], motivated by the

need to explain the quantization of the electric charges, introduced magnetic monopoles in

this framework to provide sources for the magnetic induction field i.e.

−→∇ ·−→H = gδ (x) . (1.4)

To preserve the relation with the magnetic vector potential,

15


−→H =

−→∇ ×

−→A. (1.5)

1.3.2 Assymmetry of Maxwell’s equation of Duality non-invariance

The possibility of isolating electric poles i.e. electric charges and the impossibility to iso-

late magnetic charges is a fundamental difference between electricity and magnetism. The

asymmetry between electricity and magnetism became very clear at the end of last century

with the formulation of Maxwell’s equation for electromagnetism. Maxwell’s differential

equations for electromagnetic fields [159, 160, 161],

−→∇ .−→E =ρ;

−→∇ .−→H =0;

−→∇ ×

−→E = −∂

−→H

∂t;

−→∇ ×

−→H =

∂−→E

∂t+−→j ; (1.6)

where the vector fields−→E and

−→H denotes the electric and magnetic fields, respectively,

which are induced from the presence of an electric charge density distribution ρ, and a

current density−→j and use of natural units c = ~ = 1 has been made throughout the

notations. Maxwell’s equations (1.6) in free space where ρ = 0, j = 0, these equations are

symmetric and invariant under the duality transformations−→E 7−→

−→H and

−→H 7−→ −

−→E . Thus

in the presence of sources, (1.6) are neither symmetric nor dual invariant. Introduction of

four-potential Aµ =φ,−→A

derived from the source terms ρ and−→j , the electric and

magnetic fields in the following manner,

−→E =−

−→∇φ− ∂

−→A

∂t;

−→H =−→∇ ×

−→A. (1.7)

In covariant notation, these electric and magnetic fields are components of following elec-

16


tromagnetic field tensor,

Fµν =∂νAµ − ∂νAµ; (µ, ν = 0, 1, 2, 3) . (1.8)

Such that F0i = Ei and Fij = εijkHk. Covariant forms of Maxwell’s equations may then be

written as follows,

Fµν =∂νFµν = jµ; (1.9)

Fµν =∂νFµν = 0; (1.10)

where jµ =ρ,−→j

is a four current density and Fµν is defined as,

Fµν =1

2εµνσλF

σλ; (1.11)

in which ”Fµν” represents the dual part while εµνσλ is completely antisymmetric Ricci tensor

of rank four. Equations (1.9) and (1.10) are obviously asymmetrical in Fµν and Fµν and do

not remain invariant under the duality transformation Fµν 7−→Fµν and Fµν 7−→ −Fµν . The

vector wave function−→ψ associated with electromagnetic fields is defined as,

−→ψ =−→E − i

−→H. (1.12)

Taking the divergence and curl of equation (1.7), we get the following equations,

−→∇ .−→ψ =ρ;

−→∇ ×

−→ψ =− ij − ∂

−→ψ

∂t. (1.13)

The lack of symmetries in Maxwell’s equations may be visualized in connection with the

following points,

(1) There is no magnetic charge analogy of electric charge and current source densities.

17


(2) The magnetic field appears is produced by the motion of electric charge but there is no

similar contribution of magnetic charge in producing electric field.

(3) In terms of four potential Aµ, the equation (1.7) demands the different nature of

relations, which show that the magnetic field is the effect for the rotation of the spatial part

of potential while such interpretation cannot be given to the electric field.

(4) The symmetry is also explicitly revealed in the equation (1.8) for electromagnetic field

tensor.

(5) Equation (1.6) gives no evidence for magnetic sources.

1.3.3 Duality invariance and need of monopole

Duality invariance is an old idea introduced a century ago in classical electromagnetism

[162]-[168] for the following Maxwell’s equation in vacuum i.e.

−→∇ ·−→E = 0;

−→∇ ·−→H = 0;

−→∇ ×

−→E = −∂

−→H

∂t;

−→∇ ×

−→H =

∂−→E

∂t. (1.14)

The space-time four-vector xµ = (t, x, y, z) xµ = ηµνxµ and ηµν = +1,−1,−1,−1 = ηµν

through out the text. Maxwell’s equations (1.14) are invariant not only under Lorentz and

conformal transformations but are also invariant under the following duality transforma-

tions in equation (1.3) and the duality transformation in equation (1.2), which can be

written as

−→E−→H

=⇒

0 1

−1 0

−→E−→H

. (1.15)

Consequently, Maxwell’s equations may be solved by introducing the concept of vector

18


potential in either two ways [169, 170].

Case-I : The conventional choice is being used as in equation (1.7) so, the dual sym-

metric and Lorentz covariant Maxwell’s equations (1.14) are written in as

∂νFµν = 0;

∂νFµν = 0; (1.16)

where F µν = ∂νAµ−∂µAν = Aµ,ν−Aν,µ is anti-symmetric electromagnetic field tensor, Fµν =

12εµνλωFλω (∀µ, ν, λ, ω = 0, 1, 2, 3) is the dual of electromagnetic field tensor and εµνλω is the

four index Levi-Civita symbol. εµνλω = +1∀ (µνλω = 0123) for cyclic permutation, εµνλω =

−1 for any two permutations and εµνλω = 0 if any two indices are equal. Using equation

(1.7), the electric and magnetic fields as the components of anti-symmetric electromagnetic

field tensors F µν and Fµν given by

F 0j = Ej; F jk = εjklHl (∀j, k, l = 1, 2, 3) ;

F oj = Hj; F jk = εjklEl (∀j, k, l = 1, 2, 3) ; (1.17)

where εjkl is three index Levi-Civita symbol and εjkl = +1 for cyclic, εjkl = −1 for anti-cyclic

permutations and εjkl = 0 for repeated indices. The duality symmetry is lost if electric

charge and current source densities enter to the conventional inhomogeneous Maxwell’s

equations given by in equation (1.6) so, the covariant form of Maxwell’s equation (1.6)

is described as in equations (1.9) and (1.10). The pair(−→∇ ·−→H = 0;

−→∇ ×

−→E = −∂

−→H∂t

)of

Maxwell’s equations (1.6) is described by ∂νF µν = 0 in equations (1.9) and (1.10). It has

become kinematical while the dynamics is contained in another pair−→∇ ·−→E = ρ; ∇×

−→H =

−→j + ∂

−→E∂t

of Maxwell’s equations (1.6) which described as ∂νF µν = jµ in equation (1.9)

and (1.10) and also reduces to following wave equation in the presence of Lorentz gauge

condition ∂µAµ = 0 i.e.

19


Aµ =jµ; (1.18)

where = ∂2

∂t2− ∂2

∂x2− ∂2

∂y2− ∂2

∂z2is the D’ Alembertian operator. So, a particle of mass m

electric charge e moving with a velocity uν in an electromagnetic field is subjected by a

Lorentz force given by

md2xµdτ 2

=dpµdτ

= fµ =eFµνuν ; (1.19)

where xµ is the four-acceleration, fµ is four-force and pµ is four-momentum of a particle.

Equation (1.19) is reduced to

~f =d−→pdt

= md2−→xdt2

= e[−→E +−→u ×

−→H]

; (1.20)

where −→p ,−→f ,−→x , −→u are respectively the three vector forms of momentum, force, displace-

ment and velocity of a particle. The Lorentz force equation of motion (1.19 - 1.20) are also

not invariant under duality transformations (1.2) and (1.15).

Case-II : On the other hand, let us introduce [169, 170] the another alternative way

instead of equation (1.7) to write

−→H = −∂

−→B∂t−−→∇ϕ;

−→E = −

−→∇ ×

−→B ; (1.21)

where a new potential Bµ =(ϕ,−→B)

is introduced [169]-[173] as an alternative to Aµ .

Thus, the source free (homogeneous) Maxwell’s equation are same as those equations

(1.14) but the inhomogeneous Maxwell’s equation (1.6) are changed to

20


−→∇ ·−→E = 0;

−→∇ ·−→H = %;

−→∇ ×

−→H =

∂−→E

∂t;

−→∇ ×

−→E = −

−→k − ∂

−→H

∂t; (1.22)

subjected by the introduction of a new four current source density kµ =(%,−→k)

. In equa-

tion (1.22) we see that the pair(−→∇ ·−→E = 0;

−→∇ ×

−→H = ∂

−→E∂t

)becomes kinematical while

the dynamics is contained in the second pair(−→∇ ·−→H = %;

−→∇ ×

−→E = −

−→k − ∂

−→H∂t

). Equation

(1.22) may also be written in following covariant forms

∂νFµν = 0;

∂νF µν = kµ; (1.23)

where F µν = ∂νBµ − ∂µBν;˜F µν = F µν;kµ =(%,−→k)

and kµ =(%, −−→k)

. Equation

(1.22) may also be obtained on applying the transformations (1.2) and (1.15) to equation

(1.6) followed by following duality transformations for potential, current and antisymmet-

ric electromagnetic field tensors as

Aµ −→ Bµ;Bµ −→ −Aµ;

jµ −→ kµ;kµ −→ −jµ;

F µν −→ F µν ;F µν −→ −F µν . (1.24)

As such, we may identify the potential Bµ =(ϕ,−→H)

as the dual of potential Aµ and

the current kµ =(%,−→k)

as the dual of current jµ. Correspondingly, the differential

equations (1.21) are identified as the dual Maxwell’s equations. So, accordingly, the elec-

trodynamics of a charged particle with the charge dual to the electric charge (i.e. magnetic

21


monopole). Applying the the electromagnetic duality to the Maxwell’s equations, estab-

lished the connection between electric and magnetic charge (monopole) [174, 175] , in

the same manner as an electric charge e interacts with electric field and the dual charge

(magnetic monopole) g interacts with magnetic field as,

e −→ g; g −→ −e; (1.25)

where g is described as the dual electric charge (charge of magnetic monopole). Hence,

the dual electrodynamics as the dynamics of pure magnetic monopole. Consequently, the

corresponding dynamical variables associated there in are described as the dynamical vari-

ables in the theory of magnetic monopole. So, the new electromagnetic field tensor Fµν in

place of F µν as

F µν 7−→ Fµν = ∂νBµ − ∂µBν (µ, ν = 1, 2, 3) ; (1.26)

which reproduces the following definition of magneto-electric fields of monopole as

F0i = H i;

Fij = −εijkEk. (1.27)

Hence the covariant form of Maxwell’s equations (1.20) for magnetic monopole may now

be written as

Fµν,ν = ∂νFµν = kµ;

Fµν,ν = ∂νFµν = 0. (1.28)

Where kµ =(%, −−→k)

is the four-current density due to the presence of the magnetic

22


charge g. Accordingly, the wave equation (1.23) for pure monopole is described as

Bµ = kµ; (1.29)

in presence of Lorentz gauge condition ∂µBµ =0. Accordingly, the classical Lagrangian for-

mulation in order to obtain the field equation (dual Maxwell’s equations) and equation of

motion for the dynamics of a dual charge (magnetic monopole) interacting with electro-

magnetic field. So, the Lorentz force equation of motion for a dual charge (i.e magnetic

monopole) may now be written from the duality equations (1.2) and (1.15) as

d−→pdt

=−→f = m

−→x =g

(−→H −−→u ×

−→E)

; (1.30)

where −→p = m−→x = m−→u is the momentum, and

−→f is a force acting on a particle of charge

g, mass m and moving with the velocity −→v in electromagnetic fields. Equation (1.28) can

be generalized to write it in the following four vector formulation as

md2xµdτ 2

=dpµdτ

= fµ = mxµ = gFµνuν ; (1.31)

where uν is the four velocity, pµis four momentum, fµ is four force and xµ is the

four-acceleration of a particle carrying the dual charge (namely magnetic monopole). The

need of monopole are

• Dirac quantization condition

One of the defining advances in quantum theory was Dirac’s work on developing a

relativistic quantum electromagnetism. Before his formulation, the presence of elec-

tric charge was simply inserted into the equations of quantum mechanics (QM), but

in 1931 Dirac [1] showed that a discrete charge naturally falls out of QM. The form

of Maxwell’s equations and still have magnetic charges. Consider a system consisting

23


of a single stationary electric monopole (electron) and a single stationary magnetic

monopole. Classically, the electromagnetic field surrounding them has a momentum

density given by the Poynting vector, and it also has a total angular momentum, which

is proportional to the product (qeqm), and independent of the distance between them.

Quantum mechanics dictates however, that angular momentum is quantized in units

of ~ so therefore, the product (qeqm) must also be quantized. This means that if

even a single magnetic monopole existed in the universe and the form of Maxwell’s

equations is valid, all electric charges would then be quantized. Although it would

be possible simply to integrate over all space to find the total angular momentum.

Dirac’s considered a point-like magnetic charge whose magnetic field behaves as qmr2

and is directed in the radial direction. Because the divergence of B is equal to zero

almost everywhere, except for the locus of the magnetic monopole at r = 0 one can

locally define the vector potential such that the curl of the vector potential A equals

the magnetic field B.

The concept of local gauge invariance (gauge theory) below provides a natural ex-

planation of charge quantization, without invoking the need for magnetic monopoles

but only if the U (1) gauge group is compact, in which case we will have magnetic

monopoles anyway. If we maximally extend the definition of the vector potential for

the southern hemisphere, it will be defined everywhere except for a semi-infinite line

stretched from the origin in the direction towards the northern pole. This semi-infinite

line is called the Dirac string and its effect on the wave function is analogous to the ef-

fect of the solenoid in the Aharonov - Bohm effect. The quantization condition comes

from the requirement that the phases around the Dirac string are trivial, which means

that the Dirac string must be unphysical. The Dirac string is merely an artifact of the

coordinate chart used and should not be taken seriously. The Dirac monopole is a

singular solution of Maxwell’s equation because it requires removing the world line

from space time in more complicated theories, it is superseded by a smooth solution

such as the ’t Hooft–Polyakov monopole.

• Dirac String

A gauge theory like electromagnetism is defined by a gauge field which associates a

24


group element to each path in space time. For infinitesimal paths, the group element

is close to the identity, while for longer paths the group element is the successive

product of the infinitesimal group elements along the way. In electrodynamics, the

group is U (1) unit complex numbers under multiplication. For infinitesimal paths,

the group element is (1 + iAµdxµ) which implies that for finite paths parametrized by

s, the group element is

∏s

(1 + ieAµ

dxµ

dsds

)= exp

(ie

ˆA · ds

). (1.32)

So that the phase a charged particle gets when going in a loop is the magnetic flux through

the loop. Dirac’s monopole solution in fact describes an infinitesimal line solenoid ending at

a point, and the location of the solenoid is the singular part of the solution, the Dirac string.

Dirac strings link monopoles and anti monopoles of opposite magnetic charge, although in

Dirac’s version, the string just goes off to infinity. The string is unobservable, so you can

put it anywhere and by using two coordinate patches, the field in each patch can be made

non singular by sliding the string to where it cannot be seen.

• String theory

In our universe quantum gravity provides the regulator. When gravity is included,

the monopole singularity can be a black hole, and for large magnetic charge and

mass, the black hole mass is equal to the black hole charge, so that the mass of the

magnetic black hole is not infinite. If the black hole can decay completely by Hawking

radiation, the lightest charged particles cannot be too heavy. The lightest monopole

should have a mass less than or comparable to its charge in natural units. So in

a consistent holographic theory, of which string theory is the only known example,

there are always finite-mass monopoles. For ordinary electromagnetism, the mass

bound is not very useful because it is about same size as the Planck mass.

25


1.3.4 Dirac Monopole and their properties

The Dirac monopole [1] is an object which occurs in the theory of charged fields, both

classical and quantized and in the quantum theory of charged particles. For the particle

formulation, the characteristic structure of Dirac monopoles only appears at the quantum

level whereas for the field theory Dirac monopoles already exist as classical objects and

does not change when the theory is quantized. Dirac proposed [1] a procedure that may

be used to symmetrizing [176], which starts from the extended equations by postulating

the existence of magnetic monopoles, the generalized Dirac Maxwell’s (GDM) equations

[177, 178] are expressed in S. I. units (c = ~ = 1) as

−→∇ .−→E =ρe;

−→∇ .−→H =ρm;

−→∇ ×

−→E =− ∂

−→H

∂t−−→jm;

−→∇ ×

−→H =

∂−→E

∂t+−→je ; (1.33)

where ρe and ρm are respectively the electric and magnetic charge densities,−→je and

−→jm

are the corresponding current densities. GDM equations (1.6) are invariant not only under

Lorentz and conformal transformations but also invariant under the duality transformations

in equation (1.2) and (1.3), which can be written as −→E−→H

⇒ 0 1

−1 0

−→E−→H

; (1.34)

together with

je → jm, jm → −je ⇐⇒

je

jm

⇒ 0 1

−1 0

je

jm

. (1.35)

Under these transformation Lorentz equation of motion written in following form is also

invariant,

26


md2x

dt2=(eFµν − gFµν

)uν ; (1.36)

or

mdv

dt=e(−→E +−→u ×

−→H)

+ g(−→H −−→u ×

−→E)

; (1.37)

where m is the mass of the particle, e is the electric charge, uν is four-velocity of particle,

space-time four vector is defined as xµ= t,−→x and g is magnetic charge. Electric and

magnetic four-current are related as jµ = euµ and kµ = guµ. As such the duality invari-

ance is an intrinsic property of Maxwell’s Lorentz theory of electrodynamics in presence of

monopole.

Thus, not only the Maxwell’s equation allow one to predict mathematically the behavior of

magnetic monopole, but also has a nice pictorial representation at the intuitive level of how

a magnetic monopole look [1]. We can immediately realize that the end of long needle or

a thin solenoid behaves like magnetic pole in all most all the region of space around it.

Dirac attempted to find a fundamental quantum mechanical principle that would explain

the quantization of electric charge. Dirac [1] examines the connection between the physical

meaning of the phase factor of a wave function and the gauge principle in quantum me-

chanics. Dirac [1] published his work about the possibility of adding magnetic monopole

of electromagnetism and put forward an idea of magnetic monopole as generalization of

electrodynamics. This hypothesis of the existence of magnetic monopoles provides an ex-

planation for the quantization of electric charge and maintains symmetries in the Maxwell’s

equations. Dirac gives an interesting result is that the product of a magnetic monopole

charge(g) with the electron electric charge(e) must be quantized [1] i.e.

eg =1

2n, n = 1, 2, 3, .... (1.38)

where e and g are respectively the electric and magnetic charges and n is the principle

27


quantum number. This condition implies that in the presence of magnetic monopole, elec-

tric charge must be integral multiple of a fundamental unit. This quantization condition

demands the existence of free magnetic pole having the pole strength,

g =e

2α; (1.39)

where α is fine structure constant. In deriving this condition it was assumed that a particle

has either electric charge or magnetic charge (not both). This is so - called Abelian magnetic

monopole.

In spite of many good points, Dirac theory encounters with many difficulties. In this theory

if the magnetic field,

−→H =

gr

r2; (1.40)

produced by magnetic charge g located at the origin is described by vector potential−→A (r) ,

then

−→H 6=−→∇ ×

−→A ; (1.41)

along the line going for monopole to infinity. Such a line may be curved or planer is referred

as Dirac string in literature. For the straight string, S(n) we may write

A(n)(r) =g

r.−→r × n

r − (−→r .n);

=g

r.[−→r × n (−→r .n)][r2 − (−→r .n)

2] . (1.42)

For these vector potentials and−→H 6=

−→∇ ×

−→A along the singular line −→r = cn

| An |=∞; (1.43)

and hence in Dirac theory,

28


(1) A string of arbitrary shape ends at each monopole.

(2)−→A (r) is singular along string from monopole location to a infinity.

(3) Charged particles can never pass through string.

These conditions are referred to as Dirac’s veto that has been the source of troubles as the

history of theory shows. It seems to be unnatural and undesirable conditions, since strings

are unphysical objects. Besides this, singular electromagnetic potential is not present in

usual field theory and this veto gave rise to difficulty [179] in scattering of electrons and

monopoles. It has been shown by Seo [180] that due to Dirac’s veto, the fermion Hamil-

tonian in presence of an abelian monopole is not Hermitian when acting on full function

space. Though Dirac’s veto lead to remarkable quantization condition (1.38), it has been

the source of criticism and controversies and as such a covariant theory could not be for-

mulated as a standard field theory of these fields.

The first attempt to construct the theory of monopole free from Dirac’s veto was made in-

dependently by Mandelstom [181] and Cabibbo [182] by developing gauge independent

method using two vector potentials and then utilizing the treatment of Quantum Electro

Dynamics. The method of Mandelstom does not use potentials but instead quantize the

field directly. In Cabibbo-Ferrari theory the introduction of second potential is compen-

sated by an enlargement of the group of gauge transformations. The introduction of this

potential does not lead to an increase of the number of independent variables which de-

scribe the free field. The theory developed by Keon [183] on the basis of idea of Cabbibo

and that by Ezawa and Tze [184] without using Dirac veto, lack in action principle in the

presence of particles carrying both electric and magnetic charges.

The group H (θ) for θ ∈ M0, where M0 is a two dimensional sphere of radius a in isotopic

space. The group consisting of these elements in G = SO (3) which leaves the given θ

invariant, is just the group of rotations about θ axis and it is isomorphic to SO (2) or equiv-

alently U (1). Since the group H (θ) for θ ∈ M0 are all isomorphic, we shall denote any

of them by H. The original symmetry G is spontaneously broken down to H by θ. Thus

after symmetry breaking, we are left with U (1) gauge theory which has the characteristics

of Maxwell’s electromagnetic theory. We must emphasize the following ansatz for seeking

solution for monopole [185]-[188],

29


θ(b) =rber2

H (aer) ;

Aib =− εbijrjer2

[1−K (aer)] ;

A0b =0; (1.44)

where H and K are dimensionless functions to be determined by equation of motion. Now

the energy of the system is given by,

E =4πa

e

∞

0

dζ

ζ2[ζ2(dk

dζ

)2

+1

2

(dH

dζ−H

)2

+1

2

(K2 − 1

)2+K2H2 +

π

4e2(H2 − ζ2

)2]; (1.45)

where ζ = aer. The conditions for E to be satisfactory with respect to the variations of H

and K are,

ζ2d2H

dζ2= KH2 +K

(K2 − 1

);

ζ2d2H

dζ2= 2KH2 +

λ

e2H(H2 − ζ2

). (1.46)

In this total energy solution, which will be interpreted as classical mass, can be obtained

form equation (1.45) in the following form,

Mass =4πa

ef

(λ

e2

); (1.47)

where f(λe2

)is the value of integral (1.45). This energy solution which is topologically non-

trivial, is called pt Hooft - Polyakov monopole [44, 45] which differs from Dirac monopole

is a point like one. The field tensor for pt Hooft-Polyakov monopole has the form,

30


F ij = ∂iAj − ∂jAi + (extra term) . (1.48)

In Dirac monopole the extra term is singular and involves Dirac string, while in the t’ Hooft-

Polyakov monopole the extra term is smooth and involves scalar field. t’ Hooft - Polyakov

model obviously leads the following consequences

• Monopoles are intrinsic apart of the grand unified theories (GUT’s) [189] Since all

the current unified theories of fundamental interactions envisage a gauge invariant

theory with a gauge group G spontaneously broken to a subgroup H. All such theories

exhibit monopole solution with non-trivial topological properties as demonstrated by

Dokes-Tomras [190] for

G = SU (5)

H = SU (3)× SU (2)× U (1) (Locally)

where due to unbroken symmetry group consisting U (1) factor, the general argument

implies that such models contain stable solutions which carry U (1) magnetic charge

and super heavy masses of the order of scale of symmetry breaking.

• Beside stable, when smoothed out, the monopole is super heavy and strongly inter-

acting. All the reasons and the fact that Dirac’s monopole serves as an approximate

dynamical theory of t’ Hooft-Polyakov monopole, the recent interest in the subject of

monopole has been enhanced due to the following more reasons

(I) Magnetic condensation of vaccum

Magnetic condensation of vaccum would guarantees the absolute color confinement

through dual Meissner effect. The related problem is to carry out the study of multi-

ple structure of non-Abelian monopoles. This problem has not been taken up properly

except some efforts made by Cho [191] to analyze magnetic symmetry of restricted

31


quantum chromo dynamics.

(II) Baryon asymmetry

One of the exciting of GUT’s is the explanation of excess of matter over antimatter.

It was suggested by Dokos and Tomaras [190] that grand unified monopole acts as

a catalyst in baryon number non conserving process with cross section of order 1M2x

.

When Mx ≈ 105Gev (Unificationmass). Such that

M +N −→M + e+ +Mesons

where M is the monopole and N = P , n are independent of Mx. It represents dominant

catalyst decay mode excepted in many GUT’s in connection with strong fermion number

non-conservation in monopole fermion interaction [192]. On the basis of the arguments

Rubakov [193] and Callan [194] have shown that monopole catalysis baryon decay as the

result of chiral charge condensation induced by chiral anomaly. As such, the proton decay

takes place around the magnetic monopole through a mechanism different from X − boson

exchange. Its rate is neither dependent on coupling constant nor suppressed by any power

of boson mass. These processes have typical strong interaction cross-section of the order of

(1Gev)−2. It opened new prospects for monopole search and proton decay experiments and

a new area of developing strong interaction calculation method. Actually SU (5) monopoles

attracted much attention only after Rubakov’s discovery.

1.3.5 GUT’s and pt Hooft - Polyakov Monopoles

A Grand Unified Theory (GUT) is a candidate model in particle physics in which at high

energy, the three gauge interactions of the Standard Model which define the electromag-

netic, weak, and strong interactions are merged into one single interaction characterized by

one larger gauge symmetry and thus one unified coupling constant [125]. The experimen-

32


tally verified Standard Model of particle physics is based on three independent interactions,

symmetries and coupling constants. Models that do not unify all interactions using one sim-

ple Lie group as the gauge symmetry, but do so using semi simple groups can exhibit similar

properties and are sometimes referred to as Grand Unified Theories as well [126]. Unifying

gravity with the other three interactions would provide a theory of everything rather than

a GUT. Nevertheless, GUT’s are often seen as an intermediate step towards a theory of ev-

erything. The new particles predicted by models of grand unification cannot be observed

directly at particle colliders because their masses are expected to be of the order of the so-

called GUT scale, which is predicted to be just a few orders of magnitude below the Planck

scale and thus far beyond the reach of currently foreseen collision experiments. Instead,

effects of grand unification might be detected through indirect observations such as proton

decay, electric dipole moments of elementary particles, or the properties of neutrinos. Some

grand unified theories predict the existence of magnetic monopoles.

All GUT models which aim to be completely realistic are quite complicated even compared

to the Standard Model because they need to introduce additional fields and interactions or

even additional dimensions of space. The main reason for this complexity lies in the diffi-

culty of reproducing the observed fermion masses and mixing angles. Due to this difficulty,

and due to the lack of any observed effect of grand unification so far, there is no generally

accepted GUT model. The fact that the electric charges of electrons and protons seem to

cancel each other exactly to extreme precision is essential for the existence of the macro-

scopic world but this important property of elementary particles is not explained in the

Standard Model of particle physics. While the description of strong and weak interactions

within the Standard Model is based on gauge symmetries governed by the simple symmetry

groups SU (3) and SU (2) which allow only discrete charges, the remaining component, the

weak hyper charge interaction is described by an abelian symmetry U (1) which in princi-

ple allows for arbitrary charge assignments. The observed charge quantization namely the

fact that all known elementary particles carry electric charges which appear to be exact

multiples of 13

of the elementary charge has led to the idea that hyper charge interactions

and possibly the strong and weak interactions might be embedded in one Grand Unified

interaction described by a single, larger simple symmetry group containing the Standard

33


Model [127]. This would automatically predict the quantized nature and values of all ele-

mentary particle charges. Since this also results in a prediction for the relative strengths of

the fundamental interactions which we observe in particular the weak mixing angle, Grand

Unification ideally reduces the number of independent input parameters but is also con-

strained by observations.

Grand Unification is reminiscent of the unification of electric and magnetic forces by Maxwell’s

theory of electromagnetism in the 19th century, but its physical implications and mathemat-

ical structure are qualitatively different. The unification of matter particles SU (5) is the

simplest GUT. The smallest simple Lie group which contains the standard model, and upon

which the first Grand Unified Theory was based is SU (5) ⊃ SU (3)× SU (2)× U (1) . Such

group symmetries allow the reinterpretation of several known particles as different states of

a single particle field. However it is not obvious that the simplest possible choices for the ex-

tended Grand Unified symmetry should yield the correct inventory of elementary particles.

The fact that all currently known matter particles fit nicely into three copies of the smallest

group representations of SU (5) and immediately carry the correct observed charges is one

of the first and most important reasons why people believe that a Grand Unified Theory

might actually be realized in nature [128]. The two smallest irreducible representations of

SU (5) are 5 and 10. In the standard assignment, the 5 contains the charge conjugates of

the right-handed down-type quark color triplet and a left-handed lepton isospin doublet,

while the 10 contains the six up-type quark components, the left-handed down-type quark

color triplet, and the right-handed electron. This scheme has to be replicated for each of

the three known generations of matter. It is notable that the theory is anomaly free with

this matter content [129]-[131].

The unification of forces is possible due to the energy scale dependence of parameters in

quantum field theory called renormalization group running, which allows parameters with

vastly different values at collider energies to converge at much higher energy scales. The

renormalization group running of the three gauge couplings in the Standard Model has

been found to nearly, but not quite, meet at the same point if the hyper charge is nor-

malized so that it is consistent with SU (5) or SO (10) GUT’s, which are precisely the GUT

groups which lead to a simple fermion unification. In this case, the coupling constants of

34


the strong and electroweak interactions meet at the grand unification energy also known

as the GUT scale ΛGUT ≈ 1016Gev. It is commonly believed that this matching is unlikely to

be a coincidence and is often quoted as one of the main motivations to further investigate

Super symmetric theories despite the fact that no Super symmetric partner particles have

been experimentally observed. Also most model builders simply assume super symmetry

because it solves the hierarchy problem i.e. it stabilizes the electroweak Higgs mass against

radiative corrections [125]-[131].

Though the physicists were fascinated since Dirac’s work on monopole, the fresh interest

in this subject was enhanced by the remarkable observations of pt Hooft [44] that classi-

cal solutions having the properties of magnetic monopoles may be found in Yang - Mill’s

theory with spontaneous symmetry breaking with a suitable identified of electromagnetic

fields. Furthermore, pt Hooft - Polyakov [44, 45] showed that the existence of monopoles

follow from quite general ideas about the unification of fundamental interactions. A deeply

held belief of many particle theorists is that the observed strong and electroweak gauge

interactions actually become unified at extremely short distances into a single gauge in-

teraction with just one single gauge coupling constant. Any such grand unified theory of

particle physics necessarily contains magnetic monopoles. The implications of this discov-

ery are rich and surprising and are still being explored. Monopoles associated with the

spontaneous symmetry breakdown of grand unified theory are super heavy (≈ 1016 Gev)

and complicated extended objects having a definite size in side of which massive fields play

a role in in providing a smooth structure and outside they rapidly vanish leaving the field

configuration like that of Dirac’s monopole. Magnetic monopoles have been the object of

intense interest ever since it was shown that they can arise as classical solutions in spon-

taneously broken gauge theories. In theoretical physics, the pt Hooft - Polyakov [44, 45]

monopole is a topological soliton similar to the Dirac monopole but without any singulari-

ties. It arises in the case of a Yang - Mill’s theory with a gauge group G, coupled to a Higgs

field which spontaneously breaks it down to a smaller group H via the Higgs mechanism.

It was first found independently by Gerard pt Hooft and Alexander Polyakov. It was shown

by pt Hooft and Polyakov that the specific boundary condition applied to the coupled Yang

- Mill’s - Higgs field leads to a finite action configuration featuring a nonzero magnetic

35


charge, a magnetic monopole. This nontrivial solution of the coupled field equations can

be obtained whenever the electromagnetic group U(1) is a subgroup of the gauge group

with a compact covering group such as SU(2), SU(3) and SO(3), but not SU(2)×U(1). It is

extended magnetic monopole without a Dirac string generated by the charged non Abelian

fields. They showed that magnetic monopoles are obligatory in the low energy theory if

U(1) is embedded in a large, simple group G. The pt Hooft - Polyakov monopole is a topo-

logical soliton similar to the Dirac monopole but without any singularities. It arises in the

case of a Yang - Mills theory with a gauge group G. Actually this is a topologically stable

system of Higgs and gauge fields with magnetic charge g = n(1e

). An amazing property of

this monopole is that its mass is finite. Of course its magnetic properties only display at a

distance far from the core, within the core gauge fields are mixed together. This is so called

non - Abelian magnetic monopole coupled to a Higgs field which spontaneously breaks it

down to a smaller group H via the Higgs mechanism. It was first found independently by

Gerard pt Hooft and Alexander Polyakov. pt Hooft and Polyakov showed that the existence

of magnetic monopole follows from the quite general idea of unification of fundamental

interactions. In the mean time it became clear that monopole can be understand better in

non - Abelian gauge theories and that the reason for not seeing monopoles so far lies else-

where then in their possible inconsistencies with the relativity and quantum mechanics.

Spontaneously broken non - Abelian gauge theories may support a classical solution which

is asymptotically equivalent to a monopole magnetic field and also leads to charge quan-

tization. In particular pt Hooft and Polyakov [44, 45] have demonstrated for the system

described by the Lagrange function,

L =− 1

4GaµνGaµν +

1

2Dµ−→φ Dµ

−→φ − V

(−→φ)

; (1.49)

36


where

V(−→φ)

=λ

4

(φ21 + φ2

1 + φ21 − a2

)2; (1.50)

Gµνa =∂µW ν

a − ∂νW µa − εabcW

µb W

νc ; (1.51)

(Dµφ)a =∂µφa − eεabcW µb φc

= (∂µφa + ieW µb τa)φa; (1.52)

where a = 1, 2, 3. The quantities φa , Gµνa and (Dµφ)a all transforms as vectors with respect

to local SO(3) rotations. These equation leads to the equation of motion,

(DνGµν)a =− eεabcφb (Dµφ)c ; (1.53)

(DµDµφ)a =− λφa

(−→φ .−→φ − a2

). (1.54)

The energy density is

θ00 =1

2

((Eia

)2+(Eia

)2+((D0φ

)a

)2+((Diφ

)a

)2)+ V (φ) . (1.55)

It is obvious that θ00 ≥ 0 and vanishes if and only if

Gaµν =0; (1.56)

(Dµφ)a =0; (1.57)

V (φ) =1

4λ(−→φ .−→φ − a2

)2. (1.58)

The component of scalier triplet, φ3 has a constant, non-vanishing vacuum expectation

value < φ3 >= a . The remaining components φ1,2 are absorbed into the longitudinal

components of the charged vector fields W µ1,2 which have charge e. The neutral electric

field W µ3 remains mass less and may be identified with the photon. The original symme-

try G is spontaneously broken down down to H by φ. After symmetry breaking, we are

left with a U(1) gauge theory which has all the characteristics of Max well’s electromag-

netic theory. The pt Hooft - Polyakov [44, 45] monopole carries one Dirac unit of magnetic

charge. These monopoles are not elementary particles like Dirac’s monopoles but compli-

37


cated extended objects having a definite size inside of which massive fields play a role in

providing a smooth structure and outside they rapidly vanish leaving the field configuration

identical to Dirac’s monopoles. Wu and Yang [195] introducing fiber bundle formulation

into gauge theories, reformulated Dirac’s theory to avoid any singularities in−→A by dividing

the space surrounding monopole into few regions whose vector potentials are connected

through gauge transformations. The pt Hooft - Polyakov monopole was known numerically

but there is simplified model introduced by Prasad and Sommerfield [196, 197] which has

an explicit stable monopole solution. Such solution satisfying Bogomonly condition [198]

are named as Bogomonly-Prasad-Sommerfield (BPS) monopoles. These static monopoles in

R3 - space have been extensively studied in recent years and it became clear that they have

remarkable properties which are best understood as a special case of self - duality equa-

tions in four space for solutions independent of one of the variables. The mass of monopole

solution with a smooth internal structure is calculable to have the following lower limit of

Prasad and Sommerfield

M ≥a | g |; (1.59)

where

a =(φ21 + φ2

2 + φ23

); (1.60)

where φ1 , φ2 and φ3 are components of iso-vector Higgs fields in SU(2) gauge theory. The

condition (1.60) shows that mass of pt Hooft monopole is not arbitrary like that of Dirac

monopole.

