generalized gravity in clifford spaces and grand unification

8/8/2019 Generalized Gravity in Clifford Spaces and Grand Unification

1/21

Generalized Gravity in Clifford Spaces,Vacuum Energy and Grand Unication

Carlos CastroCenter for Theoretical Studies of Physical Systems

Clark Atlanta University, Atlanta, GA. 30314; [email protected]

August, 2010

Abstract

Polyvector-valued gauge eld theories in Clifford spaces are used toconstruct a novel Cl (3, 2) gauge theory of gravity that furnishes modifiedcurvature and torsion tensors leading to important modifications of thestandard gravitational action with a cosmological constant. Vacuum so-lutions exist which allow a cancellation of the contributions of a verylarge cosmological constant term and the extra terms present in the mod-ied eld equations. Generalized gravitational actions in Clifford-spacesare provided and some of their physical implications are discussed. It isshown how the 16 fermions and their masses in each family can be ac-commodated within a Cl (4) gauge eld theory. In particular, the Higgselds admit a natural Clifford-space interpretation that differs from the

one in the Chamseddine-Connes spectral action model of Noncommuta-tive geometry. We nalize with a discussion on the relationship with thePati-Salam color-avor model group SU (4) C SU (4) F and its symmetrybreaking patterns. An Appendix is included with useful Clifford algebraicrelations.

1 Introduction

Clifford algebras are deeply related and essential tools in many aspects inPhysics. The Extended Relativity theory in Clifford-spaces ( C -spaces ) is a nat-ural extension of the ordinary Relativity theory [1] whose generalized polyvector-valued coordinates are Clifford-valued quantities which incorporate lines, areas,volumes, hyper-volumes.... degrees of freedom associated with the collectiveparticle, string, membrane, p-brane,... dynamics of p-loops (closed p-branes) inD -dimensional target spacetime backgrounds.

Dedicated to the memory of Gustavo Ponce

1


2/21

C-space Relativity naturally incorporates the ideas of an invariant length(Planck scale), maximal acceleration, non-commuting coordinates, supersym-metry, holography, higher derivative gravity with torsion; it permits to studythe dynamics of all (closed) p-branes, for different values of p, on a unied foot-ing [1]. It resolves the ordering ambiguities in QFT [2]; the problem of time inCosmology and admits superluminal propagation (tachyons) without violationsof causality [3], [1]. The relativity of signatures of the underlying spacetimeresults from taking different slices of C-space [4], [1]. Ideas very close to theextended Relativity in Clifford spaces have been considered by [6] and [7].

The conformal group in spacetime emerges as a natural subgroup of theClifford group and Relativity in C-spaces involves natural scale changes in thesizes of physical objects without the introduction of forces nor Weyls gaugeeld of dilations [1]. A generalization of Maxwell theory of Electrodynamicsof point charges to a theory in C-spaces involves extended charges coupled toantisymmetric tensor elds of arbitrary rank and where the analog of photons aretensionless p-branes. The Extended Relativity Theory in Born-Clifford PhaseSpaces with a Lower and Upper Length Scales and the program behind a CliffordGroup Geometric Unication was advanced by [8].

Furthermore, there is no EPR paradox in Clifford spaces [9] and Clifford-space tensorial-gauge elds generalizations of Yang-Mills theories and the Stan-dard Model allows to predict the existence of new particles (bosons, fermions)and tensor-gauge elds of higher-spins in the 10 TeV regime [10], [11]. Clifford-spaces can also be extended to Clifford-Superspaces by including both orthogo-nal and symplectic Clifford algebras and generalizing the Clifford super-differentialexterior calculus in ordinary superspace to the full edged Clifford-Superspaceoutlined in [12]. Clifford-Superspace is far richer than ordinary superspace andClifford Supergravity involving polyvector-valued extensions of Poincare and

(Anti) de Sitter supergravity (antisymmetric tensorial charges of higher rank)is a very relevant generalization of ordinary supergravity with applications inM -theory.

Grand-Unication models in 4D based on the exceptional E 8 Lie algebrahave been known for sometime [14]. The supersymmetric E 8 model has morerecently been studied as a fermion family and grand unication model [15].Supersymmetric non-linear sigma models of Exceptional Kahler coset spaces areknown to contain three generations of quarks and leptons as (quasi) Nambu-Goldstone superelds [16]. A Chern-Simons E 8 Gauge theory of Gravity wasproposed [17] as a unied eld theory (at the Planck scale) of a Lanczos-LovelockGravitational theory with a E 8 Generalized Yang-Mills eld theory which isdened in the 15D boundary of a 16D bulk space. In particular, it was discussedin [12] how an E 8 Yang-Mills in 8D, after a sequence of symmetry breaking

processes E 8 E 7 E 6 SO (8, 2), leads to a Conformal gravitational theoryin 8D based on gauging the conformal group SO (8, 2) in 8D. Upon performinga Kaluza-Klein-Batakis [18] compactication on CP 2 , involving a nontrivialtorsion, leads to a Conformal Gravity-Yang-Mills unied theory based on theStandard Model group SU (3) SU (2) U (1) in 4D. Furthermore, it was shown[12] how a conformal (super) gravity and (super) Yang-Mills unied theory in

2


3/21

any dimension can be embedded into a (super) Clifford-algebra-valued gaugeeld theory by choosing the appropriate Clifford group.

A candidate action for an Exceptional E 8 gauge theory of gravity in 8 Dwas constructed [19]. It was obtained by recasting the E 8 group as the semi-direct product of GL(8, R ) with a deformed Weyl-Heisenberg group associatedwith canonical-conjugate pairs of vectorial and antisymmetric tensorial gener-ators of rank two and three. Other actions were proposed, like the quartic E 8group-invariant action in 8D associated with the Chern-Simons E 8 gauge the-ory dened on the 7-dim boundary of a 8D bulk. To nalize, it was shown howthe E 8 gauge theory of gravity can be embedded into a more general extendedgravitational theory in Clifford spaces associated with the Cl(16) algebra.

Quantum gravity models in 4 D based on gauging the (covering of the)GL (4, R) group were shown to be renormalizable by [20] however, due to thepresence of fourth-derivatives terms in the metric which appeared in the quan-tum effective action, upon including gauge xing terms and ghost terms, theprospects of unitarity were spoiled. The key question remains if this novelgravitational model based on gauging the E 8 group may still be renormalizablewithout spoiling unitarity at the quantum level.

Most recently it was shown in [21] how a Conformal Gravity and U (4) U (4) Yang-Mills Grand Unication model in four dimensions can be attainedfrom a Clifford Gauge Field Theory in C -spaces (Clifford spaces) based on the(complex) Clifford Cl (4, C ) algebra underlying a complexied four dimensionalspacetime (8 real dimensions). Upon taking a real slice, and after symmetrybreaking, it leads to ordinary Gravity and a Yang-Mills theory based on theStandard Model group SU (3) SU (2) U (1) in four real dimensions. Otherapproaches to unication based on Clifford algebras can be found in [13].

