The Nonsymmetric Kaluza-Klein Theory and Modern Physics. A Novel Approach
aa r X i v : . [ phy s i c s . g e n - ph ] S e p The Nonsymmetric Kaluza–Klein Theory and Modern PhysicsA Novel Approach
M. W. KalinowskiPracownia Bioinformatyki, Instytut Medycyny Doświadczalnej i Klinicznej PAN,ul. Pawińskiego 5, 02-106 Warszawa, Polande-mail: [email protected], [email protected] 15, 2018
To the memory of my teacher Professor Stanisław Szpikowski
Abstract
We consider in the paper the Nonsymmetric Kaluza–Klein Theory finding a condition fora color confinement in the theory. We consider also a Kerner–Wong–Kopczyński equation inthis theory. The Nonsymmetric Kaluza–Klein Theory with a spontaneous symmetry breakingand Higgs’ mechanism is examined. We find a mass spectrum for a broken gauge bosonsand Higgs’ particles. We derive a generalization of Kerner–Wong–Kopczyński equation in thepresence of Higgs’ field. A new term in the equation is a generalization of a Lorentz forceterm for a Higgs’ field. We consider also a bosonic part of GSW (Glashow–Salam–Weinberg)model in our theory, getting masses for W , Z bosons and for a Higgs’ boson agreed with anexperiment. We consider Kerner–Wong–Kopczyński equation in GSW model obtaining someadditional charges coupled to Higgs’ field. Introduction
In this paper we consider the Nonsymmetric Kaluza–Klein Theory in a non-Abelian case and theNonsymmetric Kaluza–Klein Theory with Higgs’ mechanism and spontaneous symmetry breakingin a new setting. The paper gives a comprehensive review of a subject with many new featureswhich are shortly summarized at the end of the introduction. Moreover, it cannot be consideredas a review paper because it contains new achievements in this rapidly developing subject.The subject of the paper is specialized of course, but it could be very interesting for a wideaudience because geometrization and unification of fundamental physical interactions are very in-teresting. This idea gives a justification for some phenomenological theories which are completelyarbitrary. There is no physics without mathematics, especially without geometry—differentialgeometry. Even Maxwell–Lorentz electrodynamics happens post factum geometrized in fibre bun-dle formalism. In the case of ordinary Kaluza–Klein Theory the geometrization and unificationhave been achieved. Unfortunately, without “interference effects”. We consider in the paper someadditional versions of the Nonsymmetric Kaluza–Klein Theory. In particular, except of a realversion we consider also Nonsymmetric Hermitian Theory in two realizations, complex and hy-percomplex. They are natural (Hermitian) metrization of a fiber bundle over a space-time. The1onsymmetric Kaluza–Klein (Jordan–Thiry) Theory (a real version) has been developed in thepast (see Refs [1]–[5]). The theory unifies gravitational theory described by NGT (NonsymmetricGravitational Theory, see Ref. [6]) and Yang–Mills’ fields (also electromagnetic field). In the caseof the Nonsymmetric Jordan–Thiry Theory this theory includes scalar field. The NonsymmetricKaluza–Klein Theory can be obtained from the Nonsymmetric Jordan–Thiry Theory by simplyputting this scalar field to zero. In this way it is a limit of the Nonsymmetric Jordan–ThiryTheory.The Nonsymmetric Jordan–Thiry Theory has several physical applications in cosmology, e.g.:(1) cosmological constant, (2) inflation, (3) quintessence, and some possible relations to the darkmatter problem. There is also a possibility to apply this theory to an anomalous accelerationproblem of Pioneer 10/11 (see Refs [7], [8]).In this paper a scalar field Ψ = 0 ( ρ = 1). Moreover, the extension to Jordan–Thiry Theoryin any nonsymmetric version is still possible and will be done elsewhere. The scalar field canplay a role as a dark matter—quintessence with weak interactions with ordinary matter. On theclassical level, this is only a gravitational interaction with the possibility to change a strength ofgravitational interaction via a change of gravitational constant. On a quantum level due to anexcitation of a quantum vacuum a very weak nongravitational interaction with ordinary matteris possible, i.e. a scattering of scalarons with ordinary matter particles and also a scattering ofskewons with those particles.The theory unifies gravity with gauge fields in a nontrivial way via geometrical unifications oftwo fundamental invariance principles in Physics: (1) the coordinate invariance principle, (2) thegauge invariance principle. Unification on the level of invariance principles is more important anddeeper than on the level of interactions for from invariance principles we get conservation laws(via the Noether theorem). In some sense Kaluza–Klein theory unifies the energy-momentumconservation law with the conservation of a color (isotopic) charge (an electric charge in anelectromagnetic case).Let us notice that an idea of geometrization and simultaneously unification of fundamentalinteractions is quite old. GR is 100 years old and Kaluza–Klein Theory is almost 100 yearsold. Both ideas: a geometrization of physical interactions and a unification are well