1.3.6 Why dyons

In spite of the enormous potential importance of Dirac’s monopole and the fact that

they have been intensively studied recently there has been presented no reliable theory,

which is as conceptually transparent and practically tractable as the usual electrodynam-

ics. However the problem raised by the Dirac’s veto were eventually solved when Wu and

Yang [195] introduced the fibre bundle formulation into gauge theories still there remain

38


following paradoxes.

1.3.6.1 Wrong connection between spin and statistics

If the monopole exists, then classical physics tells that a system of pole g and electric

charge e has an angular momentum of magnitude eg directed from charge to pole i.e.

−→J =

−→L −egr. (1.61)

This is the sum of orbital angular momentum of the particle and the angular momentum

of the electromagnetic fields. In quantum mechanics that spin adds to orbital and intrinsic

angular momentum, so that for eg =(n+ 1

2

), an otherwise integral spin system will have

net half integral total angular momentum. This holds equally good in the SU (2) gauge field

formulation of charge pole interactions. In fact this spin may be used to derive the gauge

field shown by Jackiw and Rebbi [199] and demonstrating that in an SU (2) quantum gauge

field theory, with iso - spin symmetry broken spontaneously by triplet of scalar mesons,

iso-spinors degrees of freedom and converted into spin degrees of freedom in the field of

magnetic monopole consequently in solution - monopole sector of quantum theory total

angular momentum(−→J)

is the sum of conventional orbital plus spin(−→M)

and iso - spin(−→I)

;

−→J =

−→L +

−→M +

−→I . (1.62)

As such, perhaps an object whose half integral spin comes from charge pole contribution

obeys Fermi - Dirac statistics, so that a fermion may be made out of bosons.

It led Goldhaber [18] to prove a theorem which states that if electric charge e can combine

with magnetic monopole g to from the cluster with half integral values for the product eg

then there must entities with wrong connection between spin and statistics, considering

electric and magnetic charge on the same particle (a dyons) could solve this problem.

They showed that dyon (provides carrying electric and magnetic charges) wave function

(which is diagonal in both angular momentum and parity) lead to correct spin statistics

39


relationship. Such particle (carrying electric and magnetic charges) were named as dyons

by Schwinger [23]-[29] suggesting that quark and dyons.

1.3.6.2 Witten Effect

In nature CP is violated (but not weakly), it has been shown by Witten [49] that if the

θ defined by the argument that rotations in gauge space through θ should be accompanied

by multiplication with the phase −φgθ2π

, which can be thought as rotation in magnetic space

by −φθ2π

. Witten [49] showed that CP non conserving theories, if a non-zero angle θ of the

world is only mechanism for CP violation, the electric charge of the monopole is exactly

calculated and is − eθ2π

plus an integer i.e.

q = ne− θe

2π. (1.63)

It shows that in CP non conserving theories the electric charge of pt Hooft monopole will

not ordinarily be integral. One may therefore, suspect that the monopole, if they exists,

have charges that are almost nut not quite integers. The derivation of the monopole from

integral charges would be proportional to the strength of CP violation. It shows that

monopoles are necessarily dyons. It is an existing result leading to deep insight into the

fermion structure of monopoles [200].

1.3.6.3 Recent conflict with existence with pure monopoles

J. Pantaleone [201] have analyzed the experimental results of Fairbank’s [202] et. al.

about the reported evidence for object with electric charges and showed that of Cabrera

[4] if both result are true, and showed that if both result are true, the condition is puzzling

since it conflicts with either the Dirac’s quantization condition or colour force confining

quarks or the exact gauge symmetry as SU (3)c×U (1)em. If we assume that the monopoles

are necessarily dyons, such difficulty is automatically removed.

1.3.6.4 Non-abelian dyons

Julia and Zee [48] extending t’ Hooft model, showed the possibility of constructing

40


classical solutions having both electric and magnetic charges. Such solutions have also

been reported by Carmeli [203] by means of null tetrad notations. Julia and Zee [48]

demonstrated that t’ Hooft monopole has an integral degrees of freedom which allows

it to readily exchange changes with ordinary particles. It has been pointed out that a

dyon is energetically not allowed to decay into magnetic monopole emmiting charge vector

mesons. At classical level the charge of dyon is not quantized and the dyons are not much

massive then magnetic monopole. Now it has been clear that a theory, which describes

electromagnetic fields in terms of single potential, can not avoid controversial. Dirac string

variables have been tried to avoid by means of two four potential [15]. But these theories

have failed to formulate a self consistent and manifestly covariant theory of monopoles and

dyons.

1.3.7 Schwinger- Zwanziger Dyons

The electromagnetic phenomena are the best understood of all nature’s manifestations,

there are still great mysteries in this and related areas. Here are four of the them [29]

1. The fundamental electromagnetic equations of Maxwell show an intrinsic symmetry

between electric and magnetic quantities which, incidentally, is unique to the four

dimensions of space and time.

2. The unit of electric charge is universal. It is observed, with precision, to be identical

on all charged particles, despite wide variations in other characteristics.

3. A new periodic table is coming into being through the artificial creation of sub nuclear

particles and their tentative grouping into families. Two approximate but significant

properties have been recognized, isotopic spin and hyper charge, which serve also to

specify the electric charge of the particle.

4. The behavior of all particles with respect to strong, electromagnetic, and weak inter-

actions has seemed consistent with a general symmetry property in which the inter-

change of left and right, symbolized by P (parity), is combined with the exchange of

positive and negative charge, C .

41


In long began ago when Dirac [1] pointed out that according to the quantum laws of

atomic physics, the existence of a magnetic charge would lead to be quantization of elec-

tric charge in which only integral multiple of the fundamental unit could occur [1]. The

law of reciprocal electric and magnetic charge quantization is such that the unit of mag-

netic charge, deduced from the known unit of electric charge, it is quite large. It should

be very difficult to separate opposite magnetic charges in which normally magnetically

neutral matter. Thus, through the unquestioned quantitative asymmetry between electric

and magnetic charge, thin qualitative relationship be upheld. Now the condition of charge

quantization are less stringent, and fractional electric charges becomes a physically possi-

bility, consonant with integral charge necessarily carried by magnetically neutral particles.

Such dual-charged particle supply a physical realization for the constituents used in the

empirical models of so called hadrons [177], which are the strongly interacting nuclear

particles. Furthermore, in the introduction of particles with definite ratios between electric

and magnetic charges, a mechanism for CP -violation has made its appearance. Electric

and magnetic charges like electric and magnetic fields, behaves oppositely and magnetic

charges, when both types are considered together. If the dual-charged particles, and their

antiparticles, realized a certain ratio between electric and magnetic charge, but not its

negative value, the rule of CP -violation is broken. This physical model requires further

elaboration to explain, rather paradoxically, why the observed violation of CP -invariance

is so remarkably weak. The same refinement is also relevant in the establishment of the

detailed correspondence with the empirical mass spectrum that relates to the meaning of

isotopic spin and hyper charge.

A quicker, more heuristic derivation of the Dirac quantization condition eg = 2πn, fol-

lows by invoking the quantization of angular momentum. The orbital angular momentum−→L = −→r ×m−→r of a particle of mass m and charge q in the presence of a magnetic monopole−→B (−→r ) = gr

4πr3is not conserved. Where g is the magnetic charge. Indeed, using the Lorentz

force law,

42


d−→L

dt=−→r ×m−→r

=−→r ×(q−→r ×

−→B)

=qg

4πr3−→r ×

(−→r ×−→r )=d

dt

(qg−→r4πr

); (1.64)

whence the conserved quantity is instead

−→J =

−→L − qg−→r

4πr. (1.65)

A dyon is a particle which possesses both electric and magnetic charge. Consider two dyons

of charges (q = e, g) and (qp = ep, gp) imposing the quantization of the angular momentum

of the resulting electromagnetic field yields the following condition,

egp − epg = 2πn for n ∈ Z; (1.66)

the existence of the electron (that is, a particle with charges(e, 0) does not tell us anything

about the electric charge of a monopole (e, g), although it does tell us something about the

difference between the electric charges of two such monopoles: (e, g) and (ep, g). Indeed,

tells us immediately that g (e− ep) = 2πn for some integer n. If g has the minimum magnetic

charge g = 2πe

, then the difference between the electric charges of the dyons (e, g) and (ep g)

is an integer multiple of the electric charge of the electron e − ep = ne for some integer n.

But we cannot say anything further about the absolute magnitude of either e or ep.

1.3.8 Julia-Zee Dyons

Julia and Zee [48] extending t’ Hooft model showed the possibilities of constructing

43


classical solutions having both electric and magnetic charges. They demonstrated that t’

Hooft monopole has an internal degrees of freedom which allows it to exist in the states

of non zero electric charge and to readily exchange charge with ordinary particles. It was

shown that a dyon is energetically not allowed to decay into magnetic monopole by emitting

charged vector mesons. It has also been asserted that at classical level the charge of dyon

is not quantized and the dyons are not much massive than magnetic monopoles. It is more

natural for monopoles and dyons to exist in non Abelian gauge field-theoretical models

because non vanishing commutators themselves are present as electric and magnetic source

terms. In other words, non-Abelian monopoles and dyons are self induced and inevitable.

The simplest non Abelian dyons known as the Julia-Zee dyons. Here we introduce the

work B. Julia and A. Zee [48] on the existence of dyons in the simplest non-Abelian gauge

field theory, the Georgi-Glashow theory. The physical significance of such solutions is that,

unlike the Dirac monopoles and Schwinger dyons the Julia Zee dyons carry finite energies.

Julia and Zee [48] showed that there is a generalization of the t’ Hooft - Polyakov [44, 45]

monopole which is electrically charged. Julia and Zee found a 1- parameter family of

solutions, all of which have the same magnetic charge g = −2π, but a variable electric

charge whose strength q is seen from the form of the asymptotic electric field

e =q

4πr2x. (1.67)

These solutions were found numerically for a number of values of q, the energy increasing

with | q |. Dyons are therefore more massive than monopoles but there is no simple formula

for the dependence of mass on electric charge. The quantum-mechanical excitations of the

fundamental monopole have been shown to include dyons, first introduced by Julia and

Zee in non Abelian gauge theories. As a result, the monopoles and dyons have become the

intrinsic part of grand unified theories.

1.3.9 Quaternion

Quaternions (or division algebra) mean a set of four and introduce new methods in

physics and mathematics. Quaternion represents the natural extension of complex num-

bers and form an algebra under addition and multiplication. They were first described

44


by Irish mathematician Sir William Rowan Hamilton [62]-[64] and [204, 205] applied to

mechanics in three-dimensional space. Quaternions have the same properties as complex

numbers but differ in the way that commutative law is not valid which gave the possibility

of developing the fundamental laws in physics [206]-[216]. A striking feature of quater-

nions is that the product of two quaternions is non commutative, meaning that the product

of two quaternions depends on which factor is to the left of the multiplication sign and

which factor is to the right. The algebra of quaternion H is a four - dimensional algebra

over the field of real numbers R and a quaternion φ is expressed in terms of its four base

elements as

φ = φµeµ =φ0 + e1φ1 + e2φ2 + e3φ3 (µ = 0, 1, 2, 3); (1.68)

where φ0, φ1, φ2, φ3 are the real quarterate of a quaternion and e0, e1, e2, e3 are called

quaternion units and satisfies the following relations,

e0eA = eAe0 = eA;

eAeB = −δABe0 + fABCeC (∀A,B,C = 1, 2, 3); (1.69)

where δAB is the delta symbol and fABC is the Levi Civita three index symbol having

value (fABC = +1) for cyclic permutation, (fABC = −1) for anti cyclic permutation and

(fABC = 0) for any two repeated indices. As such we may write the following relations

among quaternion basis elements

[eA, eB] = 2 fABC eC ;

eA, eB = −2 δABe0;

eA( eB eC) = (eA eB ) eC ; (1.70)

where brackets [ , ] and , are used respectively for commutation and the anti commu-

tation relations while δAB is the usual Kronecker Dirac - Delta symbol. H is an associative

but non commutative algebra. Alternatively, a quaternion is defined as a two dimensional

45


algebra over the field of complex numbers C. We thus have

φ = (φ0 + e1φ1) + e2 (φ2 − e1φ3) . (1.71)

The quaternion conjugate φ is defined as

φ = φµeµ =φ0 − e1φ1 − e2φ2 − e3φ3. (1.72)

In practice φ is often represented as a 2× 2 matrix where e0 = I, ej = −iσj (j=1, 2, 3) and

σj are the usual Pauli spin matrices. Hence an quaternion can be decomposed in terms of

its scalar (Sc(x)) and vector (V ec(x)) parts as

Sc(φ) =1

2(φ + φ );

V ec(x) =1

2(φ − φ ). (1.73)

The norm of a quaternion is expressed as

N (φ) =φφ = φφ =| φ |2= φ20 + φ2

1 + φ22 + φ2

3. (1.74)

Since there exists the norm of a quaternion, we have a division i.e. every φ has an inverse

of a quaternion and is described as

φ−1 =φ

| φ |. (1.75)

While the quaternion conjugation satisfies the following property

φ1φ2 =φ1 φ2. (1.76)

The norm of the quaternion is positive definite and obey the composition law

N (φ1φ2) =N (φ1)N (φ2) . (1.77)

46


Quaternion is also written as

φ =(φ0,−→φ)

(1.78)

−→φ = e1φ1 + e2φ2 + e3φ3 is its vector part and φ0 is its scalar part. So the sum and product of

two quaternions are described as

(α0,−→α ) +

(β0,−→β)

=(α0 + β0,

−→α +−→β),

(α0,−→α ) .

(β0,−→β)

=(α0β0 −−→α .

−→β , α0

−→β + β0.

−→α). (1.79)

Quaternion elements are non-Abelian in nature and thus represent a non commutative divi-

sion ring. Quaternions were the first example of hyper complex numbers having significant

impacts on Physics and Mathematics. Quaternion is an important fundamental mathemat-

ical tool and is appropriate for four dimension world. Quaternions themselves occupy a

unique place in mathematics in that they are the most general quantity that satisfies the

division.

1.3.10 Octonion

The octonions form the widest normed algebra after the algebra of real numbers, com-

plex numbers and quaternions. The octonions, also known as Cayley-Graves [217, 218]

numbers, are an algebraic structure defined on the 8-dimensional real vector space such

that two octonions can be added, multiplied and divided, except that multiplication is nei-

ther commutative or associative. Otherwise, all the other expected properties hold such

as distributivity. The octonions have extremely interesting algebraic, combinatorial and

geometric properties described for further details [219, 220] Octonions are a super set of

quaternions in the same way that quaternions are a super set of complex numbers. So,

• Scalars are represented by 1 number.

• Complex numbers are represented by 2 numbers (1 real and 1 imaginary).

• Quaternions are represented by 4 numbers (1 real and 3 imaginary).

47


• Octonions are represented by 8 numbers (1 real and 7 imaginary).

We might expect this sequence to continue with an element consisting of 16 numbers,

but such an algebra does not exist, and the sequence ends with octonions. There are

algebras, such as matrices and multi vectors, which can have more than 8 dimensions but

these don’t have the same properties that division always exists and norms preserved by

multiplication. The octonions are an 8 - dimensional algebra with basis 1, e1, e2, e3, e4, e5

e6, e7. An octonion x is expressed [221, 222] as a set of eight real numbers

x =e0x0 + e1x1 + e2x2 + e3x3 + e4x4 + e5x5 + e6x6 + e7x7

=e0x0 +7∑

A=1

eAxA; (1.80)

where eA(A = 1, 2, ..., 7) are imaginary octonion units and e0 is the multiplicative unit

element. Set of octets (e0, e1, e2, e3, e4, e5, e6, e7) are known as the octonion basis elements

and satisfy the following multiplication rules

e0 = 1; e0eA = eAe0 = eA

eAeB = −δABe0 + fABCeC . (A,B,C = 1, 2, ....., 7). (1.81)

The structure constants fABC is completely antisymmetric and takes the value 1 for follow-

ing combinations,

fABC = +1;∀(ABC) = (123), (471), (257), (165), (624), (543),(736). (1.82)

It is to be noted that the summation convention is used for repeated indices. Here the

octonion algebra O is described over the algebra of real numbers having the vector space of

48


dimension 8. As such we may write the following relations among octonion basis elements

[eA, eB] = 2 fABC eC ;

eA, eB = −2 δABe0;

eA( eB eC) 6= (eA eB ) eC ; (1.83)

where brackets [ , ] and , are used respectively for commutation and the anti commu-

tation relations while δAB is the usual Kronecker Dirac - Delta symbol. Octonion conjugate

is defined as

x =e0x0 − e1x1 − e2x2 − e3x3 − e4x4 − e5x5 − e6x6 − e7x7

=e0x0 −7∑

A=1

eAxA; (1.84)

where we have used the conjugates of basis elements as e0 = e0 and eA = −eA. Hence an

octonion can be decomposed in terms of its scalar (Sc(x)) and vector (V ec(x)) parts as

Sc(x) =1

2(x + x );

V ec(x) =1

2(x − x ) =

7∑A=1

eAxA. (1.85)

Conjugates of product of two octonions and its own are described as

(x y) = y x; (x) = x; (1.86)

while the scalar product of two octonions is defined as

〈x , y〉 = 12(x y + y x) = 1

2(x y + y x) =

7∑α=0

xα yα. (1.87)

49


The norm N(x) and inverse x−1(for a nonzero x) of an octonion are respectively defined as

N(x) = xx = x x =7∑

α=0

x2α.e0;

x−1 =x

N(x)=⇒x x−1 = x−1 x = 1. (1.88)

The norm N(x) of an octonion x is zero if x = 0, and is always positive otherwise. It also

satisfies the following property of normed algebra

N(x y) = N(x)N(y) = N(y)N(x). (1.89)

Equation (1.83) shows that octonions are not associative in nature and thus do not form

the group in their usual form. Non - associativity of octonion algebra O is provided by

the associators (x, y, z) = (xy)z − x(yz) ∀x, y, z ∈ O defined for any three octonions. If

the associators is totally antisymmetric for exchanges of any three variables, i.e. (x, y, z) =

−(z, y, x) = −(y, x, z) = −(x, z, y), then the algebra is called alternative. Hence, the oc-

tonion algebra is neither commutative nor associative but, is alternative. It is well-known

that octonions form an alternative algebra [223]; associativity holds when two of the three

terms are equal. However, Albuquerque and Majid [224] have proved that the octonions

are associative up to a natural transformation.

1.3.11 Split - Octonion

The split octonions are a non associative extension of the quaternions (or the split-

quaternions). They differ from the octonions in the signature of quadratic form. The split

octonions have a split signature (4, 4) whereas the octonions have a positive definite signa-

ture (8, 0). The split octonions form the unique split octonion algebra over the real numbers

and there are corresponding algebras over any field. The Cayley algebra of octonions over

the field of complex numbers is visualized as the algebra of split octonions with its following

basic elements [225]-[227],

50


u0 =1

2(1 + ie7) , u?0 =

1

2(1− ie7) ;

u1 =1

2(e1 + ie4) , u?1 =

1

2(e1 − ie4) ;

u2 =1

2(e2 + ie5) , u?2 =

1

2(e2 − ie5) ;

u3 =1

2(e3 + ie6) , u?3 =

1

2(e3 − ie6) ; (1.90)

where(i =√−1)is usual complex imaginary number and commutes with all seven octo-

nion imaginary units eA (A = 1, 2, ...., 7). Gunaydin and Gursey [225]-[227] pointed out

that the automorphism group of octonion is G and its subgroup which leaves imaginary oc-

tonion unit e7 invariant or equivalently the idempotents u0 and u?0 is SU (3) where the units

uj and u?j(i = 1, 2, 3) transform respectively like a triplet and anti triplet and are associated

with colour and anti colour triplet of SU (3) group. Now introduce a convenient realiza-

tion for the split octonion basis elements(u0, uj, u

?0, u

?j

)through the use of Pauli matrices in

terms of quaternions basis elements, e0 −→ σ0 = 1 and ej −→ −iσj as

u0 =

0 0

0 1

, u?0 =

1 0

0 0

;

uj =

0 0

ej 0

, u?j =

0 −ej0 0

; (1.91)

where 1, e1, e2, e3 are quaternion units satisfying the multiplication rule ejek = −δjk + εjklel.

As such for an arbitrary split octonion A we have [228],

A = au?0 + bu0 + xju?j + yjuj;

=

a −−→x−→y b

; (1.92)

where a and b are scalars and−→x and−→y are three vectors. Thus the product of two octonions

51


in terms of 2× 2 Zorn’s matrix realization is expressed as,

a −→x−→y b

c −→u−→v d

=

ac+−→x · −→v a−→u + d−→x −−→y ×−→v

c−→y + b−→v +−→x ×−→u −→y · −→u + bd

; (1.93)

where (×) denotes the usual vector product, ej(j = 1, 2, 3) with ej × ek = εjklel and ejek =

−δjk. As such, the split octonions to the vector matrices given by equation (1.91). Octonion

conjugate of equation (1.92) in terms of 2× 2 Zorn’s matrix realization is defined as,

A = bu?0 + au0 − xju?j − yjuj;

=

b −→x

−−→y b

. (1.94)

The norm of A is defined as,

N (A) =AA = AA;

= (ab+−→x · −→y ) · 1;

=n (A) 1. (1.95)

where 1 is the identity element of the algebra given by 1 = 1u?0 + 1u0 and the expression

n (A) = (ab+−→x · −→y ) defines the quadratic form which admits the composition n(−→A ·−→B)

=

n(−→A)n(−→B)

for all−→A,−→B ∈ O. The Euclidean or Minikowaski four-vector in split octo-

nion formulation in terms of 2× 2 Zorn’s vector matrix realization. So, the space-time four

differential operator and its conjugates are then be written as

52


=

∂4 −−→∇

−→∇ ∂4

=

i∂0 −−→∇

−→∇ i∂0

;

=

∂4−→∇

−−→∇ ∂4

=

i∂0−→∇

−−→∇ i∂0

. (1.96)

53

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71

Chapter 2

GENERALIZED GRAVI -

ELECTROMAGNETISM

Chapter 2 Generalized Gravi - Electromagnetism

'

&

$

%

ABSTRACT

A self consistent and manifestly covariant theory for the dynamics of four charges (masses)

(namely electric, magnetic, gravitational, Heavisidian) in terms of quaternion variables has

been developed in simple, compact and consistent manner. Starting with an invariant La-

grangian density and its quaternionic representation, we have obtained the consistent field

equation for the dynamics of four charges. It has been shown that the present reformula-

tion reproduces the dynamics of individual charges (masses) in the absence of other charge

(masses) as well as the generalized theory of dyons (gravito - dyons) in the absence gravito -

dyons (dyons).

73


2.1 Introduction

General relativity has several features that raise peculiar difficulties in applications of

practical interest. These features include the non-linearity, over-determination and under-

determination of the field equations, as well as the absence of a background geometry that

might be used to define physically interesting quantities like momentum or energy. Due to

these technical and conceptual difficulties, one must use the approximation scheme in or-

der to make the relevant theory physically interesting and practicable. Such approximation

in general relativity and gravitations may be applied up to the limit of linear gravity [1, 2].

However, now a days the question of existence of monopole [3, 4] and dyons [5]-[9] has

become a challenging new frontier and the object of more interest in high energy physics.

Magnetic monopoles [3, 4] were advocated to symmetrize Maxwell’s equations in a mani-

fest way that the mere existence of an isolated magnetic charge implies the quantization of

electric charge. Dirac showed [3, 4] that the quantum mechanics of an electrically charged

particle of charge e and a magnetically charged particle of charge g, is consistent only if

eg = 2πn, n being an integer. Schwinger-Zwanziger [5]-[9] generalized this condition to

allow for the possibility of particles (dyons) that carry both electric and magnetic charge.

So, a quantum mechanical theory can have two particles of electric and magnetic charges

(e1, g1) and (e2, g2) only if e1g2− e2g1 = 2πn. The angular momentum in the field of the two

particle system can be calculated readily with the magnitude e1g2−e2g14πc

. This has an integer

or half-integer value, as expected in quantum mechanics, only if e1g2 − e2g1 = 2πn~c. The

fresh interests in this subject are enhanced with the idea of t’ Hooft and Polyakov [10, 11]

that the classical solutions having the properties of magnetic monopoles may be found in

Yang-Mills gauge theories. The Dirac monopoles were elementary particles but the t’ Hooft-

Polyakov [10, 11] monopoles are complicated extended object having a definite mass and

finite size inside of which massive fields play an important role in providing a smooth struc-

ture and outside it they vanish rapidly leaving the field configuration identical to abelian

Dirac monopole. Julia and Zee [12] extended the theory of non-Abelian monopoles of t’

Hooft-Polyakov [10, 11] to the theory of non Abelian dyons (particles carrying simultane-

ously electric and magnetic charges). The quantum mechanical excitation of fundamental

74


monopoles include dyons which are automatically arisen from the semi-classical quantiza-

tion of global charge rotation degree of freedom of monopoles. In view of the explanation

of CP -violation in terms of non-zero vacuum angle of world [13, 14], the monopoles are

necessary dyons and Dirac quantization condition permits dyons to have analogous electric

charge. Accordingly, a self consistent and manifestly covariant theory has been developed

[15, 16] for the generalized electromagnetic fields of dyons. Despite of the potential impor-

tance of monopoles [3, 4] and [10, 11] and dyons [12], the formalism necessary to describe

them has been clumsy and not manifestly covariant. So, a self consistent and manifestly

covariant theory of generalized electromagnetic fields of dyons (particle carrying electric

and magnetic charges) and those for generalized fields of gravito-dyons, has been con-

structed [17]-[21] in terms of two four-potentials to avoid the use of controversial string

variables. On the other hand, the quaternionic formulation [22]-[29] of electrodynamics

has a long history [30]-[41], stretching back to Maxwell himself [30]-[32] who used real

(Hamilton) quaternion in his original manuscript ‘on the application of quaternion to elec-

tromagnetism’ and in his celebrated book “Treatise on Electricity and Magnetism”. Quater-

nion analysis has since been rediscovered at regular intervals and accordingly the Maxwell’s

equations of electromagnetism were rewritten as one quaternion equations [42]-[56]. Negi

and coworkers [57]-[60], have also studied the quaternionic formulation for generalized

electromagnetic fields of dyons (particles carrying simultaneous existence of electric and

magnetic charges) in unique, simpler and compact notations. Kravchenko and co-authors

[61]-[64], discussed the Maxwell’s equations in homogeneous media, chiral media and

inhomogeneous media. Accordingly, they have developed the quaternionic reformulation

of the time-dependent Maxwell’s equations along with the classical solution of a moving

source, i.e. electron. In the series of papers Negi and coworkers [15, 16] and [65]-[67]

have derived the generalized Dirac-Maxwell (GDM) equations in presence of electric and

magnetic sources in an isotropic (homogeneous) medium. They have also analyzed the

other quantum equations of dyons in consistent and manifest covariant way [65]. This the-

ory has been shown to remain invariant under the duality transformations in isotropic ho-

mogeneous medium. Quaternion analysis of time dependent Maxwell’s equations has been

developed by them [15] in presence of electric and magnetic charges and the solutions for

75


the classical problem of moving charge (electric and magnetic) are obtained consistently.

The time dependent generalized Dirac-Maxwell’s (GDM) equations of dyons are also dis-

cussed [66] in chiral and inhomogeneous media and the solutions for the classical problem

are obtained. The quaternion reformulation of generalized electromagnetic fields of dyons

in chiral and inhomogeneous media has also been analyzed [16]. The monochromatic fields

of generalized electromagnetic fields of dyons have also discussed [67] in slowly changing

media in a consistent manner. The quaternion analysis has also been used [68] to combine

the complex description of dyons and gravito-dyons and accordingly the unified quater-

nionic angular momentum for generalized fields of dyons and gravito-dyons along with

their commutation relations has been analyzed in unique and consistent manner. Keeping

all these facts in mind, in this chapter [69], attempts are made to develop a self-consistent

and manifestly covariant theory for the dynamics of simultaneously existing four charges

(masses) (namely electric, magnetic, gravitational, Heavisidian) in simple, compact and

consistent manner. Starting with the structural symmetry between linear gravity and elec-

tromagnetism, the chirality quantization condition for both dyons and gravito-dyons, has

been reproduced. Accordingly, the quaternionic form of unified charge and unified fields

of dyons and gravito-dyons, which reproduces the dynamics of Maxwell’s field equations

associated with electric, magnetic, gravitational and Heavisidian charges, has been devel-

oped in a consistent way. Considering an invariant Lagrangian density and its quaternionic

representation, the consistent field equation and equation of motion for the dynamics of

four charges are obtained. It is shown that the Euler Lagrangian equations provides the

field equations associated with each charge (namely electric, magnetic, gravitational and

Heavisidian) after taking care of usual method of variation with respect to potential. These

corresponding equations may immediately be generalized to the unified field equations of

a particle simultaneously containing four charges namely electric, magnetic, gravitational

and Heavisidian charges (masses). It has been shown that the present reformulation repro-

duces the dynamics of individual charges (masses) in the absence of other charge (masses)

as well as the generalized theory of dyons (gravito-dyons) in the absence gravito-dyons

(dyons) and vice versa.

76


2.2 Linearization of Gravitational fields

Einstein’s equations may be written in a very simple way, which leads straight to the

analogy with Maxwell’s equation (weak field) [70, 71]. So, let us start full non linear

equations (for brevity we have considered the natural units c = ~ = 1)

Gµν =8πGTµν ; (2.1)

Then,

gµν = ηµν + hµν ; (2.2)

where ηµν is the Minikowaski metric tensor, and |hµν | << 1 is a small deviation from it.

Then we define,

hµν = hµν −1

2ηµνh ; h = hαα. (2.3)

So, it expands the field equation (2.1) in power of hµν keeping only the linear terms, then

hµν = 16πGTµν ; (2.4)

after imposing the Lorentz gauge condition hµα

,α = 0, equation (2.4) constitutes the lin-

earized Einstein’s field equations. The analogy with the corresponding Maxwell equations

Aν = 4πjν is evident. The solution of equation (2.4) may be written exactly in terms of

retarded potentials

hµν =− 4G

ˆTµν (t− |x− xp| , x)

|x− xp|d3xp; (2.5)

77


The role of the electromagnetic vector potential Aν is played here by the tensor potential

hµν , while the role of the four-current jν is played by the stress-energy tensor T µν . The

explicit expression for the tensor potential hµν is then

h00

=4φ;

h0l

=− 2Al. (2.6)

where φ is the Newtonian or gravito-electric potential is given as

φ =− GM

r; (2.7)

while−→A is the gravito-magnetic vector potential in terms of the total angular momentum

of the system−→S is given by

Al =GSnxk

r3εlnk. (2.8)

It follows that T 00 = ρ is the mass energy density. Hence the total mass M of the system is

expressed as

ˆρd3x = M ; (2.9)

while T i0 = ji represents the mass-current density, and the total angular momentum of the

system is

Si =2

ˆεijkx

pjT k0d3xp. (2.10)

In terms of the potentials φ,−→A , the Lorentz gauge conditions becomes

78


−→∇ ·−→A +

∂φ

∂t= 0. (2.11)

It is then straightforward to define the gravito-electric and gravito-magnetic fields−→G and

−→H as [70],

−→G = −

−→∇φ− ∂

−→A

∂t;

−→H =

−→∇ ×

−→A. (2.12)

Using equations (2.4), (2.11) and (2.12), the definitions of mass density and current, the

complete set of Maxwell’s equations called gravielectromagnetic (GEM) fields [70], is de-

scribed as

−→∇ ·−→G = −ρg;

−→∇ ·−→H = 0;

−→∇ ×

−→G = −∂

−→H∂t

;

−→∇ ×

−→H =

−→jG +

∂G

∂t. (2.13)

Einstein’s field equations in the form correspond to a solution that describes the field around

a rotating object in terms of gravito-electric and gravito-magnetic potentials. So, the metric

tensor can be read from the corresponding space-time invariant

ds2 = (1 + 2φ) dt2 + 4dt (dr · A)

− (1− 2φ) δijdxidxj. (2.14)

Hence the gravitational field is understood in analogy with electromagnetism. For instance,

79


the gravito-magnetic field of the earth as well as other weakly gravitating and rotating mass

may be considered as a dipolar field,

−→H = −4G

3r (r · S)− Sr2

2r5; (2.15)

In fact, using the present formalism, the geodesic equation for a particle in the field of a

weakly gravitating rotating object, can be cast in the form of an equation of motion under

the action of a Lorentz force. The geodesic equation is given as,

d2xα

ds2+ Γαµβ

dxµ

ds

dxβ

ds= 0; (2.16)

If a particle in non-relativistic motion dx0

dsw 1, hence the velocity of the particle becomes

vi w dxi

ds. Limiting ourselves to static fields, where gαβ,0 = 0, it is easy to verify that the

geodesic equation may be written as

d−→vdt

=−→G +−→v ×

−→H ; (2.17)

which shows that the free fall, in the field of a massive rotating object, can be looked at a

motion under the action of the Lorentz force produced by the GEM fields.

2.3 Analogy between electromagnetic and linear gravita-

tional fields

The foregoing analysis leads the striking resemblance between Newton’s law of gravity

and Coulomb’s law of electrostatics which has already been noted since the beginning of

Physics. It is some what less well known that the post-Newtonian laws of gravity also corre-

sponds quite closely to the Maxwellian laws of electromagnetism [72]. In 1893, Heaviside

[73] assuming wave propagation for the gravitational field as for electromagnetic case,

80


wrote Maxwell’s equations for the field of gravitating bodies. In 1967, Dowker and Roche

[74] discussed the analogy between gravitation and electromagnetism while Cattani [75],

on introduction of a new field (calling Heavisidian fields) depending upon the velocities of

gravitational charge in the same way as magnetic field depends on the velocities of electric

charge, described gravito-dynamics in terms of Maxwellian covariant equation for linear

gravity. Following Dower and Roche [74], let us consider a gravitational field determined

by the metric in equation (2.2). Where ηµν = diag (−1,−1,−1, 1) and hµν is small. Then

the classical Hamiltonian for a (spinless) particle of mass m in this field is described [76]

as follows in non-relativistic limit,

H =1

2m(p−mh)2 +

1

2mh00; (2.18)

where h = − (h10, h20, h30) = (−h?1, h?2, h?3). Arguing now that H, a Schrodinger quantum

Hamiltonian, yields the statement that the change in phase of the wave function of the

particle when taken along a curve V1 is

δβ =m

h

ˆV1

hµdxµ; hµ =

(−→h , h00

). (2.19)

In complete comparison with the electromagnetic case Aµ ≈ hµ. Incorporation of the spin

effects with equation (2.19), one gets [74],

δβ p = δβ +m

h

˛V2

HµνdTµν +

1

4

˛RµναβJ

αβdT µν ; (2.20)

analogues to the following equation in electromagnetism

δβ p = δβ +e

h

˛V2

FµνdTµν ; (2.21)

where dT µν is an element of closed surface V2, Fµν is the electromagnetic field tensor, e

is the electric charge, Hµν = ∂µhν − ∂νhµ, Rµναβ is the curvature tensor and Jαβ are the

81


generators of the Lorentz group in the (j, 0) representation where j corresponds to spin.

Equation (2.20) and (2.21) describe the following analogies,

e −→m;

h −→A;

e −→hJαβ;

Fµν −→Rµναβ. (2.22)

Thus one can write Einstein’s linear gravitational equation in analogy with electromag-

netism. On the other hand, some authors [76]-[82] write down the Maxwell’s equation

for linear gravitational field calling it Maxwellian gravity and corresponding fields as gravi-

electric (g-electric) and gravi-magnetic (g-magnetic) fields. The correspondence between

the four-vector potential Aµ =φ,−→A

and the metric gµν discussed above is also evident

when the Lorentz force equation is compared with the geodesic equation. Like electromag-

netism four-potential aµ = a0,−→a satisfying the following definition of gravitational

(g-electric) and Heavisidian (g-magnetic) fields similar to equation (2.12) i.e.