Having presented some of the relevant issues behind the role of Clifford

algebras we outline the contents of this work. In 2 we construct a novel Cl (3, 2)gauge theory of gravity that furnishes modified curvature and torsion tensorsleading to important modifications of the standard gravitational action witha cosmological constant. Vacuum solutions exist which allow a cancellation of the contributions of very large cosmological constant term with the extra termspresent in the modied eld equations. Generalized gravitational actions in C -spaces are provided and some of their physical implications are discussed. In thenal section we describe how the 16 fermions and their masses in each familycan be accommodated within a Cl (4) gauge eld theory. In particular, how theHiggs elds admit a natural C -space interpretation that differs from the one inthe Chamseddine-Connes spectral action model of Noncommutative geometry[37]. We nalize with a discussion on the relationship with the Pati-Salamcolor-avor model SU (4) C SU (4) F and its symmetry breaking patterns. An

Appendix is included with useful Clifford algebraic relations.

3


4/21

2 Extended-Gravity in Clifford Spaces as a Gauge

Field Theory based on the Clifford Group

A model of Emergent Gravity with the observed Cosmological Constant from aBF-Chern-Simons-Higgs Model was recently revisited [24] which allowed to showhow a Conformal Gravity, Maxwell and SU (2) SU (2) U (1) U (1) Yang-Mills Unication model in four dimensions can be attained from a CliffordGauge Field Theory in a very natural and geometric fashion. In particular [21]a Conformal Gravity-Maxwell model can be constructed from a Clifford gaugeeld theory based on a Cl (1, 3) algebra. Chamseddine [27] has studied theU (2, 2) gravity model using the Cl (2, 2) algebra generators.

The Cl (3, 1) algebra-valued anti-Hermitian one-form was dened as

A = i a 1 + b 5 + ea a + f

a 5 a 5 +

14

ab ab dx

. (2.1)

where 5 = 1234 , (5)2 = 1. The elds ea , f a are related to the physicalvielbein eld (tetrad) V a that gauges the translation P a symmetry, and the physical eld V a 5 that gauges the conformal boosts K a transformations, asfollows

ea a + f a 5 a 5 = V

a P a + V

a 5 K a e

a =

12

(V a + V a 5

); f a =

i2

(V a V a 5

);(2.2)

The above relations are found after recurring to a Clifford algebra realization of the translation and conformal boost generators given by

P a = 12a (1+ i 5); K a = 12

a (1 i 5); ( 5)2 = 1; a = 1 , 2, 3, 4 (2.3)

such that [ P a , P b] = [K a , K b] = 0 and [P a , K b] ab D + M ab . The Lorentzgenerators and dilations are realized as M ab = 12

ab and D = i5 . The elda was identied with the Maxwell eld and b with the Weyl gauge eld of dilations.

The problem (caveat) with the above realization of the momentum operatorP a = 12 a (1 + i5) is that it constrains P a to be nilpotent P 1P 1 = P 2P 2 =P 3P 3 = P 4P 4 = 0, and also P a P b = 0 for any pair of indices a = b, due to theconditions {a , 5} = 0 and (1+ i5)(1 i5) = 1+( 5)2 = 1 1 = 0. Therefore,such realization (2.3) leads to a trivial commutator [ P a , P b] = P a P b P bP a =0 0 = 0. The same results apply to the conformal boosts generators as well

K 1K 1 = ........ = K 4K 4 = 0 and K a K b = 0. In the case of the Cl (2, 2) algebraone has ( 5)2 = 1, and the operator realizations P a = a (1 + 5), K a =a (1 5) lead to the same conclusions. The same occurs for the other Cliffordalgebras Cl (1, 3), Cl (4, 0), Cl (0, 4), irrespective of the signature.

To solve this problem, while obtaining an extended gravitational theory inC -spaces from a gauge eld theory, we shall recur to the Cl (3, 2) algebra and

4


5/21

write the gauge connection AM = AI M I , I = 1 , 2, 3, ......, 32 in terms of the 32Cl (3, 2) algebra generators

I : 1 ; a = 1 , 2 , 3 , 4 , 5 ; a 1 a 2 =12

a 1 a 2 =12

[a 1 , a 2 ];

a 1 a 2 a 3 =13!

a 1 a 2 a 3 ; ........, a 1 a 2 ......a 5 =15!

a 1 a 2 ......... a 5(2.4)

The decomposition of the connection AM = AI M I contains Hermitian andanti-Hermitian components. By suitably introducing i factors in the appropriateterms one may render all the components Hermitian or anti-Hermitian if desired.

It is common practice to split the de Sitter/Anti de Sitter algebra gaugeconnection in 4 D into a (Lorentz) rotational piece a 1 a 2 a 1 a 2 where a1 , a 2 =1, 2, 3, 4; , = 1 , 2, 3, 4, and a momentum piece a 5

a 5 = 1

lV a

P a , where V a is

the physical vielbein eld, l is the de Sitter/Anti de Sitter throat size, and P ais the momentum generator with a = 1 , 2, 3, 4. One may proceed in the samefashion in the Clifford algebra Cl (3, 2), Cl (4, 1), .... case. The poly-momentumgenerator corresponds to those poly-rotations with a component along the 5-thdirection in the internal space.

In odd dimensions there is no chirality operator. The spinorial representa-tions in D = 2 n +1 and D = 2 n are both 2 n dimensional. One cannot representthe 2 5 = 32 generators of the Clifford algebra Cl (5) in terms of 32 independent(22 22 = 4 4) matrices because only 16 matrices (out of the 32) would havebeen independent . For example, the 6 generators of the semi-simple algebraso(4) = su (2) su (2) cannot be represented in terms of 6 independent (2 2)matrices. Nevertheless, they can be represented in terms of 6 independent (4 4)

matrices of the form

L i =12

( i 1 22); Rj =

12

(1 22 j ); i, j = 1 , 2, 3. (2.5)

given in terms of the tensor products of the unit 2 2 matrix 1 and the threePauli spin 2 2 matrices i , i = 1 , 2, 3 obeying i j = ij + i ijk k . Thecommutators are

[L i , L j ] = i ijk Lk ; [R i , R j ] = i ijk R k ; [L i , R j ] = 0 (2.6)

Similarly, one may represent the 32 generators of the Cl (3, 2) = Cl (3, 1)L Cl (3, 1)R algebra in terms of 16 + 16 = 32 independent 8 8 matrices LA , R A(instead of 4 4 matrices) obtained from the tensor products of the 16 (4 4)matrices A with the unit 2 2 matrix 1 as follows