establishedcontemporarily.This unification has been achieved in higher than four-dimensional world, i.e. ( n + 4)-dimen-sional, where n = dim G , G is a gauge group for a Yang–Mills’ field, which is a semisimple Liegroup (non-Abelian). In an electromagnetic case we have G = U(1) and a unification is in 5-dimensional world (see also [9]). The unification has been achieved via a natural nonsymmetricmetrization of a fiber bundle. This metrization is right-invariant with respect to an action of agroup G . We present also an Hermitian metrization of a fiber bundle in two versions: complex andhypercomplex. The connection on a fiber bundle of frames over a manifold P (a bundle manifold)is compatible with a metric tensor (nonsymmetric or Hermitian in complex or hypercomplexversion). In the case of G = U(1) the geometrical structure is biinvariant with respect to anaction of U(1), in a general non-Abelian case this is only right-invariant.In the paper we do not mention some “modern” Kaluza–Klein developments for the reasondescribed in Conclusion of Ref. [9] which we do not repeat here.Let us notice the following fact. We use a notion of a nonsymmetric metric as an abuse ofnomination for a metric is always symmetric. This will not cause any misunderstanding. It issimilar to an abuse of nomination in the case of Minkowski metric in Special Relativity for a2etric is always positive definite.The unification is nontrivial for we can get some additional effects unknown in conventionaltheories of gravity and gauge fields (Yang–Mills’ or electromagnetic field). All of these effects,which we call interference effects between gravity and gauge fields are testable in principle inexperiment or an observation. The formalism of this unification has been described in Refs [1]–[5], [9] (without Hermitian versions).The theory considered here is non-Abelian and even if there are some formulations similarto those from Ref. [9] one should remember that the theory described in Ref. [9] is an Abeliantheory with U(1) group. The difference is profound not only because a higher level of mathematicalcalculations but also because of completely new features which appear in a non-Abelian theory.If we can use similar formulations as in Ref. [9] it means that a geometrical language is correctto describe a physical reality.It is possible to extend the Nonsymmetric (non-Abelian) Kaluza–Klein Theory to the case of aspontaneous symmetry breaking and Higgs’ mechanism (see Ref. [1]) by a nontrivial combinationof Kaluza principle (Kaluza miracle) with dimensional reduction procedure. This consists in anextension of a base manifold of a principal fiber bundle from E (a space-time) to V = E × M ,where M = G/G is a manifold of classical vacuum states.In this paper we consider a condition for a color confinement in the theory. We solve the con-straints in the case of non-Abelian Nonsymmetric Kaluza–Klein Theory getting an exact form ofan induction tensor for Yang–Mills’ fields in the theory. We find a formula for a non-Abelian chargein the theory in comparison to 4-momentum in gravitation theory. We derive the Lagrangian forYang–Mills’ field and an energy-momentum tensor in terms of H aµν only. We consider also Non-symmetric Kaluza–Klein Theory with Higgs’ fields and spontaneous symmetry breaking. We solveconstraints in the theory getting Lagrangian for Yang–Mills’ field, kinetic energy Lagrangian for aHiggs field and Higgs potential in terms of gauge fields and Higgs fields only. We derive pattern ofmasses for a massive intermediate bosons and Higgs’ particles. We derive also a generalization ofKerner–Wong–Kopczyński equation for a test particle. In such an equation there is a new chargefor a test particle which couples a Higgs’ field to the particle. This is similar to a Lorentz forceterm in an electromagnetic case. This term is also similar to a new term coupled a Yang–Mills’field to a test particle via a color (isotopic) charge in ordinary Kerner–Wong–Kopczyński equation(see Ref. [3]).The Nonsymmetric Kaluza–Klein Theory is an example of the geometrization of fundamentalinteraction (described by Yang–Mills’ and Higgs’ fields) and gravitation according to the Einsteinprogram for geometrization of gravitational and electromagnetic interactions. It means an exam-ple to create a Unified Field Theory. In the Einstein program we have to do with electromagnetismand gravity only. Now we have to do with more degrees of freedom, unknown in Einstein times,i.e. GSW (Glashow–Salam–Weinberg) model, QCD, Higgs’ fields, GUT (Grand Unified Theories).Moreover, the program seems to be the same.We can paraphrase the definition from Ref. [10]: Unified Field Theory: any theory whichattemps to express gravitational theory and fundamental interactions theories within a single uni-fied framework. Usually an attempt to generalize Einstein’s general theory of relativity alone toa theory of gravity and classical theories describing fundamental interactions . In our case thissingle unified framework is a multidimensional analogue of geometry from Einstein Unified FieldTheory (treated as generalized gravity) defined