−→G =

∂−→a∂t

+−→∇a0;

−→H =−→∇ ×−→a ; (2.23)

where−→G is the gravitational (gravito-electric) and

−→H is the Heavisidian (gravito-magnetic)

fields. The existence of gravi-magnetic field in terms of directly from (2.22) and (2.19)

analogy, and arosed from moving masses in same way that a magnetic charge produces

a magnetic field. In the absence of vector-potential a, we may have Newtonian gravita-

tional field G = grad a0. The foregoing analysis now resembles with electromagnetism

and we can describe the parallel covariant formulation for this theory. Gravitational and

Heavisidian fields given by equation (2.23) satisfy the following Maxwellian differential

equations in equation (2.13), where ρG and jG are respectively known as gravitational

82


charge and current source densities. ρG and jG are also related in terms of charge and

velocity as ρG = c√Gµ and jG = c

√Gµν where c = 3 × 1010cm/sec (light velocity),

G = 6.67 × 10−8cm3g−1s−2, µ is the mass density and v is particle velocity. For brevity,

we take the universal constants c = ~ = G = 1 in our formulation. Let us introduce fol-

lowing form of gravito-Heavisidian field tensor to represent the gravi-electric (gravito) and

gravi-magnetic (Heavisidian) fields as

Eµν = aµ,ν − aν,µ;

Edµν =

1

2εµνρσE

ρσ. (2.24)

where aµ = a0,−→a . The components of the gravito Heavisidian field tensor satisfies

gravito-Heavisidian fields (2.23) as E0i = Gi and Eij = εijkHk (i, j, k = 1, 2, 3). Thus the

covariant form of Maxwell’s equation (2.13) for linear gravity may be written as,

Eµν,ν = −jµ;

Edµν,ν = 0. (2.25)

For a particle of gravitational mass m and inertial mass mi (usually these two masses are

treated on equal footing), in Einstein’s linearized theory of gravity, the Lorentz force equa-

tion governing the geodesic equation corresponds to

−→f = mi

−→x = −m

[−→G +

−→U ×

−→H]

; (2.26)

or in tensorial form

fµ = mixµ = −mEµνuν (ν = 0, 1, 2, 3) ; (2.27)

83


minus sign with Maxwell’s equation (2.25) and Lorentz force equation (2.26,2.27) arises

due to the fact that gravity is attractive. The incorporation of magnetic monopoles in elec-

tromagnetism is most easily performed by symmetrizing Maxwell’s equation when they are

expressed in terms of field tensor and its dual part. The striking similarity of Maxwell’s

equations with Einstein’s lineared equations, of gravity discussed above, suggests the exis-

tence of dual mass (g-monopole) that is the gravitational analogue of magnetic monopole

[79]. There are many arguments suggested for the existence of dual mass playing the role

of Heavisidian monopole (gravitational analogue of magnetic charge). Without going into

details let us speculate that point like “gravitopoles” [78] (i.e. the Heavisidian monopole)

exists in gravito-Heavisidian fields like Dirac monopole [83] in electromagnetism, so that

equation−→∇ ·−→H = 0 is amended to read

−→∇ ·−→H = hδ3 (x) ; (2.28)

when a gravitopoles are present at the origin. In other words, one can describe that the

g-magnetic (Heavisidian) field−→H is exerted by g-monopole (Heavisidian monopole) with

Heavisidian charge h. Like electromagnetism gravitational charge quantization condition

resembles to Dirac quantization condition

mh =1

2n; (2.29)

where m is the gravitational charge (mass) and h is Heavisidian charge (mass). The postu-

lation of Heavisidian monopole (g-monopole) increases the structural symmetry between

the linear equations for gravitational field and extended Maxwell’s equation. Adopting the

same procedure for the case of Dirac monopole parallel descriptions and ambiguities of

magnetic monopoles arises with the theory of Heavisidian monopole.

2.4 Postulation of Heavisidian (gravi - magnetic) monopoles

The postulation of Heavisidian monopoles (gravi-magnetic) [85, 86] has increased the

84


structural symmetry between the linear equations for gravitational field and extended

Maxwell’s equations. The linear equation for gravitational field with Heavisidian monopole

are [84, 85, 86]

−→∇ ·−→G = −ρg;

−→∇ ·−→H = −ρh;

−→∇ ×

−→G = −∂

−→H∂t

+−→jH ;

−→∇ ×

−→H =

−→jG +

∂G

∂t; (2.30)

where ρG is gravitational charge (mass) density,−→jG is gravitational current density, ρH is

Heavisidian charge density,−→jH is Heavisidian current density. For the sake of simplicity we

have expressed these equation (2.30) in Gaussian-like unit system. From these equation it

is clear that when

−→G −→

−→H ,

−→H −→ −

−→G ;

ρg −→ ρh , ρh −→ −ρg; (2.31)

with the current densities following that the linear equation for the gravitational field are

invariant under the following duality transformations [88];

−→G =−→G cos θ +

−→H sin θ;

−→H =−

−→G sin θ +

−→H cos θ;

ρG =ρG cos θ + ρH sin θ;

ρH =− ρG sin θ + ρH cos θ;

−→jG =

−→jG cos θ +

−→jH sin θ;

−→jH =−−→jG sin θ +

−→jH cos θ; (2.32)

where the angle θ is real. The gravito - Heavisidian force on a particle of mass m1 in the

85


presence of the gravitational field−→G and the Heavisidian field

−→H is

m1d−→vdt

=m1

(−→G −−→v ×

−→H), (2.33)

where −→v is the velocity of the particle. Equation (2.32) of duality transformation suggest

that the gravito-Heavisidian force on a particle with Heavisidian charges h1 in the presence

of the gravitational field−→G and Heavisidian field

−→H would be

h1d−→vdt

=h1

(−→H +−→v ×

−→G). (2.34)

Schwinger has achieved the charge quantization condition by introducing the concept of

particles that possess both an electric and a magnetic charge (or dyons). Following the

same lines we introduce the gravitational dyons (particles that possess both the gravita-

tional charge and the Heavisidian charge). If a particle with gravitational charge m1 and

Heavisidian charge h1 is moving with non - relativistic velocity −→v in the field of a stationary

particle with gravitational charge and Heavisidian charge h2, the equation of motion from

equation (2.33) and (2.34) is

(m1 + h1)d−→vdt

= m1

(−→G −−→v ×

−→H)

+ h1

(−→H +−→v ×

−→G)

; (2.35)

where−→G and

−→H are gravitational and Heavisidian field strengths of the stationary gravita-

tional dyon at the position of moving particle. If −→r is the vector position of m1 relative to

the fixed particle (which is sitting at the origin), then we have

−→G = m2

−→rr3

,−→H = h2

−→rr3

; (2.36)

From equation (2.35) and (2.36) and one has

86


(m1 + h1)d−→vdt

= (m1m2 + h1h2)−→rr3

+ (m1h2 − h1m2) v ×−→rr3

; (2.37)

The associated moment equation is

−→r × (m1 + h1)d−→vdt

= (m1h2 − h1m2)−→r × (−→v ×−→r )

r3

= (m1h2 − h1m2)d

dt

(−→rr

); (2.38)

The total conserved angular - momentum vector is given by

−→J = −→r × (m1 + h1)

−→v − (m1h2 − h1m2)

(−→rr

). (2.39)

If we quantize the component of the angular momentum along the line connecting the two

particles, we obtain the gravitational-charge quantization condition

(m1h2 − h1m2) = n; (2.40)

where n is an integer. Although the Heavisidian monopole have not been found experimen-

tally, the consideration of a hypothetical Heavisidian charge h leads to the following mass

quantization condition;

mh = n. (2.41)

It is clear that the forces between Heavisidian charges are enormous and the Heavisidian

monopoles are the most strongly interacting form of matter. If we assume that the funda-

mental gravitational charge is equal to the rest mass of an electron (me), the gravitational

charge quantization condition then becomes

87


Gmeh = n;

or

Gh2 = n2 1

Gm2e

;

where G is gravitational constant. Then

Gm2e =1.75× 10−45;

Gh2 =n25.7× 1044.

We deduce a unit of Heavisidian charge by making n = 1 i.e.

Gh20 = 5.7× 1044.

In a natural unit (~ = c = 1) . It is clear that the forces between Heavisidian charges are

enormous. The above equation suggests that Heavisidian monopoles are the most strongly

interacting form of matter. The huge mass of the Heavisidian monopole is probably one of

the reasons why it has not been found yet.

2.5 Generalized fields of gravito - dyons

The existence of a dual mass associated with the gravi-magnetic (Heavisidian) field

playing the role of monopole in the linear theory of gravitation describe a theory of particles

carrying simultaneously gravitational and Heavisidian charges (i.e., gravito-dyons). The

generalized charge of a gravito-dyon is defined as

q = m− ih(i =√−1)

(2.42)

where m and h are masses (charges) associated with gravitational (gravi-electric) and

88


Heavisidian (gravi-magnetic) fields. We write the following tensorial form of Maxwell-

Dirac equation of gravito - dyon as,

fµν,ν = −j(G)µ ;

Nµν,ν = −j(H)µ ; (2.43)

wherej(G)µ

and

j(H)µ

are respectively, gravitational and Heavisidian four - current den-

sities, then

fµν =Lµν −Kdµν ;

Nµν =Kµν + Ldµν ;

Lµν =Cµ,ν − Cν,µ;

Kµν =Dµ,ν −Dν,µ;

Ldµν =1

2εµνρσD

ρσ;

Kdµν =

1

2εµνρσC

ρσ; (2.44)

where Cµ and Dµ are respectively, gravitational (gravi-electric) and Heavisidian (gravi-

magnetic) four-potential. The symbol (d) denotes the dual part, εµνρσ is a 4-index Levi-

Civita symbol, µ,ν,ρ,σ are space-time indices having values 0,1,2,3 and fµν and Nµν repre-

sents the field tensor of gravitational and Heavisidian fields. The theory of gravito-dyons de-

scribes the existence of generalized charge, potential, current and generalized field tensor

as complex quantities with their real and imaginary parts as gravitational (gravi-electric)

and Heavisidian (gravi-magnetic) constituents i.e.

Vµ =Cµ − iDµ;

jµ =j(G)µ − ij(H)

µ ;

gµν =fµν − iNµν ; (2.45)

89


where Vµ, jµ and gµν represents the generalized four-potential, generalized four-current

and generalized field tensor of gravito-dyon. Equation (2.45) can be combined into the fol-

lowing covariant field (Maxwell-Dirac) equation for generalized gravito-Heavisidian fields

of gravito-dyons as,

gµν,ν = −jµ. (2.46)

Equation (2.46) is the generalized field equation (i.e. Maxwell-Dirac equation) in covariant

form of gravito-dyons. This equation is invariant under Lorentz transformation, gauge

transformations and duality transformations and also follows the conservation laws i.e.

(fµν , Nµν) = (fµν cos θ +Nµν sin θ,−fµν sin θ +Nµν cos θ) ;

(Cµ, Dµ) = (Cµ cos θ +Dµ sin θ,−Cµ sin θ +Dµ cos θ) ;(j(G)µ , j(H)

µ

)=(j(G)µ cos θ + j(H)

µ sin θ,−j(G)µ sin θ + j(H)

µ cos θ)

; (2.47)

where we have used the constancy condition

h

m= Dµ

Cµ=

j(H)µ

j(G)µ

= − tan θ. (2.48)

A gauge-covariant and rotationally symmetric theory of the angular momentum operator

for gravito-dyon has been described by Rajput [87, 88] in terms of structural symmetry

between generalized electromagnetic fields of dyons and generalized gravito-Heavisidian

fields of gravito-dyons. The duality transformation holds good for generalized fields of

gravito-dyons in terms of structural symmetry.

90


2.6 Quaternion charge and unified fields of dyons and grav-

ito - dyons

Let us describe the property of quaternion algebra with the use of natural units (c =

~ = G = 1) in order to reformulate the unified theory of generalized electromagnetic

fields (associated with dyons) and generalized gravito - Heavisidian fields (associated with

gravito - dyons) of linear gravity. So, we define the unified charge [68] as

Q = (e, g, m, h) = e− ig − jm− kh; (2.49)

where e, g, m and h are respectively described as the electric, magnetic, gravitational and

Heavisidian charges (masses). In equation (2.49), i, j and k are the quaternion units satisfy

the following properties

ij = −ji = k (say); (2.50)

and

i(ij) = (ii)j = −j;

(ij)j = i(jj) = −i;

ik = −ki = −j;

kj = −jk = −i;

i2 = j2 = k2 = −1. (2.51)

Complex structure (e, g) represents the generalized charge of dyons of electromagnetic

fields while (m, h) denotes the generalized charge of gravito - dyons. The norm of unified

quaternion charge (2.49) is

91


N(Q) = QQ = (e2 + g2 +m2 + h2); (2.52)

where

Q = (e, −g, −m, −h) = e+ ig + jm+ kh. (2.53)

Unified quaternion valued four-potential may then be defined as

Vµ = Aµ − i Bµ − j Cµ − k Dµ ; (2.54)

where Aµ is the four-potential associated with the dynamics of electric charge, Bµ is

used for magnetic charge, Cµ describes the gravitational charge (mass) while Dµ has

been associated with the gravi-magnetic (Heavisidian) charge (mass). Then the various

potentials of equation (2.54) are written in the following quaternionic forms,

A = A0 − iA1 − jA2 − kA3;

B = B0 − iB1 − jB2 − kB3;

C = C0 − iC1 − jC2 − kC3;

D = D0 − iD1 − jD2 − kD3. (2.55)

Similarly one may define the quaternion valued unified field tensor as

=µν = Fµν − iMµν − jfµν − kNµν ; (2.56)

where

92


Fµν = Aµ,ν − Aν,µ − iεµνρσBρσ;

Mµν = Bµ,ν −Bν,µ − iεµνρσAρσ;

fµν = Cµ,ν − Cν,µ − iεµνρσDρσ;

Nµν = Dµ,ν −Dν,µ − iεµνρσCρσ; (2.57)

are the field tensors respectively associated with the dynamics of the electric, magnetic,

gravitational and Heavisidian charges (masses). These field tensors satisfy the following

Maxwellian field equations

Fµν,ν = j(e)µ ;

Mµν,ν = j(m)µ ;

fµν,ν = j(G)µ ;

Nµν,ν = j(H)µ . (2.58)

In equations (2.56 - 2.58), the field tensors Fµν and Mµν are associated with dyons, while

fµν and Nµν are associated with gravito - dyons. As such the quaternion valued current may

then be defined as

Jµ = j(e)µ − ij(m)µ − jj(G)

µ − kj(H)µ . (2.59)

2.7 Lagrangian Formulation of dyons and gravito - dyons

The Lagrangian density for the unified charged particle containing the electric, mag-

netic, gravitational and Heavisidian charges and the rest mass of the unified particle M ,

may be written in the following form [69],

93


L =−M − 1

4

[α

(Aµ,ν − Aν,µ)2 − (Bµ,ν −Bν,µ)2 − (Cµ,ν − Cν,µ)2 − (Dµ,ν −Dν,µ)2]

−2β [(Aµ,ν − Aν,µ) (Bµ,ν −Bν,µ) + (Cµ,ν − Cν,µ) (Dµ,ν −Dν,µ)]

−2γ [(Aµ,ν − Aν,µ) (Cµ,ν − Cν,µ) + (Bµ,ν −Bν,µ) (Dµ,ν −Dν,µ)]

−2∆ [(Aµ,ν − Aν,µ) (Dµ,ν −Dν,µ) + (Bµ,ν −Bν,µ) (Cµ,ν − Cν,µ)]

+ (αAµ − βBµ + γCµ −∆Dµ) j(e)µ − (αBµ − βAµ + γDµ −∆Cµ) j(m)µ

+ (αCµ − βDµ + γAµ −∆Bµ) j(G)µ − (αDµ − βCµ + γBµ −∆Aµ) j(H)

µ

=LP + LF + LI ; (2.60)

where LP is the free particle Lagrangian, LF is the field Lagrangian and LI is the interaction

Lagrangian and α, β, γ, ∆ are real positive arbitrary uni modular parameters satisfying the

following conditions

α− iβ − jγ − k∆ = e−θn = cos θ − n sin θ; (2.61)

and

α + iβ + jγ + k∆ = eθn = cos θ + n sin θ. (2.62)

From equations (2.61) and (2.62), we get

α2 + β2 + γ2 + ∆2 = 1. (2.63)

As such, we may write the constancy condition [57]-[60] as,

tan θ = ge

= hm

= BµAµ

=Dµ

Cµ=j(m)µ

j(e)µ

=j(H)µ

j(G)µ

. (2.64)

94


The action integral of the above unified system may be written as,

S =

ˆLdt = SP + SF + SI ; (2.65)

where the action part SP depends upon the properties of the particle, SF depends on the

properties of the field and SI depends on the parameters of the particle and field both. In

deriving the equation of motion of the particle, we vary the trajectory of the particle without

changing the field parameters and as such the action SF does not affect the motion. On

the other hand, in order to find the field equations we take the variation with respect to

field parameters assuming the trajectory of the particle fixed. For deriving the equation of

motion, we write the concerned part of action in the following form [69],

S = SP + SI = −t2ˆ

t1

Mdt

+

t2ˆ

t1

(αAµ − βBµ + γCµ −∆Dµ) j(e)0

dxµdt

dt

−t2ˆ

t1

(αBµ − βAµ + γDµ −∆Cµ) j(m)0

dxµdt

dt

+

t2ˆ

t1

(αCµ − βDµ + γAµ −∆Bµ) j(G)0

dxµdt

dt

−t2ˆ

t1

(αDµ − βCµ + γBµ −∆Aµ) j(H)0

dxµdt

dt; (2.66)

where j(e)µ , j(m)

µ , j(G)µ and j

(H)µ are expressed in terms of j(e)0 , j(m)

0 , j(G)0 and j

(H)0 and dxµ

dt.

Equation (2.66) may also be written as,

95


S = −Mˆ b

a

dS

+

ˆ b

a

[(αAµ − βBµ + γCµ −∆Dµ) j(e)0 dxµ

−ˆ b

a

(αBµ − βAµ + γDµ −∆Cµ) j(m)0 dxµ

+

ˆ b

a

(αCµ − βDµ + γAµ −∆Bµ) j(G)0 dxµ

−ˆ b

a

(αDµ − βCµ + γBµ −∆Aµ) j(H)0 dxµ; (2.67)

where the first term is an integral along the world line of the particle between two events a

and b i.e. the presence of the particle at its initial and final positions at time t1 and t2. The

variation of the action S may be written as,

δS = δˆ b

a

−MdS

+

ˆ b

a

[(αAµ − βBµ + γCµ −∆Dµ) j(e)0 dxµ

− (αBµ − βAµ + γDµ −∆Cµ) j(m)0 dxµ

+ (αCµ − βDµ + γAµ −∆Bµ) j(G)0 dxµ

− (αDµ − βCµ + γBµ −∆Aµ) j(H)0 dxµ] = 0. (2.68)

Taking the variation of the terms one by one, we get

I = δ

ˆ b

a

−MdS =

ˆ b

a

Muµdδxµ; (2.69)

where uµ = dxµdS

(four - velocity). Integrating equation (2.69) by parts, we get

I = |Muµδxµ|ba −ˆ b

a

Mduµδxµ; (2.70)

96


where first term is zero, since the integral is varied between fixed limits, i.e.

(δxµ)b = (δxµ)a = 0. (2.71)

Thus

I = −´ baMduµδxµ = −

ˆ b

a

MduµdS

dS δxµ. (2.72)

Now. it is convenient to take the following variations in order to solve the equations of

motion of various charges, i.e.

δ

ˆ b

a

αAµj(e)0 dxµ =

ˆ b

a

αj(e)0 (Aν,µ − Aµ,ν)uνδxµdS =

ˆ b

a

αj(e)0 FµνuνδxµdS;

δ

ˆ b

a

βBµj(e)0 dxµ =

ˆ b

a

βj(e)0 (Bν,µ −Bµ,ν)uνδxµdS =

ˆ b

a

βj(e)0 MµνuνδxµdS;

δ

ˆ b

a

γCµj(e)0 dxµ =

ˆ b

a

γj(e)0 (Cν,µ − Cµ,ν)uνδxµdS =

ˆ b

a

γj(e)0 fµνuνδxµdS;

δ

ˆ b

a

∆Dµj(e)0 dxµ =

ˆ b

a

∆j(e)0 (Dν,µ −Dµ,ν)uνδxµdS =

ˆ b

a

∆j(e)0 NµνuνδxµdS; (2.73)

δ

ˆ b

a

αBµj(m)0 dxµ =

ˆ b

a

αj(m)0 (Bν,µ −Bµ,ν)uνδxµdS =

ˆ b

a

αj(m)0 MµνuνδxµdS;

δ

ˆ b

a

βAµj(m)0 dxµ =

ˆ b

a

βj(m)0 (Aν,µ − Aµ,ν)uνδxµdS =

ˆ b

a

βj(m)0 FµνuνδxµdS;

δ

ˆ b

a

γDµj(m)0 dxµ =

ˆ b

a

γj(m)0 (Dν,µ −Dµ,ν)uνδxµdS =

ˆ b

a

γj(m)0 NµνuνδxµdS;

δ

ˆ b

a

∆Cµj(m)0 dxµ =

ˆ b

a

∆j(m)0 (Cν,µ − Cµ,ν)uνδxµdS =

ˆ b

a

∆j(m)0 fµνuνδxµdS; (2.74)

97


δ

ˆ b

a

αCµj(G)0 dxµ =

ˆ b

a

αj(G)0 (Cν,µ − Cµ,ν)uνδxµdS =

ˆ b

a

αj(G)0 fµνuνδxµdS;

δ

ˆ b

a

βDµj(G)0 dxµ =

ˆ b

a

βj(G)0 (Dν,µ −Dµ,ν)uνδxµdS =

ˆ b

a

βj(G)0 NµνuνδxµdS;

δ

ˆ b

a

γAµj(G)0 dxµ =

ˆ b

a

γj(G)0 (Aν,µ − Aµ,ν)uνδxµdS =

ˆ b

a

γj(G)0 FµνuνδxµdS;

δ

ˆ b

a

∆Bµj(G)0 dxµ =

ˆ b

a

∆j(G)0 (Bν,µ −Bµ,ν)uνδxµdS =

ˆ b

a

∆j(G)0 MµνuνδxµdS; (2.75)

and

δ

ˆ b

a

αDµj(H)0 dxµ =

ˆ b

a

αj(H)0 (Dν,µ −Dµ,ν)uνδxµdS =

ˆ b

a

αj(H)0 NµνuνδxµdS;

δ

ˆ b

a

βCµj(H)0 dxµ =

ˆ b

a

βj(H)0 (Cν,µ − Cµ,ν)uνδxµdS =

ˆ b

a

βj(H)0 fµνuνδxµdS;

δ

ˆ b

a

γAµj(H)0 dxµ =

ˆ b

a

γj(H)0 (Aν,µ − Aµ,ν)uνδxµdS =

ˆ b

a

γj(H)0 FµνuνδxµdS;

δ

ˆ b

a

∆Bµj(H)0 dxµ =

ˆ b

a

∆j(H)0 (Bν,µ −Bµ,ν)uνδxµdS =

ˆ b

a

∆j(H)0 MµνuνδxµdS. (2.76)

Substituting equations (2.72-2.76) into equation (2.68), we get the following equations of

motion for the dynamics of fields of a particle containing four charges as,

Md2xµdt2

= e Fµνuν ;

Md2xµdt2

= gMµνuν ;

Md2xµdt2

= mfµνuν ;

Md2xµdt2

= hNµνuν . (2.77)

Here M is the effective mass given by [69]

98


M =m − κ− 1

2h; (2.78)

where κ = +1 for electromagnetic fields and κ = −1 for gravito-Heavisidian fields. The field

equations (2.77) may also be obtained from the action (2.65) by taking the trajectory of the

unified particle fixed and considering the variation of the field parameters (i.e. potentials)

only. So, the parameters associated with the unified charged particle such as the free

particle action and four - current density Jµ are treated to be constant i.e. δSP = 0. From

equation (2.60), we may write the field and interaction parts of the Lagrangian required

for the variation as,

LF + LI = −1

4[α(FµνF

µν −MµνMµν − fµνfµν −NµνN

µν)]

−2β [FµνMµν + fµνNµν ]− 2γ [Fµνfµν +MµνNµν ]− 2∆ [FµνNµν + fµνMµν ]

+ (αAµ − βBµ + γCµ −∆Dµ) j(e)µ − (αBµ − βAµ + γDµ −∆Cµ) j(m)µ

+ (αCµ − βDµ + γAµ −∆Bµ) j(G)µ − (αDµ − βCµ + γBµ −∆Aµ) j(H)

µ . (2.79)

Now considering the term wise variation in equation (2.79), we get

δS1 =

ˆ−α4δF 2

µνdΩ =

ˆ−α2FµνδFµνdΩ;

δS2 =

ˆ−α4δM2

µνdΩ =

ˆ−α2MµνδMµνdΩ;

δS3 =

ˆ−α4δf 2

µνdΩ =

ˆ−α2fµνδfµνdΩ;

δS4 =

ˆ−α4δN2

µνdΩ =

ˆ−α2NµνδNµνdΩ; (2.80)

99


δS5 =

ˆβ

2δ(FµνMµν + fµνNµν)dΩ

=

ˆβ

2[FµνδMµν +MµνδFµν ] + [ fµνδNµν +Nµνδfµν ] dΩ; (2.81)

δS6 =

ˆγ

2δ(Fµνfµν +MµνNµν)dΩ

=

ˆγ

2[Fµνδfµν + fµνδFµν ] + [MµνδNµν +NµνδMµν ] dΩ; (2.82)

δS7 =

ˆ∆

2δ(FµνNµν +Mµνfµν)dΩ

=

ˆ∆

2[FµνδNµν +NµνδFµν ] + [Mµνδfµν + fµνδMµν ] dΩ; (2.83)

δS8 =

ˆαj(e)µ δAµ −

ˆβj(e)µ δBµ +

ˆγj(e)µ δCµ −

ˆ∆j(e)µ δDµ

−ˆαj(m)

µ δBµ +

ˆβj(m)

µ δAµ −ˆγj(m)

µ δDµ +

ˆ4j(m)

µ δCµ

+

ˆαj(G)

µ δCµ −ˆβj(G)

µ δDµ +

ˆγj(G)

µ δAµ −ˆ

∆j(G)µ δDµ

−ˆαj(H)

µ δDµ +

ˆβj(H)

µ δCµ −ˆγj(H)

µ δBµ +

ˆ4j(H)

µ δAµ. (2.84)

δS1 may be calculated after integrating it by parts and applying Gauss theorem as

δS1 = −αˆ∂Fµν∂xµ

δAµdΩ. (2.85)

Similarly, other variations may be calculated by taking into account the equation

δS = δS1 + δS2 + δS3 + δS4 + δS5 + δS6 + δS7 + δS8 = 0; (2.86)

100


from which we get

ˆ (j(e)µ δAµ + j(m)

µ δBµ + j(G)µ δCµ + j(H)

µ δDµ

)α dΩ

−ˆ (

∂Fµν∂xµ

δAµ −∂Mµν

∂xµδBµ −

∂fµν∂xµ

δCµ −∂Nµν

∂xµδDµ

)α dΩ

−ˆ (

j(e)µ δBµ + j(m)µ δAµ + j(G)

µ δDµ + j(H)µ δCµ

)α dΩ

−ˆ (

∂Fµν∂xµ

δAµ −∂Mµν

∂xµδBµ −

∂fµν∂xµ

δCµ −∂Nµν

∂xµδDµ

)α dΩ|

−ˆ

(j(e)µ − ij(m)µ − jj(G)

µ − kj(H)µ )δ(Aµ + iBµ + jCµ + kDµ)β dΩ

−ˆ

(Fµν,ν − iMµν,ν − jfµν,ν − kNµν,ν)δ(Aµ + iBµ + jCµ +Dµ)β dΩ

−ˆ

(j(e)µ − ij(m)µ − jj(G)

µ − kj(H)µ )δ(Bµ + iAµ + jDµ + kCµ)β dΩ

−(Fµν,ν − iMµν,ν − jfµν,ν − kNµν,ν)δ(Bµ + iAµ + jDµ + kCµ)β dΩ

=0; (2.87)

which yields the Maxwellian field equations given by equation (2.58) for the dynamics of

electric, magnetic, gravitational and Heavisidian charges (masses). These equations thus

provide the following form of unified field equations of a particle simultaneously contains

these four charges (masses)

=µν,ν = Jµ; (2.88)

where =µν is the unified field tensor defined by equation (2.56) and Jµ is the unified current

density given by equation (2.59).

2.8 Equation of motion of a unified charge in quaternion

form

From equation (2.74), we may write the unified equation of motion for a particle simul-

101


taneously contains four charges (masses) in terms of quaternion as

Md2xµdt2

=Re(Q=µν

)uν ; (2.89)

where xµ = d2xµdt2

is particle acceleration, uν is the four - velocity, and Q is the quaternion

conjugate of the unified charge given by equation (2.53). The right hand side of the equa-

tion (2.89) may also be written as

Re(Q=µν

)uν = (eFµν + gMµν +mfµν + hNµν)u

ν ; (2.90)

which can be reduced to

Re [Q, =µν ]uν =1

2

(Q=µν + Q=µν

)uν

= (eFµν + gMµν +mfµν + hNµν)uν

=e[−→E +−→v ×

−→H]

+ g[−→H −−→v ×

−→E]

+m[−→G −−→v ×

−→H]

+ h[−→H +−→v ×

−→G]

; (2.91)

where−→E is the electric field,

−→H is the magnetic field,

−→G is the gravitational field and

−→H is

Heavisidian field. Equation (2.91) represent the combination of four Lorentz forces due to

the presence of electric, magnetic, gravitational and Heavisidian charges.

2.9 Euler’s Lagrangian equation of motion of unified charges

The Lagrangian density for unified fields of a particle containing simultaneously the four

charges namely electric, magnetic, gravitational and Heavisidian may also be expressed as

102


L =− 1

4FµνF

µν − 1

4MµνM

µν − 1

4fµνf

µν − 1

4NµνN

µν

+Aµj(e)µ +Bµj

(m)µ + Cµj

(G)µ +Dµj

(H)µ . (2.92)

This Lagrangian density may also be written in terms of Grassmann product as

L = =1

8

[=µν ,=µν

]+[Vµ, Jµ

]; (2.93)

where[Vµ, Jµ

]= 1

2

[VµJµ + V µJµ

], with V µ (the quaternion conjugate of the unified four-

potential), Jµ (the quaternion conjugate of unified four-current density) and =µν (the

quaternion conjugate of unified field tensor) are defined as follows,

V µ = Aµ + iBµ + jCµ + kDµ;

Jµ = j(e)µ + ij(m)µ + jj(G)

µ + kj(H)µ ;

=µν = Fµν + iMµν + jfµν + kNµν . (2.94)

Then

[Vµ, Jµ

]=

[Aµj

(e)µ +Bµj

(m)µ + Cµj

(G)µ +Dµj

(H)µ

]. (2.95)

Similarly, we have

1

8

[=µν ,=µν

]=

1

4[FµνF

µν +MµνMµν + fµνf

µν +NµνNµν ] . (2.96)

From the Lagrangian density (2.92), we get the Euler’s Lagrangian equations in the follow-

ing forms,

103


∂L

∂Aµ− ∂ν

∂L

∂(∂νAµ)= 0;

∂L

∂Bµ

− ∂ν∂L

∂(∂νBµ)= 0;

∂L

∂Cµ− ∂ν

∂L

∂(∂νCµ)= 0;

∂L

∂Dµ

− ∂ν∂L

∂(∂νDµ)= 0. (2.97)

This equation provides the field equations associated respectively with the dynamics of

electric, magnetic, gravitational and Heavisidian charges (masses) given by equation (2.58)

after taking care the usual method of variations with respect to potential. These equation

may immediately be generalized to equation (2.88) as unified field equations of a particle

simultaneously containing four charges namely electric, magnetic, gravitational and Heav-

isidian charges (masses).

2.10 Discussion and Conclusion

Equation (2.1) represents the Einstein’s equation in non linear fields. The metric ten-

sor of Einsteins field equation in terms of perturbation and Minkowski metric is defined as

equation (2.2). Equation (2.4) represents the linearized Einstein’s field equation. Christof-

fel symbol is described in terms of perturbation. The explicit expression for the tensor

potential hµν is expressed in equation (2.6). The total angular momentum of the system(−→S)

with gravito-magnetic vector potential(−→A)

represents equation (2.8). Equation (2.9)

describes the total mass of the system with mass energy density T 00 = ρ. Equation (2.10)

demonstrates the total angular momentum of the system with mass energy density T 0i = ji.

The equation (2.11-2.13) describes the Lorentz gauge condition for gravito-electric and

gravito-magnetic fields and Maxwell’s equations in gravi-electromagnetic (GEM) fields. The

metric tensor in terms of space-time is invariant defined as equation (2.14). The geodesic

equation for a particle in the field of a weakly gravitating, rotating object, can be cast in

the form of an equation of motion under the action of a Lorentz force is represented in

equation (2.16). Equation (2.17) is defined as the motion under the action of the Lorentz

104


force produced by the GEM fields. Analogy between electromagnetic and linear gravita-

tional field has been developed in section 2.3 and simultaneous Maxwell’s equations for

Einstein’s linear gravity has been described by equation (2.13). Simultaneous Lorentz force

equation (2.26) and (2.27) and covariant field equation (2.24) has been written in symmet-

rical and consistent manner. Striking similarity between Maxwell’s equation and Einstein’s

linearized gravitational equation (2.13) has been pointed out to demonstrate the existence

of dual mass playing the role of gravitational monopole analogous to magnetic monopole

with electromagnetism. Quantization condition (2.29) resembles with Dirac quantization

condition eg = n2. Postulation of Heavisidian monopole, avoiding the use of string variables

and keeping in mind the structural symmetry between these two fields, a self consistent and

manifestly covariant theory of generalized gravito- Heavisidian fields of gravito-dyon has

been developed in section 2.4. Corresponding field equations, equation of motion and other

relevant equation of gravito-dyon have been investigated on the parallel lines of dyons as-

sociated with generalized electromagnetic fields. It has been shown that on passing from

generalized electromagnetic fields of dyon to generalized gravito-Heavisidian fields the sign

of the field equations is changed. Like chirality quantization condition of dyons, leading

to a large force between positive and negative unit magnetic charges, the corresponding

quantization condition (mjhk −mkhj = 0,±1...) for gravito-Heavisidian fields leads to the

enormous force fµ = m0xµ = Re (q?Gµν)uν between Heavisidian charges and suggests

that Heavisidian monopole are the most strongly interacting particles. The linear equation

for gravitational field with Heavisidian monopole are given in equation (2.30) and their

duality conditions are formulated in equation (2.31) and (2.32) which shows that the field

equation (2.30) is dual invariant. The gravito-Heavisidian force on a particle with Heav-

isidian charges h1 in the presence of the gravitational field−→G and Heavisidian field

−→H is

expressed by equation (2.33) and its dual force equation is given by equation (2.34). After

the introduction of gravitational dyon (particle that possess both the gravitational charge

and the Heavisidian charge), the equation of motion of a particle with gravitational charge

m1 and Heavisidian charge h1 is moving with non-relativistic velocity −→v in the field of a

stationary particle with gravitational charge m2 and Heavisidian charge h2, has been de-

scribed by equation (2.35). The gravitational and Heavisidian field strengths of the station-

105


ary gravito-dyon has been expressed by the equation (2.36). Equation (2.39) is the total

conserved angular momentum of the gravito-dyon and thus the equation (2.40) describes

the quantization condition of gravitational - Heavisidian monopole while equation (2.41)

relates the mass quantization condition. It is emphasized that the forces between Heavisid-

ian charges are enormous and the Heavisidian monopoles are the most strongly interacting

form of matter. In section 2.5, we have discussed the charge of gravito-dyons, tensorial form

of Maxwell’s Dirac equation of gravito - Heavisidian fields, gravitational and Heavisidian

four-current density, generalized four-potential, generalized four-current and generalized

field tensor of gravito-dyon from equations (2.42) to (2.47). A gauge - covariant and ro-

tationally symmetric theory of the angular momentum operator for gravito-dyon has been

rediscovered in terms of structural symmetry between generalized electromagnetic fields

of dyons and generalized gravito - Heavisidian fields of gravito-dyons. It has been shown

that the duality transformation holds good for generalized fields of gravito-dyons in terms

of structural symmetry. The generalized theory of gravito-dyons is discussed consistently

as gravitational analogous of the generalized electromagnetic fields of dyons. The duality

transformation holds good for generalized fields of gravito-dyons and demonstrates struc-

tural symmetry. In section 2.6, we have described the property of quaternion algebra with

the use of natural units (c = ~ = G = 1) in order to reformulate the unified theory of

generalized electromagnetic fields (associated with dyons) and generalized gravito - Heavi-

sidian fields (associated with gravito-dyons) of linear gravity. Then the equation (2.49) has

been investigated as the equation of unified charge. The quaternion valued four-potential

has been given by equation (2.54), while the various potentials of equation (2.54) are

expressed in terms of equation (2.55). Equation (2.56) describes the quaternion valued

unified field tensor which is the combined form of field tensors respectively associated with

the dynamics of the electric, magnetic, gravitational and Heavisidian charges (masses) dis-

cussed by equation (2.57). It is shown that the field tensors (2.57) satisfy the Maxwell field

equations (2.58). Equation (2.59) has been used for the quaternion valued unified current

as the combination of currents associated with various charges. The Lagrangian density for

the unified charged particle containing the electric, magnetic, gravitational and Heavisidian

charges and the rest mass of the unified particle M , has been given by equation (2.60). The

106


constancy condition of dyons and gravito-dyons is described by equation (2.64). The action

integral of a unified system such that properties of particle, properties of the field for both

the dyons has been described by equation (2.65). The Lagrangian density which contains

both types of generalized fields of dyons and gravito-dyons is given by (2.60) lies in fact

that the individual components of four-currents and four-potentials have been embodied in

the corresponding generalized quantities through the unknown parameters α, β, γ and ∆

which satisfy the constraints described by equations (2.61 - 2.62) and (2.63). The action

integral (2.65) has been described free from all the singularities and the string variables

in order to represent the properties of particle, field and the parameters associated with

both particle and field. Rather, it has been described in terms of four-potentials associated

with four charges namely electric, magnetic, gravitational and Heavisidian charges usual

method of the variation of this action (2.65) leads to equation of motion (2.77) for the

dynamics of different charges. It has also shown that the field equations (2.77) may also

be obtained from the action (2.65) by taking the trajectory of the unified particle fixed and

considering the variation of the field parameters (i.e. potentials) only. So, the parameters

associated with unified charged particle, such as the free particle action and four-current

density Jµ, are treated to be constant i.e. δSP = 0. Correspondingly, we have developed

the Lagrangian of the variation with unified fields, interaction part and variation of inte-

gral of four different types of charges which are given in equations (2.79 - 2.84). Equation

(2.87) yields the Maxwell field equations given by equation (2.58) for the dynamics of

electric, magnetic, gravitational and Heavisidian charges (masses). It has already been

shown that the Lagrangian density (2.60), equation of motion (2.77) and the unified field

equations (2.88) are invariant under the rotation in charge space or its combination with

space and time reflections and also under the reflection in charge space combined with

time reversal or space reflection. It means that the system possess strong symmetry under

rotation in charge space. Equation (2.89 - 2.91) are defined as the unified field equation

for a particle simultaneously containing four charges (e, g, m, h) in terms of quaternions.