LA =12

(A 1 22); R B =12

(1 22 B ); A, B = 1 , 2, 3, ......, 16. (2.7a)

5


6/21

such that

[LA , LB ] = f ABC LC ; [R A , R B ] = f ABC R C ; [LA , R B ] = 0 (2.7b)where f ABC are the structure constants of the Cl (3, 1) algebra. We must empha-size that in this section we will not be concerned with nding matrix represen-tations of the Cl (3, 2) gauge algebra but with the Cl (3, 2) algebra commutatorsper se. Therefore, one may assign

5 = P 0 ; a 5 = l P a , a = 1 , 2, 3, 4; a 1 a 2 5 = l2 P a 1 a 2 , a1 , a 2 = 1 , 2, 3, 4

a 1 a 2 a 3 5 = l3 P a 1 a 2 a 3 , a1 , a 2 , a3 = 1 , 2, 3, 4

a 1 a 2 a 3 a 4 5 = l4 P a 1 a 2 a 3 a 4 , a1 , a 2 , a 3 , a 4 = 1 , 2, 3, 4; (2.8)

In this way the 16 components of the ( noncommutative ) poly-momentum opera-

tor P A = P 0 , P a , P a 1 a 2 , P a 1 a 2 a 3 , P a 1 a 2 a 3 a 4 are identied with those poly-rotationswith a component along the 5-th direction in the internal space. A length scalel is needed to match dimensions.

P 0 does not transform as a Cl (3, 2) algebra scalar, but as a vector. P a doesnot transform as a Cl (3, 2) vector but as a bivector. P a 1 a 2 does not transformas Cl (3, 2) bivector but as a trivector, etc.... What about under Cl (3, 1) trans-formations ? One can notice [ ab , 5] = [ab , P 0] = 0 when a, b = 1 , 2, 3, 4.Thus under rotations along the four dimensional subspace, 5 = P 0 is inert, itbehaves like a scalar from the four-dimensional point of view. This justies thelabeling of 5 as P 0 . The commutator

[ab , c5] = [ab , l P c ] = ac b5 + bc a 5 = ac l P b + bc l P a (2.9)

so that c5 = lP c does behave like a vector under rotations along the four-dimsubspace. Thus this justies the labeling of c5 as lP c , etc...

To sum up, one has split the Cl (3, 2) gauge algebra generators into twosectors. One sector represented by M which comprises poly-rotations along thefour -dim subspace involving the generators

1 ; a 1 ; a 1 a 2 ; a 1 a 2 a 3 ; a 1 a 2 a 3 a 4 , a1 , a 2 , a 3 , a 4 = 1 , 2, 3, 4. (2.10)

and another sector represented by P involving poly-rotations with one coordi-nate pointing along the internal 5-th direction as displayed in (2.8).

Thus their commutation relations are of the form

[P , P ] M ; [M , M ] M ; [M , P ] P . (2.11)

which are compatible with the commutators of the Anti de Sitter, de Sitteralgebra SO (3, 2), SO (4, 1) respectively. To sum up, we have decomposed theCl (3, 2) gauge connection one-form in C -space as

AM dX M = A I M I dX M = AM A + E AM P A dX M ; A M , P A P (2.12)

6

7/21

where X = X M M is a C -space poly-vector valued coordinate

X = s 1 + x + x 1 2 1 2 + x 1 2 3 1 2 3 + ....... (2.13)

In order to match dimensions in each term of (2.3) a length scale parameter mustbe suitably introduced. In [1] we introduced the Planck scale as the expansionparameter in (2.3). The scalar component s of the spacetime poly-vector valuedcoordinate X was interpreted by [5] as a Stuckelberg time-like parameter thatsolves the problem of time in Cosmology in a very elegant fashion.

Denoting the derivatives with respect to the poly-vector valued coordinatesby M , the analog of the Abelian U (1) eld strength sector is R 0MN = [M 0N ].The other relevant components of the Cl (3, 2)-valued gauge eld strengths F AMN that are written as R AMN , for reasons that will become clear below, are givenby

R aMN = [M aN ] + mnM

rN < [ mn , r ]

a > + mnpqM rstN < [ mnpq , rst ]

a > .(2.14)

where the brackets < [ mn , r ] a >, < [ mnpq , rst ] a > in (2.14) indicatethe scalar part of the product of the Cl (3, 2) algebra elements; i.e it extractsthe Cl (3, 2) invariant contribution. For example,

< [ mn , r ] a > = < mr n a > + < nr m a > = mr an + nr am .

(2.15)The commutation relations among the gamma generators of any rank and in anydimension are provided in the Appendix. The C -space version of the curvaturetwo-form is

R a 1 a 2MN = [M a 1 a 2N ] +

mM

rN < [ m , r ]

a 1 a 2 > + mnM rsN < [ mn , rs ]

a 1 a 2 > +

mnpM rstN < [ mnp , rst ]

a 1 a 2 > + mnpqM rstuN < [ mnpq , rstu ]

a 1 a 2 > +

mnpqkM rstuvN < [ mnpqk , rstuv ]

a 1 a 2 > . (2.16)

To evaluate the torsion component T 0MN P 0 = R 5MN 5 requires writing

R 5MN = [M 5N ] + f 5BC

BM

C N = [M

5N ] +

mnM rN < [ mn , r ]

5 > + mnpqM rstN < [ mnpq , rst ]

5 > . (2.17)

To evaluate the torsion component T aMN P a = 1l Ra 5MN a 5 requires writing

T a

MN = Ra 5MN = [M

a 5N ] +

mM

rN < [ m , r ]

a 5> +

mnM rsN < [ mn , rs ]

a 5 > +

mnpM rstN < [ mnp , rst ]

a 5 > + mnpqM rstuN < [ mnpq , rstu ]

a 5 > +

mnpqkM rstuvN < [ mnpqk , rstuv ]

a 5 > . (2.18)

7

8/21

For example, when M, N are both vector indices one arrives at the modifiedtorsion

T a = R a 5 = [ a 5 ] +

m r < [ m , r ]

a 5 > + mn rs < [ mn , rs ]

a 5 > +

mnp rst < [ mnp , rst ]

a 5 > + mnpq rstu < [ mnpq , rstu ]

a 5 > +

mnpqk rstuv < [ mnpqk , rstuv ]

a 5 > . (2.19)

Form ( 2.19) one can see that the Cl (3, 2)-algebraic expression for the torsionT a contains many more terms than the standard expression for the torsion inRiemann-Cartan spacetimes

T a dx dx = R a 5 dx

dx = l (d a 5 + ab

b5 ) =

d V a + ab V b. (2.20)

The vielbein one-form is V a = V a dx = l a 5 dx and the spin connectionone-form is ab = ab dx (it is customary to denote the spin connection by abinstead).