on principal fiber bundles with base manifolds: E or E × V and structural groups G or H . Thus the definition from an old dictionary (paraphrased3y us) is still valid.Summing up, Nonsymmetric Kaluza–Klein Theory connects old ideas of unitary field theo-ries (unified field theories, see Refs [11, 12] for a review) with modern applications. This is ageometrization and unification of a bosonic part of four fundamental interactions.The paper has been divided into four sections. In the first section we give some elements ofgeometry used in the paper. In the second section we give some elements of the Nonsymmetric(non-Abelian) Kaluza–Klein Theory in some new setting. We give also a condition for the di-electric confinement of a color charge. We consider in details a non-Abelian charge, color chargein static situations. We consider two versions of the Nonsymmetric Kaluza–Klein Theory: 1. thereal version and 2. the Hermitian version (complex and hypercomplex). We shortly present thesecond version. In the third section we give some elements of Nonsymmetric Kaluza–Klein Theorywith spontaneous symmetry breaking and Higgs’ mechanism. In this section we consider also twoversions of the Nonsymmetric Kaluza–Klein Theory (real and Hermitian, complex and hypercom-plex). We derive a pattern of masses for broken intermediate bosons and Higgs’ bosons. We writedown a generalization of Kerner–Wong–Kopczyński equation in this case, getting a coupling ofHiggs’ field to a test particle. In other words, we derive an analogue of a Lorentz force term fora Higgs’ field.In the fourth section a bosonic part of GSW (Glashow–Salam–Weinberg) model, accordingto Manton (6-dimensional model) has been extended to the Nonsymmetric Kaluza–Klein Theory.We get a realistic pattern of masses of W ± , Z and Higgs’ boson. In particular, we get a mass ofa Higgs boson agreed with an experiment, which is impossible in a pure Manton model. We haveas before the G θ W = 30 ◦ (sin θ W = 0 . ξ = 0 and g µν = η µν (in Minkowski space) we calculate a small deviation δ of a bare Weinberg angle (equalto π ) as a 1-loop and 2-loop corrections using a ∆r (or ∆R ) theory known in literature. InAppendix A we give some details of calculations concerning solutions of constraints in the theory.In Appendix B we give some elements of Manton model in a connection to our approach. InAppendix C we consider the Kerner–Wong–Kopczyński equation in GSW model. We derive someexplicit influence of new charges coupled to Higgs’ field (from the SM model) on a movement of atest particle. The existence of those new charges and their influence on a test particle movementcan be tested in experiment. In Appendix D we give formulas for interactions between gravityand Higgs’ field and Yang–Mills’ fields in Hermitian Kaluza–Klein Theory in an application tobosonic part of GSW model. These are “interference effects” between nonsymmetric gravity andGSW model in a unified theory. They can be considered as effects of unification. In Appendix Ewe calculate a correction δ to a Weinberg angle (equal to π ) as radiation corrections to a bareangle using ∆r theory.In Conclusions we give some prospects for further research, in particular, how to treat fermionsin the Nonsymmetric Kaluza–Klein Theory with a spontaneous symmetry breaking and we givea sketch of a program of quantization of the Nonsymmetric Kaluza–Klein Theory.Summing up, the paper contains many novel features (without repetition of heavy calculationsfrom Refs [1]–[5], [9]:1. Hermitian versions (complex and hypercomplex) in the case of U(1) and a general non-Abelian semisimple group G also in the case with spontaneous symmetry breaking andHiggs’ mechanism. 4. Solutions of constraints appearing in the theory (also in all considered versions).3. Detailed calculations of a classical dielectric model of confinement of color (a non-Abeliangauge charge).4. Spectrum of masses for broken gauge bosons and scalar (Higgs’) particles in a general case.5. An application to bosonic part of GSW model, where we get masses for W ± , Z and Higgs’boson agreed with experiment. In the last case this is possible only for an Hermitian complexversion on S and invokes some new research connecting the theory to Kählerian structures.6. A Kerner–Wong–Kopczyński equation in GSW model with some additional charges coupleda test particle in a motion to Higgs’ field (this one from the Standard Model).7. Additional (non-classical) interaction of a Higgs’ field (from the Standard Model) withgravity (described by NGT) and also additional Higgs’ phenomena in SM.8. A possibility to tune a cosmological constant to the value obtained from observational data.For we have not any traces of GUT or supersymmetry from LHC results we do not considerextensions of our Kaluza–Klein Theory in these directions. Thus we stop (temporarily) on 20-dimensional unification of electro-weak interactions (a bosonic part) and nonsymmetric gravity(NGT) and on 12-dimensional unifications of strong interactions (a bosonic part of QCD) withnonsymmetric gravity (NGT). The inclusion of fermions is under consideration and the work