The Lagrangian density for unified fields of a particle containing simultaneously the four

charges namely electric, magnetic, gravitational and Heavisidian may also be expressed by

equation (2.92) and in terms of Grassmann product, which is expressed by equation (2.93).

107


The quaternion conjugate of unified four-potential, unified four-current density and unified

field tensor is written as equation (2.94). The Euler’s Lagrangian equations (2.97) provides

the field equations associated respectively with the dynamics of electric, magnetic, gravi-

tational and Heavisidian charges (masses) given by equation (2.58) after taking care the

usual method of variations with respect to potential. These equation may immediately

be generalized to equation (2.88) as unified field equations of a particle simultaneously

containing four charges namely electric, magnetic, gravitational and Heavisidian charges

(masses). As such, the present foregoing analysis of dyons and gravito-dyons in terms of

quaternionic analyticity has been described in compact and consistent way which is mani-

festly covariant and free from the arbitrary string variables. It has been shown the present

formulation reproduces the dynamics of individual charges (masses) in the absence of other

charges (masses) as well as the generalized theory of dyons (gravito-dyons) in the absence

of gravito-dyons (dyons) or vice versa. It is also emphasized that the Heavisidian monopole

(or gravito-dyons) are the so heavy which may be one of the reason not to found them

experimentally but needs to the better insight into the theory for further investigations.

108

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115



fields”, J. Math. Phys., 25 (1984), 351.

116

Chapter 3

ABELIAN GAUGE AND TWO POTENTIAL

THEORY OF DYONS AND

GRAVITO-DYONS

Chapter 3 Abelian Gauge and Two Potential Theory of Dyons and Gravito Dyons

'

&

$

%

ABSTRACT

The manifestly covariant, dual symmetric and gauge invariant two potential theory of gener-

alized electromagnetic fields of dyons and gravito-dyons has been developed consistently from

U(1)×U(1) gauge symmetry. Corresponding field equations and equation of motion are de-

rived from Lagrangian formulation adopted for U(1)×U(1) gauge symmetry for the justifica-

tion of two four potentials of dyons and gravito-dyons.

118


3.1 Introduction

The asymmetry between electricity and magnetism has become clear at the end of 19th

century with the formulation of Maxwell’s equations and physicists were fascinated about

the idea of magnetic monopoles. Dirac [1, 2] put forward the idea of magnetic monopoles

to symmetrize Maxwell’s equations and showed that the quantum mechanics of an elec-

trically charged particle of charge e and a magnetically charged particle of charge g is

consistent only if eg = 2πn, n being an integer. Schwinger-Zwanziger [3]-[8] generalized

this condition to allow for the possibility of particles (dyons) that carry both electric and

magnetic charge. A quantum mechanical theory can have two particles of electric and

magnetic charges (e1, g1) and (e2, g2) only if e1g2 − e2g1 = 2πn . Julia and Zee [9] con-

structed the theory of non Abelian dyons. It is now widely recognized that the standard

model, which combines the gauge theory of strong interactions with the model of elec-

tro weak interaction, is a gauge theory that contains monopole and dyon solutions. The

quantum mechanical excitation of fundamental monopoles include dyons which are auto-

matically arisen [10] from the semi-classical quantization of global charge rotation degree

of freedom of monopoles. In view of the explanation of CP-violation in terms of non-zero

vacuum angle of world [11, 12, 13], the monopoles are necessary dyons and Dirac quan-

tization condition permits dyons to have analogous electric charge. Renewed interests in

the subject of monopole has gathered [14]-[27] enormous potential importance in connec-

tion current grand unified theories, supersymmetric gauge theories and super strings. But

unfortunately the experimental searches [21]-[27] for these elusive particles have proved

fruitless as the monopoles are expected to be super heavy and their typically masses are

about two orders of magnitude heavier than the super heavy X bosons mediating proton

decay. In spite of the enormous potential importance of monopoles (dyons) and the fact

that these particles have been extensively studied, there is still the lack of reliable theory

which is as conceptually transparent and predictably tractable as the usual electrodynamics

and the formalism necessary to describe them has been clumsy and not manifestly covari-

ant. Rather, the concept of electromagnetic (EM) duality has been receiving much attention

[18]-[27] in gauge theories, field theories, super symmetry and super strings. So, keeping

119


in view the recent potential importance of monopoles (dyons) and the applications of elec-

tromagnetic duality, in this chapter [28], we have made an attempt to revisit the analogous

consistent formulation of dual electrodynamics subjected by the magnetic monopole only.

Gauge formulation has been adopted accordingly to derive the dual Maxwell’s equation,

equation of motion and Bianchi identity for dual electric charge (i.e. magnetic monopole)

from the minimum action principle. Accordingly, we have discussed the dual symmetric

and manifestly covariant formulation of generalized fields of dyons in order to obtain the

generalized Dirac-Maxwell’s (GDM) field equations and Lorentz force equation of motion of

dyons in terms of two four potentials. Two potential theory of magnetic monopoles (dyons)

has been justified from U (1) × U (1) gauge symmetry. Consequently, the gauge symmetric

and dual invariant manifestly covariant theory has been discussed consistently from the

U e (1) × U g (1) gauge symmetry. It has been emphasized that the two U (1) gauge groups

act in different manner whereas the first U e (1) act on the Dirac spinors while the other

group U g (1) act on Dirac iso-spinors. We have also developed accordingly the consistent

Lagrangian formulation for the justification of two gauge potentials of dyons. Accordingly,

the analogous gauge formulation has been adopted in order to derive the dual Maxwell’s

equation, equation of motion and Bianchi identity for dual electric charge (i.e. magnetic

monopole) and dual gravitational charge (i.e. Heavisidian monopole) from the minimum

action principle. Likewise, we have discussed the dual symmetric and manifestly covariant

formulation of generalized fields of dyons and gravito-dyons in order to obtain the gen-

eralized Dirac-Maxwell’s (GDM) field equations and Lorentz force equation of motion of

dyons and gravito-dyons in terms of two four potentials. Two potential theory of mag-

netic monopoles (dyons) and gravi-Heavisidian (gravito-dyons) have been justified from

U (1)×U (1) gauge symmetry. Consequently, the gauge symmetric and dual invariant man-

ifestly covariant theory has been reformulated consistently from the U (1) × U (1) gauge

symmetry for the dyons and gravito-dyons as a separate case. It is shown that Um (1) acts

on the Dirac spinors while the second unitary abelian group Uh (1) acts on Dirac iso-spinors

due to presence of Heavisidian monopole. Furthermore, the consistent Lagrangian formu-

lation has been developed in order to obtain consistent field equations for the justification

of two gauge potentials of dyons and gravito-dyons.

120


3.2 Gauge symmetry of Dual Electromagnetism

Let us define a Dirac field ψ with dual (magnetic) charge g for which the free Dirac

Lagrangian is

L0 = ψ (iγµ∂µ +m)ψ(i =√−1)

; (3.1)

where γ0 =

1 0

0 −1

and γj =

0 τ j

−τ j 0

are 4 × 4 complex Dirac matrices with 0, 1

and τ j are respectively the 2× 2 null, unit and Pauli matrices ∀j = 1, 2, 3. Also the gamma

matrices satisfy the property

γµ, γν =2ηµν ; (3.2)

and ψ = ψ†γ0 with (†) denotes the Hermitian conjugation for the fields of dyons. So, the

Dirac Lagrangian (3.1) is clearly invariant under the global gauge transformation

ψ 7−→ ψp 7−→ exp igαψ;

ψ 7−→ ψp 7−→ ψ exp −igα ; (3.3)

where α is independent of space-time. So, like electromagnetism, we elevate this symmetry

to invariance under local gauge transformation

ψ 7−→ ψp 7−→ exp igα (x)ψ;

ψ 7−→ ψp 7−→ ψ exp −igα (x) ; (3.4)

so that the Lagrangian (3.1) transforms as

121


ψ (iγµ∂µ +m)ψ 7−→ψ (iγµ∂µ − gγµ∂µα +m)ψ. (3.5)

Since the extra term gγµ∂µα looks like a gauge transformation of potential, we may couple

the gauge field Bµ with ψ so that the Lagrangian has the local gauge symmetry. We, thus,

write the Lagrangian (3.1) as

L =ψ (iγµ∇µ +m)ψ; (3.6)

where the covariant derivative in equation (3.5) is given by

∇µ = ∂µ − igBµ. (3.7)

Hence the Lagrangian (3.1) is invariant under the combined gauge transformation

ψ (x) 7−→ ψp (x) 7−→ exp igα (x)ψ (x) ;

ψ (x) 7−→ ψp(x) 7−→ ψ (x) exp −igα (x) ;

Bµ 7−→ B pµ = Bµ + ∂µα (x) ; (3.8)

and the covariant derivative is transformed as

∇pµψ

p 7−→ exp igχ (x) (∇µψ). (3.9)

As such, the Lagrangian for total dual quantum electrodynamics is subjected by

L = −1

4FµνFµν + ψ (iγµ∂µ +m)ψ −Bµ j

µ(g); (3.10)

122


where

jµ(g) =gψ (x) γµψ (x) . (3.11)

Accordingly, we get

[∇µ,∇ν ]ψ (x) =− igFµνψ (x) ; (3.12)

and with the use of Jacobi identity,

[∇µ, [∇ν ,∇λ]] + [∇ν , [∇λ,∇µ]] + [∇λ, [∇µ,∇ν ]] = 0; (3.13)

we get the Bianchi identity,

∇µFνλ+ ∇νFλµ+ ∇λFµν = 0; (3.14)

which is the kinematical statement equivalent to Fµν,ν=∂νFµν = 0 for dual electrodynamics

analogous to the kinematical statement ∂νF µν = 0 for usual electrodynamics. As such dual

symmetry of electrodynamics requires the existence of dual electric charge (i.e magnetic

monopole).

3.3 Gauge symmetry of dual gravito-dynamics

Let us define a Dirac field χ with dual Heavisidian monopole [29, 30, 31] (gravi-

magnetic) charge h for which the free Dirac Lagrangian of gravi-Heavisidian field is given

by

123


L0 =χ (iγµ∂µ +m)χ; (3.15)

and χ = χ†γ0 with (†) denotes the Hermitian conjugation of gravi-Heavisidian fields. So,

the Dirac Lagrangian (3.15) is clearly invariant under the global gauge transformation

χ 7−→ χp 7−→ exp ihβχ;

χ 7−→ χp 7−→ χ exp −ihβ ; (3.16)

where β is independent of space-time. So, like gravi - Heavisidian, we elevate this symmetry

to invariance under local gauge transformation

χ 7−→ χp 7−→ exp ihβ (x)χ;

χ 7−→ χp 7−→ χ exp −ih (x) ; (3.17)

so that the Lagrangian (3.15) transforms as

χ (iγµ∂µ +m)χ 7−→χ (iγµ∂µ − hγµ∂µβ +m)χ. (3.18)

Since the extra term hγµ∂µβ looks like a gauge transformation of potential, we may couple

the gauge field Dµ with χ so that the Lagrangian has the local gauge symmetry. We thus,

write the Lagrangian (3.15) as

L =χ (iγµ4µ +m)χ; (3.19)

where the covariant derivative in equation (3.18) is given by 4µ = ∂µ − ihDµ. Hence the

Lagrangian (3.15) is invariant under the combined gauge transformation

124


χ (x) 7−→ χp (x) 7−→ exp ihβ (x)χ (x) ;

χ (x) 7−→ χp (x) 7−→ χ (x) exp −igβ (x) ;

Dµ 7−→ Dpµ = Dµ + ∂µβ (x) ; (3.20)

and the covariant derivative is transformed as

4pµχ

p 7−→ exp ihχ (x) (4µχ) . (3.21)

As such, we express the Lagrangian for total dual gravi-Heavisidian field as

L = −1

4NµνN µν + χ (iγµ∂µ +m)χ−Dµj

µ(h); (3.22)

where

jµ(h) =hχ (x) γµχ (x) . (3.23)

Accordingly, we get

[4µ,4ν ]χ (x) =− ihNµνχ (x) ; (3.24)

and with the use of Jacobi identity of gravi-Heavisidian field i.e.

[4µ, [4ν ,4λ]] + [4ν , [4λ,4µ]] + [4λ, [4µ,4ν ]] = 0; (3.25)

we get the Bianchi identity of gravi-Heavisidian as

125


4µNνλ+ 4νNλµ+ 4λNµν = 0. (3.26)

which is equivalent to Nµν,ν=∂νNµν = 0 for dual gravi-Heavisidian fields analogous to the

kinematical statement ∂ν fµν = 0 for usual gravi-Heavisidian fields due to the presence

of gravitational charge. As such, dual symmetry of gravi-Heavisidian fields requires the

existence of dual Heavisidian (gravi-magnetic charge) (i.e. Heavisidian monopole).

3.4 Dual symmetric covariant formulation of dyons

The electric field−→E and magnetic field

−→H described the GDM equations (1.22) in Chap-

ter 1 may now be identified as the generalized electromagnetic fields of dyons by combin-

ing equations (1.7) and (1.21) in terms of the components of two four-potentials Aµ and

Bµ [32]-[44] respectively associated with electric and magnetic charge as

−→E =− ∂

−→A

∂t−−→∇A0 −

−→∇ ×

−→B ;

−→H =− ∂

−→B

∂t−−→∇B0 +

−→∇ ×

−→A. (3.27)

Generalized Dirac Maxwell’s (GDM) equations (1.22) are invariant not only under Lorentz

and conformal transformations but are also invariant under the duality transformations be-

tween electric E and magneticH given by equation (3.27), where E =(e,−→E , ρe,

−→je , A0,

−→A)

andH =(g,−→H, ρg,

−→jg , B0,

−→B)

. The generalized anti-symmetric dual invariant electromag-

netic field tensors for dyons are then be written as

F µν = ∂νAµ − ∂µAν − 1

2εµνλσ (∂λBσ − ∂σBλ) = F µν − Fµν ;

Mµν = ∂νBµ − ∂µBν +1

2εµνλσ (∂λAσ − ∂σAλ) = Fµν + F µν . (3.28)

The generalized electromagnetic fields (3.27) of dyons produce the components of gener-

126


alized field tensors (3.28) as

F 0j = Ej ; M jk = εjlkHl (∀j, k, l = 1, 2, 3) ;

M0j = Hj ; F jk = εjlkEl (∀j, k, l = 1, 2, 3) . (3.29)

Hence, the covariant form of dual symmetric GDM equations (1.22) is described [32]-[44]

as

Fµν,ν ⇒ Fµν,ν = j(e)µ ;

Mµν,ν ⇒ Fµν,ν = j(g)µ . (3.30)

Therefore, we may write the Lagrangian density which follows the minimum action princi-

ple for generalized electromagnetic fields of dyons as

LGEM = −1

4FµνF

µν − 1

4MµνM

µν+ Aµjµ(e)+ Bµj

µ(g). (3.31)

This Lagrangian yields the GDM field equations (3.30) and the Lorentz force equation of

motion for dyons as

=µ =Fµνjν(e) + Fµνjν(g). (3.32)

On the other hand, the dual parts of field tensors giving rise ∂νF µν = 0 and ∂νFµν = 0

describe the Bianchi identities like equation (3.14) for electric and magnetic charges. If we

define the four currents in terms of charge and velocity as

jν(e) = euν ; jν(g) = guν , (3.33)

127


the dual invariant Lorentz force expression (3.32) for dyon reduces [32]-[44] to

−→= = m

d2−→xdt2

= e(−→E +−→u ×

−→H)

+ g(−→H −−→u ×

−→E). (3.34)

where −→u is the velocity of a particle.

3.5 Dual symmetric covariant formulation of gravito-dyons

Here we identity GDM equation (2.30) the field equation of gravito-dyons. So, the

gravitationl (gravi-electric) field−→G and Heavisidian (gravi-magnetic field

−→H in Chapter

2 are now be identified as the generalized gravito-Heavisidian fields of gravito-dyons in

terms of the components of two four-potentials Cµ and Dµ respectively associated

with gravitational and Heavisidian charges described as

−→G =− ∂

−→C

∂t−−→∇C0 −

−→∇ ×

−→D ;

−→H =

∂−→D

∂t+−→∇D0 +

−→∇ ×

−→C . (3.35)

These fields satisfy the generalized Dirac Maxwell’s (GDM) equations (2.30) in Chapter

2 of gravi-Heavisidian fields which are invariant not only under Lorentz and conformal

transformations but are also invariant under the following duality transformations between

gravitational (gravi-electric) G and Heavisidian (gravi-magnetic) H quantities in equation

(2.31) in Chapter 2, where G ≡(m,−→G, ρm,

−→jm, C0,

−→C)

andH ≡(h,−→H , ρh,

−→jh, D0,

−→D)

. The

generalized anti-symmetric dual invariant gravi-Heavisidian field tensors for gravito-dyons

are now written as

fµν = ∂νCµ − ∂µCν − 1

2εµνλσ (∂λDσ − ∂σDλ) = fµν − N µν ;

Nµν = ∂νDµ − ∂µDν +1

2εµνλσ (∂λCσ − ∂σCλ) = N µν + fµν . (3.36)

128


So, the generalized gravi-Heavisidian fields of gravito-dyons (3.35) may now be written in

terms of the components of generalized field tensors (3.36) as

f 0j = Gj ; N jk = εjlkHl (∀j, k, l = 1, 2, 3) ;

N0j = Hj ; f jk = εjlkGl (∀j, k, l = 1, 2, 3) . (3.37)

Hence, the covariant form of dual symmetric GDM equations of gravi-Heavisidian fields

(2.30) in Chapter 2 is described as [45]-[47]

fµν,ν ⇒ fµν,ν = j(m)µ ;

Nµν,ν ⇒ Nµν,ν = j(h)µ . (3.38)

Henceforth we may write the Lagrangian which follows the minimum action principle for

generalized gravi-Heavisidian fields of gravito-dyons as

LGEM = −1

4fµνf

µν − 1

4NµνN µν+ Cµj

µ(m)+ Dµjµ(h). (3.39)

This Lagrangian yields the GDM field equation of gravito-dyons given by equation (3.38).

The Lorentz force equation of motion for gravito-dyons is now be described as

<µ =fµνjν(m) +Nµνjν(h). (3.40)

The dual parts of field tensors giving rise ∂ν fµν = 0 and ∂νN µν = 0 describe the Bianchi

identities like equation (3.26) for gravitational (gravi-electric) and Heavisidian (gravi-

magnetic) charges. If we define the four currents in terms of charge and velocity of gravi-

Heavisidian field as

129


jν(m) = muν ; jν(h) = huν ; (3.41)

the dual invariant Lorentz force expression (3.40) for grvito-dyon reduces to

−→< = m

d2−→xdt2

= m(−→G −−→u ×

−→H)

+ h(−→H +−→u ×

−→G)

; (3.42)

where −→u is the velocity of a particle.

3.6 U (1)× U (1) gauge formulation of dyons

Let us start with the Lagrangian density (3.1) within the introduction of four-spinor

Dirac field Ψ for dyons instead of two component spinor ψ as

Ψ =

Ψ1

Ψ2

; (3.43)

where Ψ1 and Ψ2 are two component spinors. Here Ψ1 is identified as the Dirac spinor for

a electric charge ( like electron) while the other spinor Ψ2 has been identified as the Dirac

iso-spinors acting on the magnetic monopole. Thus the Ψ may be visualized as the bi-spinor

for dyons in terms of its electric and magnetic counterparts. Each spinor Ψ1 and Ψ2 satisfy

the free particle Dirac equation

L0 =Ψ (iγµ∂µ +m) 1Ψ;

=(Ψ1,Ψ2

) (iγµ∂µ +m) 0

0 (iγµ∂µ +m)

Ψ1

Ψ2

;

=α=2∑α=1

Ψα (iγµ∂µ +m) Ψα; (3.44)

130


where 1 is 2× 2 unit matrix. So, the Unitary transformations taking part for the invariance

of free particle Dirac equation for bi-spinor Ψ are the global U = U (e) (1) × U (g) (1) two

component spinors Ψ1 and Ψ2. In this case Ψ1 acts on unitary gauge group U (e) (1) whereas

the iso-spinor Ψ2 acts on the other unitary group U (g) (1) with the symbols (e) and (g) are

used for the electric and magnetic charges. Thus equation (3.44) is invariant under global

gauge transformation

U = U (e) × U (g) = exp(iΛjτ

jba

); (3.45)

where

τ jba = τ 1ba =

1 0

0 0

and τ jba = τ 2ba =

0 0

0 1

; (3.46)

and

[τ jba , τ

kba

]= εjkl τ

lba =0; (3.47)

because we have j, k, l = 1, 2. Accordingly, the spinor transforms as

Ψ1 7−→ Ψp1 7−→

[U (e)

]Ψ1 = exp iΛ1Ψ1;

Ψ2 7−→ Ψp2 7−→

[U (g)

]Ψ2 = exp iΛ2Ψ2;

Ψ1 7−→ Ψp1 7−→Ψ

p1

[U (e)

]−1= Ψ

p1 exp iΛ1 ;

Ψ2 7−→ Ψp2 7−→Ψ

p1

[U (g)

]−1= Ψ

p2 exp iΛ2 ;

Ψ 7−→ Ψp 7−→ UΨ =

exp (iΛ1) 0

0 exp (iΛ2)

Ψ1

Ψ2

;

Ψ 7−→ Ψp 7−→ ΨU−1 =

(Ψ1, Ψ2

) exp (−iΛ1) 0

0 exp (−iΛ2)

. (3.48)

131


In equations (3.45) and (3.47), Λj(∀j = 1, 2) are independent of space and time in order

to represent global gauge transformations. If we elevate this symmetry to invariance under

local gauge transformations where Λj (x)(∀j = 1, 2) in equations (3.45) and (3.47), the

Lagrangian (3.44) transforms as

Ψ (iγµ∂µ +m) 1Ψ −→ Ψ (iγµDµ +m) 1Ψ; (3.49)

where partial derivative ∂µ has been replaced by the covariant derivative Dµ as

Dµ = Dbµa = ∂µδ

ba + βµjτ

jba ; (3.50)

and τ jba are given by equation (3.46) along with

βµj −→βµ1 = Aµ;

βµj −→βµ2 = Bµ. (3.51)

These are now be identified as the gauge potentials respectively associated with the dynam-

ics of electric and magnetic charges with the following gauge transformations

βµ1 = Aµ 7−→ Apµ 7−→

[U (e)

]Aµ[U (e)

]−1+

1

e

[U (e)

]∂µ[U (e)

]−1;

βµ2 = Bµ 7−→ B pµ 7−→

[U (g)

]Bµ

[U (g)

]−1+

1

g

[U (g)

]∂µ[U (g)

]−1; (3.52)

where

[U (e)

]=⇒ exp iΛ1 (x) ;[

U (g)]

=⇒ exp iΛ2 (x) . (3.53)

132


As such, we may write the DµΨ as

DµΨ =

∂µ − ieAµ 0

0 ∂µ − igBµ

Ψ1

Ψ2

; (3.54)

which transforms as

DµΨ −→ DpµΨp −→

exp iΛ1 (x) 0

0 exp iΛ2 (x)

(∂µ − ieAµ) Ψ1

(∂µ − igBµ) Ψ2

;

=U (DµΨ) . (3.55)

Hence, we get

[Dµ, Dν ] Ψ (x) =

−ieFµν 0

0 −igFµν

Ψ1

Ψ2

; (3.56)

and it leads to the Jacobi identity

[Dµ, [Dν , Dλ]] + [Dν , [Dλ, Dµ]] + [Dλ, [Dµ, Dν ]] = 0; (3.57)

along with the Bianchi identities

DµFνλ+DνFλµ +DλFµν = 0;

DµFνλ+DνFλµ +DλFµν = 0. (3.58)

As such, the total Lagrangian for generalized fields of dyons is described as

133


L =− 1

4FµνF

µν − 1

4FµνFµν + ψ (iγµ∂µ +m)ψ − Aµjµ(e) −Bµj

µ(g); (3.59)

where

jµ(e) =eΨ1γµΨ1; (3.60)

and

jµ(g) =gΨ2γµΨ2; (3.61)

are the four-currents associated respectively with electric and magnetic charges on dyons.

These four-currents obtained from the Dirac spinor Ψ1 and the Dirac iso-spinor Ψ2 satisfy

the following conserved relations

∂µjµ(e) = jµ(e),µ = 0; (3.62)

and

∂µjµ(g) = jµ(g),µ = 0. (3.63)

Like other electric and magnetic dynamical variables, one can introduce duality transfor-

mations between the electric and magnetic gauges corresponding to orthogonal transfor-

mations in group space i.e.

134


Λ1 =⇒Λ1 cosϑ+ Λ2 sinϑ;

Λ2 =⇒Λ2 cosϑ− Λ1 sinϑ; (3.64)

and with the use of constancy condition [38]-[42] as

g

e=Bµ

Aµ=j(g)µ

j(e)µ

=Λ2

Λ1

=FµνFµν

=Mµν

F µν= − tan θ = Constant; (3.65)

we get

[Dµ, Dν ] Ψ (x) =

−ieFµν 0

0 −igMµν

Ψ1

Ψ2

; (3.66)

where Fµν and Mµν are the generalized dual invariant electromagnetic fields of dyons and

satisfy independently the Bianchi identity (3.58). Hence, the Lagrangian density (3.59)

may now be written as

L =− 1

4FµνF

µν − 1

4MµνM

µν + ψ (iγµ∂µ +m)ψ − Aµjµ(e) −Bµjµ(g); (3.67)

where Fµν and Mµν transform as

U [Dµ, Dν ]U−1Ψ (x) =⇒

−ie [U (e)]Fµν

[U (e)

]−10

0 −ig[U (g)

]Mµν

[U (g)

]−1 Ψ1

Ψ2

;

=⇒Fµν 7−→[U (e)

]−1Fµν

[U (e)

];

=⇒Mµν 7−→[U (g)

]−1Mµν

[U (g)

]. (3.68)

So, it is observed that the Lagrangian density (3.67) reproduces the dual symmetric and

135


Lorentz covariant generalized Dirac Maxwell’s (GDM) field equations (3.31) and Lorentz

force equation (3.32) of motion for two potential theory of dyons. Thus, with the use of

Jacobi identity (3.57), we get the Bianchi identity for generalized electromagnetic field

tensors Fµν and Mµν as

DµFνλ+DνFλµ +DλFµν = 0;

DµMνλ+DνMλµ +DλMµν = 0. (3.69)

As such, the classical theory of dyons has been verified and the incorporation of two four

potentials in generalized electromagnetic fields of dyons has been justified in the frame

work of U (1)×U (1) gauge theory where the first unitary Abelian gauge group U (e) (1) acts

on the Dirac spinors due to the presence of the electric charge while second unitary Abelian

gauge group U (g) (1) acts on the Dirac iso-spinors due to the presence of the magnetic

charge on dyons. The activation of gauge group U (g) (1) on Dirac iso-spinor is advantageous

so that it may further be extended to enlarge the gauge group to describe the non-Abelian

correspondence of monopoles (dyons) in current grand unified and supersymmetric gauge

theories associated with dyons.

3.7 U (1)× U (1) gauge formulation of gravito-dyons

In order to develop U (1) × U (1) gauge formulation of gravito-Heavisidian dyons like

electromagnetic dyons we define the Dirac spinor χ of equation (3.15) as

χ =

χ1

χ2

; (3.70)

where χ1 and χ2 are four component Dirac spinors. Here χ1 is identified as the Dirac spinor

for a gravitational (gravi-electric) charge while the other spinor χ2 has been identified as

the Dirac iso-spinors acting on the Heavisidian (gravi-magnetic) monopole. So, the χ may

be visualized as the bi-spinor for gravito-dyons in terms of its gravitational and Heavisidian

136


counterparts. Each spinor χ1 and χ2 satisfy the free particle Dirac equation of gravito-dyons

and the free particle Lagrangian density may now be written as

L0 =χ (iγµ∂µ +m) 1χ;

= (χ1, χ2)

(iγµ∂µ +m) 0

0 (iγµ∂µ +m)

χ1

χ2

;

=

β=2∑β=1

χβ (iγµ∂µ +m)χβ; (3.71)

where 1 is 2× 2 unit matrix. So, the Unitary transformations responsible for the invariance

of free particle Dirac equation with bi-spinor χ are the global U = U (m) (1) × U (h) (1) in

terms of two component spinors χ1 and χ2 . Here χ1 acts on unitary gauge group U (m) (1)

whereas the iso-spinor χ2 acts on the other unitary group U (h) (1) with the symbols (m)

and (h) are respectively used for the gravitational (gravi-electric) and Heavisidian (gravi-

magnetic) charges. Thus equation (3.71) is invariant under global gauge transformation

U = U (m) × U (h) = exp(iΘjτ

jba

); (3.72)

where j, k, l = 1, 2. Accordingly, the spinor χ transforms for gravitational (gravi-electric)

and Heavisidian (gravi-magnetic) spinors as

137


χ1 7−→ χp1 7−→

[U (m)

]χ1 = exp iΘ1χ1;

χ2 7−→ χp2 7−→

[U (h)

]χ2 = exp iΘ2χ2;

χ1 7−→ χp1 7−→χp

1

[U (m)

]−1= χp

1 exp iΘ1 ;

χ2 7−→ χp2 7−→χp

2

[U (h)

]−1= χp

2 exp iΘ2 ;

χ 7−→ χp 7−→ Uχ =

exp (iΘ1) 0

0 exp (iΘ2)

χ1

χ2

;

χ 7−→ χp 7−→ χU−1 =(χ1, χ2

) exp (−iΘ1) 0

0 exp (−iΘ2)

. (3.73)

In equations (3.72) Θj( j = 1, 2) is independent of space and time for global gauge trans-

formations. If we extend this symmetry to its invariance under local gauge transformations,

the Lagrangian (3.71) of gravito-Heavisidian monopole transforms as

χ (iγµ∂µ +m) 1χ 7−→χ (iγµDµ +m) 1χ; (3.74)

where partial derivative ∂µ is replaced by the covariant derivative Dµ i.e.

Dµ = Dbµa =∂µδba + β p

µjτjba ; (3.75)

and τ jba are given by equation (3.46) along with

β pµj 7−→β p

µ1 = Cµ;

β pµj 7−→β p

µ2 = Dµ. (3.76)

These are the gauge potentials respectively associated with the dynamics of gravitational

(gravi-electric) and Heavisidian (gravi-magnetic) charges with the following gauge trans-

138


formations i.e.

β pµ1 = Cµ 7−→ C p

µ 7−→[U (m)

]Cµ[U (m)

]−1+

1

m

[U (m)

]∂µ[U (m)

]−1;

β pµ2 = Dµ 7−→ Dp

µ 7−→[U (h)

]Dµ

[U (h)

]−1+

1

h

[U (h)

]∂µ[U (h)

]−1; (3.77)

where

[U (m)

]=⇒ exp iΘ1 (x) ;[

U (h)]

=⇒ exp iΘ2 (x) . (3.78)

As such, we may write the Dµχ as

Dµχ =

∂µ − imCµ 0

0 ∂µ − ihDµ

χ1

χ2

; (3.79)

which transforms as

Dµχ 7−→ Dpµχ

p 7−→

exp iΘ1 (x) 0

0 exp iΘ2 (x)

(∂µ − imCµ)χ1

(∂µ − ihDµ)χ2

;

=U (Dµχ) . (3.80)

Hence, the commutation relation becomes

[Dµ,Dν ]χ (x) =

−imfµν 0

0 −ihNµν

χ1

χ2

; (3.81)

which leads to the Jacobi identity of gravito-Heavisidian fields

139


[Dµ, [Dν ,Dλ]] + [Dν , [Dλ,Dµ]] + [Dλ, [Dµ,Dν ]] = 0; (3.82)

along with the Bianchi identities of gravito-Heavisidian fields

Dµfνλ +Dνfλµ +Dλfµν = 0;

DµNνλ +DνNλµ +DλNµν = 0. (3.83)

As such, the total Lagrangian for generalized fields of gravito-dyons is described as

L = −1

4fµνf

µν − 1

4NµνN µν+ χ (iγµ∂µ +m)χ− Cµj

µ(m) −Dµjµ(h); (3.84)

where

jµ(m) =mχ1γµχ1; (3.85)

and

jµ(h) =hχ2γµχ2; (3.86)

are the four-currents associated respectively with gravitational (gravi-electric) and Heavi-

sidian (gravi-magnetic) charges on gravito-dyons. These four-currents obtained from the

Dirac spinor χ1 and the Dirac iso-spinor χ2 satisfy the following conserved relations

∂µjµ(m) = jµ(m)

,µ = 0; (3.87)

140


and

∂µjµ(h) = jµ(h),µ = 0. (3.88)

The gravitational (gravi-electric) and Heavisidian (gravi-magnetic) dynamical variables re-

spect the duality transformations between the gravitational (gravi-electric) and Heavisidian

(gravi-magnetic) gauges corresponding to orthogonal transformations in group space i.e.