For example, when M is a bivector index and N is a scalar index, there is acurvature term of the form

R a 1 a 2 1 2 0 = [ 1 2 a 1 a 20 ] +

m 1 2

r0 < [ m , r ] a 1 a 2 > +

mn 1 2 rs0 < [ mn , rs ] a 1 a 2 > + mnp 1 2

rst0 < [ mnp , rst ] a 1 a 2 > +

mnpq 1 2 rstu0 < [ mnpq , rstu ] a 1 a 2 > + mnpqk 1 2

rstuv0 < [ mnpqk , rstuv ] a 1 a 2 > .

(2.21)where the bivector derivative in C -space is

1 2 =

x 1 2(2.22)

The components R a 1 a 2 1 2 0 must not be confused with the components of themodified curvature tensor

R a 1 a 2 = [ a 1 a 2 ] +

m

r < [ m , r ]

a 1 a 2 > + mn rs < [ mn , rs ]

a 1 a 2 > +

mnp rst < [ mnp , rst ]

a 1 a 2 > + mnpq rstu < [ mnpq , rstu ]

a 1 a 2 > +

mnpqk

rstuv < [ mnpqk , rstuv ]

a 1 a 2

> . (2.23)The standard curvature tensor is given by

R a 1 a 2 = [ a 1 a 2 ] +

mn

rs < [ mn , rs ]

a 1 a 2 > . (2.24)

which clearly differs from the modied expression in (2.23).

8

9/21

Since the indices m,n,r,s in general run from 1 , 2, 3, 4, 5 the standard cur-vature two-form becomes

R a 1 a 2 dx dx = d a 1 a 2 + a 1c

ca 2 55 a 1 5 a 2 5 =

d a 1 a 2 + a 1c ca 2 55

1l2

V a 1 V a 2 ; a 5 =1l

V a (2.25)

where the vielbein one-form is V a = V a dx . In the l limit the last terms1l 2 V

a 1 V a 2 in (2.25) decouple and one recovers the standard Riemmanian

curvature two-form in terms of the spin connection one form a 1 a 2 = a 1 a 2 dx

and the exterior derivative operator d = dx . From (2.25) one infers thata vacuum solution R a 1 a 2 = 0 in de Sitter/ Anti de Sitter gravity leads to therelation

R a 1 a 2 () d a 1 a 2 + a 1c ca2 = 1

l255 V a 1 V a 2 (2.26)

which is tantamount to having a constant Riemannian scalar curvature in 4 DR() = (12/l 2) and a cosmological constant = (3/l 2); the positive (nega-tive) sign corresponds to de Sitter (anti de Sitter space) respectively ; i.e. thede Sitter/ Anti de Sitter gravitational vacuum solutions are solutions of theEinstein eld equations with a non-vanishing cosmological constant.

A different approach to the cosmological constant problem can be taken asfollows. The modified curvature tensor in (2.23) is

R a 1 a 2 = Ra 1 a 2 + extra terms =

d a 1 a 2

+ a 1c

ca 2

551l2 V

a 1

V a 2

+ extra terms (2.27)vacuum solutions R a 1 a 2 = 0 imply

d a 1 a 2 + a 1c ca 2 =

1l2

55 V a 1 V a 2 extra terms. (2.28)

Consequently, as a result of the extra terms in the right hand side of (2.28)obtained from the extra terms in the denition of R a 1 a 2 in (2.23), it could bepossible to have a cancellation of a cosmological constant term associated toa very large vacuum energy density (LPlanck )4 ; i.e. one would have aneffective zero value of the cosmological constant.

For instance, one could have a cancellation (after neglecting the terms of higher order rank in eq-(2.28) ) to the contribution of the cosmological constantas follows

m n < [ m , r ]

a 1 a 2 > + m 5 r 5 < [ m 5 , r 5]

a 1 a 2 > = 0

a 1 a 2 55 a 1 5 a 2 5 = 0 . (2.29a)

9


10/21

Since the Cl (3, 2) algebra corresponds to the Anti de Sitter algebra SO (3, 2)case one has

55 = 1 V a

l= a 5 = i a (2.29b)

Hence, one can attain a cancellation of a very large cosmological constant termin (2.29) if a 5 = i a . In the de Sitter case, 55 = 1 and one would haveinstead the condition a 5 = a . Having an imaginary value for a in the Antide Sitter case ts into a gravitational theory involving a complex Hermitianmetric G = g( ) + ig[ ] which is associated to a complex tetrad E a =

12 (ea + if a ) such that G = ( E a )E b ab and the elds are constrained to

obey ea = V a ; if a = iV a = l a . For further details on complex metrics(gravity) in connection to Borns reciprocity principle of relativity [22], [23]involving a maximal speed and maximum proper force see [24] and referencestherein.

It is desirable to solve the full-edge eld equations in C -space and after-wards verify whether or not such condition (2.29) is consistent with the solutionsto the full set of eld equations. Most likely, it would be necessary to includeall the higher order rank terms in eq-(2.28). In order to solve the C -space grav-itational eld equations one must evaluate all of the remaining components of the Cl (3, 2) curvature (eld strength) R AMN and torsion T AMN in C -space, whereM, N are poly-vector valued coordinate indices X M = s, x , x , .....,x as-sociated with the C -space corresponding to the Cl (3, 1) four-dim spacetimealgebra. These expressions are very complicated. Once the expressions forR AMN , T AMN are known one can construct many actions in C -space that are in-variant under the internal Cl (3, 2) gauge transformations as well as invariantunder the Cl (3, 1) transformations associated with the poly-vector valued coor-dinates X M of the underlying C -space base manifold. The integration measure

in C -space isDX = ds ( dx ) ( dx ) ( dx ) dx (2.30)

a quadratic curvature/torsion invariant action in C -space, up to numerical fac-tors required to match units, is given by

S = DX |det G | AB R AM 1 N 1 R BM 2 N 2 + T AM 1 N 1 T BM 2 N 2 GM 1 M 2 GN 1 N 2 .(2.31)

The C -space metric GMN has for components

G 1 2 ...... n 1 2 ...... n = g 1 1 g 2 2 ....... g n n + signed permutations(2.32a)

The components G1

2

...... n 1

2

...... n in C -space can also be written as a de-terminant of the n n matrix whose entries are g I J as follows

G 1 2 ...... n 1 2 ...... n =1n! i 1 i 2 .....i n j 1 j 2 ....j n

g i 1 j 1 g i 2 j 2 ....... g i n j n .(2.32b)

10


11/21

and the range of indices is i1 , i2 , .....,i n I = 1 , 2, .....,D and j 1 , j 2 , .....,j n J = 1 , 2, .....,D . One must also include in the C -space metric GMN the (Clifford)scalar-scalar component G00 (that could be related to the dilaton eld) andthe pseudo-scalar/pseudo-scalar component G 1 2 ..... D 1 2 ...... D (that couldbe related to the axion eld). The expression for det G involves the product of the determinants associated with G 1 2 ...... n 1 2 ...... n for n = 0 , 1, 2, ......, D .