isin progress together with an approach to quantization.From technical point of view we get also some additional results:9. An exact formula (a covariant one) for L aµν and L aµν (an induction tensor).10. A Lagrangian for a Yang–Mills’s field in terms of H aµν and g µν only.11. An exact formula (a covariant one) for a torsion in higher dimension Q aµν ( Γ ) with aninterpretation as a polarization of gauge field induced by g µν and ℓ ab .12. In the case of spontaneous symmetry breaking and Higgs’ mechanism, i.e. for a Kaluza–KleinTheory with a dimensional reduction we get analogous formulas for L a ˜ n ˜ b , L aµ ˜ b in terms of aHiggs’ field Φ a ˜ a and a covariant derivative gauge ∇ µ Φ a ˜ a of the field. Those formulas are covariant.We get also similar interpretations of exact formulas for torsion in higher dimensions.Let us notice that we consider geodetic equations with respect to Levi-Civita connectiongenerated by a symmetric part of any nonsymmetric tensor on P as equations of motion from avariational principle. Let us now describe the notation and definitions of geometric quantities used in the paper. Weuse a smooth principal bundle which is an ordered sequence P = ( P, F, G, E, π ) , (1.1)5here P is a total bundle manifold, F is typical fibre, G , a Lie group, is a structural group, E isa base manifold and π is a projection. In our case G = U(1), E is a space-time, π : P → E .We have a map ϕ : P × G → P defining an action of G on P . Let a, b ∈ G and ε be a unitelement of the group G , then ϕ ( a ) ◦ ϕ ( b ) = ϕ ( ba ), ϕ ( ε ) = id, where ϕ ( a ) p = ϕ ( p, a ). Moreover, π ◦ ϕ ( a ) = π . For any open set U ⊂ E we have a local trivialization U × G ≃ π − ( U ). For any x ∈ E , π − ( { x } ) = F x ≃ G , F x is a fibre over x and is equal to F . In our case we suppose G = F ,i.e. a Lie group G is a typical fibre. ω is a 1-form of connection on P with values in the algebraof G , G . Let ϕ ′ ( a ) be a tangent map to ϕ ( a ) whereas ϕ ∗ ( a ) is the contragradient to ϕ ′ ( a ) at apoint a . The form ω is a form of ad-type, i.e. ϕ ∗ ( a ) ω = ad ′ a − ω, (1.2)where ad ′ a − is a tangent map to the internal automorphism of the group G ad a ( b ) = aba − . (1.3)We may introduce the distribution (field) of linear elements H r , r ∈ P , where H r ⊂ T r ( P ) is asubspace of the space tangent to P at a point r and v ∈ H r ⇐⇒ ω r ( v ) = 0 . (1.4)So T r ( P ) = V r ⊕ H r , (1.5)where H r is called a subspace of horizontal vectors and V r of vertical vectors. For vertical vectors v ∈ V r we have π ′ ( v ) = 0. This means that v is tangent to the fibres.Let v = hor( v ) + ver( v ) , hor( v ) ∈ H, ver( v ) ∈ V r . (1.6)It is proved that the distribution H r is equal to choosing a connection ω . We use the operationhor for forms, i.e. (hor β )( X, Y ) = β (hor X, hor Y ) , (1.7)where X, Y ∈ T ( P ).The 2-form of a curvature is defined as follows Ω = hor dω = Dω, (1.8)where D means an exterior covariant derivative with respect to ω . This form is also of ad-type.For Ω the structural Cartant equation is valid Ω = dω + [ ω, ω ] , (1.9)where [ ω, ω ]( X, Y ) = [ ω ( X ) , ω ( Y )] . (1.10)Bianchi’s identity for ω is as follows DΩ = hor dΩ = 0 . (1.11)6he map f : E ⊃ U → P such that f ◦ π = id is called a section ( U is an open set).From physical point of view it means choosing a gauge. A covariant derivative on P is definedas follows DΨ = hor dΨ. (1.12)This derivative is called a gauge derivative . Ψ can be a spinor field on P .In this paper we use also a linear connection on manifolds E and P , using the formalism ofdifferential forms. So the basic quantity is a one-form of the connection ω AB . The 2-form ofcurvature is as follows Ω AB = dω AB + ω AC ∧ ω CB (1.13)and the two-form of torsion is Θ A = Dθ A , (1.14)where θ A are basic forms and D means exterior covariant derivative with respect to connection ω AB . The following relations are established connections with generally met symbols ω AB = Γ ABC θ C Θ A = Q ABC θ B ∧ θ C Q ABC = Γ ABC − Γ ACB Ω AB = R ABCD θ C ∧ θ D , (1.15)where Γ ABC are coefficients of connection (they do not have to be symmetr in indices B and C ), R ABCD is a tensor of a curvature, Q ABC is a tensor of a torsion in a holonomic frame. Covariantexterior derivation with respect to ω AB is given by the formula D Ξ A = d Ξ A + ω AC ∧ Ξ C DΣ AB = dΣ AB + ω AC ∧ Σ CB − ω CB ∧ Σ AC . (1.16)The forms of a curvature Ω AB and torsion Θ A obey Bianchi’s identities DΩ AB = 0 DΘ A = Ω AB ∧ θ B . (1.17)All quantities introduced here can be found in Ref. [13].In this paper we use a formalism of a fibre bundle over a space-time E with an electromagneticconnection α and traditional formalism of differential geometry for linear connections on E and P .In order to simplify the notation we do not use fibre bundle formalism of frames over E and P .A vocabulary connected geometrical quantities and gauge fields (Yang–Mills fields) can be foundin Ref. [14].In Ref. [15] we have also a similar vocabulary (see Table I, Translation of terminology). More-over, we consider a little different terminology. First of all we distinguished between