Θ1 =⇒ Θ1 cosϑ+ Θ2 sinϑ;

Θ2 =⇒ Θ2 cosϑ−Θ1 sinϑ; (3.89)

and with the use of constancy condition for gravi-Heavisidian fields [38]-[42] as

h

m=Dµ

Cµ=j(h)µ

j(m)µ

=Θ2

Θ1

=Nµνfµν

=Nµν

fµν= − tan θ = Constant; (3.90)

we get

[Dµ,Dν ]χ (x) =

−imfµν 0

0 −ihNµν

χ1

χ2

; (3.91)

where fµν and Nµν are the generalized dual invariant gravi-Heavisidian fields of gravito-

dyons and satisfy independently the Bianchi identity (3.83). Hence, the Lagrangian density

(3.84) may now be written as

L = −1

4fµνf

µν − 1

4NµνN

µν+ χ (iγµ∂µ +m)χ− Cµjµ(m) −Dµj

µ(h); (3.92)

where fµν and Nµν transform as

141


U [Dµ, Dν ]U−1χ (x) =⇒

−im [U (g)]fµν[U (g)

]−10

0 −ih[U (h)

]Nµν

[U (h)

]−1 χ1

χ2

;

=⇒fµν −→[U (g)

]−1fµν[U (g)

];

=⇒Nµν −→[U (h)

]−1Nµν

[U (h)

]. (3.93)

So, it is observed that the Lagrangian density (3.92) reproduces the dual symmetric and


force equation (3.40) of motion for two potential theory of gravito-dyons. Thus, with the

use of Jacobi identity (3.82), we get the Bianchi identity for generalized gravi-Heavisidian

field tensors fµν and Nµν as

Dµfνλ +Dνfλµ +Dλfµν = 0;

DµNνλ +DνNλµ +DλNµν = 0. (3.94)

So, the classical theory of gravito-dyons has been verified and the incorporation of two four

potentials in generalized gravito-Heavisidian fields of gravito-dyons has been justified in

the frame work of U (1) × U (1) gauge theory where the first unitary Abelian gauge group

U (m) (1) acts on the Dirac spinors due to the presence of the gravitational (gravi-electric)

charge while second unitary Abelian gauge group U (h) (1) acts on the Dirac iso-spinors

due to the presence of the Heavisidian (gravi-magnetic) charge on gravito-dyons. The

activation of gauge group U (h) (1) on Dirac iso-spinor is advantageous so that it may further

be extended to enlarge the gauge group to describe the non-Abelian correspondence of

gravito-dyons in current grand unified and supersymmetric gauge theories associated with

gravito-dyons.


Section 3.2, briefly describes the gauge symmetry of dual electromagnetism. Equation

142


(3.1) is the free Lagrangian of Dirac field in presence of dual magnetic charge. The usual

gamma matrices satisfy the property given by equation (3.2). The Dirac Lagrangian (3.1)

is invariant under the global gauge transformation (3.3). In the case of electromagnetism,

this symmetry (3.3) is invariant under local gauge transformation (3.4), so that the La-

grangian (3.1) transforms as equation (3.5). In equation (3.5) the extra term gγµ∂µα looks

like a gauge transformation of potential, which may couple the gauge field Bµ with ψ,

so that the Lagrangian (3.1) has the local gauge symmetry. Then the Dirac Lagrangian

(3.1) is invariant under the combined gauge transformation (3.8) and the transformation

of covariant derivative (3.7) is described by equation (3.9). Equation (3.10) is the La-

grangian for the case of total dual quantum electrodynamics. The Jacobi identity is defined

by equation (3.13), while the Bianchi identity is expressed by equation (3.14), which is

the kinematical statement equivalent to Fµν,ν=∂νFµν = 0 for dual electrodynamics analo-

gous to the kinematical statement ∂νF µν = 0 for usual electrodynamics. As such, the dual

symmetry of electrodynamics requires the existence of dual electric charge (i.e magnetic

monopole). Likewise, in section 3.3 we have extended the gauge symmetry to dual gravito

dynamics. The free Dirac Lagrangian of gravi-Heavisidian field in presence of dual Heav-

isidian monopole is given by equation (3.15), which is invariant under the global gauge

transformation (3.16) and also elevates this symmetry to invariance under local gauge

transformation (3.17). Under the transformation (3.17), the Lagrangian (3.15) transforms

as equation (3.18), in which the extra term hγµ∂µβ looks like a gauge transformation of

potential, which may couple the gauge field Dµ with χ, so that the Lagrangian (3.15)

has the local gauge symmetry. Lagrangian (3.15) is invariant under the combined gauge

transformation (3.20). The Lagrangian for total dual gravi-Heavisidian field is expressed

by equation (3.22). In gravi-Heavisidian field the Jacobi identity is described by equation

(3.25), whereas the Bianchi identity is expressed by equation (3.26), which is equivalent

to Nµν,ν=∂νNµν = 0 for dual gravi-Heavisidian fields analogous to the kinematical state-

ment ∂ν fµν = 0 for usual gravi-Heavisidian field. It is shown that the dual symmetry of

gravi-Heavisidian requires the postulation of existence of dual Heavisidian (gravi-magnetic

charge) (i.e. Heavisidian monopole). Equation (3.27) describes the electric and magnetic

fields in terms of electric and magnetic four-potentials Aµ and Bµ. The generalized

143


anti-symmetric dual invariant electromagnetic field tensors of dyons are described by equa-

tion (3.28), whose components are expressed as the components of generalized electro-

magnetic field tensor by equation (3.28). Equation (3.30) describe the covariant form

of dual symmetric generalized Dirac-Maxwell’s equation. The Lagrangian density (3.31)

follows the minimum action principle for generalized electromagnetic fields of dyons and

leads to the equation (3.32) which is the Lorentz force equation of motion for dyons. On

the other hand, the dual parts of field tensors giving rise ∂νF µν = 0 and ∂νFµν = 0 de-

scribe the Bianchi identities like equation (3.14) for electric and magnetic charges. Equa-

tion (3.33) is the four-currents of electric and magnetic charges and equation (3.34) is

the dual invariant Lorentz force of dyon. The gravitational (gravi-electric) field−→G and

Heavisidian (gravi-magnetic) field−→H are expressed by equation (3.35) in terms of gravita-

tional and Heavisidian four-potential Cµ and Dµ. Equation (3.36) are the generalized

anti-symmetric dual invariant gravi-Heavisidian fields tensors for gravito-dyons and equa-

tion (3.37) are the components of generalized field tensors. The covariant form of dual

symmetric generalized Dirac-Maxwell’s equation of gravito-Heavisidian fields has been de-

scribed by equation (3.38). The Lagrangian (3.39) follows the minimum action principle

for generalized gravi-Heavisidian fields of gravito-dyons, which yields the GDM field equa-

tion of gravito-dyons represented by equation (3.38) and yields the equation (3.40) which

is the Lorentz force equation of motion for gravito-dyons. The dual parts of field tensors

giving rise ∂ν fµν = 0 and ∂νN µν = 0 describe the Bianchi identities like equation (3.26) for

gravitational (gravi-electric) and Heavisidian (gravi-magnetic) charges. Equation (3.41)

gives rise the gravitational and Heavisidian four-currents in terms of charge and velocity

of gravi-Heavisidian fields and thus provides equation (3.42) which is the dual invariant

Lorentz force equation for gravito-dyon. Equation (3.43) describes the combination of

two Dirac spinors which Ψ1 and Ψ2 which acts on the Dirac equation. Here Ψ1 is iden-

tified as the Dirac spinor for a electric charge ( like electron) while the other spinor Ψ2

has been described as the Dirac iso-spinors acting on the magnetic monopole so that the

resultant Ψ may be visualized as the bi-spinor for dyons in terms of its electric and mag-

netic counterparts. These two component spinors satisfy the free particle Dirac equation

(3.44). The Unitary transformations taking part for the invariance of free particle Dirac

144


equation for bi-spinor Ψ are the global U = U (e) (1) × U (g) (1) in terms of two component

spinors Ψ1 and Ψ2. Here it should be noted that the Ψ1 acts on unitary gauge group U (e) (1)

whereas the iso-spinor Ψ2 acts on the other unitary group U (g) (1) with the symbols (e)

and (g) are respectively used for the electric and magnetic charges. Thus equation (3.44)

is invariant under global gauge transformation (3.45). Equation (3.45) and (3.47) are

independent of space and time for global gauge transformations. If we elevate this sym-

metry to invariance under local gauge transformation where Λj (x) (∀j = 1, 2) in equation

(3.45) and (3.47), the Lagrangian (3.44) transforms as equation (3.49), in which partial

derivative ∂µ has been replaced by the covariant derivative Dµ in equation (3.50). The

gauge potentials (3.51) are associated with the dynamics of electric and magnetic charges

and related with the gauge transformation (3.52). Equation (3.56) is the commutation

relation of the covariant derivative of dyons, which leads the Jacobi identity (3.57) along

with the Bianchi identity (3.58). Equation (3.59) describes the total Lagrangian for gener-

alized fields of dyons, where equation (3.60) and (3.61) are the four-currents associated

respectively with electric and magnetic charges on dyons. These electric and magnetic four-

current obtained from the Dirac spinor Ψ1 and the Dirac iso-spinor Ψ2 satisfy the conserved

relations (3.62) and (3.63). Equation (3.64) describes the electric and magnetic gauges

corresponding to orthogonal transformation in group space, with the use of constancy con-

dition (3.65). Likewise, the Lagrangian density (3.67) reproduces the dual symmetric and


force equation (3.32) of motion for two potential theory of dyons. Equation (3.69) repro-

duces the Bianchi identity for generalized electromagnetic field tensors Fµν and Mµν . As

such, the classical theory of dyons has been verified and it is shown that the incorporation

of two four-potentials in generalized electromagnetic fields of dyons justifies the interac-

tion of two photons of U (1) × U (1) gauge theory where the first unitary Abelian gauge

group U (e) (1) acts on the Dirac spinors leading to the ordinary photon due to the pres-

ence of the electric charge while second unitary abelian gauge group U (g) (1) acts on the

Dirac iso-spinors and provides the existence of another photon due to the presence of the

magnetic charge on dyons. The activation of gauge group U (g) (1) on Dirac iso-spinor is

advantageous so that it may further be extended to enlarge the gauge group to describe

145


the non-Abelian correspondence of monopoles (dyons) in current grand unified and super-

symmetric gauge theories and the physics beyond the standard model. Similar to the case

of dyon discussed in section 3.6, we have reformulated the U (1) × U (1) gauge formula-

tion of gravito-dyons in section 3.7. Equation (3.70) is the four spinor Dirac field χ for

gravito-dyons in terms of two component spinors χ1 and χ2. In equation (3.70) χ1 is iden-

tified as the Dirac spinor for a gravitational (gravi-electric) charge while the other spinor

χ2 has been identified as the Dirac iso-spinors acting on the Heavisidian (gravi-magnetic)

monopole. Thus the Dirac spinor χ has been described as the bi-spinor for gravito-dyons in

terms of its gravitational and Heavisidian counterparts. The two spinors χ1 and χ2 satisfy

the free particle Dirac equation of gravito-dyons as equation (3.70). The Unitary transfor-

mations taking part for the invariance of free particle Dirac equation for bi-spinor χ are

the global U = U (m) (1) × U (h) (1) two component spinors χ1 and χ2 . In this case χ1 acts

on unitary gauge group U (m) (1) whereas the iso-spinor χ2 acts on the other unitary group

U (h) (1) with the symbols (m) and (h) are used for the gravitational (gravi-electric) and

Heavisidian (gravi-magnetic) charges. The free particle Lagrangian (3.71) of gravito-dyons

is invariant under global gauge transformation (3.72). Equation (3.73) describes the spinor

transformations of gravitational (gravi-electric) and Heavisidian (gravi-magnetic) spinors.

Equation (3.72) is independent of space and time for global gauge transformations. If we

elevate this symmetry to invariance under local gauge transformations where Θj(∀j = 1, 2)

in equations (3.47) and (3.72), the Lagrangian (3.71) of gravito-Heavisidian monopole

transforms as equation (3.74), in which partial derivative ∂µ is replaced by the covari-

ant derivative Dµ as (3.75). It is shown that the whole theory is investigated in terms of

two gauge potentials (3.76) respectively associated with the dynamics of gravitational and

Heavisidian charges with the gauge transformation (3.77). As such the commutation rela-

tion (3.81) of the covariant derivative of gravito-dyons leads to the Jacobi identity (3.82)

along with the Bianchi identity (3.83) of gravito-Heavisidian fields. The total Lagrangian

for generalized fields of gravito-dyons has been described by equation (3.84), whereas the

equations (3.85) and (3.86) are the four currents associated respectively with gravitational

(gravi-electric) and Heavisidian (gravi-magnetic) charges on gravito-dyons. These gravi-

tational and Heavisidian four-current have been obtained from the Dirac spinors χ1 and

146


χ2 and satisfy the conserved relations (3.87) and (3.88). The gravitational (gravi-electric)

and Heavisidian (gravi-magnetic) dynamical variables, lead to the duality transformations

between the gravitational (gravi-electric) and Heavisidian (gravi-magnetic) gauges corre-

sponding to orthogonal transformations in group space (3.89) with the use of constancy

condition (3.90) for gravi-Heavisidian fields. Equation (3.91) is the generalized dual in-

variant gravi-Heavisidian fields of gravito-dyons, which satisfies independently the Bianchi

identity (3.83). Then it is shown that the Lagrangian density (3.92) reproduces the dual

symmetric and Lorentz covariant and dual invariant generalized Dirac Maxwell’s (GDM)

field equations (3.38) and Lorentz force equation (3.40) of motion for two potential theory

of gravito-dyons. Thus, with the use of Jacobi identity (3.82), the Bianchi identity for gen-

eralized gravi-Heavisidian field tensors fµν and Nµν as equation (3.94) has been retained.

Finally, the classical theory of gravito-dyons has been verified and the incorporation of two

four potentials in terms of two photon theory of generalized gravito-Heavisidian fields of

gravito-dyons has been justified in the frame work of U (1)× U (1) gauge theory where the

first unitary Abelian gauge group U (m) (1) acts on the Dirac spinors leads to gravito-photons

due to the presence of the gravitational (gravi-electric) charge while second unitary Abelian

gauge group U (h) (1) acts on the Dirac iso-spinors gives the existence of Heavisidian photons

to the presence of the Heavisidian (gravi-magnetic) charge on gravito-dyons. The activa-

tion of gauge group U (h) (1) on Dirac iso-spinor is important and advantageous so that it

may further be studied to look the role of Heavisidian monopole in the non-Abelian gauge

theories and physics beyond the standard model since Heavisidian monopoles are already

described as so heavy and most interacting form of matter.

147

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151

Chapter 4

GENERALIZATION OF SCHWINGER

ZWANZIGER DYON TO QUATERNION

Chapter 4 Generalization of Schwinger Zwanziger Dyon to Quaternion

'

&

$

%

ABSTRACT

Postulating the existence of magnetic monopole in electromagnetism and Heavisidian

monopoles in gravitational interactions, a unified theory of gravi - electromagnetism has been

developed on generalizing the Schwinger - Zwanziger formulation of dyon to quaternion in

simple and consistent manner. Starting with the four Lorentz like forces on different charges,

we have generalized the Schwinger - Zwanziger quantization parameters in order to obtain

the angular momentum for unified fields of dyons and gravito - dyons (i.e. Gravi - electro-

magnetism). The octonion covariant derivative has been discussed as the gauge covariant

derivative of generalized fields of dyons. The generalized four-potential, current and fields of

gravito-dyons in terms of split octonion variable, the U (1) abelian and SU (2) non-Abelian

gauge structure of dyons and gravito-dyons are also discussed consistently. It is shown that the

generalized four-current is not conserved but only the Noetherian four-current is considered

to be conserved one. It is concluded that the presence analysis reproduces the theory of elec-

tric (gravitational) charge (mass) in the absence of magnetic (Heavisidian) charge (mass) on

dyons (gravito-dyons) or vice versa.

153


4.1 Introduction

The question of existence of monopole [1, 2] and dyons [3, 4, 5, 6, 7] has become

a challenging new frontier and the object of more interest in High Energy Physics. Dirac

showed [1, 2] that the quantum mechanics of an electrically charged particle of charge e

and a magnetically charged particle of charge g is consistent only if eg = 2π n, n being an

integer. Schwinger - Zwanziger [3]-[7] generalized this condition to allow for the possibil-

ity of particles (dyons) that carry both electric and magnetic charge. A quantum mechanical

theory can have two particles of electric and magnetic charges (e1, g1) and (e2, g2) only if

e1g2 − e2g1 = 2π n. The angular momentum in the field of the two particle system can be

calculated readily with the magnitude e1g2−e2g14πc

. This has an integer or half - integer value,

as expected in quantum mechanics, only if e1g2 − e2g1 = 2πnc. The fresh interests in this

subject have been enhanced by ’t Hooft - Polyakov [8, 9] with the idea that the classical

solutions having the properties of magnetic monopoles may be found in Yang - Mills gauge

theories. Julia and Zee [10, 11] extended the ’ t Hooft-Polyakov theory [8, 9] of monopoles

and constructed the theory of non - Abelian dyons. The quantum mechanical excitation

of fundamental monopoles include dyons which are automatically arisen from the semi-

classical quantization of global charge rotation degree of freedom of monopoles. In view of

the explanation of CP - violation in terms of non-zero vacuum angle of world [12, 13], the

monopoles are necessary dyons and Dirac quantization condition permits dyons to have

analogous electric charge. Accordingly, a self consistent and manifestly covariant theory

has been developed [14, 15] for the generalized electromagnetic fields of dyons.

On the other hand, the analogy between linear gravitational and electromagnetic fields

leads to the asymmetry in Einstein’s linear equation of gravity and suggests the existence

of gravitational analogue of magnetic monopole [16, 17, 18, 19]. Cattani [16] has also

derived the covariant field equations for linear gravitation like Maxwell’s equations on in-

troducing a new (called the Heavisidian ) field ( like magnetic field ) which depends upon

the velocity of gravitational charges (masses). Avoiding the use of arbitrary string vari-

able, the manifestly covariant and consistent theory of gravito-dyons has also been devel-

oped [20, 21] in terms of two four-potentials leading to the structural symmetry between

154


generalized electromagnetic fields of dyons and generalized gravito - Heavisidian fields of

gravito-dyons. Extending this recently, a consistent theory for the dynamics of four charges

(masses) (namely electric, magnetic, gravitational, Heavisidian) have also been formulated

[22, 23] in simple, compact and consistent manner. Considering an invariant Lagrangian

density and its quaternionic representation, the consistent field equations for the dynamics

of four charges have already been derived [24] and it has been shown that the present

reformulation reproduces the dynamics of individual charges (masses) in the absence of

other charge (masses) as well as the generalized theory of dyons (gravito-dyons) in the

absence gravito-dyons (dyons).

Keeping above facts in mind and the interest on Gravi-electromagnetism, as well as

postulation of the existence of magnetic monopole in electromagnetism and Heavisidian

monopoles in gravitational interactions, in this chapter [25], a unified theory of gravi -

electromagnetism has been developed on generalizing the Schwinger - Zwanziger formu-

lation of dyon to quaternion in simple and consistent manner. This chapter comprises six

sections. In section 4.2 we have undertaken the study of Schwinger - Zwanziger dyon in

electromagnetic field. Starting with the force experienced by an electric charge, the gen-

eralized Dirac Maxwell’s equations are obtained and shown to be manifestly covariant and

dual invariant. It is shown that the angular momentum of dyons lead to the chirality quan-

tization condition as the extension of Dirac quantization condition. Similar techniques are

applied in section (4.3) where we have undertaken consistently study of dual invariant and

manifestly covariant theory of Schwinger - Zwanziger dyon in Gravito-Heavisidian fields.

Starting with the four Lorentz like forces on different charges, in section 4.4 we have gen-

eralized the Schwinger - Zwanziger quantization parameters as quaternion valued in order

to obtain the angular momentum for unified fields of dyons and gravito-dyons. Taking

the unified charge as quaternion, in section 4.5 we have reformulated manifestly covariant

and consistent theory for the dynamics of four charges namely electric, magnetic, gravita-

tional and Heavisidian associated with gravi-electromagnetism. It has been shown that the

combined theory describes the interaction of particles in terms of four coupling parameters

which we name as the different chirality parameters associated to electric, magnetic, gravi-

tational, and Heavisidian charges. On applying the various quaternion conjugations, it has

155


also been emphasized that the combined theory of gravitation and electromagnetism repro-

duces the dynamics of generalized electromagnetic (gravito - Heavisidian) fields of dyons

(gravito-dyons) in the absence generalized gravito Heavisidian (electromagnetic) fields of

gravito-dyons( dyons) or vice versa. Section 4.6 describes the octonionic gauge theory of

dyons and it has been shown that the octonion gauge formalism may hope for the better

understanding of these two type of forces of dyons in non-Abelian and abelian limits. Oc-

tonion formalism has been shown to be simple, compact, consistent and unique one. It

reproduces the theories of electromagnetism and gravitation separately in the absence of

each other. The O- derivative, reduces to partial derivative in the absence of split octonion.

Split octonions are described in terms of U (1)×SU (2) gauge group simultaneously to give

rise the abelian (point like) and non - Abelian (extended structure) of dyons. This gauge

group plays the role of U (1)×SU (2) Salam Weinberg theory of electro-weak interaction in

the absence of Heavisidian gravity and taking the magnetic charge on dyons vanishing. It

has been shown that the enlarged gauge group U (1)× SU (2) explains the built in duality

to reproduce abelian and non - Abelian gauge structure of dyons. Similarly in section 4.7,

the same techniques of octonion gauge formalism are applied to the case of gravito-dyons

with the same consideration.

4.2 Schwinger - Zwanziger dyons in Electromagnetic fields

Let us consider the case of a Schwinger - Zwanziger dyon which is described as a particle

of mass M carrying simultaneously the electric charge e1 and magnetic charge g1. Let this

particle moves with velocity −→v so that the force−→Fe experienced by an electric charge e1 is

described as,

−→Fe = M d−→v

dt= e1

[−→E +−→v ×

−→H]

; (4.1)

where−→E and

−→H are respectively the electric and magnetic fields and let us take throughout

the notation the system of natural units for which c = = 1. Accordingly the force−→Fg ex-

perienced by a magnetic monopole can be written by applying the duality transformations

156


given in Chapter 1 as equation (1.2) i.e.−→E −→

−→H ,−→H −→ −

−→E and e1 −→ g1 as,

−→Fg = g1

[−→H −−→v ×

−→E]

; (4.2)

Schwinger - Zwanziger [3] - [7] generalized equations (4.1) and (4.2) as the equation of

motion for the force experienced by a particle (dyon) carrying simultaneous existence of

electric and magnetic charges as

−→F = M

d−→vdt

=−→Fe +

−→Fg = e1

[−→E +−→v ×

−→H]

+ g1

[−→H −−→v ×

−→E]

; (4.3)

where we may write the following forms of the electric and magnetic field strengths at the

point with −→r of magnitude r,

−→E = e2

−→rr3, and

−→H = g2

−→rr3

; (4.4)

in which the electric e2 and magnetic g2 charges for a stationary body are located at origin.

Substituting−→E and

−→H from equation (4.4) into the equation (4.3), we get the following

expression for equation of motion for Schwinger - Zwanziger [3] - [7] dyon as

−→F = M

d−→vdt

= α12

−→rr3

+ β12(−→v ×−→r )

r3; (4.5)

where

α12 =(e1e2 + g1g2) and β12 = (e1g2 − g1e2); (4.6)

are respectively known as electric and magnetic coupling parameters of dyons. Taking the

vector product of equation (4.5) with −→r , we get

157


−→r ×Md−→vdt

= α12

−→(r ×

−→r)

r3+ β12

−→r × (−→v ×−→r )

r3; (4.7)

and using vector multiplication rule−→(r×−→r) = 0 and d

dt(−→r ×M−→v ) = −→r ×M d−→v

dt, the equation

(4.7) reduces to ,

d

dt(−→r ×M−→v ) = β12

−→r × (−→v ×−→r )

r3= β12

d

dt(−→rr

); (4.8)

where we have used the identity−→r ×(−→v ×−→r )

r3= d

dt(−→rr

). So that we may define following

expression for conserved angular momentum for dyon as

−→J =−→r ×M−→v − β12

−→rr

; (4.9)

which gives rise the component of the angular momentum−→J along the direction of −→r as

Jr = r ·−→J = β12r ·

−→rr

= β12; (4.10)

where r is the unit vector along the vector −→r and we have used−→(r ×

−→r) = 0. Thus the

quantization of the component of the angular momentum Jr along the line of particle leads

to Schwinger - Zwanziger [3] - [7] charge quantization ( or chirality quantization) in the

units of Plank’s constant

β12 =(e1g2 − g1e2) = ν; (4.11)

where ν may be an integer or half integer. But in most cases, it has been taken as integer

as its half integral values were already excluded by Dirac in his seminal paper [1, 2]. On

substituting e1 = e; g1 = 0; e2 = 0 and g2 = g in equation (4.11) for interaction of two

dyons with charges (e, 0) and (0, g), we get the Dirac quantization condition [1, 2]

158


eg = ν; (4.12)

in the units of Planck constant. It is clear that if we do not consider dyon Dirac quantization

condition is not dual invariant.

4.3 Schwinger - Zwanziger dyons in gravito - Heavisidian

fields

Analogy between electromagnetic and linear gravitational field equations suggests [16]

- [19] the structural symmetry between these two forces of nature. Accordingly, on postu-

lating the existence of Heavisidian monopole [26] - [29] leads the symmetry between the

linear equations of gravito-Heavisidian fields and the generalized Maxwell’s Dirac equa-

tions in electromagnetic Fields. Likewise, generalized Maxwell’s Dirac equations (1.33),

we may write the the following form [20] - [23] and [26] - [29] of linear equations for

generalized gravito - Heavisidian fields in presence of Heavisidian monopole as in equation

(2.30). Similarly, the force−→Fm the force experienced by a gravitational (gravi - electric)

charge (mass) m1 may then be expressed as in equation (2.33). Accordingly GDM type

equations of gravito - Heavisidian fields are invariant not only under Lorentz and confor-

mal transformations but also invariant under the following duality transformations between

gravitational (gravi-electric) G and Heavisidian (gravi-magnetic)M quantities i.e

G =⇒G cosϑ+M sinϑ;

M =⇒M cosϑ− G sinϑ; (4.13)

where G =(m1,−→G , ρg,

−→jg

)andM =

(h1,−→H , ρh,

−→jh

)with h1 denotes the Heavisidian (gravi

- magnetic) charge (mass). For a particular value of ϑ = π2, equations (4.13) reduces to

159


G 7−→M M 7−→ −G. (4.14)

So, on applying the duality symmetry on gravitational and Heavisidian fields and charges

(masses) like the electromagnetism, we may write the net force experienced by Heavisidian

(gravi - magnetic) charge (mass) h1 as in equation (2.34), Following Schwinger - Zwanziger

[3] - [7], here also we may adopt the same process for gravito-dyons. Hence the gravito-

dyons are considered as the particles carrying simultaneously the existence of gravitational

(gravi - electric) and Heavisidian (gravi - magnetic) charges (masses). Thus, we may write

the net force−→F acting on gravito - dyon [26] - [29] as in equation (2.35). Where following

expressions for stationary gravitational (gravi - electric) and Heavisidian (gravi - magnetic)

field strengths at the point with −→r of magnitude r may be used with gravitational charge

(mass) m2 and Heavisidian charge (mass) h2 as in equation (2.36). Substituting−→G and

−→H

from equation (2.36) into the equation (2.35), we get the following expression for equation

of motion for Schwinger - Zwanziger gravito-dyon as in equation (2.37), where

γ12 =(m1m2 + h1h2) and δ12 = (m1h2 − h1m2); (4.15)

are respectively known as and gravitational (gravi - electric) and Heavisidian (gravi - mag-

netic) coupling parameters of gravito-dyons. Taking the vector product of equation (2.37)

with −→r , we get

−→r × (m1 + h1)d−→vdt

= γ12

−→(r ×

−→r)

r3+ δ12

−→r × (−→v ×−→r )

r3. (4.16)

Using vector multiplication rule−→(r ×

−→r) = 0 and d

dt[−→r × (m1 + h1)

−→v ] = −→r × (m1 + h1)d−→vdt

,

the equation (4.16) reduces to ,

d

dt[−→r × (m1 + h1)

−→v ] = δ12

−→r × (−→v ×−→r )

r3= δ12

d

dt(−→rr

); (4.17)

where−→r ×(−→v ×−→r )

r3= d

dt(−→rr

). As such, we may write the following expression for conserved

160


angular momentum for gravito - dyon as in equation (2.39) which gives rise the component

of the angular momentum−→J along the direction of −→r as

Jr = r ·−→J = δ12r ·

−→rr

= δ12; (4.18)

where r is the unit vector along the vector −→r and−→(r ×

−→r) = 0. Thus the quantization of

the component of the angular momentum Jr along the line of particle leads to Schwinger

- Zwanziger charge (mass) quantization ( or chirality quantization) condition for gravito-

dyons in the units of Plank’s constant

δ12 =(m1h2 − h1m2) = n; (4.19)

where n may be an integer or half integer as may be the case of gravito-dyons in gravito-

Heavisidian fields. On substituting m1 = m; h1 = 0; m2 = 0 and h2 = h in equation (4.19)

for interaction of two gravito-dyons with charges (masses) (m, 0) and (0, h), we get the

Dirac quantization condition [1],[2] for gravito-dyons as in equation (2.41). Which can be

dual invariant only if we consider the case of gravito-dyons in linear gravitational fields.

4.4 Generalization of Schwinger - Zwanziger dyons to Quater-

nions

Let us assume that a particle of mass M carries simultaneous existence of four charges

namely electric (e1), magnetic (g1), gravitational (gravi - electric) (m1) and Heavisidian

(gravi - magnetic) (h1). Let this particle moves with velocity −→v so that it experiences a

combined force which is the sum of the forces exerted independently due to individual

charges i.e.

−→F =

−→Fe +

−→Fg +

−→Fm +

−→Fh; (4.20)

161


where−→Fe ,−→Fg,−→Fg and

−→Fh are respectively given by equations (4.1), (4.2), (2.33) and (2.34).

Now substituting the values of electric (−→E ), magnetic (

−→H ), gravitational (gravi-electric)

(−→G) and Heavisidian (gravi-magnetic) (

−→H ) field strengths given by equations (4.4), and

(4.5) into equation (4.20), we get

−→F =W12

−→rr3

+ (X12 + Y12 + Z12)−→v ×

−→rr3

; (4.21)

where

W12 =(e1e2 + g1g2 +m1m2 + h1h2); (4.22)

X12 =(e1g2 − e2g1 +m1h2 −m2h1); (4.23)

Y12 =(e1m2 −m1e2 − h2g1 + h1g2); (4.24)

Z12 =(e1h2 − h1e2 + g1m2 −m1g2); (4.25)

are different four coupling parameters. W12, X12, Y12,Z12 may also be identified as electric,

magnetic, gravitational and Heavisidian parameters [22, 23]. According to the Newton’s

second law−→F = M d−→v

dt, equation (4.21) is written as

Md−→vdt

= W12

−→rr3

+ (X12 + Y12 + Z12)−→v ×

−→rr3. (4.26)

Taking the vector product of equation (4.26) with −→r , we get

−→r ×Md−→vdt

= W12

−→(r ×

−→r)

r3+ (X12 + Y12 + Z12)

−→r × (−→v ×−→r )

r3. (4.27)

Using vector multiplication rule−→(r ×

−→r) = 0 , d(−→r ×M

−→v)

dt= −→r × md−→v

dt, and the identity

−→r ×(−→v ×−→r )r3

= ddt

(−→rr

) , equation (4.27) reduces to,

162


d

dt(−→r ×M−→v ) = (X12 + Y12 + Z12)

d

dt(−→rr

). (4.28)

Thus by adopting the above procedure, we get the following expression for angular mo-

mentum−→J as

−→J = −→r ×M −→v − (X12 + Y12 + Z12)(

−→rr

); (4.29)

which gives rise the component of the angular momentum−→J along the direction of −→r as,

Jr = (X12 + Y12 + Z12). (4.30)

This is called the residual component of unified angular momentum and leads the following

form of generalized Schwinger - Zwanziger quantization condition i.e.

Jr = (X12 + Y12 + Z12) = n~; (4.31)

where n is an integer and h is Plank’s constant.

4.5 Quaternion Formulation of gravito - Heavisidian fields

A unified theory of generalized electromagnetic and Heavisidian fields may then be de-

veloped consistently by generalizing Schwinger - Zwanziger dyon to a quaternion possess-

ing a quartet (e, g, m, h) of four charges. Quaternion (e, g, m, h) charge thus represents the

theory of gravi - electromagnetism for the particles carrying simultaneously electric, mag-

netic, gravitational and Heavisidian charges. So, let us generalize two types of Schwinger

- Zwanziger dyonic charges (e, g) and (m, h) to a quaternion charge of gravi - electromag-

netic fields as [20, 21] as in equation (2.49), The quaternion conjugation of equation (2.49)

is defined as

163


Q = (e+ ig) + (m+ ih)j =e+ ig + jm+ kh. (4.32)

Thus, on using the quaternion multiplication rule (2.51), the interaction of between two

quaternions (a and b) with charges Qa = (ea, ga,ma, ha) = ea − iga − jma − kha and Qb =

(eb, gb,mb, hb) = eb − igb − jmb − khb leads [22, 23] to

QaQb = (ea + iga + jma + kha) (eb − igb − jmb − khb)

=αab + βab + γab + δab; (4.33)

where

αab =(eaeb + gagb +mamb + hahb); (4.34)

βab =(eagb − ebga +mahb −mbha); (4.35)

γab =(eamb −maeb − hbga + hagb); (4.36)

δab =(eahb − haeb + gamb −magb). (4.37)

Hence, for a = 1 and b = 2, equations (4.34 - 4.37) are same as equations (4.22 - 4.25)

for four different chirality parameters i.e α12 = W12, β12 =X12, γ12 =Y12, and δ12 = Z12.

So, the equations (4.22) and (4.23) immediately reduces to W12 = e1e2 + g1g2 and X12 =

(e1g2 − e2g1) for the interaction of two Schwinger - Zwanziger dyons from two quaternion

charges (e1, g1, 0, 0) and (e2, g2, 0, 0) with the vanishing of other parameters. Similarly, for

the Schwinger - Zwanziger case for interaction of gravito-dyons i.e. gravitational charge

and Heavisidian monopole, we get W12 = m1m2 + h1h2 and X12 = m1h2 − h1m2 from

quaternions (0, 0,m1, h1) and (0, 0,m2, h2). Similarly, we may be speculate a new kind

of dyon i.e. electric charge and Heavisidian monopole obtained from a quaternion like

(e, 0, 0, h) where we may get W12 = e1e2 + h1h2 and Z12 = e1h2 − h1e2. Similarly a new

kind of dyon i.e., electric charge and gravitational charge may also be speculated from the

164


quaternion like (e, 0,m, 0) for which we get W12 = e1e2 + m1m2 and Y12 = e1m2 − m1e2.

Also there are the possibilities of other dyons like magnetic charge and gravitational charge

from a quaternion (0, g,m, 0) for which W12 = m1m2 + g1g2 and Z12 = g1m2 − m1g2 as

well as for the purely hypothetical dyons like magnetic and Heavisidian monopoles we

get W12 = h1h2 + g1g2 and Y12 = h1g2 − g1h2 . As such, the quaternion generalization of

Schwinger - Zwanziger quantization condition of dyons extends the possibilities of six types

of dyons like (e, g), (m, h), (e, h), (e,m), (g, m) and (g, h). Let us try to obtain the six kinds

of Schwinger - Zwanziger dyons from a quaternion (2.49).

• Applying the j−conjugation i.e. i → i , j → −j , k → −k as k = ij on quaternion

(2.49), we get,

qj = e− ig + jm+ kh; (4.38)

and adding this equations (4.38) to quaternion (2.49), we get

e− ig =1

2(q + qj); (4.39)

which refers to generalized charge (as complex quantity) as the order pair of (e, g) for

Schwinger - Zwanziger dyon moving in electromagnetic fields.

• Similarly, on subtracting equation (4.38) from equation (2.49), we get

m− ih =1

2j(q − qj); (4.40)

which refers to generalized charge (as complex quantity) as the order pair of (m, h) for

Schwinger - Zwanziger dyon moving in gravito - Heavisidian fields.

165


• Now, applying i−conjugation i.e i → −i , j → j , k → −k as k = ij on quaternion

(2.49), we get

qi = e+ ig − jm+ kh; (4.41)

and adding the this equations (4.41) to (2.49) we find

e− jm =1

2(q + qj); (4.42)

which defines the generalized charge (as complex quantity) as the order pair of (e, m) for

of a Schwinger - Zwanziger dyon obtained from electric and gravitational charges.

• Furthermore, on subtracting equation (4.41) from equation (2.49), we get

(g − jh) =1

2 i(q − qi); (4.43)

which describes the generalized charge (as complex quantity) as the order pair of (g, h) for

of a Schwinger - Zwanziger dyon obtained from magnetic and Heavisidian charges.

• Applying the transformation for i→ −i, j → j, k → −k in equation (2.49), we get,

qk = e+ ig + jm− kh; (4.44)

166


and adding equations (2.49) and (4.44), we get

e− kh =1

2(q + qk); (4.45)

which gives the generalized charge (as complex quantity) as the order pair of (e, h) for of

a Schwinger - Zwanziger dyon obtained from electric and Heavisidian charges.