To simplify matters, we may just concentrate in the ordinary 4 D spacetimeactions comprised of the vector coordinates x . Even in this case, one nds clearmodifications to the standard gravitational actions due to the Cl (3, 2) algebraicstructure. One can introduce an SO (3, 2)-valued scalar multiplet 1 , 2 , ....., 5and construct an SO (3, 2) invariant action of the form

S = M d4x 5 R ab R cd + a R bc R d5 + ........ abcd 5 . (2.33)as described above the modified curvature two-form R ab dx dx is given bythe standard expression R ab () dx dx + 1l 2 V

a dx V b dx plus the addition

of many extra terms as shown in (2.23). Also the modied torsion R a 5 dx dx in (2.18) is given by the standard torsion expression plus extra terms.Therefore, by a simple inspection, the action (2.33) after setting a = 0 , 5 =5o = constant which breaks the SO (3, 2) symmetry down to the Lorentz groupsymmetry SO (3, 1), contains many more terms than the Macdowell-Mansouri-Chamseddine-West gravitational action given by (after supressing spacetimeindices for convenience)

S = 5o d4x ( R ab () + 1l2 V a V b ) ( R cd () + 1l2 V cV d ) abcd . (2.34)it is comprised of the topological invariant Gauss-Bonnet term R ab ()R cd () abcd ;the Einstein-Hilbert term 1l 2 R

ab ()V cV d abcd , and the cosmological constantterm 1l 4 V

aV b V c V d abcd .

A quadratic Cl (3, 2) gauge invariant action in a 4 D spacetime involving themodied curvature R A and torsion terms T A in eqs-(2.18, 2.23), is given by

d4x |g| [ (R 0 )2 + ( R a )2 + ( R a 1 a 2 )2 + ........ (R a 1 a 2 a 3 a 4 )2 +(R 5 )2 + ( R a 5 )2 + ( R a 1 a 5 )2 + ........ (R a 1 a 2 a 3 5 )2 + ( R a 1 a 2 a 3 a 4 5 )2 ] (2.35)

The modications to the ordinary scalar Riemmanian curvature R() isgiven in terms of the inverse vielbein V

aby the expression R a 1 a 2

V [

[a 1V ]

a 2 ]which

is comprised of R(), plus the cosmological constant term , plus the extra termsstemming from the additional connection pieces in (2.23)

a 1 a 2 , a 1b1 b2 b1 b2 a 2 , ......., a 1b1 b2 b3 b4

b1 b2 b3 b4 a 2 (2.36)

11


12/21

one of many SO (3, 2) invariant actions in ordinary spacetime, linear in thecurvature is

S =1

22 d4x |g| R a 1 a 2 V [[a 1 V ]a 2 ]; g = V a V b ab , |g| = |det g |.(2.37)

where 2 = 8 G N , GN is the Newtonian gravitational constant and the compo-nents of the curvature two-form are antisymmetric under the exchange of indicesby construction R a 1 a 2 = R a 1 a 2 , R a 1 a 2 = R a 2 a 1 . The action (2.37) containsclear modifications to the Einstein-Hilbert action with a cosmological constantdue to the extra terms (2.36) stemming from the higher rank connection ele-ments.

The components of the gauge connection a 5 1 2 , ab 1 2 in C -space must notbe identied in general with the ordinary torsion and curvature two-form inRiemann-Cartan spaces, despite the correspondence

a 5 1 2 dx 1 2 T a 1 2 dx

1dx 2 (2.38)

a 1 a 2 1 2 dx 1 2 R a 1 a 2 1 2 dx

1dx 2 (2.39)

where dx 1 2 is a bivector differential form involving areas in C -space. If anidentication is made in eqs-(2.38, 2.39) the generalized gravitational action inC -space (given by the Clifford space scalar-curvature version of the Einstein-Hilbert Lagrangian) can be decomposed into sums of terms involving higherpowers of the ordinary curvature and torsion [1]. Such actions are comprisedof higher derivatives. The Lanczos-Lovelock gravitational actions are based onhigher powers of the curvature tensor but with the key feature that the eldequations do not contain terms of higher derivatives than order two for themetric tensor. Gravitational actions in Noncommutative spaces can also beconstructed based on star products and the results obtained for poly-vectorvalued gauge eld theories in Noncommutative C -spaces [25]. NoncommutativeClifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomor-phisms in Clifford-spaces would require quantum Hopf algebraic deformations of Clifford algebras. Generalized poly-vector-valued supersymmetry algebras in C spaces based on antisymmetric tensor-spinorial coordinates have been recentlystudied in [26]. These novel algebraic structures and the study of generalizedsuper-gravitational theories in supersymmetric Clifford spaces deserve furtherinvestigation.

One of the most salient features of the Cl (3, 2) algebra modications togravity is the very plausible cancellation mechanism of a very large vacuumenergy density as described in eqs-(2.27-2.29). This procedure is very differentthan the other approaches to the resolution to the cosmological constant problembased on scaling/conformal symmetry [28], for example.

12

13/21

3 Yang-Mills, Fermion Masses and Unication

It was recently shown [21] how an unication of Conformal Gravity and aU (4) U (4) Yang-Mills theory in four dimensions could be attained from aClifford Gauge Field Theory in C -spaces (Clifford spaces) based on the (com-plex) Clifford Cl (4, C ) algebra underlying a complexied four dimensional space-time (8 real dimensions). Tensorial Generalized Yang-Mills in C -spaces (Clif-ford spaces) based on poly-vector valued (anti-symmetric tensor elds) gaugeelds AM (X ) and eld strengths F MN (X ) have been studied in [1], [10] whereX = X M M is a C -space poly-vector valued coordinate. A Clifford geometricbasis of the standard model has been advanced by [29]. In this last section wedescribe how the 16 fermions in each family and their masses can be accom-modated within a Cl (4) gauge eld theory and how the Higgs elds admit anatural C -space interpretation that differs from the one in the Chamseddine-Connes spectral action model of Noncommutative geometry [37].

The 16 fermions (quarks and leptons) of the rst generation can be arrangedinto the 16 entries of the 4 4 matrix associated with the A = 1 , 2, 3, ....., 16degrees of freedom corresponding to the Cl (4) gauge algebra as follows

A (A )mn

e u r ub uge dr db dg

ce ucr ucb ucg

ec dcr dcb dcg

, A (A )mn

e u r ub uge dr db dg ce ucr ucb u

cg

ec dcr dcb dcg

(3.1)where we have omitted the spacetime spinorial indices = 1 , 2, 3, 4 in each oneof the entries of the above 4 4 matrices whose row and column indices are

m, n = 1 , 2, 3, 4. In particular, e, e denote the electron and its neutrino. Thesubscripts r,b,g denote the red, blue, green color of the up and down quarks,u, d . The superscript c denotes their anti-particles. The Dirac adjoint of eachspacetime spinor entry inside the second 4 4 matrix is denoted by e, e , u r , ....and is dened as usual = (o ) . One must not confuse the gammamatrices ( ) associated with the spacetime Dirac Cl (3, 1) algebra and theinternal Cl (4) gauge algebra matrices ( a )mn , ( a 1 a 2 )mn , .......... ; a = 1 , 2, 3, 4.

By writing mn = C (C )mn , mn = A (A )mn , and attaching an extraindex i = 1 , 2, 3,....,n f indicating the fermion family, the fermionic matter ki-netic terms is given by the expression involving a trace over the 4 4 matrixindices as

Lm =

n f

i =1

mn

i

( np

i + gAnp

) pm

i ; Anp

= AB

(B )np

. (3.2)

and can be rewritten as

Lm =n f

i =1

Ai AC (i

C i ) +

n f

i =1

g Ai A

B

C i < A B C > =

13

14/21

n f

i =1

Ai ( AC i + g hABC A

B )

C i . (3.3)

where the indices i = 1 , 2, 3, ...n f extend over the number of generations (fami-lies) and A,B,C = 1 , 2, 3, ....., 16. g is the coupling constant. hABC is the scalarpart of the Clifford product < A B C > which can be written in terms of the(anti) commutators structure constants of the Cl (4) gauge algebra as follows

hABC =12

(f ABC + dABC ); [A , B ] = f ABC C ; {A , B } = dABC C

(3.4)Given the denition

mn = A (A )

mn A =

mn (

A )nm . (3.5)

one can infer that each of the quantities A for A, 1, 2, 3, ......., 16 is a linearsuperposition of all the 16 fermions in each single family. Hence, a Cl (4) gaugeinvariant mass term corresponding to a degenerate mass M for all members of a single fermion family can be written as the trace over the 4 4 matrix indicesas

M nm mn = M ( e e + e e + u r u r + d r d r + ........ ) =

M A C < A C > = M

AC A C (3.6)

where the scalar part of the Clifford product is < A C > = AC . Themass degeneracy can be lifted by introducing the interaction terms involvingthe (complex) Higgs scalars

hABC A B C (3.7)

such that the vacuum expectation values of the Higgs scalars vev valuedin the adjoint representation of the Cl (4) gauge group will break the Cl (4)symmetry and lead to a mass splitting M + 1 , M + 2 , .......,M + 16 . The sare the eigenvalues of the 16 16 mass matrix M AC appearing in

M AC A C (3.8)

and which is dened as M AC = hABC vev .A Hermitian 16 16 matrix M AC has real eigenvalues and can be diagonal-

ized by a unitary matrix. A priori there is no reason why the matrix M AC isHermitian unless one chooses judiciously the vacuum expectation values of the

 vev in the denition M AC = hABC vev . If one has an initialmassless family M = 0, by judiciously choosing the 16 parameters associatedwith the vevs vev , B = 1 , 2, 3, ......, 16, in order to render a Hermi-tian matrix M AC one may nd 16 real eigenvalues 1 , 2 , ......, 16 to coincidewith the fermion masses of e, e , u rbg , drbg and their anti-particles. To matchthe masses requires a Renormalization group ow of the values of the observed

14

15/21

fermion masses, at a given scale, to the scale at which the Cl (4) symmetry isbroken. Some of the eigenvalues have to be degenerate since the masses of thered, blue, green up quark and anti-quark are equal. Similarly, the masses of the red, blue, green down quark and anti-quark are equal. The electron andpositron mass are also equal. The neutrino is considered also to be masive.At the moment we will not introduce Majorana neutrino mass terms that leadto a very small neutrino mass for the left handed neutrino eL via the see-sawmechanism.

Yukawa couplings among all generations of the form

Y ij hABC Ai B C j ; i, j = 1 , 2, .....,n f (3.9)

will also lead to fermionic mass terms for all fermion generations after the sym-metry breaking vev = 0 and a diagonalization procedure similar to theconstruction of the CKM quark matrix in the standard model.

Next we are going to discuss the relation to the Pati-Salam model [30]. Insection 2 the Cl (3, 2) gauge eld theory model of gravity was constructed. TheCl (5, 0) algebra is isomorphic to the direct sum of the algebras Cl (4, 0)Cl (4, 0),and which in turn, is isomorphic to M (2, H ) M (2, H ), where M (2, H ) is thematrix algebra of the 2 2 matrices with quaternionic entries. The group Cl (4) Cl (4) admits a correspondence with U (4) U (4) as shown in [21], and each factorU (4) = SU (4) U (1). A unied theory of the strong, weak and electromagneticinteractions based on the avor-color symmetric group SU (4) C SU (4) F wasadvanced by Pati and Salam [30]. Another version of the Pati-Salam model isbased on the group SU (4) C SU (2) L SU (2) R . Therefore, the Cl (4) algebrais relevant in as much as it is connected to the SU (4) U (1) algebra becausethe algebra SU (4) is a key ingredient in the Pati-Salam model.

The procedure of the symmetry breaking patterns is very elaborate in gen-eral [33]. For instance, the symmetry breaking patterns for SU (N ) gauge the-ories with Higgs scalars in totally antisymmetric and symmetric representa-tions of degree k were studied by [34] by solving the extremum conditions of the SU (N ) invariant Higgs potential for the elds H a 1 a 2 ....a k . Antisymmetrictensors H [a 1 a 2 ....a k ] are the ones appearing in the components of the Cliffordpoly-vector A associated with the Clifford gauge group Cl (4).