a gaugepotential and a connection on a fibre bundle. In our terminology a gauge potential A µ θ µ is in aparticular gauge e (a section of a bundle), i.e. A µ θ µ = e ∗ ω (1.18)7here A µ θ µ is a 1-form defined on E with values in a Lie algebra G of G . In the case of a strengthof a gauge field we have similarly F µν θ µ ∧ θ ν = e ∗ Ω (1.19)where F µν θ µ ∧ θ ν is a 2-form defined on E with values in a Lie algebra G of G .Using generators of a Lie algebra G of G we get A = A aµ θ µ X a = e ∗ ω and F = F aµν θ µ ∧ θ ν X a = e ∗ Ω (1.20)where [ X a , X b ] = C cab X c , a, b, c = 1 , , . . . , n, n = dim G (= dim G ) , (1.21)are generators of G , C cab are structure constants of a Lie algebra of G , G , [ · , · ] is a commutatorof Lie algebra elements.In this paper we are using Latin lower case letters for 3-dimensional space indices. Here we areusing Latin lower case letters as Lie algebra indices. It does not result in any misunderstanding. F aµν = ∂ µ A aν − ∂ ν A aµ + C abc A bµ A cν . (1.22)In the case of an electromagnetic connection α the field strength F does not depend on gauge(i.e. on a section of a bundle).Finally it is convenient to connect our approach using gauge potentials A aµ with usually met(see Ref. [16]) matrix valued gauge quantities A µ and F µν . It is easy to see how to do it if weconsider Lie algebra generators X a as matrices. Usually one supposes that X a are matrices of anadjoint representation of a Lie algebra G , T a with a normalization conditionTr( { T a , T b } ) = 2 δ ab , (1.23)where {· , ·} means anticommutator in an adjoint representation.In this way A µ = A aµ T a , (1.24) F µν = F aµν T a . (1.25)One can easily see that if we take F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] (1.26)from Ref. [16] we get F µν = ( F aµν ) T a , (1.27)where F aµν is given by (1.22). From the other side if we take a section f , f : U → P , U ⊂ E ,and corresponding to it A = A aµ θ µ X a = f ∗ ω (1.28) F = F aµν θ µ ∧ θ ν X a = f ∗ Ω (1.29)and consider both sections e and f we get transformation from A aµ to A aµ and from F aµν to F aµν in the following way. For every x ∈ U ⊂ E there is an element g ( x ) ∈ G such that f ( x ) = e ( x ) g ( x ) = ϕ ( e ( x ) , g ( x )) . (1.30)8ue to (1.2) one gets A ( x ) = ad ′ g − ( x ) A ( x ) + g − ( x ) dg ( x ) (1.31) F ( x ) = ad ′ g − ( x ) F ( x ) (1.32)where A ( x ) , F ( x ) are defined by (1.28)–(1.29) and A ( x ) , F ( x ) by (1.20). The formulae (1.31)–(1.32) give a geometrical meaning of a gauge transformation (see Ref. [14]). In an electromagneticcase G = U(1) we have similarly, if we change a local section from e to f we get f ( x ) = ϕ ( e ( x ) , exp( iχ ( x ))) ( f : U ⊃ E → P )and A = A + dχ .Moreover, in the traditional approach (see Ref. [16]) one gets A µ ( x ) = U ( x ) − A µ ( x ) U ( x ) + U − ( x ) ∂ µ U ( x ) (1.33) F µν ( x ) = U − ( x ) F µν U ( x ) , (1.34)where U ( x ) is the matrix of an adjoint representation of a Lie group G .For an action of a group G on P is via (1.2), g ( x ) is exactly a matrix of an adjoint representationof G . In this way (1.31)–(1.32) and (1.33)–(1.34) are equivalent.Let us notice that usually a Lagrangian of a gauge field (Yang–Mills field) is written as L YM ∼ Tr( F µν F µν ) (1.35)where F µν is given by (1.25)–(1.26). It is easy to see that one gets L YM ∼ h ab F aµν F bµν (1.36)where h ab = C dac C cbd (1.37)is a Cartan–Killing tensor for a Lie algebra G , if we remember that X a in adjoint representationare given by structure constants C cab .Moreover, in Refs [1, 3] we use the notation Ω = H aµν θ µ ∧ θ ν X a . (1.38)In this language L YM = π h ab H aµν H bµν . (1.39)It is easy to see that e ∗ ( H aµν θ µ ∧ θ ν X a ) = F aµν θ µ ∧ θ ν X a . (1.40)Thus (1.39) is equivalent to (1.36) and to (1.35). (1.35) is invariant to a change of a gauge. (1.39)is invariant with respect to the action of a group G on P .Let us notice that h ab F aµν F bµν = h ab H aµν H bµν , even H aµν is defined on P and F aµν on E . Inthe non-Abelian case it is more natural to use H aµν in place of F aµν .Eventually we connect the general fibre bundle formalism and Cartan calculus with a formalismof linear connections on E , P and E × G/G . 9et M be an m -dimensional pseudo-Riemannian manifold with metric g of arbitrary signature.Let T ( M ) be a tangent bundle and O ( M, g ) the principal fiber bundle of frames (orthogonalframes) over M . The structure group O ( M, g ) is a group GL ( m, R ) or the subgroup of GL ( m, R ) O ( m − p, p ) which leaves the metric invariant. Let Π be the projection of O ( M, g ) onto M . Let X be a tangent vector at x in O ( M, g ). The canonical or soldering form e θ is an R m -valued form on O ( M, g ) whose A -th component e θ A at x of X is the A -th component of Π ′ ( X ) in the frame x . Theconnection form e ω = ω AB X BA is a 1-form on O ( M, g ) which takes its values in the Lie algebra gl ( m, R ) of Gl ( m, R ) or in o ( m − p, p ) of O ( m − p, p ) and satisfies the structure equation d e ω + 12 [ e ω, e ω ] = e Ω = e Hor d e ω (1.41)where e Hor is understood in the sense of e ω and e Ω = e Ω AB X