• Similarly on subtracting the equation (4.44) from equation (2.49), we find,

q − qk = −2(ig + jm); (4.46)

and applying the quaternion property j = −ik, the equation (4.46) reduces to ,

(g − km) =i(q − qk)

2; (4.47)

which describes the generalized charge (as complex quantity) as the order pair of (g, m)

for of a Schwinger - Zwanziger dyon obtained from magnetic and gravitational charges.

4.6 Octonion gauge theory of dyons

Let us consideration Pµi andQµi are defined as the electric and magnetic gauge potential

Pµi =eAµi;

Qµi =gBµi; . (4.48)

167


In order to reformulation the quantum equation of dyons by means of split octonion real-

ization, we write the O - derivative as follows,

Oqµ = O·µ+ [=µ,O] ; (4.49)

where we used

=µ =− Pµiu?i −Qµiui

=− Paµu?a −Qaµua

=− eAaµu?a − gBaµua

=

O2 eAµ · e

−gBµ · e O2

=− ea

(eAaµ + gBa

µ

)+ iea+3

(eAaµ − gBa

µ

)=− eaReal

(q?V a

µ

)+ iea+3Real

(qV a

µ

); (4.50)

then we have

Oqµ =[−eaRe

(q?V a

µ

)+ iea+3Re

(qV a

µ

),O]

=O·µ − ea[−Re

(q?V a

µ

),O]

+[iea+3Re

(qV a

µ

),O]

; (4.51)

where q? is the complex conjugate of generalized charge q of dyons and Vµ is defined as

gauge potential in SU (2) non - Abelian gauge space. The gauge potential Vµ and gauge

field strength Gµν of dyons may be expressed in terms of split octonion as

168


Vµ =Vµu0 + V aµ u

a + Vµu?0 + V a

µ ua?

=

Vµe0 −V a

µ ea

V aµ e

a Vµe0

; (4.52)

and

Gµν =Gµνu0 + Gµνu?0 + εaµνua +Ha

µνu?a

=

Gµνe0 −εaµνea

Haµνe

a Gµνe0

; (4.53)

where Gµν is defined as the generalized field tensor for dyon and defined as

Gµν = Gµνe0 +

1

i

a=3∑a=1

Gaµνe

a ∀ (a = 1, 2, 3) . (4.54)

In equation (4.52), (4.53) and (4.54), we have expressed the generalized four - potential Vµ

and generalized field tensor Gµν of dyons in terms of abelian U (1) and non - Abelian SU (2)

gauge coupling strengths where the real quaternion unit e0 is associated with U (1) abelian

gauge and the pure imaginary unit quaternion is related with the SU (2) non - Abelian

Yang-Mill’s field. Equation is invariant under local and global phase transformation. The

non - Abelian SU (2) Yang - Mill’s gauge field strength Gaµν may be expressed in

Gaµν =Gaµν + q?εabcVbνVcµ;

=∂νVaµ − ∂µVaν + q?εabcVbνVcµ. (4.55)

These equation yields the correct field equation for non - Abelian gauge theory of dyons

and rise to its extended structure. Hence the O−derivative of generalized field tensor Gµν

gives by the equation of dyons leads the following field equation i.e.

169


Gµνqν =Gµν,ν + [=ν , Gµν ]

=Jµ (u0 + u?0) + J aµ

(ua + ua

?)=Jµ; (4.56)

where Jµ is a four - current of dyons in split octonion realization. Then

Gµνqν =Gµν,ν + [Γν , Gµν ]

=Gµν,ν + iq? [Vν , Gµν ]

=Jµe0 +1

i

a=3∑a=1

J aµ e

a = Jµ. (4.57)

In deriving equation we have used equation . Thus the generalized four current of dyons

in split octonion realization leads to the abelian and non - Abelian nature of current in this

theory. The O−derivative of generalized four-current is given by

Jµ =Jµ (u0 + u?0) + J aµ

(ua + ua

?)=

Jµe0 −J aµ e

a

J aµ e

a Jµe0

; (4.58)

which gives to

Jµqµ =0; (4.59)

which shows that the resemblance with Noetherian current and describes the analogue of

continuity equation in abelian gauge theory.

170


4.7 Octonion and unified fields of gravito-dyons

The idea of Dowker and Roche [19] of dual mass playing the role of magnetic charge

(Heavisidian monopole), the gravito - dyons may also be expressed as the particle carrying

gravitational mass m and Heavisidian mass h having generalized mass q is given by in

equation (2.42). Let the space-time interval,

ds2 =1

ntr (ηµνdx

µdxν) ; (4.60)

where

ηµν =ηabµν (x)

(a, b = 1, 2, 3, ......, n) ; (4.61)

is a matrix of internal space. Then we can write ds2 as,

ds2 =1

4tr (ηµνdx

µdxν) =1

2gµνdx

µdxν ; (4.62)

where ′tr′ acting upon Zorn matrix and Gµν is given as,

Gµν =εµν0 +Hµν0; (4.63)

where εµν0 is the diagonal metric and Hµν0 is the non - diagonal metric. Both εµν0 and Hµν0

are non - symmetric. As such, the ds2 of equation may be considered as the line element in

curved space - time i.e.

171


ηµν =Gµν (u0 + u?0) + εµν .u? +Hµν .u

=

Gµνe0 εµν .e

Hµν .e Gµνe0

; (4.64)

where η+µν = ηµν , εµν and Hµν represents the Yang - Mill’s field strength. The superscript′+′ denotes the Hermitian conjugate in internal space. The Christoffel symbol (or non -

symmetric connection) may also be written in the form of split octonion realization as,

Γρµν = Γρµν (u0 + u?0) +δρµ (Pν .u? +Qν .u) ; (4.65)

where Pν .u? + Qν .u are the O - affinity of the theory. The four index Ricci tensor Rσρµν is

defined as,

Rσρµν =Rσ

ρµν (u?0 + u0) + δσρPµν ; (4.66)

where Pµν is the O - curvature (space - time curvature) and is expressed as,

Pµν ==µ,ν −=ν,µ − [=µ,=ν ] ;

= (PµiQνi − PνiQµi)u?0 + (QµiPνi −QνiPµi)u0

+ (Pνk,µ − Pµk,ν − 2εijkQµiQνj)u?k

+ (Qνk,µ −Qµk,ν − 2εijkPµiPνj)uk (i, j, k = 1, 2, 3) . (4.67)

TheO−space - time curvature in this theory obviously retains the usual form of Riemannian

space - time geometry. Thus we can express the generalized four - potential of gravito

- dyons by means of octonions gauge formulation given by equation. Similarly one can

construct the generalized field tensor of gravito - dyons in terms of split octonion as,

172


Gµν =Gµν (u0 + u?0) + εµν .u? +Hµν .u

=

Gµνeo −εµν .eHµν .e Gµνeo

; (4.68)

where

Gµν =Vµ,ν − Vν,µ;

Gaµν =V a

µ,ν − V aν,µ + q?εabcV

bν V

cµ ; (4.69)

which has the same form of equation for the case of non - Abelian gauge theory associated

with generalized electromagnetic fields. The forms of different gauge potentials associated

with gravito - dyons may be expressed as,

Vµ =Vµu0 + V aµ u

a + Vµu?0 + V a

µ ua?

=

Vµe0 −V a

µ ea

V aµ e

a Vµe0

; (4.70)

Aµ =Aµu0 + Aaµua + Aµu

?0 + Aaµu

a?

=

Aµe0 −Aaµea

Aaµea Aµe

0

; (4.71)

Bµ =Bµu0 +Baµu

a +Bµu?0 +Ba

µua?

=

Bµe0 −Ba

µea

Baµea Bµe

0

; (4.72)

173


Cµ =Cµu0 + Caµu

a + Cµu?0 + Ca

µua?

=

Cµe0 −Ca

µea

Caµe

a Cµe0

; (4.73)

Dµ =Dµu0 +Daµu

a +Dµu?0 +Da

µua?

=

Dµe0 −Da

µea

Daµea Dµe

0

. (4.74)

As such the O - derivative of generalized field tensor Gµν leads to,

Gµνqν =Gµν,ν + [=ν , Gµν ]

=− Jµ (u0 + u?0)− J aµ

(ua + ua

?);

Gµνqν =− Jµ; (4.75)

where Jµ is the generalized split octonion current associated with gravito - dyons and given

by equation (4.58). Hence like previous can also, the derivative of the generalized current

vanishes and demonstrating that, the conservation of generalized four - current follow the

continuity equation of abelian and non - Abelian gauge theories.


Starting from the Lorentz force of an electric charge as equation (4.1), applying the

condition of duality transformation in equation (4.1), we have obtained the Lorentz force

for magnetic charge (monopole) as equation (4.2), whereas equation (4.3) has been de-

rived as the Lorentz force equation for the force experienced by a particle (dyon) carrying

simultaneous existence of electric and magnetic charges. At the point −→r the electric and

magnetic field strengths is expressed by equation (4.4). Equation (4.5) is the equation of

motion for Schwinger - Zwanziger dyon in terms of electric and magnetic coupling param-

eters. The quantization of the component of the angular momentum Jr along the line of

174


particle leads to Schwinger - Zwanziger charge quantization ( or chirality quantization)

in the units of Plank’s constant as defined by equation (4.11). It should be noted that

Dirac quantization condition leads in dual invariance while the chirality quantization con-

dition (4.11) is dual invariant. Dual invariant chirality quantization condition reduces to

the Dirac quantization condition (4.12) on inserting e1 = e; g1 = 0; e2 = 0 and g2 = g in

equation (4.11) for interaction of two dyons with charges (e, 0) and (0, g). From the equa-

tion (4.12), it is clear that monopoles are dyons only when the dual invariance has been

taken into account. Analogy between the linear gravity and the electromagnetism in terms

of duality transformation leads to the existence of gravi-magnetic (Heavisidian) fields and

postulation of Heavisidian monopoles. Applying the duality symmetry on gravitational and

Heavisidian fields and charges (masses) like the electromagnetism, we have obtained the

Schwinger - Zwanziger condition for gravito-dyons. As such, the gravito-dyons are consid-

ered as the particles carrying simultaneously the existence of gravitational (gravi - electric)

and Heavisidian (gravi - magnetic) charges (masses). Accordingly, we have derived the

gravitational (gravi - electric) and Heavisidian (gravi - magnetic) coupling parameter as

equation (4.15). Equation (4.19) leads the quantization of the component of the angular

momentum Jr along the line of particle yields the dual invariant to Schwinger - Zwanziger

charge (mass) quantization ( or chirality quantization) condition for gravito-dyons in the

units of Plank’s constant. If we apply m1 = m; h1 = 0; m2 = 0 and h2 = h in equation

(4.19) for interaction of two gravito-dyons with charges (masses) (m, 0) and (0, h), one

may get the Dirac quantization condition for gravito-dyons. In section 4.4, we have consid-

ered that a particle of massM carries simultaneous existence of four charges namely electric

(e1), magnetic (g1), gravitational (gravi - electric) (m1) and Heavisidian (gravi - magnetic)

(h1), and if this particle moves with velocity −→v , it experiences a combined force (4.20)

which is the sum of the forces exerted independently on individual charges. As such the

equation (4.21) has been considered as the combined force experienced by the quaternion

charge and thus contain four coupling parameters W12,X12,Y12 and Z12 namely the electric,

magnetic, gravitational (gravi - electric) and Heavisidian (gravi - magnetic) represented by

equations (4.22 - 4.25). As such we have derived the unified force equation (4.26) which

contains the four charges in terms of the coupling parameters W12,X12,Y12 and Z12 con-

175


sequently the expression for the angular momentum are derived in equation (4.27) and

(4.28). As such the unified angular momentum has been derived in equation (4.29) in

the direction(−→r ) for unified fields of dyons and gravito-dyons where as its residual com-

ponent leads for the Schwinger - Zwanziger quantization quantization condition (4.31). In

section 4.5, we have generalized the two types of Schwinger - Zwanziger dyonic charges

(e, g) and (m, h) to a quaternion charge of gravi - electromagnetic fields and its quaternion

conjugation is defined by equation (4.32). Equation (4.33) is the interaction of between

two quaternions (a and b) with charges Qa = (ea, ga,ma, ha) = ea − iga − jma − kha and

Qb = (eb, gb,mb, hb) = eb−igb−jmb−khb in terms of different chirality parameters described

by equations (4.34 - 4.37). In case of the Schwinger - Zwanziger generalization of gravito-

dyons we have also obtained W12 = m1m2 +h1h2 and X12 = m1h2−h1m2 from quaternions

(0, 0,m1, h1) and (0, 0,m2, h2) in the absence of electromagnetic dyons. Here, we may spec-

ulate a new kind of dyon i.e. electric charge and Heavisidian monopole obtained from a

quaternion like (e, 0, 0, h) where W12 = e1e2 +h1h2 and Z12 = e1h2−h1e2. Similarly, another

kind of dyon (i.e., electric charge and gravitational charge) may also be speculated from

the quaternion like (e, 0,m, 0) giving rise to W12 = e1e2 + m1m2 and Y12 = e1m2 − m1e2.

Yet there is another possibility to the existence of a dyons (i.e., magnetic charge and

gravitational charge) from a quaternion (0, g,m, 0) which leads W12 = m1m2 + g1g2 and

Z12 = g1m2 −m1g2. There also exist a purely hypothetical dyons (like magnetic and Heav-

isidian monopoles) where we get W12 = h1h2 + g1g2 and Y12 = h1g2 − g1h2. As such, the

quaternion generalization of Schwinger - Zwanziger quantization condition of dyons leads

to the possibilities of six types of dyons namely (e, g), (m, h), (e, h), (e,m), (g, m) and

(g, h). Applying the j - conjugation of quaternion charge i.e.(i −→ i, j −→ −j, k −→ k) as

k = ij we get the electromagnetic fields of dyons in equation (4.38), as the order pair of

(e, g) (i.e. Schwinger - Zwanziger dyon) in equation (4.39). Similarly, equation (4.40)

represents the generalized charge (m, h) for Schwinger - Zwanziger dyon moving in grav-

ito - Heavisidian fields. Accordingly, equation (4.41) is defined as the i−conjugation of

quaternion for (i −→ −i, j −→ j, k −→ −k) as k = ij and leads to the generalized charge

as the order pair of (e, m) for of a Schwinger- Zwanziger dyon obtained from from electric

and Heavisidian charges in equation (4.42). Also, equation (4.43) represents the general-

176


ized charge as the order pair of (g, h) for of a Schwinger - Zwanziger dyon obtained from

gravitational and Heavisidian charges. Equations (4.45) and equation (4.47) represent the

transformation for i −→ −i , j −→ j , k −→ −k giving rise to the generalized charge

as the order pair of (e, h) for a Schwinger - Zwanziger dyon obtained from electric and

Heavisidian charges. Finally, the generalized charge as the order pair of (g, m) for of a

Schwinger - Zwanziger dyon has been obtained from magnetic and gravitational charges.

Equation (4.48) represents the Zorn matrix of Yang - Mill’s type in terms of electric and mag-

netic gauge potentials. Equation (4.51) defines the covariant derivative for the generalized

fields of dyons. Equation (4.52), (4.53) and (4.54) represent the split octonion realization

of gauge potential and gauge field strength of dyons in terms of abelian U (1) and non -

Abelian SU (2) gauge coupling strengths where the real quaternion unit is associated with

U (1) abelian gauge and the pure imaginary unit quaternion is related with the SU (2) non

- Abelian Yang - Mill’s field. In equation (4.52) and (4.53), we have expressed the gener-

alized four-potential Vµ and generalized field tensor Gµν of dyons in terms of abelian U (1)

and non - Abelian SU (2) gauge coupling strengths where the real quaternion unit e0 is

associated with U (1) abelian gauge and the pure imaginary unit quaternion is related with

the SU (2) non - Abelian Yang-Mill’s field. The gauge potential and non - Abelian gauge

field tensor are related by equation (4.55). Thus, equation (4.56) shows the Noetherian

current which is considered as the analogue of continuity equation in abelian gauge theory.

These equation yield the correct field equation for non - Abelian gauge theory of dyons and

give rise to its extended structure. Equation (4.60) represents the space-time interval of

gravi - Heavisidian field. Equation (4.62) is described as the line element on the octonionic

space in terms of octonionic metric tensor so that the line element has been established in

terms of Yang - Mill’s field strength in curved space-time in equation (4.64). The non - sym-

metric connection (Christoffel symbol) has also been analyzed in terms of split octonion

realization by equation (4.65). Equation (4.66) and equation (4.67) are expressed as the

Ricci tensor and O - curvature and thus retains the usual form of Riemannian space-time

geometry in this formalism. The generalized field tensor of gravito-dyons in terms of split

octonion has been described by equation (4.68). Equations (4.70 - 4.74) define the dif-

ferent gauge potential associated with gravito-dyons in terms of split octonion formalism.

177


The O−derivative of generalized field tensor of gravito-dyon or Maxwell’s Dirac equation

for linear gravitational field (4.75) leads generalized split octonion current associated with

gravito-dyons. Here also the conservation of generalized four-current follows the continuity

equation of abelian and non - Abelian gauge theories. As such, the split octonion gauge for-

malism demonstrates the structural symmetry between generalized electromagnetic fields

of dyons and that of generalized gravito - Heavisidian fields of gravito-dyons. In case of

split octonions, the automorphism group is described as G2 (an exceptional Lie group).

Thus octonion transformations are isomorphic to the rotation group O3. Under the SU(3)

subgroup of split G2 leaving u0 and u?0 invariant, the three split octonions (u1, u2, u3) trans-

form like a isospin triplet (quarks) and the complex conjugate octonions transform like a

unitary anti - triplet (anti quarks). The abelian and non - Abelian gauge structures of dyons

and gravito-dyons are discussed in terms of split octonion variables are described here in

simple, compact and consistent way. It is shown that the field equations derived here are

invariant under octonion gauge transformations. From the foregoing analysis one can ob-

tain the independent theories of electromagnetism and gravitation in the absence of each

other. The justification behind the use of octonions is to obtain the simultaneous structure

of SU(2)×U(1) gauge theory of dyons and gravito-dyons in simple and compact manner. As

such, the well - known SU(2) non - Abelian and U(1) Abelian gauge structure of dyons and

gravito-dyons are reformulated in terms of compact gauge formulation. The O- derivative

may be considered as the partial derivative if we do not incorporate the split octonion vari-

able into account. Split octonions are described here in terms of U(1)×SU(2) gauge group

simultaneously to give rise the Abelian (point like) and non - Abelian (extended structure)

of dyons. This gauge group plays the role of U(1)× SU(2) Salam Weinberg theory of elec-

tro - weak interaction in the absence of Heavisidian monopoles. The enlarged gauge group

SU(2) × U(1) explains the built in duality to reproduce Abelian and non - Abelian gauge

structure of dyons and those for gravito-dyons.

178

Bibliography

[1] P. A. M. Dirac, “Quantized singularities in the electromagnetic field”, Proc. Roy.

Soc. London, A133 (1931), 60.

[2] P. A. M. Dirac, “The Theory of Magnetic Poles”, Phys. Rev., 74 (1948), 817.


(1966), 1087.

[4] J. Schwinger, “Electric and Magnetic Charge Renormalization”, Phys. Rev., 151

(1966), 1055.



B137 (1965), 647.



[8] G.’ t Hooft, “Magnetic monopoles in unified gauge theories”, Nucl. Phys., B79

(1974), 276.

[9] A. M. Polyakov,“Particle spectrum in quantum field theory”, JETP Lett., 20

(1974), 194.

[10] B. Julia and A. Zee, “Poles with both magnetic and electric charges in non -


[11] M. Prasad and C. Sommerfield, “Exact Classical Solution for thept Hooft Monopole

and the Julia - Zee Dyon”, Phys. Rev. Lett., 35 (1975), 760.

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[12] E. Witten, “Dyons of charge ej/2p”, Phys. Lett., B86 (1979), 283 .

[13] J. Preskill, “Magnetic monopoles”, Ann. Rev. Nucl. Sci., 34 (1984), 461 .

[14] Jivan Singh , P. S. Bisht and O. P. S. Negi, “Generalized electromagnetic fields in

a chiral medium”, J. Phys. A: Math. and Theoretical, 40 (2007), 11395.

[15] Jivan Singh , P. S. Bisht and O. P. S. Negi, “Quaternion analysis for generalized

electromagnetic fields of dyons in an isotropic medium”, J. Phys. A: Math. and

Theoretical, 40 (2007), 9137.

[16] D. D. Cattani, “Linear Equations for the Gravitational Field”, Nuovo Cimento, B60

(1980), 67.

[17] L. Motz, “Gauge invariance and the quantization of mass of gravitational

charge”, Nuovo Cimento, B12 (1972), 239 .

[18] L. Motz, “The Quantization of Gravitational Charge and the Value of the Fine -

Structure Constant”, Nuovo Cimento, A37 (1977), 13.

[19] J. S. Dowker and R. A. Roche, “The Gravitational Analogues of Magnetic

Monopoles”, Proc. of Phys. Soc., 92 (1967), 1.

[20] P. S. Bisht, O. P. S. Negi and B. S. Rajput, “Null tetrad formulation of non - Abelian

dyons”, Inter. J. of Theor. Phys., 32 (1993), 2099 .


Fields”, Prog. Theor. Phys., 85 (1991), 157 .


Dyons”, Russian Physics Journal, 49 (2006), 1274.

[23] P. S. Bisht, O. P. S. Negi and B. S. Rajput, “Quaternion formulation for unified of

dyons and gravito - dyons”, Indian J. Pure & Appl. Math., 24 (1993), 543.



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[25] O. P. S. Negi, H. Dehnen, Gaurav Karnatak and P. S. Bisht, “Generalization of

Schwinger - Zwanziger Dyon to Quaternion”, Inter. J. of Theor. Phys., 50 (2011),

1908.

[26] A. Singh, “Expression of electromagnetic-potential equations in quaternionic

form”, Letter Nuovo Cimento, 32 (1981), 171.

[27] A. Singh, “Quaternionic Form of Linear Equations for the Gravitational Field

with Heavisidian Monopoles”, Letter Nuovo Cimento, 32 (1981), 5.

[28] A. Singh, “Heavisidian monopoles and quantization of gravitational charge”, 32

(1981), 231.

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Letter Nuovo Cimento, 35 (1982), 205.

181

Chapter 5

VALIDITY OF EHRENFEST’S THEOREM

AND ENERGY MOMENTUM TENSOR FOR

DYONS

Chapter 5 Validity of Ehrenfests theorem and Energy Momentum Tensor for Dyons

'

&

$

%

ABSTRACT

The validity of Ehrenfest theorem with its classical correspondence has been justified for the

manifestly covariant equations of dyons. We have also developed accordingly the Lagrangian

formulation for electromagnetic fields in a minimum coupled source giving rise to conserved

current of dyons. Applying the Gupta subsidiary condition we have extended the validity of

the Ehrenfest theorem for U (1) × U (1) abelian gauge theory of dyon. It is shown that the

expectation value of the quantum equation of motion reproduces the classical equation of

motion, which has been generalized the Ehrenfest theorem in quantum field theory. Finally,

we have discussed the energy momentum tensor and conservation laws for generalized fields of

dyons.

183


5.1 Introduction

In classical mechanics the dynamical state of each particle is defined by their

position and momentum, at a given instant. In quantum mechanics the dynamical states of

the system is represented by its wave function [1]. The expectation values of displacement

and momentum obey time evolution equations, which are analogous to those of classical

mechanics. This well result is called Ehrenfest’s theorem [2]. The value of the quantum

equation of motion taken with the states in the physical subspace reproduces the classi-

cal equations of motion. In a paper Zwanziger calculated the physical states in quantum

electrodynamics in terms of observable fields [3]. T. Thiemann and O. Winkler [4] also

establish the “Ehrenfest Property” of compact gauge states which are labelled by a point, a

connection and an electric field, in the classical phase space. Recently, Parthasarthy [5] has

generalized the validity of Ehrenfest theorem in abelian and non-Abelian quantum field

theories. The first consistent manifestly covariant quantization of electromagnetic fields

was formulated by Gupta [6] and Bleuler [7] describing the photon in the Fermi gauge

and the Lorentz condition not being an operator identity but a restriction which is imposed

on the physical states. Since the Lorentz condition is not consistent with the canonical

commutation relations, the former is regarded as a supplement condition which holds only

for certain physical states. According to the Gupta-Bleuler formalism, this subspace is a

non-negative subspace so that elements of physical subspace, called physical states, which

can be probabilistically interpretable. Correspondingly, the photon propagator does not

satisfy the Lorentz condition [8]. Cahill [9] is also suggested that the Fermi’s form of the

subsidiary condition is the correct one because it does not require the use of an indefinite

metric and because it is equivalent to the requirement that physical states be invariant un-

der a certain class of local gauge transformations. The scattering matrix of the relativistic

quantum electrodynamics, which is usually obtained with the Gupta-Bleuler method, was

deduced with a gauge-independent treatment, in which no use is made of any subsidiary

condition. The commutation rules between the components of the incoming electromag-

netic potentials are replaced by the weaker commutation rules between the components of

the incoming-electromagnetic- field strengths [10]. The covariant quantization of the elec-

184


tromagnetic field is one of the most peculiar problems of Quantum Field Theory because

of the masslessness of the photon. In spite of its vectorial nature, only the two transverse

components of the photon are observable, and the third freedom yields the Coulomb inter-

action between charged particles. It is well known, in the theory of Melo and coworkers

[11], the state vectors do not necessarily have positive norm, and the space spanned by

them is an indefinite metric Hilbert space. Akito Suzuki [12] using the Gupta subsidiary

condition and select the physical subspace in a unique and simple manner. The question

of existence of monopole [13]-[16] and dyons [17]-[24] has become a challenging new

frontier and the object of more interest in the recent years of high energy physics. Keeping

in view the recent potential importance of monopoles and dyons along with the fact that

despite the potential importance of monopoles, the formalism necessary to describe them

has been clumsy and not manifestly covariant, Negi and coworkers has already developed

a self consistent quantum field theory of generalized electromagnetic fields associated with

dyons (particles carrying electric and magnetic charges) [25, 26, 27]. In spite of the enor-

mous potential importance of monopoles (dyons) and the fact that these particles have been

extensively studied, there has been presented no reliable theory which is as conceptually

transparent and predictably tactable as the usual electrodynamics and the formalism nec-

essary to describe them has been clumsy and not manifestly covariant. On the other hand,

the concept of electromagnetic (EM) duality has been receiving much attention [28]-[38]

in gauge theories, field theories, super symmetry and super strings. In the recent paper it is

also developed a unified theory of gravi-electromagnetism on generalized the Schwinger-

Zwanziger [39] formulation on dyon in quaternion in simple and consistent manner.

In this chapter [40], we have made an attempt to show the validity of Ehrenfest

theorem for charged particle, dual charged particle in the case of separate electric and mag-

netic charge and dyons along with the energy momentum tensor for generalized fields of

dyons. Starting from the basic definition of Ehrenfest theorem in section 5.2 and in section

5.3 we have shown the validity of Ehrenfest theorem in the case of a Dirac particle moving

in the electromagnetic field carrying electric charge and it is shown that the Ehrenfest the-

orem is valid for a Dirac particle moving in an electromagnetic field. Section 5.4 describes

the validity of Ehrenfest theorem of magnetic charge (i.e. monopole) and the generaliza-

185


tion of Ehrenfest theorem for magnetic monopole has been obtained. In section 5.5 starting

from the Hamiltonian of Dirac fields in presence of dyon (particle carrying simultaneously

the electric and magnetic charge) we have discussed the validity of Ehrenfest theorem in a

static case for electric and magnetic charge. It is also shown that the equation of motion

of dyons may be visualized as the generalization of Ehrenfest theorem for dyons moving

in generalized electromagnetic fields. In section 5.6 we have developed accordingly the

Lagrangian formulation for the electromagnetic fields in a minimum coupled source which

justify the conserved Dirac current for dyons. Applying the Gupta subsidiary condition, we

have also reproduced the classical equation of motion and the validity of Ehrenfest theorem

to abelian quantum field theory has been checked and verified. It is shown that the expec-

tation value of the quantum equation of motion reproduces the classical equation of motion

which is the generalized form of the Ehrenfest theorem in quantum field theory. Section 5.7

describes the energy momentum tensor of generalized fields of dyons and energy momen-

tum conservation laws are discussed consistently for dyons. Here we have also discussed

the momentum operator, Hamiltonian and Poynting vector for generalized electromagnetic

fields in a manifest and consistent way.

5.2 Basics of Ehrenfest Theorem

Let us define the time derivative of the expectation value of a quantum mechanical

operator in terms of the commutator of that operator with the Hamiltonian of the system

[41]. Using the Heisenberg equation of motion as

d

dt

⟨A⟩

=1

i

⟨[A, H

]⟩+

⟨∂A

∂t

⟩. (5.1)

Here for brevity we have considered the natural units (c = ~ = 1). In equation (5.1) A is

the quantum mechanical operator and 〈A〉 is its expectation value. For the case of massive

particle moving in a potential, then the Hamiltonian is defined as

186


H (x, p, t) =p2

2m+ V (x, t) ; (5.2)

where x is just the location of the particle. Suppose the instantaneous change in momentum

p, using Ehrenfest’s theorem, we have

d

dt〈p〉 =

1

i〈[p, H]〉+

⟨∂p

∂t

⟩;

=1

i〈[p, V (x, t)]〉 . (5.3)

Since the operator p commutes with itself and has no time dependence. By expanding the

right-hand side of equation (5.3) and replacing p by −i∇, then

d

dt〈p〉 =

ˆψ?V (x, t)∇ψdx3 −

ˆψ?∇(V (x, t)ψ)dx3. (5.4)

Applying the product rule on the second term in equation (5.4), then we have

d

dt〈p〉 =

ˆψ?V (x, t)∇ψdx3 −

ˆψ? (∇V (x, t))ψdx3 −

ˆψ?V (x t)∇ψdx3;

=

ˆψ? (∇V (x, t))ψdx3;

= 〈−∇V (x , t)〉 = 〈F 〉 . (5.5)

This result manifests as Newton’s second law in the case of having so many particles that

the net motion is given exactly by the expectation value of a single particle. Similarly, the

instantaneous change in the position (x) expectation value is defined as,

187


d

dt〈x〉 =

1

i〈[x, H]〉+

⟨∂x

∂t

⟩;

=1

i

⟨[x,

p2

2m+ V (x , t)

]⟩;

=1

i

⟨[x,

p2

2m

]+ V (x, t)

⟩;

=1

m〈p〉 . (5.6)

This result is again in accord with the classical equation. As such, equations (5.5) and (5.6)

are known as Ehrenfest’s theorem.

5.3 Validity of Ehrenfest’s Theorem for Charged particles

Dirac Hamiltonian for the electromagnetic field (electric case) is described as [42]

H = −→α · π + βm0 + eφ; (5.7)

where α, β are usual arbitrary constants given by Dirac in relativistic quantum mechanics

and the momentum of charged particle gets modified to

π = p− eA. (5.8)

In equation (5.7), where m0 is the rest mass of the particle carrying electric charge, e

is the electric charge and in equation (5.8) A is the vector part of electric four-potential

Aµ =−→A, iφ

where as the φ is the scalar part of Aµ. In the case of weak magnetic

field, the vector part of electric four-potential is vanishes i.e. A = 0 so equation (5.8)

reduces to π = p. The Heisenberg equation of motion (5.1) for a dynamical variable F now

reduces to

188


dF

dt=

1

i~

[F , H

]. (5.9)

Assuming dynamical variable F corresponding to x and using equation (5.7) for H we get

〈v〉 =d−→xdt

=1

i[−→x , H]

= −→α =−→pm

; (5.10)

which can be expressed as ddt〈−→x 〉 =

⟨−→pm

⟩given by equation (5.6) and thus verifies the

Ehrenfest theorem. In equation (5.10) 〈v〉 ≡ d−→xdt

= −→α is known as relativistic velocity

operator. Now identifying the dynamical variable−→F as the momentum operator, we get

d−→πdt

=∂π

∂t+

1

i[H, π]

=−→αi

[π, π]− (e∇φ)− e∂A∂t

; (5.11)

where [π, π] can not be zero but its commutation relations reduce to

[πi, πj] = ie

(∂Aj∂xi− ∂Ai∂xj

)(∀i, j = 1, 2)

= ieεijkHk (∀ijk = 1, 2, 3) . (5.12)

Hence equation (5.11) reduces to

d−→πdt

= e

[(−∇φ− ∂A

∂t

)+(−→α ×−→H)] ;

= e[−→E +−→v ×

−→H]

= Fe. (5.13)

Equation (5.13) is the equation for Lorentz force acting on a charged particle moving the

189


electromagnetic field. In case of weak magnetic field[i.e.−→H = 0⇒ ∇× A⇒ A = 0⇒ π = −→p

],

equation (5.13) reduces to

d−→πdt

= e−→E = −e∇

−→φ ; (5.14)

which can further be expressed as

d

dt〈p〉 = 〈−∇ (eφ)〉 =

⟨−→Fe

⟩; (5.15)

which implies the Ehrenfest theorem. As such the Ehrenfest theorem provides its validity

in case of a Dirac particle moving in electromagnetic field carrying electric charge and

equation (5.13) is the generalized form of Ehrenfest theorem for a Dirac particle moving in

an electromagnetic field.

5.4 Validity of Ehrenfest Theorem for dual charged parti-

cles

Taking into account the electromagnetic duality among electric and magnetic parame-

ters already discussed in Chapter 1, we may write the following expression for the Dirac

Hamiltonian for a dual charge (i.e. magnetic monopole) moving in electromagnetic field as

H = −→α · π + βm0 + gψ; (5.16)

where ψ is the dual (magnetic) scalar potential and g is the dual (magnetic) charge while

π = p− gB. (5.17)

Here−→B is the vector part of magnetic (dual) four-potential i.e. Bµ =

−→B , i−→ψ. So,

190


dynamical quantity F in equation (5.9) corresponds to 〈x〉 and can further be described as

〈v〉 =d−→xdt

=1

i

[−→x , H]= α =

⟨ pm

⟩; (5.18)

which verifies the Ehrenfest theoremd〈−→x 〉dt

=⟨pm

⟩for the dynamics of dual charge (mag-

netic monopole). Likewise identifying the dynamical quantity F corresponding the momen-

tum operator we get

d−→πdt

=∂π

∂t+

1

i[H,π]

=−→αi

[π, π]− (g∇ψ)− g∂B∂t

; (5.19)

where like previous case [π, π] can not be zero but its commutation relations reduce to,

[πi, πj] = −ig(∂Bj

∂xi− ∂Bi

∂xj

)(∀i, j = 1, 2)

= −igεijkEk (∀ijk = 1, 2, 3) . (5.20)


d−→πdt

= g

[(−∇ψ − ∂A

∂t

)−(−→α ×−→E)]

= g[−→H −−→v ×

−→E]

= Fg; (5.21)

which is force acting on pure magnetic monopole moving in an electromagnetic field.

In the absence of electric field (in order to keep in mind the electromagnetic duality)

i.e.(−→E −→ 0⇒ ∇×B ⇒ B = 0⇒ π = −→p

). So, equation (5.21) may then be generalized

to

191


d−→πdt

= g−→H = −∇ (gψ) =

−→Fg; (5.22)

which shows the validity of Ehrenfest theorem of magnetic charge (i.e. monopole) and the

equation (5.21) is the generalization of Ehrenfest theorem for magnetic monopole.

5.5 Validity of Ehrenfest’s Theorem for dyons

Let us write the Hamiltonian of Dirac particle in generalized electromagnetic fields in

presence of (particles simultaneously carrying the electric and magnetic charges) namely

dyons as

H = −→α · −→π + βm0 +∑j

ajϕj (∀j = 1, 2)

= −→α · −→π + βm0 + a1ϕ1 + a2ϕ2; (5.23)

where

π =−→p −∑j

ajVj;

a1 = e; ϕ1 = φ;

a2 = g; ϕ2 = ψ;

V1 =−→A ; V2 =

−→B . (5.24)


H = −→α · −→π + βm0 + eφ+ gψ; (5.25)

where π is the modified momentum operator of dyons expressed as

192


−→π = p− eA− gB. (5.26)

Hence we may obtain the velocity operator for dyons as

〈v〉 =d 〈−→x 〉dt

⇒ 1

i[−→x , H]

⇒ −→α ⇒ 1

m〈p〉 ; (5.27)

which is the same as equation (5.10) and (5.18) and may be regarded as relativistic velocity

operator of dyons. So, the momentum operator of dyon is described as

dπ

dt=∂π

∂t+

1

i[H, π]

=1

i[−→α · π, π] +

1

i[βm0, π] +

1

i[eφ, π] +

1

i[gψ, π]− e∂A

∂t− g∂B

∂t; (5.28)

which is reduced to

d−→πdt

= e[−→E +−→v ×

−→H]

+ g[−→H −−→v ×

−→E]

; (5.29)

where

−→E =−

−→∇φ− ∂

−→A

∂t−−→∇ ×

−→B ;

−→H =−

−→∇ψ − ∂

−→B

∂t+−→∇ ×

−→A. (5.30)

Let us decompose−→E and

−→H in terms of longitudinal and transverse components i.e.