The Pati-Salam SU (4) SU (2) L SU (2) R group arises from the symmetrybreaking of one of the SU (4) factors in SU (4) SU (4) given by SU (4) SU (2)L SU (2) R U (1) Z , see [31], [36] and references therein. This requirestaking the following vacuum expectation value (VEV) of the Higgs scalar

< > v1

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

. (3.10)

15

16/21

Taking the VEV of the other Higgs scalar

< > v21 0 0 00 1 0 00 0 1 00 0 0 3

(3.11)

leads to a breaking of SU (4) SU (3) c U (1) B L . Therefore, an overall break-ing of SU (4) SU (4) contains the Patti-Salam (PS) model in the intermediatestage as follows

SU (4) SU (4) [SU (4) SU (2) L SU (2) R ]P S U (1) Z

SU (3) c U (1) B L SU (2) L SU (2) R U (1)Z . (3.12)The Higgs Potential V (, ) involving quadratic and quartic powers of the

elds is of the form

V = m21 T r(2) + 1 [T r (2)]2 + 2 T r(4) m22 T r (2) + 3 [T r (2)]2 +

4 T r(4) + 5 T r(22) + 6 T r() . (3.13)

A further symmetry breaking

U (1) B L SU (2) R U (1) Z U (1) Y . (3.14)

requires additional Higgs elds leading to the Standard Model

SU (3) c SU (2) L U (1)Y SU (3) c U (1) EM . (3.15)

There is another symmetry-breaking branch that leads to the Standard Modeland which does not contain the PS model. This requires breaking one of theSU (4) factors as

SU (4) SU (4) SU (3) c SU (4) U (1) B L . (2.34)

leading to a partial unication model based on SU (4) U (1) B L . which canbe broken down to the minimal left-right model via the Higgs mechanism [31],[36].

To nalize we show how the Higgs elds admit a natural C -space interpreta-tion that differs from the one in the Chamseddine-Connes spectral action modelof Noncommutative geometry [37]. The C -space analog of the fermionic kineticterms (3.3) is

nf

i =1

Ai M ( AC i M + g hABC A

BM )

C i . (3.16)

where M is the derivative with respect to a C -space poly-vector valued index(/s ), (/x ), (/x ), ..... and M = 1 , , , ..... are the generatorscorresponding to the Cl (3, 1) spacetime algebra.

16

17/21

One may notice that the Yukawa self-coupling terms (within each fam-ily) furnishing mass terms for the quarks and leptons are contained in thehABC A AB0 C , hABC A AB 1 2 .... 4 C pieces (after taking the VEV of the Higgsscalars) associated to the C -space fermionic kinetic terms A M (DM )AC C .This is due to the fact that two sets of Higgs scalar elds can be identied withthe Cl (3, 1) scalar part A = AA0 and pseudo-scalar part A = AA =AA5 , respectively. The kinetic terms for the Higgs elds ( D )(D ) and(D )(D ) are contained in the F A0N F 0N A and F A5N F 5N A components, respec-tively, associated to the Cl (4) gauge elds kinetic F AMN F MN A terms in C -space.As usual, the Cl (4) eld strength in C -space is dened as

F AMN = [M AAN ] + A

BM A

C N < [B , C ]

A > = [M AAN ] + ABM A

C N f

ABC .

(3.17)Inserting the VEV of the Higgs scalars into their kinetic terms, after redeningthe elds such that the new elds have zero VEV, yields the mass terms fromthe gauge elds associated to the broken gauge symmetries.

To nalize, there are models were the 16 fermions of a single family t intothe 16 dimensional chiral spinor representation of the SO (10) gauge unicationgroup. All the 16 fermions can be assembled into a column of 16 entries. Thechirality operator in the internal group space SO (10) (associated with the Cl (10)algebra) must not be confused with the usual Dirac chirality operator 5 of the Cl (3, 1) spacetime algebra and which implies denite parity (left-handedor right-handed currents) in weak interactions [32]. A thorough study of thesymmetry breaking patterns of SO (10) and its descent into the SU (5) group of Georgi-Glashow; the Pati-Salam SU (4) c SU (2) L SU (2) R and other groupswas performed by [32]. In our Clifford algebra model based on Cl (4), the 16fermions of a single family are assembled into the 4 4 matrix entries as shownin (3.1). More work remains to be done to verify whether or not the Cliffordalgebraic approach to unication is feasible.

APPENDIXWe begin rstly by writing the commutators [ A , B ]. For pq = odd one has

[35]

[ b1 b2 .....b p , a 1 a 2 ......a q ] = 2 a 1 a 2 ......a qb1 b2 .....b p

2 p!q!2!( p 2)!(q 2)!

[a 1 a 2[b1 b2 a 3 ....a q ]b3 .....b p ] +

2 p!q!4!( p 4)!(q 4)!

[a 1 ....a 4[b1 ....b 4 a 5 ....a q ]b5 .....b p ] ......

(A.1)for pq = even one has

[ b1 b2 .....b p , a 1 a 2 ......a q ] = ( 1) p12 p!q!

1!( p 1)!(q 1)![a 1[b1

a 2 a 3 ....a q ]b2 b3 .....b p ]

( 1) p12 p!q!3!( p 3)!(q 3)!

[a 1 ....a 3[b1 ....b 3 a 4 ....a q ]b4 .....b p ] + ...... (A.2)

17


18/21

The anti-commutators for pq = even are

{ b1 b2 .....b p , a 1 a 2 ......a q } = 2 a 1 a 2 ......a qb1 b2 .....b p

2 p!q!2!( p 2)!(q 2)!

[a 1 a 2[b1 b2 a 3 ....a q ]b3 .....b p ] +

2 p!q!4!( p 4)!(q 4)!

[a 1 ....a 4[b1 ....b 4 a 5 ....a q ]b5 .....b p ] ......

(A.3)and the anti-commutators for pq = odd are

{ b1 b2 .....b p , a 1 a 2 ......a q } = ( 1) p12 p!q!

1!( p 1)!(q 1)![a 1[b1

a 2 a 3 ....a q ]b2 b3 .....b p ]

( 1) p12 p!q!3!( p 3)!(q 3)!

[a 1 ....a 3[b1 ....b 3 a 4 ....a q ]b4 .....b p ] + ...... (A.4)

For instance,

[ b , a ] = 2 ab ; [ b1 b2 , a 1 a 2 ] = 8 [a 1[b1

a 2 ]b2 ] . (A.5)

[ b1 b2 b3 , a 1 a 2 a 3 ] = 2 a 1 a 2 a 3b1 b2 b3 36 [a 1 a 2[b1 b2

a 3 ]b3 ] . (A.6)

[ b1 b2 b3 b4 , a 1 a 2 a 3 a 4 ] = 32 [a 1[b1

a 2 a 3 a 4 ]b2 b3 b4 ] + 192

[a 1 a 2 a 3[b1 b2 b3

a 4 ]b4 ] . (A.7)

[ b1 b2 b3 b4 b5 , a 1 a 2 a 3 a 4 a 5 ] = 2 a 1 a 2 a 3 a 4 a 5b1 b2 b3 b4 b5 400 [a 1 a 2[b1 b2

a 3 a 4 a 5 ]b3 b4 b5 ] + 1200

[a 1 a 2 a 3 a 4[b1 b2 b3 b4

a 5 ]b5 ] .

(A.8)

[ b1 b2 b3 , a 1 a 2 ] = 1 2 [a 1[b1

a 2 ]b2 b3 ]. (A.9)

[ b1 b2 b3 b4 , a 1 a 2 a 3 ] = 24 [a 1[b1

a 2 a 3 ]b2 b3 b4 ] + 48

[a 1 a 2 a 3[b1 b2 b3 b4 ] (A.10)

etc...

AcknowledgmentsWe thank M. Bowers for her assistance.

18


19/21

References

[1] C. Castro, M. Pavsic, Progress in Physics 1 (2005) 31; Phys. Letts B 559(2003) 74; Int. J. Theor. Phys 42 (2003) 1693.

[2] M. Pavsic, Class. Quan. Grav. 20 (2003) 2697.

[3] M. Pavsic, Found. Phys 33 (2003) 1277.

[4] M. Pavsic, Found. Phys 31 (2001) 1185.

[5] M.Pavsic, The Landscape of Theoretical Physics: A Global View, From Point Particles to the Brane World and Beyond, in Search of a Unifying Principle (Kluwer Academic Publishers, Dordrecht-Boston-London, 2001).M. Pavsic, Int. J. Mod. Phys A 21 (2006) 5905; Found. Phys. 37 (2007)1197; J.Phys. A 41 (2008) 332001.

[6] W. Pezzaglia, Classication of Multivector Theories and the Modica-tion of the Postulates of Physics gr-qc/9306006; Dimensionally Demo-cratic Calculus and Principles of Polydimensional Physics gr-qc/9912025;Polydimensional Relativity, a Classical Generalization of the Automor-phism Invariance Principle gr-qc/9608052.

[7] D. Hestenes, Found. Phys 12 (1982) 153; Gauge Gravity and ElectroweakTheory arXiv : 0807.0060 [gr-qc].

[8] C. Castro, Foundations of Physics 35 , no.6 (2005) 971.

[9] C.Castro, Adv. Stud. Theor. Phys 1 , no. 12 (2007) 603.J. Christian, Disproof of Bells Theorem by Clifford Algebra Valued LocalVariables arXiv : quant-ph/0703179.

[10] C. Castro, Annals of Physics 321 , no.4 (2006) 813.

[11] S. Konitopoulos, R. Fazio and G. Savvidy, Europhys. Lett. 85 (2009) 51001.G. Savvidy, Fortsch. Phys. 54 (2006) 472.

[12] C. Castro, IJGMMP 6 no. 3 (2009) 1-33.

[13] Frank (Tony) Smith, The Physics of E 8 and Cl (16) = Cl (8) Cl (8)www.tony5m17h.net/E8physicsbook.pdf (Carterville, Georgia, June 2008,367 pages).

[14] I. Bars and M. Gunaydin, Phys. Rev. Lett 45 (1980) 859;N. Baaklini, Phys. Lett B 91 (1980) 376;S. Konshtein and E. Fradkin, Pisma Zh. Eksp. Teor. Fiz 42 (1980) 575;M. Koca, Phys. Lett B 107 (1981) 73;R. Slansky, Phys. Reports 79 (1981) 1.

19


20/21

[15] S. Adler, Further thoughts on Supersymmetric E 8 as family and grandunication theory, arXiv.org : hep-ph/0401212.

[16] K. Itoh, T. Kugo and H. Kunimoto, Progress of Theoretical Physics 75 ,no. 2 (1986) 386.

[17] C. Castro, IJGMMP 4 no. 8 (2007) 1239.

[18] N. Batakis, Class and Quantum Gravity 3 (1986) L 99.

[19] C. Castro, IJGMMP 6 , no. 6 (2009) 911.

[20] C. Y. Lee, Class. Quantum. Grav. 9 (1992) 2001; C. Y. Lee and Y. Neeman,Phys. Letts. B 242 (1990) 59.

[21] C. Castro, IJMPA 25 , no. 1 (2010) 123.

[22] M. Born, Proc. Royal Society A 165 , 291 (1938). Rev. Mod. Physics 21 ,463 (1949).

[23] S. Low: Jour. Phys A Math. Gen 35 , 5711 (2002). J. Math. Phys. 38 ,2197 (1997).

[24] C. Castro, Phys Letts B 668 (2008) 442.

[25] C. Castro, J. Phys A : Math. Theor. 43 (2010) 365201.

[26] C. Castro, A Clifford algebra realization of Supersymmetry and its Polyvec-tor extension in Clifford Spaces submitted to Advances in Clifford Alge-bras.

[27] A. Chamseddine, Comm. Math. Phys 218 (2001) 283. J. Moffat, J. Math.Phys 36 , no. 10 (1995) 5897.

[28] C. Wetterich, Warping with dilation symmetry and self tuning of thecosmological constant [arXiv : 1003.3809].E. Mottola, New Horizons in Gravity : The Trace Anomaly, Dark Energyand Condensate Stars arXiv : 1008.5006.L. Nottale, Fractal Space-Time and Microphysics : Towards a Theory of Scale Relativity (World Scientic, Singapore 1993). L. Nottale, Int. J. Mod.Phys A 4 , 5047 (1989).C. Castro, Phys. Lett. B 675 , 226 (2009).

[29] G. Trayling and W. Baylis, J. Phys. A 34 (2001) 3309. J. Chisholm andR. Farwell, J. Phys. A 22 (1989) 1059. G. Trayling, hep-th/9912231.

[30] J. Pati and A. Salam, Phys. Rev. Lett 31 (1973) 661; Phys. Rev. D 8(1973) 1240; Phys. Rev. D 10 (1974) 275.

[31] S. Rajpoot and M. Singer, J. Phys. G : Nuc. Phys. 5 , no. 7 (1979) 871.

20


21/21

[32] S. Rajpoot, Phys. Rev. D 22 , no. 9 (1980) 2244.

[33] L. Fong Li, Phys. Rev. D 9 , no. 6 (1974) 1723.[34] P. Jetzer, J. Gerard and D. Wyler, Nuc. Phys. B 241 (1984) 204.

[35] K. Becker, M. Becker and J. Schwarz, String Theory and M-Theory pp543-545, Cambridge Univ Press 2007

[36] T. Li, F. Wang and J. Yang, The SU (3) c SU (4) U (1) B L modelswith left-right unication arXiv : 0901.2161.

[37] A. Chamseddine and A. Connes, Noncommutative Geometry as a Frame-work for Unication of all Fundamental Interactions including Gravity. PartI [ arXiv : 1004.0464] A. Chamseddine and A. Connes, The Spectral Ac-tion Principle, Comm. Math. Phys. 186 , (1997) 731. A. Chamseddine, An

Effective Superstring Spectral Action, Phys.Rev. D 56 (1997) 3555.

21

generalized gravity in clifford spaces and grand unification

Documents