193


−→E =−→EL +

−→ET ;

−→H =−→HL +

−→HT ; (5.31)

where

−→EL =−

−→∇φ;

−→ET =− ∂

−→A

∂t−−→∇ ×

−→B ;

−→HL =−

−→∇ψ;

−→HT =− ∂

−→B

∂t+−→∇ ×

−→A. (5.32)

If the particles are stationary, then the transverse part of electric and magnetic field are

vanishing i.e.

ET = 0 ⇒ ∂−→A∂t

= 0 ;−→∇ ×

−→B = 0;

HT = 0 ⇒ ∂−→B∂t

= 0 ;−→∇ ×

−→A = 0. (5.33)

Equation (5.33) immediately shows that in the absence of transverse electromagnetic field

or for static dyon−→A =

−→B = 0 and

−→E = −

−→∇φ and

−→H = −

−→∇ψ. So the equation (5.29)

reduces to

d 〈p〉dt

= ∇ (−eφ) +∇ (−gψ) ; (5.34)

which is the combination of classical values of dpdt

for electric and magnetic charges in static

cases and may be described as the combination of validity of Ehrenfest theorem for electric

and magnetic charges. Thus the equation of motion of dyon (5.29) may be visualized as

the generalization of Ehrenfest theorem for dyons moving in generalized electromagnetic

194


fields.

5.6 Abelian gauge theory and Validity of Ehrenfest Theo-

rem

Starting from the Lagrangian density of generalized electromagnetic field for a mini-

mally coupled source of electric and magnetic charges as

L = −1

4FµνF

µν − 1

4MµνM

µν + eAµjµ(e) + gBµj

µ(g); (5.35)

where jµ(e) is the four-current due to the presence of electric charge and jµ(g) is the four-

current due to the existence of magnetic charges. Lagrangian density (5.35) yields the

following field equations

∂νFµν = ej(e)µ ; (5.36)

and

∂νMµν = gj(g)µ ; (5.37)

where we have used the following subsidiary conditions

∂µAµ =0;

∂µBµ =0. (5.38)

Equations (5.36) and (5.37) are the classical equation of motion giving to the Generalized

Dirac Maxwell’s (GDM) equation of dyons. Following Parthasarathy approach [5] let us

introduce the gauge fixing two auxiliary hermitian scalar field α(x) and β(x) in case of

195


electric and magnetic couplings and accordingly, let us consider the following Lagrangian

density

L =− 1

4FµνF

µν − 1

4MµνM

µν + α(x)∂µAµ +a

2α2(x)

+β(x)∂µBµ +b

2β2(x) + eAµj

µ(e) + gBµjµ(g); (5.39)

where a , b are the arbitrary parameters. Here like Parthasarathy we assume that the gauge

field Aµ and Bµ are operators. So, the Lagrangian density (5.39) reproduces the follow-

ing two quantum equations of motion respectively associated with electric and magnetic

charges

∂µFµν − ∂να(x) =− ejν(e);

∂µAµ + aα(x) =0; (5.40)

and

∂µMµν − ∂νβ(x) =− gjν(g);

∂µBµ + bβ(x) =0. (5.41)

It should be noted that the fields in equation (5.40) and (5.41) are operators and thus act

on the various functions (state) in indefinite metric quantum mechanical Hilbert space. So,

here the method of quantization be described as the “Operator method of quantization”.

On the other hand in classical the physical meaningful degrees of freedoms contribute only

to the observables. So, one can impose the Gupta subsidiary condition on the two photons

respectively associated with electric and magnetic charges of dyons i.e.

196


α+ (x) = 0;

β+ (x) = 0. (5.42)

Here the superscript (+) describes the positive frequency part of α and β, So the quantum

mechanical Hilbert space (VHS) is identified with equation (5.42). So the expectation values

of the Parthasarathy approach [5] are generalized for equations (5.40) and (5.41) as

⟨φ∣∣ejν(e) − ∂µα(x) + ∂µFµν

∣∣φ⟩ =0; | φ >∈ VHS,

〈φ |∂µAµ + aα(x)|φ〉 =0; (5.43)

and

⟨φ∣∣gjν(g) − ∂µβ(x) + ∂µMµν

∣∣φ⟩ =0; | φ >∈ VHS,

〈φ |∂µBµ + bβ(x)|φ〉 =0. (5.44)

Now let us use α− = (α+)+ and β− = (β+)

+, we get

⟨φ∣∣ejν(e) + ∂µFµν

∣∣φ⟩ =0; | φ >∈ VHS;

〈φ |∂µAµ|φ〉 =0; (5.45)

and

⟨φ∣∣gjν(g) + ∂µMµν

∣∣φ⟩ =0; | φ >∈ VHS;

〈φ |∂µBµ|φ〉 =0. (5.46)

Comparing the equations (5.45) and (5.46) with equation (5.36) and (5.37), the expec-

197


tation values of quantum equation of motion reproduces the classical equation of motion

which is nothing but the generalization of the Ehrenfest’s theorem to abelian quantum

field theory for generalized fields of dyons as the Ehrenfest theorems establish a formal

connection between the time dependence of quantum mechanical expectation values of

observables and the corresponding classical equations of motion. As such the validity of

Ehrenfest theorem has been justified for abelian U (1) × U (1) gauge theory of dyons in

terms of two photons.

5.7 Energy Momentum Tensor of dyons

We may now define the energy momentum tensor for generalized electromagnetic fields

of dyons as

T νµ = (∂µAρ)∂L

∂(∂νAρ)+ (∂µBρ)

∂L∂(∂νBρ)

− δνµL; (5.47)

where the first part of right hand side of equation (5.47) appears due to the contribution

of electric four-potential Aµ so that we may calculate the value of ∂L∂(∂νAρ)

from the La-

grangian (5.35) as

∂L∂(∂νAρ)

=∂(−1

4FµνF

µν)

∂(∂νAρ)

=− (∂νAρ − ∂ρAν) = −F νρ; (5.48)

Similarly, the second part of equation (5.47) appears due to the contribution of magnetic

four-potential Bµ and accordingly we get

∂L∂(∂νBρ)

=∂(−1

4MµνM

µν)

∂(∂νBρ)

=− (∂νBρ − ∂ρBν) = −Mνρ. (5.49)

198


Inserting equations (5.48) and (5.49) into equation (5.47), we get

T νµ =− (∂µAρ)Fνρ − (∂µBρ)M

νρ

+δνµ

[1

4FσρF

σρ +1

4MσρM

σρ − eAσj(e)σ − gBσj(g)σ

]; (5.50)

which is not symmetric in the indices µ, ν. To make it symmetric one must add another

appropriate rank two tensor which also obeys the same conservation law. So, the equation

(5.50) reduces to

T µν =− (∂νAρ)F µρ − (∂νBρ)Mµ

ρ

+ηµν[

1

4FσρF

σρ +MσρMσρ − eAσj(e)σ − gBσj

(g)σ

]. (5.51)

Now adding the following new tensor,

Sµν = (∂ρAν)F µρ + (∂ρBν)Mµ

ρ ; (5.52)

with equation (5.51), we get the following expression for the energy momentum tensor of

generalized electromagnetic fields of dyons as

T µν =T µν + Sµν

=− F νρF µρ −MνρMµ

ρ + ηµν[

1

4FσρF

σρ +MσρMσρ − eAσj(e)σ − gBσj

(g)σ

]. (5.53)

Thus equation (5.53) provides the field equations associated respectively with the dynamics

of electric and magnetic charges of dyons after taking care the usual method of variations

with respect to potential. Equation (5.53) may further be decomposed to

199


T µν =− (∂νAρ)Fµρ − (∂νBρ)M

µρ

+ηµν[

1

4FσρF

σρ +1

4MσρM


]; (5.54)

and

Sµν = (∂ρAν)Fµρ + (∂ρBν)M

µρ. (5.55)

Hence we may write the covariant derivative of energy momentum tensor (5.53) as

∂µTµν =

[− (∂νAρ)F

µρ − (∂νBρ)Mµρ + δµν

[1

4FσρF

σρ +1

4MσρM


]]+∂µ [(∂ρAν)F

µρ + (∂ρBν)Mµρ]

=− (∂ν∂µAρ)Fµρ − (∂νAρ) ∂µF

µρ − (∂ν∂µBρ)Mµρ − (∂νBρ) ∂µM

µρ

+1

2(∂νFσρ)F

σρ +1

2(∂νMσρ)M

σρ − 2∂ν(AσJ

(e)σ

)− 2∂ν

(Bσj

(g)σ

)+∂ρ (∂µAν)F

µρ + (∂ρAν) ∂µFµρ + ∂ρ (∂µBν)M

µρ + (∂ρBν) ∂µMµρ

=− (∂ν∂µAρ)Fµρ − (∂ν∂µBρ)M

µρ

+1

2(∂νFσρ)F

σρ +1

2(∂νMσρ)M

σρ. (5.56)

If we substitute

∂µAρ =1

2(∂µAρ − ∂ρAµ) +

1

2(∂µAρ + ∂ρAµ) ;

∂µBρ =1

2(∂µBρ − ∂ρBµ) +

1

2(∂µBρ + ∂ρBµ) ; (5.57)

into equation (5.56), we get

200


∂µTµν = Tµν,µ =− 1

2(∂νFµρ)F

µρ − 1

2(∂νMµρ)M

µρ +1

2(∂νFµρ)F

µρ +1

2(∂νMµρ)M

µρ

=0; (5.58)

which represents the continuity equation of energy momentum tensor of dyonic fields. Let

us write the energy momentum tensor of dyons in an alternative form as

T µν =− F νρF µρ −MνρMµ

ρ

+ηµν(

1

4FσρF

σρ +1

4MσρM

σρ − eAσJ (e)σ − gBσj

(g)σ

); (5.59)

where

ηµν =

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

. (5.60)

Equation (5.59) may further be decomposed to momentum and energy operates as

T 0ν =− F νρF 0ρ −MνρM0

ρ

+η0ν(

1

4FσρF

σρ +1

4MσρM


); (5.61)

T 00 = −F 0iF 0i −M0iM0

i +1

4FσρF

σρ +1

4MσρM

σρ − eAσj(e)σ − gBσj(g)σ . (5.62)

For i = 1, we may write

201


−F 01F 01 −M01M0

1 = −E (−E)−H (−H) = E2 +H2 = u; (5.63)

where u = E2 + H2 is known as energy density of the generalized electromagnetic field of

dyons. So, equation (5.62) reduces to

T 00 =(E2 +H2

)+

1

4FσρF

σρ +1

4MσρM

σρ − eAσj(e)σ − gBσj(g)σ ; (5.64)

If we put µ = 0 and ν = 3 in equation (5.59) we get

T 03 =− F 3ρF 0ρ −M3ρM0

ρ ; (5.65)

where

F 3ρF 0ρ =F 31F 0

1 + F 32F 02 ;

M3ρM0ρ =M31M0

1 +M32M02 . (5.66)

Then equation (5.65) becomes,

T 03 =− (H2) (−E1)− (−H1) (−E2)− (−E2) (−H1)− (E1) (−H2)

=H2E1 −H1E2 − E2H1 +H2E1

=2 (H2E1 − E2H1) = 2(−→E ×

−→H)3

; (5.67)

where T 0i = Si = 2(−→E ×

−→H)i

is known as Poynting vector for dyons showing that T 0i

is proportional to the ith component of Poynting vector−→S = 2

(−→E ×

−→H)

. As such the

conservation law becomes

202


∂νTµν =0; (5.68)

where

∂νT0ν =

∂

∂tT 00 +∇iT

0i = 0; (∀i = 1, 2, 3) (5.69)

Substituting the value of T 00 and T 0i into equation (5.69), we get

∂

∂t

[(E2 +H2

)+

1

4FσρF

σρ +1

4MσρM


]+∇.

−→S = 0;

which can further be reduced to

∂w

∂t+∇.

−→S =0. (5.70)

Equation (5.70) represents the Poynting Theorem for generalized fields of dyons.


Starting from equation (5.1) which represents the time derivative of the expectation

value of a quantum mechanical operator in terms of the commutator of the operator with

the Hamiltonian of the system. We have discussed the basics of Ehrenfest theorem in quan-

tum mechanics. Accordingly equation (5.2) describes the Hamiltonian. Equation (5.3)

gives rise to the instantaneous change in momentum which has been expanded in equation

(5.4). Equation (5.5) gives the representation of Ehrenfest’s theorem which manifests New-

ton’s second law for so many particles system. Similarly equation (5.6) describes the other

form of Ehrenfest’s theorem and it is in accord with the classical equation. In section (5.3)

we have discussed the validity of Ehrenfest theorem for charged particle. Dirac Hamil-

tonian for the electromagnetic field (electric case) is described by equation (5.7) while

203


equation (5.8) represents the modified form of momentum of the electric charge. Defining

the Heisenberg equation of motion for a dynamical variable by equation (5.9), we have

obtain the relativistic velocity operator by equation (5.10) which is in accordance of the

Ehrenfest’s theorem at classical level. Identifying equation (5.11) as time derivative of the

momentum operator for electric charge particle, we have obtain the commutation relations

among momentum operators in equation (5.12). It is shown that the time derivative of

momentum operator provides the equation (5.13) for Lorentz force acting on a charged

particle moving in the electromagnetic field. As such, the Ehrenfest theorem has been veri-

fied by equation (5.15) and it is shown that the validity of Ehrenfest theorem is consistently

applicable for Dirac particle moving in electromagnetic field carrying electric charge. In sec-

tion (5.4) we have discussed the validity of Ehrenfest theorem for dual charged particle.

Starting from the equation (5.16) describing the Dirac Hamiltonian for a dual charge (i.e.

magnetic monopole) moving in electromagnetic field. We have obtained equation (5.17) is

the modified momentum of dual charge moving in electromagnetic field. As such we have

obtained equation (5.18) for the velocity operator and verifies the validity of Ehrenfest’s

theorem for the dynamics of dual charge (i.e. magnetic monopole). Describing equation

(5.19) for the time derivative of the momentum operator for a dual charge, we have ob-

tained the commutations relations (5.20) among momentum operator for monopole. It is

shown that the rate of change of momentum of dual charged particle exerts the Lorentz

force equations (5.21) and (5.22) acting on pure magnetic monopole moving in electro-

magnetic field. It verifies the validity of Ehrenfest theorem for magnetic monopole. In

section (5.5), we have discussed the validity of Ehrenfest theorem for the particles carry-

ing simultaneously existence of electric and magnetic charges (namely dyons). As such,

we have written the Hamiltonian of Dirac particle in electromagnetic fields for dyons by

equation (5.23) whereas the momentum operator for the dyons is expressed by equation

(5.24). Accordingly we have established the momentum operator of dyons by equation

(5.26). Hence we have obtained the velocity operator for dyonic field by equation (5.27).

The rate of change of momentum operator of dyon is described by equation (5.28) which

provides the Lorentz force equation (5.29) of dyon carrying the generalized electric and

magnetic fields given by equation (5.30). Decomposing the electric and magnetic field in

204


terms of longitudinal and transverse components of electric and magnetic field by equa-

tion (5.31), we have obtained the transverse and longitudinal components of generalized

electric and magnetic fields of dyons and are expressed in terms of electric and magnetic

four-potentials as given by equation (5.32). Restricting the direction of propagation along a

particular axis we have made transverse components of generalized electromagnetic fields

vanishing by equation (5.33) so that one can write the horizontal components in terms

of gradient of potential. Hence we have obtained the combined term of Ehrenfest theo-

rem for electric and magnetic charges. It shows that the validity of Ehrenfest theorem for

dyon. Thus the equations (5.29) and (5.34) are the generalization of Ehrenfest theorem

for generalized fields of dyons. In section (5.6) we have discussed the validity of Ehrenfest

theorem for two potential abelian gauge theory of dyons. Starting with the Lagrangian den-

sity (5.35) of generalized electromagnetic field for a minimally coupled source of electric

and magnetic charges in abelian gauge, it is shown that the Lagrangian density yields the

classical equation of motion expressed by equation (5.36) and (5.37) respectively for elec-

tric and magnetic constituents of dyons whereas the subsidiary condition are imposed as

equation (5.38). Introducing the two different auxiliary Hermitian scalar fields α (x) and

β (x) for the electric and magnetic couplings of dyons, we have suitably handled the La-

grangian density by equation (5.39), this Lagrangian density yields the quantum equation

of motion given by equations (5.40) and (5.41) respectively associated with electric and

magnetic charge of dyons. Here the fields in equation (5.40) and (5.41) are described as

operators acting on the various functions (state) in indefinite metric quantum mechanical

Hilbert space. So, here the method of quantization be described as the “Operator method of

quantization” contrary to classical case where the physical meaningful degrees of freedoms

contribute only to the observables. Imposing the Gupta subsidiary condition (5.42) on the

two photons given in terms of α (x) and β (x) respectively associated with electric and mag-

netic charges of dyons. We have obtained the modified form of field equations (5.43) and

(5.44) as the expectation values for electric and magnetic charges. Comparing the equa-

tions (5.45) and (5.46) with equation (5.36) and (5.37), it is shown that the expectation

values of quantum equation of motion reproduce the classical equations of motion which is

nothing but the generalization of the Ehrenfest’s theorem to abelian quantum field theory

205


for generalized fields of dyons since the Ehrenfest theorems establish a formal connection

between the time dependence of quantum mechanical expectation values of observables

and the corresponding classical equations of motion. As such, the validity of Ehrenfest

theorem has been justified for abelian U (1)× U (1) gauge theory of dyons in terms of two

photons. In section (5.7) we have discussed the energy momentum tensor of dyons which

is given by equation (5.47) and its further consequences. It is shown that the energy mo-

mentum tensor (5.47) contains the contribution of electric and magnetic four-potentials

respectively giving rise the variation of Lagrangian in equations (5.48) and (5.49). Insert-

ing (5.48) and (5.49) into equation (5.47) we have obtained the general expression (5.50)

for energy-momentum tensor dyons and it is shown that the expression (5.50) is no more

symmetric and to make it symmetric we need the rank two tensor. Accordingly the equation

(5.50) has been converted to rank two tensor by expression (5.51) and it is concluded that

the energy momentum tensor will became symmetric unless and until we add an extra term

given by equation (5.52). As such we have obtained symmetric term energy momentum

tensor for generalized fields of dyons (5.53). It is concluded that the symmetric energy

momentum tensor (5.53) of dyons provides the field equation of dyons as the combina-

tion of field equations respectively due to the dynamics of electric and magnetic charges

after taking care the usual method of variation. In order to obtain the energy momentum

conservation laws we have decomposed the equation (5.53) of energy momentum tensor

of dyons in terms of equations (5.54) and (5.55). It is shown by equation (5.58) which

is obtained by using equations (5.56) and (5.57) that the covariant derivative of energy

momentum tensor is vanishing and thus leads the law of energy-momentum conservation.

Alternatively we have obtained the energy and momentum components of energy momen-

tum tensors given by equations (5.61) and (5.62) which is further expanded in terms of

energy and momentum densities respectively given by equations (5.64) and (5.67). Equa-

tion (5.67) describes the Poynting vector while the conservation of work energy theorem

is expressed by equation (5.68). Thus the Poynting vector is nothing but the energy flux

or the momentum of radiation flowing into or out of a volume. This leads to an increase

or decrease in energy of radiation. But in the case of generalized electromagnetic fields of

dyons not only the energy is conserved but momentum is also conserved. This shows that

206


the rate of change of total momentum in a volume is because of “momentum flux flowing in

and out of the system”. The conservation law of energy momentum tensor is described as

equation (5.68). As such equation (5.70) represents the Poynting theorem for generalized

fields of dyons.

207

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211

Chapter 6

QUATERNION-OCTONION GAUGE

FORMULATION AND UNIFICATION OF

FUNDAMENTAL INTERACTION

Chapter 6 Quaternion Octonion Gauge Formulation and Unification of fundamentalInteraction'

&

$

%

ABSTRACT

Starting with the quaternion formulation of SU (2)×U (1) gauge theory of dyons and gravito-

dyons, it is shown that the formulation characterizes the abelian and non-Abelian gauge struc-

ture of dyons and gravito-dyons in terms of purely real and imaginary units of quaternion. It is

shown that the three quaternion units explain the structure of Yang-Mill’s field while the seven

octonion units provide the consistent structure of SU (3)c gauge symmetry of quantum chromo

dynamics. Grand Unified theories are discussed accordingly in terms of quaternion and octo-

nions in terms of quaternion basis elements and pauli matrices as well as the Octonions and

Gell Mann matrices. Connection between the unitary group of GUT’s and the normed division

algebra has been established to redescribe the SU (5) gauge group. The SU (5) gauge group

and its subgroup SU (3)C × SU (2)L × U (1) has been analyzed in terms of quaternion and

octonion basis elements. It is concluded that the division algebra approach to the theory of

unification of fundamental interactions in the case of GUT’s leads to the consequences towards

the new understanding of these theories which incorporate the existence of magnetic monopole

and dyon.

213

Chapter 6 Quaternion Octonion Gauge Formulation and Unification of fundamentalInteraction

6.1 Introduction

The existence of monopole [1, 2] and dyons [3]-[7] has become a challenging new

frontier and the object of more interest in High Energy Physics. Dirac showed [1, 2] that

the quantum mechanics of an electrically charged particle of charge e and a magnetically

charged particle of charge g is consistent only if eg = 2π n, n being an integer. Schwinger -

Zwanziger [3]-[7] generalized this condition to allow for the possibility of particles (dyons)

that carry both electric and magnetic charge. A quantum mechanical theory can have two

particles of electric and magnetic charges (e1, g1) and (e2, g2) only if e1g2−e2g1 = 2π n. The

angular momentum in the field of the two particle system can be calculated readily with

the magnitude e1g2−e2g14πc

. This has an integer or half - integer value, as expected in quan-

tum mechanics, only if e1g2 − e2g1 = 2πnc. The fresh interests in this subject have been

enhanced by ’t Hooft - Polyakov [8, 9] with the idea that the classical solutions having the

properties of magnetic monopoles may be found in Yang - Mills gauge theories. Julia and

Zee [10, 11] extended the ’ t Hooft-Polyakov theory [8, 9] of monopoles and constructed

the theory of non - Abelian dyons. The quantum mechanical excitation of fundamental

monopoles include dyons which are automatically arisen from the semi-classical quantiza-

tion of global charge rotation degree of freedom of monopoles. In view of the explanation

of CP - violation in terms of non-zero vacuum angle of world [12, 13], the monopoles are

necessary dyons and Dirac quantization condition permits dyons to have analogous electric

charge. Accordingly, a self consistent and manifestly covariant theory has been developed

[14, 15] for the generalized electromagnetic fields of dyons. Several authors [16]-[19]

have developed the quaternion quantum mechanics while Adler [20]-[22] proposed the

idea of alzebric quaternion generalization of classical Yang-Mill’s fields. Several authors

[23]-[25] described the quaternionic forms of Maxwell’s equation. Quaternions also play

an important role in classical field theories [26] and extended the manifold structure. The

theoretical existence of monopoles (dyons) [1], [8, 9] and keeping in view of their recent

potential importance with the fact of that formalism necessary to describe them has been

clumsy and not manifestly covariant, the quaternionic formalism for Generalized fields

of dyons in unique, simple, compact and consistent manner. Postulation of Heavisidian

214


monopole [27, 28] immediately follows the structural symmetry [29]-[31] between gen-

eralized gravito-Heavisidian and generalized electromagnetic fields of dyons. It has also

been shown that [32, 33], in quaternion, non-abelian gauge formalism, the SL (2, c) gauge

group of gravitation and SU (2) group of Yang-Mill’s non-Abelian gauge theory play the

same role of symmetrical manner. The isomorphic SU (2)×U (1) gauge structure of general

linear quaternion gauge group of dyons and gravito-dyons is different from the well known

gauge group structure of Salam-Weinberg theory of electroweak interactions. Octonions

are widely used for the understanding of unification structure of successful gauge theory of

fundamental interaction. Recently, the quaternionic formulation of Yang–Mill’s field equa-

tions and octonion reformulation of quantum chromo dynamics (QCD) by taking magnetic

monopoles and dyons (particles carrying electric and magnetic charges) into account. It

has been shown that the three quaternion units explain the structure of Yang-Mill’s field

while the seven octonion units provide the consistent structure of SU (3)c gauge symmetry

of quantum chromo dynamics. Here we apply entirely different approach for the quater-

nion gauge theory of electroweak interactions and octonion gauge structure for quantum

chromo dynamics (QCD) compare to the approaches adopted earlier by Morita [34, 35]

and others [36, 37]. Keeping all these facts in mind, in this chapter we have made an at-

tempt to discuss the quaternion formulation of SU (2)× U (1) gauge theories of dyons and

gravito-dyons followed by the SU (3) gauge structure and grand unified theories in terms

of octonion. It has been emphasized that the three quaternion units explain the structure of

Yang- Mill’s field while the seven octonion units provide the consistent structure of SU (3)c

gauge symmetry of quantum chromo dynamics (QCD) as these are well connected with the

well known SU (3) Gellmann matrices [38]. The symmetry breaking mechanism of non-

Abelian gauge theories in terms of quaternion and octonion opens the window towards

the discovery of two type of gauge bosons associated with electric and magnetic charges.

Here, we have also developed the quaternion formulation of SU (2) × U (1) gauge theory

of dyons and gravito-dyons. It has been shown that the structure of Yang- Mill’s field is

explained by three quaternion units while the seven octonion units provide the consistent

structure of SU(3)C gauge symmetry of quantum chromo dynamics (QCD) with their inter

connectivity with the well-known SU(3) Gellmann λ matrices. In this case the gauge fields

215


describe the potential and currents associated with the generalized fields of dyons parti-

cles carrying simultaneously the electric and magnetic charges. Grand Unified theories are

discussed in terms of quaternions and octonions by using the relation between quaternion

basis elements with Pauli matrices and Octonions with Gell Mann matrices. Connection

between the unitary groups of GUT’s and the normed division algebra has been established

to re-describe the SU (5) gauge group. We have thus described the SU (5) gauge group and

its subgroup SU (3)C × SU (2)L × U (1) by using quaternion and octonion basis elements.

As such, the connection between U (1) gauge group and complex number, SU (2) gauge

group and quaternions and SU (3) and octonions is established. It is concluded that the

division algebra approach to the theory of unification of fundamental interactions as the

case of GUT’s leads to the consequences towards the new understanding of these theories

which incorporate the existence of magnetic monopole and dyon. As such it is concluded

that the division algebra approach to the theory of unification of fundamental interactions

as the case of GUT’s leads to the consequences towards the new understanding of these

theories which incorporate the existence of magnetic monopole and dyon. Three different

imaginaries associated octonion formulation may be identified with three different colors

(red, blue and green) while the Gell Mann Nishijima are described in terms of simple and

compact notations of octonion basis elements.

6.2 Quaternion formulation of SU (2)× U (1) gauge theory

of dyons

In chapter-3 we have developed the U (1)×U (1) gauge theory of dyons, where we have

two gauge fields Aµ and Bµ respectively associated with electric charge (U (1)e) and

magnetic monopole(U (1)g

). So, the generalized potential of dyons in U (1)×U (1) gauge

theory be written as

Vµ =

Aµ 0

0 Bµ

. (6.1)

216


However, the quaternion gauge theory may be expressed as the combination U (1)×SU (2)

gauge theory whereas the U (1) part is associated with its real quaternion unit (e0) and

the symmetry group SU (2) may be visualized in terms of three imaginary quaternion units

ej(∀j = 1, 2, 3) . As such the collection of tensors that is matrices in internal space with

local symmetry group SU (2) may be associated with pure imaginary quaternion limits. As

such the internal covariant derivative in U (1)×SU (2) gauge theory behaves as quaternion

covariant derivative or Q-derivative [39]. If K is a quaternion, the quaternion Q-derivative

acting on a quaternion K is described [39] as

Kqµ = K,µ + [Γµ, K] ; (6.2)

where Kµ is partial derivative and the affinity Γµ is defined [39] as

Γµ = iCµ · τ = −Cµ · e =3∑j=1

Cjµej. (6.3)

Here the affinity Γµ is the object which makes the Kqµ like a vector under transformation in

internal space. Vector Cµ plays the role of gauge potential in internal SU (2) non-Abelian

gauge space. So, the quaternionic extension of U (1)× U (1) gauge theory of dyons may be

visualized as the two fold degeneracy of U (1)×SU (2) gauge theory. Hence for the first fold

degeneracy we may write the gauge potential and gauge field strength for electric charge

in quaternionic version of U (1)× SU (2) gauge theory as

Aµ ⇒ Aµ = A0µe

0 + Ajµej (∀j = 1, 2, 3) ;

= A0µ1 +

1

iAjµτj; (6.4)

and

217


Fµν ⇒ Fµν = F 0µνe0 + F j

µνej;

= Fµν 1 +1

iF jµντj; (6.5)

where ej ⇔ −iτj and τj usual pauli-spin matrices. Similarly the another fold of degeneracy

is associated with the dynamics of dual charge (i.e. monopole) for which the gauge po-

tential and gauge field strength in quaternionic version of U (1) × SU (2) gauge theory be

written as

Bµ ⇒ Bµ = B0µe

0 +Bjµej (∀j = 1, 2, 3) ;

= B0µ1 +

1

iBjµτj; (6.6)

and

Mµν ⇒Mµν = M0µνe0 +M j

µνej;

= Mµν 1 +1

iM j

µντj. (6.7)

Hence for the U (1)× SU (2) quaternion gauge theory, the gauge potential and gauge field

strength be expressed as

Vµ =

Aµe0 − i

∑3a=1A

aµτ

a 0

0 Bµe0 − i

∑3a=1B

aµτ

a

(a = 1, 2, 3) ;

=

Aµ 0

0 Bµ

; (6.8)

218


and

Gµν =

−ie (Fµν − i∑3a=1 F

aµντ

a)

0

0 −ig(Mµν − i

∑3a=1Mµντ

a) (a = 1, 2, 3) ;

=− i

eFµν 0

0 gMµν

; (6.9)

In equations (6.8) and (6.9) we have expressed generalized four-potential Vµ and gener-

alized field tensor Gµν of dyons in terms of U (1) and non-Abelian SU (2) gauge coupling

strengths where the real quaternion unit (e0) is associated with U (1) Abelian gauge field

and the pure imaginary unit quaternions are related with the SU (2) non-Abelian Yang-

Mill’s field. Equations (6.8) and (6.9) are invariant under the local and global phase trans-

formation. Hence the Q-derivative for U (1)× SU (2) gauge theory be expressed as

Qqµ = Q,µ + [Γµ, Q] ; (6.10)

So the analogous covariant derivative for U (1)× SU (2) gauge theory is described as

Dµ =

Dµ 0

0 ∇µ

+

1iDjµτj 0

0 1i∇jµτj

; (6.11)

where

Dµ =∂µ − ieAµ;

∇µ =∂µ − igBµ; (6.12)

which gives rise

219


[Dµ, Dν ] =

(Fµν − i∑3a=1 F

aµντ

a)

0

0(Mµν − i

∑3a=1Mµντ

a)

=Gµν ; (6.13)

which yields the correct field equation for non-Abelian gauge theory of dyons and give rise

to its extended structure [40, 41], which reproduces

DνGµν =

ejµ 0

0 gkµ

=Jµ; (6.14)

where

Jµ =

e(jµe

0 − i∑3

a=1 jaµτ

a)

0

0 g(kµe

0 − i∑3

a=1 kaµτ

a) . (6.15)

Thus the generalized Q-four current of dyons in quaternion U (1)×SU (2) gauge formalism

leads to the abelian and non-Abelian nature current and leads to the Noetherian conserva-

tion of current as

Jµqµ = 0. (6.16)

Which shows the resemblance with Noetherian current analogous to the continuity equa-

tion in abelian gauge theory. In general this current is no more conserved in terms of

its ordinary derivative to Q-derivative the generalized quaternionic current becomes con-

served.

220


6.3 Quaternion formulation of SU (2)× U (1) gauge theory

of gravito-dyons

In chapter-3 we have developed the U (1) × U (1) gauge theory of gravito-dyons where

we have two gauge fields Cµ and Dµ respectively associated with gravitational charge

(U (1)m) and Heavisidian monopole (U (1)h). So, the generalized potential of gravito-dyons

in U (1)× U (1) gauge theory be written as

Vµ =

Cµ 0

0 Dµ

. (6.17)

The quaternionic extension of U (1)×U (1) gauge theory of gravito-dyons may be visualized

as the two fold degeneracy of U (1)×SU (2) gauge theory. Hence for the first fold degener-

acy we may write the gauge potential and gauge field strength for gravitational charge in

quaternionic version of U (1)× SU (2) gauge theory as

Cµ ⇒ Cµ = C0µe

0 + Cjµej (∀j = 1, 2, 3) ;

= C0µ1 +

1

iCjµτj; (6.18)

and

fµν ⇒ fµν = f 0µνe0 + f jµνej;

= fµν 1 +1

if jµντj; (6.19)

In this case the another fold of degeneracy is associated with the dynamics of dual charge

(i.e. Heavisidian monopole) for which the gauge potential and gauge field strength in

quaternionic version of U (1)× SU (2) gauge theory be written as

221


Dµ ⇒ Dµ = D0µe

0 +Djµej (∀j = 1, 2, 3) ;

= D0µ1 +

1

iDjµτj; (6.20)

and

Nµν ⇒ Nµν = N0µνe0 +N j

µνej;

= Nµν 1 +1

iN jµντj. (6.21)

Hence for the U (1)× SU (2) quaternion gauge theory, the gauge potential and gauge field

strength of gravito-dyon may be expressed as

Vµ =

Cµe0 − i

∑3a=1C

aµτ

a 0

0 Dµe0 − i

∑3a=1D

aµτ

a

(a = 1, 2, 3) ;

=

Cµ 0

0 Dµ

; (6.22)

and

Gµν =

−im (fµν − i∑3a=1 f

aµντ

a)

0

0 −ih(Nµν − i

∑3a=1Nµντ

a) (a = 1, 2, 3) ;

=− i

mfµν 0

0 hNµν

. (6.23)

We have expressed generalized four-potential Vµ and generalized field tensor Gµν of gravito-

dyons in terms of U (1) and non-Abelian SU (2) gauge coupling strengths where the real

quaternion unit (e0) is associated with U (1) Abelian gauge field and the pure imaginary

unit quaternions are related with the SU (2) non-Abelian Yang-Mill’s field. Equations (6.22)

and (6.23) are invariant under the local and global phase transformation. The covariant

222


derivative for U (1)× SU (2) gauge theory of gravito-dyons are given as

Dµ =

Dµ 0

0 ∆µ

+

1iDjµτj 0

0 1i∆jµτj

; (6.24)

where

Dµ =∂µ − imCµ;

∆µ =∂µ − ihDµ; (6.25)

which gives rise

[Dµ, Dν ] =

(fµν − i∑3a=1 f

aµντ

a)

0

0(Nµν − i

∑3a=1Nµντ

a)

=Gµν ; (6.26)

which yields the correct field equation for non-Abelian gauge theory of gravito-dyons and

give rise to its extended structure [40, 41], which reproduces

DνGµν =

mlµ 0

0 hnµ

=− Jµ; (6.27)

where

Jµ =

m(lµe

0 − i∑3

a=1 laµτ

a)

0

0 h(nµe

0 − i∑3

a=1 naµτ

a) . (6.28)

Thus the generalized Q-four current of gravito-dyons in quaternion U (1) × SU (2) gauge

formalism leads to the abelian and non-Abelian nature current and leads to the Noetherian

223


conservation of current as

Jµqµ = 0. (6.29)

Which shows the resemblance with Noetherian current analogous to the continuity equa-

tion in abelian gauge theory of gravito-dyons. In general this current is no more conserved

in terms of its ordinary derivative to Q-derivative the generalized quaternionic current be-

comes conserved.

6.4 Octonionic formulation of QCD

6.4.1 Gellmann λ matrices

In order to extend the symmetry from SU(2) to SU(3) we replace three Pauli spin matri-

ces by eight Gellmann [42] λ matrices. λj (j = 1, 2, ......8) be the 3× 3 traceless Hermitian

matrices introduced by Gell-Mann. Their explicit forms are

λ1 =

0 1 0

1 0 0

0 0 0

, λ2 =

0 −i 0

i 0 0

0 0 0

;

λ3 =

1 0 0

0 −1 0

0 0 0

, λ4 =

0 0 1

0 0 0

1 0 0

;

λ5 =

0 0 −i

0 0 0

i 0 0

, λ6 =

0 0 0

0 0 1

0 1 0

;

224


λ7 =

0 0 0

0 0 −i

0 i 0

, λ8 =1√3

1 0 0

0 1 0

0 0 −2

; (6.30)

which satisfy the following properties as

(λj)† =λj;

Trλj =0;

Tr (λjλk) =2δjk;

[λj, λk] =2Fjklλl (∀ j, k, l = 1, 2, 3, 4, 5, 6, 7, 8); (6.31)

where Fjkl are the structure constants of SU(3) group defined as

F123 = 1; F147 = F257 = F435 = F651 = F637 =1

2;

F458 =F678 =

√3

2. (6.32)

6.4.2 Relation between Octonion and Gellmann Matrices

Let us establish the relationship between octonion basis elements eA and Gellmann λ

matrices [42]. Comparing the structure constants of octonion (1.82) in chapter 1 with

structure coefficients of Gell Mann λ matrices (6.31), we find

FABC = fABC (∀ABC = 123); (6.33)

and

FABC =1

2fABC (∀ABC = 147, 246, 257, 435, 516, 637). (6.34)

Equation (6.33) leads to

[eA, eB]

[λA, λB]=

eCiλC

(∀A,B,C = 1, 2, 3);

⇒ [eA, eB] = [λA, λB] (∀ eC = iλC). (6.35)

225


On the other hand equation (6.34) gives rise to

[eA, eB]

[λA, λB]⇒ eC

2iλC(∀ABC = 516, 624, 471, 435, 673, 572);

⇒ [eA, eB]⇒ [λA, λB] (∀ eC = iλC2

). (6.36)

We may now describe correspondence between λ8 matrix and octonion units in the follow-

ing manner i.e.

λ8 =⇒− 2

i√

3[e4, e5] + [e6, e7]

=⇒− 2

i√

3(e4e5 − e5e4 + e6e7 − e7e6)

=⇒ 8e3

i√

3. (6.37)

Hence we may describe one to mapping (interrelationship) between octonion basis ele-

ments [42, 43] and Gellmann λ matrices by using equations (6.35) and (6.36) as,

eA ∝ λA ⇒ eA = kλA; (6.38)

where k is proportionality constant depending on the different values of A i.e. k = i (∀A =

1, 2, 3) and k = i2

(∀ABC = 516, 624, 471, 435, 673, 572). From equation (6.37), we also get

e3 ∝ λ8 ⇒ e3 = kλ8; (6.39)

where k = i√3

8. With these relations between the octonion units and Gellmann λ matrices,

we may develop the octonion quantum chromo dynamics in consistent way. To do this, let

us establish between octonion units elements and Gell Mann λ matrices [42, 43] i.e.

[e6, e5]

[λ6, λ5]=

[e4, e7]

[λ4, λ7]=

e12iλ1

=1

2k1

=⇒λ1 = −ie1k1; (6.40)

226


[e4, e6]

[λ4, λ6]=

[e5, e7]

[λ5, λ7]=

e22iλ2

=1

2k2

=⇒λ2 = −ie2k2; (6.41)

[e5, e4]

[λ5, λ4]=

[e6, e7]

[λ6, λ7]=

e32iλ3

=1

2k3

=⇒λ3 = −ie3k3; (6.42)

[e7, e1]

[λ7, λ1]=

[e6, e2]

[λ6, λ2]=

[e3, e5]

[λ3, λ5]=

e42iλ4

=1

2k4

=⇒λ4 = −ie4k4; (6.43)

[e4, e3]

[λ4, λ3]=

[e7, e2]

[λ7, λ2]=

[e1, e6]

[λ1, λ6]=

e52iλ5

=1

2k5

=⇒λ5 = −ie5k5; (6.44)

[e5, e1]

[λ5, λ1]=

[e7, e3]

[λ7, λ3]=

[e2, e4]

[λ2, λ4]=

e62iλ6

=1

2k6

=⇒λ6 = −ie6k6; (6.45)

[e1, e4]

[λ1, λ4]=

[e2, e5]

[λ2, λ5]=

[e3, e6]

[λ3, λ6]=

e72iλ7

=1

2k7

=⇒λ7 = −ie7k7; (6.46)

227


As such, we may get the following relationship between Gellmann l matrices and octonion

units

λA =− ieAkA (A = 1, 2, 3, 4, 5, 6, 7) ;

λ8 =− i 8√3e3. (6.47)

6.4.3 Octonionic Reformulation of QCD

The local gauge theory of color SU (3) group gives the theory of QCD. The QCD (quan-

tum chromo dynamics) is very close to Yang-Mills (non- Abelian) gauge theory. The above

mentioned SU (2) gauge symmetry describes the symmetry of the weak interactions. On the

other hand, the theory of strong interactions,quantum chromo dynamics (QCD), is based

on colour SU (3) (namely SU (3)c) group. This is a group which acts on the colour indices

of quark favours described in the form of a basic triplet i.e.

ψ =

ψ1

ψ2

ψ3

→

R

B

G

; (6.48)

where indices R, B and G are the three colour of quark flavors. Under SU(3)c symmetry,

the spinor ψ transforms as

ψ 7−→ψp = Uψ = exp iλaαa(x)ψ; (6.49)

where λ are Gellmann matrices , a = 1, 2, ......8 and the parameter α is space time de-

pendent. We may develop accordingly the octonionic reformulation of quantum chromo

dynamics (QCD) on replacing the Gellmann λ matrices by octonion basis elements eA given

by equations (6.38) and (6.39). Now calculating the value of λaαa (x) =∑8

a=1 λaαa (x)

using the relations between Gellmann l matrices and octonion units given by (6.47), we

228


get

8∑a=1

λaαa (x) = −ie1k1α1 (x)− ie2k2α2 (x)− ie3

(k3α

3 (x) + k8α8 (x)

)+ie4k4α

4 (x)− ie5k5α5 (x)− ie6k6α6 (x)− ie7k7α7 (x) . (6.50)

Now taking following transformations

k1α1 7−→ β1 ; k2α2 7−→ β2 ;

(k3α3 + k8α8) 7−→ β3 ; k4α4 7−→ β4 ;

k5α5 7−→ β5 ; k6α6 7−→ β6 ; k7α7 7−→ β7; (6.51)

then the equation (6.50) becomes

8∑a=1

λaαa (x) = −ie1β1 − ie2β2 − ie3β3 − ie4β4 − ie5β5 − ie6β6 − ie7β7. (6.52)

It may also be written in the following generalized compact form, i.e.

8∑a=1

λaαa (x) = −i

7∑q=1

eqβq (x) ; (6.53)

which may be written in terms of the following traceless Hermitian matrix form

−i7∑q=1

eqβq (x) =

α3 + α8√

3α1 − iα2 α4 − iα5

α1 + iα2 −α3 + α8√3

α6 − iα7

α4 + iα5 α6 + iα7 −2α8√3

. (6.54)

Now (6.48) becomes

ψ 7−→ ψp = Uψ = exp eqβq(x) . (6.55)

229


So we may write the locally gauge invariant SU(3)c, Lagrangian density in the following

form

Llocal =(iψγµDµψ −mψψ

)−1

4GaµνG

µνa ; (6.56)

where

Dµψ = ∂µψ + e eaAaµψ + g eaB

aµψ; (6.57)

Gaµν =∂µV

aν − ∂νV a

µ − q fabcVbµV

cν

=(∂µA

aν − ∂νAaµ − e fabcA

bµA

cν

)+(∂µB

aν − ∂νBa

µ − g fabcBbµB

cν

). (6.58)

Here in equations (6.57) and (6.58), the e and g are the coupling constants due to the

occurrence of respectively the electric and magnetic charges on dyons. On the similar

ground the two gauge fields Aµ and Bµ are present in the theory due to the occurrence

of respectively the electric and magnetic charges on dyons. As such, in the present theory

we have two kinds of color gauge groups respectively associated with the two gauge fields

of electric and magnetic charges on dyons. Hence the locally gauge covariant Lagrangian

density is written as

Llocal =(iψγµ∂µψ −mψψ

)− e

(ψγµψ

)eaA

aµ

−g(ψγµψ

)eaB

aµ −

1

4GaµνG

µνa ; (6.59)

which leads to the following expression for the gauge covariant current density of coloured

dyons

Jaµ = e(ψγµψ

)ea + g

(ψγµψ

)ea; (6.60)

230


which leads to the conservation of Noetherian current in octonion formulation of SU(3)c

gauge theory of quantum chromo dynamics (QCD) i.e.

DµJµ = 0; (6.61)

where Jµ = Jµaλa.

6.5 Quaternion-Octonion Formulation of Grand Unified The-

ory

Grand Unified Theories [44] are based on the mathematical symmetry group SU(5). The

gauge group of the Glashow-Salam -Weinberg theory SU(2)×U(1), and the SU(3) group of

the strong interaction form part of a larger symmetry. The simplest group that incorporates

the product SU(3)C×SU(2)L×U(1) as a subgroup is SU(5). An arbitrary unitary matrix can

be represented in terms of an exponential of a Hermitian matrix H [45] as,

U = eiH , H† = H; (6.62)

H is called the generating matrix for U , which can be written as

U = exp(iδH

)≈1 + iδH. (6.63)

The multiplication of two matrices U1, U2 corresponds to the sum of the infinitesimal Her-

mitian matrices as,

U2U1 ≈ 1 + i(δH2 + δH1

); (6.64)

where quadratic terms have been neglected. A complete set of linearly independent, Her-

231


mitian matrices is termed a set of generators for the unitary matrices. The unitary condition

UU † = 1 and the uni-modular condition detU = 1 leaving the 52 − 1 = 24 independent

matrices defining SU(5). U may be described as [46],

U = exp

(−i

24∑=1

AaµLa

); (6.65)

in the equation (6.65) La contains 24 generators which are Hermitian and traceless and

Aaµ is 5 × 5 matrix. Thus, the SU(5) symmetry splits into the strong force along with the

SU(2) × U(1) gauge symmetry of electroweak force. The SU(2) × U(1) gauge symmetry

also split into a SU(2) sub symmetry of the weak interaction and the U(1) sub symmetry of

the electromagnetic interaction. Here it is to be cleared that the 5× 5 matrices L such that

the colour group SU(3) acts on first three rows and columns describing by octonions, while

the SU(2) group operates on the last two rows and columns described by quaternions and

U (1) is the singlet associated with complex numbers.Thus the algebra of SU (5) illustrates

asO×Q×C. O is used for octonions, Q for quaternions and C for complex numbers. O has

the connection with SU (3), Q describes SU (2) and C is linked with U (1). This gives the

SU (3) × SU (2) × U (1) subgroup structure of SU (5) in terms of the constituents of three

division algebras namely octonions O, quaternion Q and the algebra of complex numbers

C. Let us break up a 5× 5 square matrix SU (5) into four blocks, consisting of two smaller

squares and two rectangles. The upper left-hand corner block will be a 3 × 3 sub matrix

while the lower right-hand corner block will be a 2 × 2 sub matrix. Off-diagonal upper

right-hand and lower left-hand corners consist respectively the 3× 2 and 2× 3 rectangular

matrices. The generators are described as the first eight generators are associated with the

SU(3) generators defined as [46]

La =

λa 0

0 0

; (6.66)

where a = 1, 2, .... 8 and λ are the well known 3 × 3 Gell Mann matrices. Here replace

232


Gell Mann matrices λ by octonion basis elements. For the 9th, 10th and 11th generators, we

use quaternion basis elements in the 2 × 2 block which is related to Pauli matrices σj, 1 is

the 2 × 2 unit matrix and q1, q2, q3 are the quaternion units are connected with Pauli-spin

matrices as

e0 = 1; ej = −iσj; (6.67)

L8+j =

0 0

0 iej

; (6.68)

where j = 1, 2, 3 belonging to the three generators of SU(2) gauge group. The L12 describes

the hyper charge corresponding to U (1) gauge group associated with the scalar part of a

quaternion (or complex),

L12 =1√15diag (−2,−2,−2, 3, 3) =

1√15

−2I3 0

0 3I3

. (6.69)

To define the next set of matrices, it will be convenient to define rectangular matrices A

and B as [47],

A1 =

1 0

0 0

0 0

; A2 =

0 0

1 0

0 0

; A3 =

0 0

0 0

1 0

; (6.70)

and

B1 =

0 1

0 0

0 0

; B2 =

0 0

0 1

0 0

; B3 =

0 0

0 0

0 1

. (6.71)

233


Accordingly, the 13th to 24th of SU (5) generators are expressed as,

L13,15,17 =L11+2k = e0

0 Ak

ATk 0

= e0L11+2k;

L14,16,18 =L12+2k = −e3L11+2k;

L19,21,23 =L17+2k = e0

0 Bk

BTk 0

= e0L17+2k;

L20,22,24 =L18+2k = e3L17+2k; (6.72)

where k = 1, 2, 3. Hence we may define the precise association between the vector gauge

bosons of SU(5) GUT’s, vector gauge bosons of the electro-weak model (W±µ , W

3µ , Bµ) and

the eight vector gluons Gaµ of QCD as [46],

A1,2,.....8µ =Ga

µ (∀a = 1, 2....., 8) ;

A9,10µ =W±;

A11µ =W 3

µ ;

A12µ =Bµ. (6.73)

The new vector mesons, which are only specific to SU(5) and not the Standard Model, are

expressed as,

A13,14.......18µ =Xb

µ (∀b = 1, 2, ..., 6) ;

A19,20,......24µ =Y b

µ (∀b = 1, 2, ..., 6) . (6.74)

Generalizing equations (6.72 - 6.74), then the 5 × 5 matrix representation for the SU (5)

gauge field Aµ is written in a compact form as,

234


Aµ =√

224∑i=1

AiµLi; (6.75)

which can also be expressed as,

Aaµ =

P Q

R S

; (6.76)

where

P = Gµ −2√30BµI ; Q =

X

1Y

1

X2Y

2

X3Y

3

; (6.77)

and

R =

X1 X2 X3

Y1 Y2 Y3

; S =

W3√2

+ 3B√30

W+

W− −W3√2

+ 3B√30

. (6.78)

The breaking of SU(5) into SU(3) × SU(2) × U(1) can be done in a similar way as the

breaking of SU(3) into SU(2) × U(1). The S generator in the adjoint representation of

SU(5) commutes with SU(3) × SU(2) × U(1). The 24 adjoint representation with vacuum

values in the S direction will consequently break SU(5) into SU(3)× SU(2)× U(1). Under

SU(3)× SU(2)× U(1) the decomposition of the adjoint 24 representation is [48, 49],

[24]5 =(8, 1) + (3, 2) + (3, 2) + (1, 3) + (1, 1). (6.79)

Since the gauge bosons for strong interactions due to the exchange of quarks and colors

are the eight gluon associated with the color gauge group SU (3)c and for the electro weak

235


interactions SU (2) × U (1) are the intermediate Weak bosons and photons. Following ar-

guments be made accordingly as

(i) There is a SU(3) octet of gluons Gaµ = (1, 2, ..... 8)7−→ (8, 1) .

(ii) There is an iso-vector of intermediate bosons , W iµ (i = 1, 2, 3) 7−→ (1, 3).

(iii) There is an iso-scalar field of the hyper charged boson Bµ 7−→ (1, 1).

In addition, to the above three arguments there are twelve more gauge bosons, belong-

ing to the representations (3, 2) and(3, 2

). These gauge bosons form an isospin doublet of

bosons and their antiparticles, which are colored. The simplest representation of SU(5) is

the five dimensional fundamental one ψ5 which may be represented by a column matrix

ψ5 =

a1

a2

a3

a4

a5

. (6.80)

In the SU(2) symmetry of weak interactions the quaternions occur only on the rows 4 and

5 and we see that a1, a2 and a3 are unaffected by the operation of SU(2) generators i.e.

quaternions. For the case of SU(5), the covariant derivative may be written in terms of

matrix representation as,

Dµ =∂µ −ig

2Aµ. (6.81)

Then all 24 gauge bosons Aiµ (i = 1, ..., 24) are conveniently represented by a 5× 5 matrix.

Employing these 24 generators of SU(5), we can write the equation (6.75) in the following

form,

Aµ =√

224∑a=1

AaµLa =

[8∑

a=1

GaµLa +

11∑a=9

AaµLa +BµL12 +24∑

a=13

AaµLa

]. (6.82)

236


In the equation (6.82) we substitute the value of∑8

a=1 GaµLa in terms of octonion units as,

8∑a=1

GaµLa =

7∑a=1

Gaµea +G8

µe3; (6.83)

and in the second term contains the value of L9, L10, L11 are defined in terms of quaternion

units as

8∑a=1

λaαa (x) =− i

7∑q=1

eqβq (x) ; (6.84)

and −i∑7

q=1 eqβq (x) is described in terms of the matrix form as,

−i7∑q=1

eqβq (x) =

α3 + α8√

3α1 − iα2 α4 − iα5

α1 + iα2 −α3 + α8√3

α6 − iα7

α4 + iα5 α6 + iα7 −2α8√3

; (6.85)

Hence the equation (6.82) can be written as

Aµ =

−i∑7q=1 eqβ

q (x) B

C −i∑3

i=1Wiµei

+Bµ

2√

15diag (−2,−2,−2, 3, 3) ; (6.86)

by substituting the value of P, Q, ,R, S the 5× 5 matrix becomes [46],

Aµ =

α3 + α8√3− 2Bµ√

15α1 − iα2 α4 − iα5 X

1Y

1

α1 + iα2 −α3 + α8√3− 2Bµ√

15α6 − iα7 X

2Y

2

α4 + iα5 α6 + iα7 −2α8√3− 2Bµ√

15X

3Y

3

X1 X2 X3 W 3µ + 3Bµ√

15W 1µ − iW 2

µ

Y1 Y2 Y3 W µ1 + iW 2

µ −W 3µ + 3Bµ√

15

.

(6.87)

237


The covariant derivative (6.81) associated with W 3µ and Bµ be related in terms of the cou-

pling of Aµ and Zµ by extracting the 11th and 12th generators of covariant derivative [49]

as,

Dµ =∂µ − ig

2

(W 3µL

11 +BµL12)

=∂µ − ig

2

[Aµ(sin θWL11 + cos θWL12

)+ Zµ

(cos θWL11 − sin θWL12

)](6.88)

where θW is the Weinberg angle.


Starting with the generalized potential of dyons in U (1) × U (1) gauge theory given by

equation (6.1). We have made an attempt to discuss U (1)× SU (2) gauge theory of dyons.

It is emphasized that the quaternion gauge theory may be expressed as the combination

U (1) × SU (2) gauge theory whereas the U (1) part is associated with its real quaternion

unit (e0) and the symmetry group SU (2) may be visualized in terms of three imaginary

quaternion units ej(∀j = 1, 2, 3) . As such the collection of tensors that is matrices in in-

ternal space with local symmetry group SU (2) may be associated with pure imaginary

quaternion limits. As such the internal covariant derivative in U (1) × SU (2) gauge the-

ory behaves as quaternion covariant derivative or Q-derivative. So the internal covariant

derivative in U (1)×SU (2) gauge theory defined as the quaternion derivative be expressed

as equation (6.2). Equation (6.3) describes the affinity Γµ as the object which makes the

Q-derivative like a vector under transformation in internal space. Here the vector Cµ plays

the role of gauge potential in internal SU (2) non-Abelian gauge space. The gauge poten-

tial and gauge field strength for electric charge in quaternionic version of U (1) × SU (2)

gauge theory are expressed by equations (6.4 - 6.5). Consequently the equations (6.6 -

6.7) demonstrates the dynamics of dual charge (i.e. monopole) for which the gauge po-

tential and gauge field strength in quaternionic version of U (1) × SU (2) gauge theory. In

equations (6.8) and (6.9) the generalized four-potential Vµ and generalized field tensor

Gµν of dyons are expressed as quaternion valued in terms of abelian U (1) and non-Abelian

238


SU (2) gauge coupling strengths where the real quaternion unit (e0) describes the U (1)

Abelian gauge field while the pure imaginary unit quaternions are related with the SU (2)

non-Abelian Yang-Mill’s field gauge structure. So, the equations (6.8 - 6.9) are invariant

under the local and global phase transformation. The Q-derivative for U (1)×SU (2) gauge

theory is described by equation (6.10) which is analogous to the covariant derivative for

U (1) × SU (2) gauge theory of dyons is expressed by equation (6.11) whose electric and

magnetic constituents and responsible for the components of covariant derivative are ex-

pressed by equation (6.12). It is shown by equation (6.13) is that the commutator of the

covariant derivative of U (1) × SU (2) gauge symmetry yields the field strength of dyons

and thus reproduces the field equation (6.14) for U (1) × SU (2) gauge theory of dyons.

It shows that dyons are not point particles but consists of extended structure in mass and

size. Equation (6.15) represents the generalized Q-four current of dyons in quaternion

U (1) × SU (2) gauge formalism leads to the abelian and non-Abelian nature current. The

Noetherian conservation of current has been expressed by equation (6.16) which shows the

resemblance with Noetherian current analogous to the continuity equation in abelian gauge

theory. It should be noted that usually the current is no more conserved in terms with ordi-

nary derivative but with Q-derivative. Consequently we have discussed the U (1) × SU (2)

gauge theory of gravito-dyons which is extended form in U (1) × U (1) gauge potentials

given by equation (6.17). The quaternionic extension of U (1) × U (1) gauge theory of

gravito-dyons may be visualized as the two fold degeneracy of U (1)× SU (2) gauge theory

when the first and second fold degeneracy of U (1)×SU (2) are respectively associated with

gravitational charge and Heavisidian monopole in quaternionic version of U (1) × SU (2)

gauge theory as described by equations (6.18 - 6.21). The gauge potential and gauge field

strength of gravito-dyon in terms of U (1)× SU (2) quaternion gauge theory are expressed

by equations (6.22) and (6.23). Equation (6.24) represents the covariant derivative of

gravito-dyons in U (1) × SU (2) gauge symmetry whereas equation (6.25) describes the

components of covariant derivative in gravito and Heavisidian fields. The Q-derivative of

generalized field tensor Gµν of gravito-dyons which yields the correct field equation for non-

Abelian gauge theory of gravito-dyons and gives rise the extended structure is described by

equation (6.26). It is shown that the Q-derivative of generalized field tensor of gravito-

239


dyon given by equation (6.27) represents the field equation of gravito-dyons. So, we have

obtained the generalized Q-four current of gravito-dyons in quaternion U (1)×SU (2) gauge

formalism expressed by equation (6.28). Here also the equation (6.29) shows the resem-

blance with Noetherian current analogous to the continuity equation in abelian gauge the-

ory of gravito-dyons. In general this current is no more conserved in terms of its ordinary

derivative while Q-derivative of the generalized quaternionic current leads to Noetherian

conservation law. In section 6.4.1, we have described the Gell - Mann λ matrices for SU (3)

symmetry by equations (6.30 - 6.32). Section 6.4.2, describes the inter connectivity be-

tween the properties of octonions and Gell Mann λ matrices. So, we have obtained the

relationship between the structure constants of octonion basis elements and Gell Mann λ

matrices shown in equations (6.33 - 6.34). Relation between the octonions and the Gell

Mann λ matrices has been developed in equations (6.35 - 6.47). In section 6.4.3, we have

described the octonion quantum chromodynamics by using the essential algebraic proper-

ties of octonions, Gell Mann λ matrices and related structures parallel to quaternions. The

basic triplets of SU(3)c gauge group is described in equation (6.48). The transformation

of ψ in equation (6.49) shows the generalization to the phase transformations that gives

the construction of local gauge theory which is the three components of the quark field ψ

in color space based on the concept of colors. Here strong interactions are well defined

by a local gauge theory based on the exact color group in terms of octonion units. The

value of λaαa (x) is given in equations (6.50 - 6.53) and its matrix form has been described

by equation (6.54). The transformation of octonion spinor ψ is given in equation (6.55).

The locally gauge invariant SU(3)C Lagrangian is described in equation (6.56) whereas the

covariant derivative Dµ is written in equation (6.57) describing the gauge principle in gen-

eralized form. Generalized field tensor Gaµν is given in equation (6.58) and it is shown that

the two gauge fields Aµ and Bµ describes the theory due to the presence of the electric

and magnetic charges on dyons. As such, in the present theory we have two kinds of color

gauge groups respectively associated with the two gauge fields of electric and magnetic

charges on dyons. With the help of locally gauge invariant Lagrangian written as equation

(6.59), we have found the expression for gauge covariant current density (6.60) for colored

dyons which leads to the conservation of Noethern current (6.61) in octonion formulation

240


of SU(3)C gauge theory of quantum chromodynamics. In section 6.5, an arbitrary unitary

matrix has been represented in equation (6.62) in terms of an exponential of a Hermitian

matrix which satisfies equation (6.63). The multiplication of two matrices U1 and U2 repre-

sented in equation (6.64). For SU(5) gauge group, the unitary matrix has been described

by equation (6.65), in which La contains 24 generators which are Hermitian and traceless

and Aaµ is 5 × 5 matrix. Thus, the SU(5) symmetry splits into the strong force along with

the SU(2)×U(1) gauge symmetry of electroweak force. The SU(2)×U(1) gauge symmetry

also split into a SU(2) sub symmetry of the weak interaction and the U(1) sub symmetry of

the electromagnetic interaction. Here it is to be cleared that the 5× 5 matrices L such that

the colour group SU(3) acts on first three rows and columns describing by octonions, while

the SU(2) group operates on the last two rows and columns described by quaternions and

U (1) is the singlet associated with complex numbers.Thus the algebra of SU (5) illustrates

as O×Q×C where O is used for octonions, Q for quaternions and C for complex numbers.

O-derivative has the connection with SU (3), Q describes SU (2) and C is connected to U (1)

gauge group. This gives the SU (3)× SU (2)× U (1) subgroup structure of SU (5) in terms

of the constituents of three division algebras namely octonions O, quaternion Q and the al-

gebra of complex numbers C. Let us break up a 5× 5 square matrix SU (5) into four blocks,

consisting of two smaller squares and two rectangles. The upper left-hand corner block will

be a 3 × 3 sub matrix while the lower right-hand corner is represented in terms 2 × 2 sub

matrix. Off-diagonal upper right-hand and lower left-hand corners consist respectively the

3 × 2 and 2 × 3 rectangular matrices. The generators have been chosen such that the first

eight generators are associated with the SU(3) generators, defined as equation (6.66) in

terms of the Gell Mann λ matrices. In this case replacing Gell Mann matrices λ by octonion

basis elements. For the 9th, 10th and 11th generators, we use quaternion basis elements in

the 2× 2 block which is related to Pauli matrices σj, 1 is the 2× 2 unit matrix and q1, q2, q3

are the quaternion units. Connection between quaternions and Pauli matrices is given by

equation (6.67) whereas the L9, L10 and L11- generators are described in equation (6.68) in

terms of quaternion basis elements which describe the SU(2) gauge symmetry. L12 assigned

as hyper charge defined by equation (6.69). The rectangular 3×2 matrices A and B are de-

fined by equations (6.70) and (6.71). Accordingly, the 13th to 24th generators are expressed

241


by equations (6.72). The association between the vector mesons SU(5) GUT’s vector gauge

bosons of the electroweak model(W±µ ,W

3µ , Bµ

)and eight vector gluons of QCD is described

in equations (6.73). The new vector mesons which are only specific to SU(5) and not the

standard model are expressed by equations (6.74). The compact form of Aµ is described

by equations (6.75 - 6.76) which is further reduced to equation (6.77) which components

are defined in equation (6.78). The breaking of SU(5) into SU(3) × SU(2) × U(1) can be

done in a similar way as the breaking of SU(3) into SU(2) × U(1). The S generator in

the adjoint representation of SU(5) commutes with SU(3)× SU(2)× U(1). The 24 adjoint

representation with vacuum values in the S direction will consequently break SU(5) into

SU(3)× SU(2)× U(1). Under SU(3)× SU(2)× U(1), the decomposition of the adjoint 24

representation given in equation (6.79). Since the gauge bosons for strong interactions due

to the exchange of quarks and colors are the eight gluon associated with the color gauge

group SU (3)c and for the electro weak interactions SU (2) × U (1) are the intermediate

Weak bosons and photons. Following arguments be made accordingly as

(i) There is a SU(3) octet of gluons Gaµ = (1, 2, ..... 8)7−→ (8, 1) .

(ii) There is an iso-vector of intermediate bosons , W iµ (i = 1, 2, 3) 7−→ (1, 3).

(iii) There is an iso-scalar field of the hyper charged boson Bµ 7−→ (1, 1).

In addition, to the above three arguments there are twelve more gauge bosons, belong-

ing to the representations (3, 2) and(3, 2

). These gauge bosons form an isospin doublet

of bosons and their antiparticles, which are colored. The simplest representation of SU(5)

is the five dimensional fundamental representation ψ5 which may be expressed as the col-

umn matrix given by equation (6.80). In the SU(2) symmetry of weak interactions the

quaternions occur only on the rows 4 and 5 and we see that a1, a2 and a3 are unaffected

by the operation of SU(2) generators i.e. quaternions. For the case of SU(5), the covariant

derivative may be written in terms of matrix representation as equation (6.81). By employ-

ing 24 generators of SU(5), the potential Aµ are expended in equation (6.82). The value

of∑8

a=1GaµLa in terms of octonions basis elements describes equation (6.83). Relation of

Gell Mann λ matrices with the octonion basis elements are described in equation (6.84)

and R.H.S. of equation (6.84) is expressed by equation (6.85) in terms of 3 × 3 matrices

whereas Aµ are describes in a compact form by equation (6.86) while its expended form in

242


terms of 5× 5 matrix form is expressed in equation (6.87). The covariant derivative (6.81)

associated with W 3µ and Bµ be related in terms of the coupling of Aµ and Zµ by extracting

the 11th and 12th generators giving rise to Weinberg angle as equation (6.88). The foregoing

analysis describes with the double fold degeneracy of electroweak gauge theory of standard

model whose difficulties are super nominated in grand unified theories. We have shown

that the quaternion and octonion are the better tools to expressed the theory of standard

model and the physics beyond the standard model where the monopole and dyons play

an important role. It is also concluded that the division algebra approach to the theory of

unification of fundamental interactions as the case of GUT’s leads to the consequences to-

wards the new understanding of these theories which incorporate the existence of magnetic

monopole and dyon. Three different imaginaries associated octonion formulation may be

identified with three different colors (red, blue and green) while the Gell Mann Nishijima

are described in terms of simple and compact notations of octonion basis elements.

243

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unification of strong, weak and electromagnetic interactions”, Nucl. Phys., B135

(1978), 66.

[45] G. Muller, “Gauge theory of weak interactions”, 4th edition, Springer (2009).

[46] Pushpa, P. S. Bisht, Tianjun Li and O. P. S. Negi, “Quaternion Octonion reformula-

tion of grand unified theories”, Int. J. of Theo. Phys., 51 (2012), 3328.

[47] M. Kaku, “Quantum field theory”, Oxford University Press, (1994).

[48] G. Ross, “Grand Unified Theories”, Westview Press, (1984).

[49] H. Georgi, “Unified Gauge Theories”, Proceedings, Theories and Experiments in

High Energy Physics, New York (1975), 329.

247

LIST OF PUBLICATION

1. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Generalized Gravi-Electromagnetism”,

Inter. J. of Theor. Phys., 49 (2010), 1344.

2. O. P. S. Negi, Gaurav Karnatak, H. Dehnen and P. S. Bisht, “Generalization of

Schwinger - Zwanziger Dyon to Quaternion”, Inter. J. of Theor. Phys., 50 (2011), 1908.

3. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Abelian gauge and two potential

theory of dyons and gravito-Dyons”, (Communicated).

4. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Validity of Ehrenfest Theorem and

Energy Momentum Tensor for dyons”, (Communicated).

5. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Two potential formalism for non-

Abelian gauge theory of dyons”, (Communicated).

6. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Quaternion-Octonion gauge for-

mulation and unification of fundamental interaction”, (Communicated).

7. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Generalized Gravi-Electromagnetism”,

Abstract book of “Conference on Gravitation Theories and Astronomy (CGTA)” , Bhadrawati,

Chandrapur, Maharashtra, India, during 28-30 Dec. 2009, pp 53.

8. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Quaternionic Gravi-Electromagnetism”,

Abstract book of 5th Uttarakhand State Science and Technology Congress-2010, Doon Uni-

versity, Dehradun, Uttarakhand, (India) during 10-12 Nov. 2010, pp 240.

9. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Abelian gauge and two poten-

tial theory of dyons and gravito-dyons”, Abstract book of “National Seminar on Recent

248

List of Publication

Trends In Micro and Macro Physics-2011 (NSRTMMP-2011)” , Government Post Graduate

College Gopeshwar Chamoli, Uttarakhand, (India) during 12-13 Oct. 2011, pp. 48.

10. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Validity of Ehrenfest theorem in

generalized fields of dyons”, Abstract book of “6th Uttarakhand State Science and Tech-

nology Congress-2011” , Kumaun University, S. S. J. Campus, Almora, Uttarakhand, (India)

during 14-16 Nov. 2011, pp 234.

11. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Two potential formalism for

non-Abelian gauge theory of dyons”, Abstract book of “14th International Conference of

Internal Academy of Physical Sciences (COINAPS-XIV) or Physical Sciences Interface with

Humanity”, S. V. National Institute and Technology, Surat during 22-24 Dec. 2011, pp. 122.

12. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Ehrenfest’s theorem for general-

ized field of dyons”, Abstract book of “National Conference on Recent trends in material

science and nano structure (RTMSNS-12)” , Department of Physics, Govt. P. G. College

Rudrapur, Uttarakhand, (India) during 3-4 Jan. 2012, pp 38.

13. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Quaternion-Octonion gauge for-

mulation and unification of fundamental interaction”, Abstract book of “7th Uttarak-

hand State Science and Technology Congress-2012” , Graphic Era Institute, Dehradun, Ut-

tarakhand, (India) during 22-24 Nov. 2012, pp 258.

249

CONFERENCES AND PAPER PRESENTED THEREIN

1. Gaurav Karnatak , P. S. Bisht and O. P. S. Negi, “Generalized Gravi-Electromagnetism”,

Conference on Gravitation Theories and Astronomy (CGTA), Bhadrawati, Chandrapur, Ma-

harashtra, (India) during Dec. 28-30, 2009.

2. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Gravi-Electromagnetism”, State

Level Conference on “Role of Science and Technology in the Development of Uttarakhand

State” , Department of Physics, Bageshwar, Uttarakhand, (India) during Oct. 25-26, 2010.

3. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Quaternionic Gravi-Electromagnetism”,

in 5th Uttarakhand State Science and Technology Congress-2010, Doon University, Dehradun,

Uttarakhand, (India) during Nov. 10-12, 2010.

4. Gaurav Karnatak, P. S. Bisht, O. P. S. Negi and H. Dehnen, “Validity of Ehrenfest

Theorem for Generalized Fields of dyons”, in 6th Uttarakhand State Science and Tech-

nology Congress-2011, Kumaun University, S. S. J. Campus, Almora, Uttarakhand, (India)

during Nov. 14-16, 2011.

5. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Two potential formalism for non-

Abelian gauge theory of dyons”, in 14th International Conference of Internal Academy of

Physical Sciences (COINAPS-XIV) or Physical Sciences Interface with Humanity, S. V. Na-

tional Institute and Technology, Surat during Dec. 22-24, 2011.

6. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Ehrenfest’s theorem for generalized

field of dyons”, National Conference on Recent trends in material science and nano struc-

ture (RTMSNS-12), Department of Physics, Govt. P. G. College Rudrapur, Uttarakhand,

(India) during Jan. 03-04 2012.

250

Conferences and paper presented there in

7. Gaurav Karnatak, P. S. Bisht and O. P. S. Negi, “Quaternion-Octonion gauge formu-

lation and unification of fundamental interaction”, in 7th Uttarakhand State Science and

Technology Congress-2012, Graphic Era Institute, Dehradun, Uttarakhand, (India) during

Nov. 22-24, 2012.

251

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