AB is a gl ( m, R ) ( o ( m − p, p ))-valued2-form of the curvature. X AB are generators of a Lie algebra gl ( m, R ) or o ( m − p, p ). We canwrite Eq. (1.41) using R m -valued forms and commutation relations of the Lie algebra gl ( m, R )( o ( m, m − p )) e Ω AB = d e ω AB + e ω AC ∧ e ω CB . (1.42)Taking any local section of O ( M, g ), e , one can get the coefficients of the connection, curvature,basic forms and torsion e ∗ e ω AB = ω AB e ∗ ( e Ω AB ) = e Ω AB e ∗ e θ A = θ A e ∗ e Θ A = Θ A . (1.43)The forms on the right-hand side of equations (1.43) are different in Eqs (1.13)–(1.14). We callthis formalism a linear (affine) metric, Riemannian–Levi-Civita, Einstein) connections on M .In our theory it is necessary to consider at least four principal bundles: a principal fiber bun-dle P over E with a structural group G (a gauge group), connection ω and a projection π , anoperator of a horizontality hor, a principal fiber bundle P ′ of frames over ( E, g ) with a connec-tion e ω αβ X βα = ω ′ , a structural group GL (4 , R ) ( O (1 , π , a principal fiber bundle P ′′ of frames over ( P, γ ) (a metrized fiber bundle P ) witha structural group GL ( n + 4 , R ) ( O ( n + 3 , e ω AB X BA = e ω and with an operatorof horizontality hor ′ , a projection π ′ and a principal fiber bundle of frames over G with a pro-jection Π ′′ , operator of horizontality (hor) ′′ , a connection b ω and a structural group Gl ( n, R ). Inthe case with a spontaneous symmetry breaking we need even more principal bundles of frames,i.e. a principal bundle of frames over E × G/G with additional connection ω , a projection Π , anoperator of horizontality hor. In more complicated situation we can also consider a bundle over G/G with structural group GL ( n , R ). Moreover, in order to simplify considerations, we use theformalism of linear connection coefficients on manifold ( E, g ), (
P, γ ), and a principal fiber bundleformalism for P , i.e. a principal fiber bundle over E with a structural group G , a gauge group.In the case with a spontaneous symmetry breaking we have also an additional fiber bundle witha structural group H over E × G/G . I believe this is a way to make the formalism more naturaland readable. We use tensor formalism with many kinds of indices which make some formulasvery long. Moreover, they are more readable for a non-expert.10 Elements of the Nonsymmetric Kaluza–Klein Theory in gen-eral non-Abelian case and dielectric model of a color confine-ment
Let P be a principal fiber bundle over a space-time E with a structural group G which is asemisimple Lie group. On a space-time E we define a nonsymmetric tensor g µν = g ( µν ) + g [ µν ] such that g = det( g µν ) = 0 e g = det( g ( µν ) ) = 0 . (2.1) g [ µν ] is called as usual a skewon field (e.g. in NGT, see Refs [6, 9]). We define on E a nonsymmetricconnection compatible with g µν such that Dg αβ = g αδ Q δβγ ( Γ ) θ γ (2.2)where D is an exterior covariant derivative for a connection ω αβ = Γ αβγ θ γ and Q αβδ is its torsion.We suppose also Q αβα ( Γ ) = 0 . (2.3)We introduce on E a second connection W αβ = W αβγ θ γ (2.4)such that W αβ = ω αβ − δ αβ W (2.5) W = W γ θ γ = ( W σγσ − W σσγ ) θ γ . (2.6)Now we turn to nonsymmetric metrization of a bundle P . We define a nonsymmetric tensor γ on a bundle manifold P such that γ = π ∗ g ⊕ ℓ ab θ α ⊗ θ b (2.7)where π is a projection from P to E . On P we define a connection ω (a 1-form with values in a Liealgebra g of G ). In this way we can introduce on P (a bundle manifold) a frame θ A = ( π ∗ ( θ α ) , θ a )such that θ a = λω a , ω = ω a X a , a = 5 , , . . . , n + 4 , n = dim G = dim g , λ = const . Thus our nonsymmetric tensor looks like γ = γ AB θ A ⊗ θ B , A, B = 1 , , . . . , n + 4 , (2.8) ℓ ab = h ab + µk ab , (2.9)where h ab is a biinvariant Killing–Cartan tensor on G and k ab is a right-invariant skew-symmetrictensor on G , µ = const.We have h ab = C cad C dbc = h ab k ab = − k ba (2.10)11hus we can write γ ( X, Y ) = g ( π ′ X, π ′ Y ) + λ h ( ω ( X ) , ω ( Y )) (2.11) γ ( X, Y ) = g ( π ′ X, π ′ Y ) + λ k ( ω ( X ) , ω ( Y )) (2.12)( C abc are structural constants of the Lie algebra g ). γ is the symmetric part of γ and γ is the antisymmetric part of γ . We have as usual[ X a , X b ] = C cab X c (2.13)and Ω = 12 H aµν θ µ ∧ θ ν (2.14)is a curvature of the connection ω , Ω = dω + 12 [ ω, ω ] . (2.15)The frame θ A on P is partially nonholonomic. We have dθ a = λ (cid:16) H aµν θ µ ∧ θ ν − λ C abc θ b ∧ θ c (cid:17) = 0 (2.16)even if the bundle P is trivial, i.e. for Ω = 0. This is different than in an electromagnetic case(see Ref. [3]). Our nonsymmetric metrization of a principal fiber bundle gives us a right-invariantstructure on P with respect to an action of a group G on P (see Ref. [3] for more details). Having P nonsymmetrically metrized one defines two connections on P right-invariant with respect to anaction of a group G on P . We have γ AB = g αβ ℓ ab ! (2.17)in our left horizontal frame θ A . Dγ AB = γ AD Q DBC ( Γ ) θ C (2.18) Q DBD ( Γ ) = 0 (2.19)where D is an exterior covariant derivative with respect to a connection ω AB = Γ ABC θ C on P and Q ABC ( Γ ) its torsion. One can solve Eqs (2.18)–(2.19) getting the following results ω AB = π ∗ ( ω αβ ) − ℓ db g µα L dµβ θ b L aβγ θ γ ℓ bd g αβ (2 H dγβ − L dγβ ) θ γ e ω ab ! (2.20)where g µα is an inverse tensor of g αβ g αβ g γβ = g βα g βγ = δ γα , (2.21) L dγβ = − L aβγ is an Ad-type tensor on P such that ℓ dc g µβ g γµ L dγα + ℓ cd g αµ g µγ L dβγ = 2 ℓ cd g αµ g µγ H dβγ , (2.22)12 ω ab = e Γ abc θ c is a connection on an internal space (typical fiber) compatible with a metric ℓ ab suchthat ℓ db e Γ dac + ℓ ad e Γ dcb = − ℓ db C dac (2.23) e Γ aba = 0 , e Γ abc = − e Γ acb (2.24)and of course e Q aba ( e Γ ) = 0 where e Q abc ( Γ ) is a torsion of the connection e ω ab .We also introduce an inverse tensor of g ( αβ ) g ( αβ ) e g ( αγ ) = δ γβ . (2.25)We introduce a second connection on P defined as W AB = ω AB − n + 2) δ AB W . (2.26) W is a horizontal one form W = hor W (2.27) W = W ν θ ν = ( W σνσ − W σσν ) . (2.28)In this way we define on P all analogues of four-dimensional quantities from NGT (see Refs[6, 17, 18, 19]). It means, ( n +4)-dimensional analogues from Moffat theory of gravitation, i.e. twoconnections and a nonsymmetric metric γ AB . Those quantities are right-invariant with respect toan action of a group G on P . One can calculate a scalar curvature of a connection W AB gettingthe following result (see Refs [1, 3]): R ( W ) = R ( W ) − λ (cid:0) ℓ cd H c H d − ℓ cd L cµν H dµν (cid:1) + e R ( e Γ ) (2.29)where R ( W ) = γ AB (cid:0) R CABC ( W ) + R CCAB ( W ) (cid:1) (2.30)is a Moffat–Ricci curvature scalar for the connection W AB , R ( W ) is a Moffat–Ricci curvaturescalar for the connection W αβ , and e R ( e Γ ) is a Moffat–Ricci curvature scalar for the connection e ω ab , H a = g [ µν ] H aµν (2.31) L aµν = g αµ g βν L aαβ . (2.32)Usually in ordinary (symmetric) Kaluza–Klein Theory one has λ = 2 √ G N c , where G N is a New-tonian gravitational constant and c is the speed of light. In our system of units G N = c = 1 and λ = 2. This is the same as in Nonsymmetric Kaluza–Klein Theory in an electromagnetiic case(see Refs [4, 9]). In the non-Abelian Kaluza–Klein Theory which unifies GR and Yang–Mills fieldtheory we have a Yang–Mills lagrangian and a cosmological term. Here we have L YM = − π ℓ cd (cid:0) H c H d − L cµν H dµν (cid:1) (2.33)and e R ( e Γ ) plays a role of a cosmological term. 13t is easy to see that L YM is invariant with respect to an action of a group G on P (it is gaugeinvariant). e R ( e Γ ) is also gauge invariant. L cµν plays a role of an induction tensor of Yang–Mills field (a gauge field).According to Refs [1, 3] we have Q aµν ( Γ ) = 2( H aµν − L aµν ) (2.34)where Q ABC ( Γ ) is a torsion of a connection Γ . Writing L aµν in the form L aµν = H aµν − πM aµν (2.35)we get Q aµν ( Γ ) = 8 πM aµν . (2.36)One can solve Eq. (2.22) getting the result (see Appendix A): L nωµ = H nωµ + µh na k ad H dωµ + (cid:0) H nαω e g ( αδ ) g [ δµ ] − H nαµ e g ( αδ ) g [ δω ] (cid:1) − µh na k ad e g ( δτ ) e g ( αβ ) H dδα g [ τω ] g [ βµ ] − µh na k ad e g ( δβ ) e g ( ατ ) H dβ [ ω g µ ] τ g [ δα ] + 2 µ h na h bc k ac k bd e g ( αβ ) H dα [ ω g [ µ ] β ] . (2.37)In this way we get that L aωµ = − L aµω (2.38)and simultaneously Q aµν ( Γ ) has a physical interpretation as a polarization tensor of Yang–Millstheory (a difference between an induction tensor and a gauge field strength). Moreover, it seemsfrom Eq. (2.33) that L aµν plays the role of an induction tensor. Thus one can get L YM = 18 π (cid:18) h nk H kωµ H nωµ − h cd H c H d + 2 h nk H kωµ H nδω g [ αµ ] e g ( αδ ) + µ h k nk H kωµ H nδω e g ( δα ) g [ αµ ] − k kd H kωµ H dδα e g ( δβ ) e g ( αρ ) g [ βω ] g [ ρµ ] − k kd H kωµ H dηω e g ( ηβ ) e g ( αρ ) g [ µα ] g [ βρ ] + k kd H kωµ H dηω e g ( ηδ ) e g ( αρ ) g [ δβ ] g [ ωδ ] i + µ h k nk k nd H kωµ H dηµ e g ( ρβ ) e g ( ηα ) g [ ωβ ] g [ αρ ] − k nk k nd H kωµ H dδα e g ( δη ) e g ( αρ ) g [ ηω ] g [ ρµ ] − k nk k nd H kωµ H dηω e g ( ρα ) e g ( ηβ ) g [ µα ] g [ βρ ] + k kb k bd H kωµ H dαω e g ( αβ ) g [ µα ] − k kb k bd H kωµ H dαµ e g ( αβ ) g [ ωβ ] + k pn k pk H kωµ H nωµ i + µ h k nk k nb k bd H kωµ H dαω e g ( αβ ) g [ µβ ] − k nk k nb k bd H kωµ H dαµ e g ( αβ ) g [ ωβ ] i(cid:19) . (2.39)Eq. (2.39) is written in term of H aµν only. Moreover, the form L YM , i.e. Eq. (2.33), is moreconvenient for theoretical considerations. One can say the same for L aµν . One gets Q nωµ ( Γ ) = 2 (cid:16) − µh na k ad H dωµ − (cid:0) H nαω e g ( αδ ) g [ δµ ] − H nαµ e g ( αδ ) g [ δω ] (cid:1) + 2 µh na k ad e g ( δτ ) e g ( αβ ) H dδα g [ τω ] g [ βµ ] + 2 µh na k ad e g ( δβ ) e g ( ατ ) H dβ [ ω g µ ] τ g [ δα ] − µ h na h bc k ac k bd e g ( αβ ) H dα [ ω g | µ | β ] (cid:17) (2.34*)14et us introduce the following notation: H aµν = − B a B a − E a B a − B a − E a − B a B a − E a E a E a E a (2.40) L aµν = − H a H a − D a H a − H a − D a − H a H a − D a D a D a D a . (2.41)In this way we write H aµν in terms of −→ E α = ( E a ¯ a ) = ( E a , E a , E a ), ¯ a = 1 , , , and −→ B a = ( B a ¯ a ) =( B a , B a , B a ) , L aµ in terms of −→ D a = ( D ¯ aa ) = ( D a , D a , D a ) and −→ H a = ( H ¯ aa ) = ( H a , H a , H a ) .In this way E a ¯ a = H a a D ¯ aa = L a a , −→ B a = − ( H a , H a , H a ) −→ H a = − ( L a , L a , L a ) (2.42)or B a ¯ a = − ε ¯ a ¯ b ¯ c H a ¯ b ¯ c H a ¯ c ¯ m = − ε ¯ c ¯ m ¯ e B a ¯ e (2.43) H ¯ aa = − ε ¯ a ¯ b ¯ c L a ¯ b ¯ c , L a ¯ c ¯ m = − ε ¯ c ¯ m ¯ e H ¯ ea (2.44)where ε ¯ a ¯ b ¯ c , ¯ a, ¯ b, ¯ c = 1 , , , is a usual 3-dimensional antisymmetric symbol, ε = 1 and it isunimportant for it if its indices are in up or down position. We keep these indices in up or downposition only for convenience.One gets D n ¯ e = B nf ¯ d ¯ e B f ¯ d + A nf ¯ v ¯ e E f ¯ v (2.45)where B nd ¯ p ¯ e = ε ¯ m ¯ z ¯ p (cid:16) g ¯ z g ¯ m ¯ e δ nd + µk nd g ¯ z g ¯ m ¯ e + g µ ¯ e g [ δµ ] g ¯ z e g ( ¯ mδ ) δ nd − g ω g [ δµ ] g ¯ m ¯ e e g (¯ z δ nd + µk nd g µ ¯ e e g ( ατ ) g [ µτ ] g [ δα ] g ¯ z e g ( δ ¯ m ) + µk nd g ω e g ( ατ ) g [ µτ ] g [ δα ] g ¯ m ¯ e e g ( δ ¯ z ) − µ k nc k cd g µτ g [ µβ ] g ¯ z e g ( mβ ) − µ k nc k cd g ω g [ ωβ ] g ¯ m ¯ e e g (¯ zβ ) − µk nd g ω g µτ g [ τω ] g [ βµ ] e g (¯ zτ ) e g ( ¯ mβ ) (cid:17) , (2.46) A nd ¯ m ¯ e = g g ¯ m ¯ e δ nd − µk nd g ¯ m g e + µk nd g g ¯ m ¯ e + g µ ¯ e g [ δµ ] e g (4 δ ) g ¯ m δ nd − g µ ¯ e g [ δµ ] g e g ( ¯ mδ ) δ nd − g ω g [ δω ] e g (4 δ ) e g ¯ m ¯ e δ nd + g ω g [ δω ] g e e g ( ¯ mδ ) δ nd − µk nd g µ ¯ e e g ( ατ ) g [ µτ ] g [ δα ] g ¯ m ¯ e e g ( δµ ) + µk nd g µδ e g ( ατ ) g [ µτ ] g [ δα ] g e g ( δ ¯ m ) + µk nd g ω e g ( ατ ) g [ µτ ] g [ δα ] g ¯ m ¯ e e g ( δµ ) − µk nd g ω e g ( ατ ) g [ µτ ] g [ δα ] g e e g ( δ ¯ m ) − µ k nc k cd g µ ¯ e g [ µβ ] g ¯ m e g ( β − µ k nc k cd g µ ¯ e g [ µβ ] g e g ( ¯ mβ ) − µ k nc k cd g [ ωβ ] g ¯ m ¯ e e g (4 β ) e g (4 ω ) + µ k nc k cd g ω g [ ωβ ] g e e g ( ¯ mβ ) + 2 µk nd g ω g µ ¯ e g [ τω ] g [ βµ ] e g ( ¯ mτ ) e g (4 β ) + 2 µk nd g ω g µ ¯ e g [ τω ] g [ βµ ] e g (4 τ ) e g ( ¯ mβ ) − δ nd g ¯ m g e , (2.47)15 nd = C nf ¯ d ¯ p B f ¯ p + D nfd ¯ v E f ¯ v , (2.48)where C nd ¯ p ¯ f = 12 ε ¯ p ¯ e ¯ k ε ¯ w ¯ m ¯ f (cid:16) g ¯ w ¯ k g ¯ m ¯ e δ nd + µk nd g ¯ w ¯ k g ¯ m ¯ e − g ¯ w ¯ k g µ ¯ e g [ δµ ] e g ( ¯ mδ ) δ nd + µk nd g ¯ w ¯ k g µ ¯ e g [ δµ ] e g ( ¯ mδ ) δ nd − g ω ¯ k g [ δω ] g ¯ m ¯ e e g ( ¯ wδ ) δ nd − µk nd g ω ¯ k g µ ¯ e g [ τω ] g [ βµ ] e g ( ¯ wβ ) e g ( ¯ mτ ) + µk nd g ω ¯ k e g ( ατ ) g [ ωτ ] g [ βµ ] g ¯ m ¯ e e g ( ¯ wβ ) (cid:17) , (2.49) D nd ¯ p ¯ m = 12 ε ¯ p ¯ e ¯ k (cid:16) g k g ¯ m ¯ e δ nd − g ¯ m ¯ k g e δ nd − µk nd g ¯ m ¯ k g e + µk nd g ¯ e ¯ k g k + g ¯ m ¯ k g µ ¯ e g [ δµ ] e g (4 δ ) δ nd − g k g µ ¯ e e g ( ¯ mδ ) g [ δµ ] δ nd − g ω ¯ k g [ δω ] g ¯ m ¯ e e g (4 δ ) δ nd + 2 µk nd g ω ¯ k g µ ¯ e g [ τω ] g [ βµ ] e g ( ¯ mβ ) e g (4 τ ) − µk nd g ω ¯ k g µ ¯ e g [ τω ] g [ βµ ] e g (4 β ) e g ( ¯ mτ ) + µk nd g µ ¯ e g [ δα ] g [ µτ ] e g ( ατ ) g k e g ( δ ¯ m ) − µk nd g µ ¯ e g [ δα ] g [ µτ ] e g ( ατ ) g ¯ m ¯ k e g (4 δ ) + µk nd g ω ¯ k e g ( ατ ) g [ ωτ ] g [ δα ] g ¯ m ¯ e e g (4 δ ) + µ k nc k cd g µ ¯ e g [ µβ ] g ¯ m ¯ k e g (4 β ) − µ k nc k cd g µ ¯ e g [ µβ ] g k e g ( ¯ mβ ) − µ k nc k cd g ω ¯ k g [ ωβ ] g ¯ m ¯ e e g (4 β ) + µ k nc k cd g ω ¯ k g [ ωβ ] g e e g ( ¯ mβ ) . (cid:17) (2.50)The confinement condition in this theory means D ¯ aa = 0 (2.51)with E ¯ aa = 0 and can be satisfied by special arrangement of the nonsymmetric tensor g µν . Thisgeneralizes a notion of a charge confinement from Ref. [9] and can be considered as a colorconfinement in the case of G = SU (3) c (QCD).In this case gravitation behaves as a medium which generalizes a notion of bianisotropicmedium in electromagnetic theory to non-Abelian Yang–Mills field. This is a dielectric model ofconfinement.It is easy to see that if g [ µν ] = k ab = 0 we get L aµν = H aµν . We have identities concerning H aµν and L aµν coming from Eq. (2.22): g [ µν ] L aµν = h ac ℓ cp H pµν g [ µν ] (2.52) ℓ dc g σν g αµ L dσα H cµν + ℓ cd g µσ g νβ L dβσ H cµν = 2 ℓ cd g µσ g νβ H dβσ H cµν (2.53) ℓ dc g αω g βµ L dαβ L cωµ = ℓ cd g αω g βµ L dαβ L cωµ . (2.54)The problem of a confinement emerged in QCD on a quantum level. Moreover, up to nowwe have not any realistic explanation of this problem. QCD is a quantum field theory obtainedvia quantization procedure from classical Yang–Mills’ field theory in a perturbative regime. Theconfinement is a strictly non-perturbative effect. The natural way to solve the problem is to poseit on a classical level and afterwards to quantize the new theory (classical) using non-perturbativemethods to get a quantum model of the confinement. The theory is of course highly nonlinear.Nonperturbative quantization of nonlinear theories including gravity can be achieved by using16anonical quantization as in GR (Ashtekar–Lewandowski approach) or using nonlocal approach(as we shortly described in Conclusions). Even strings models need quantization.There is no dielectric classical model of confinement in a symmetric theory, i.e. with g [ µν ] = 0 ,zero skewon field.Let us give some remarks on a confinement. According to modern ideas (see [20], [21], [22],[23]) the confinement of color could be connected to dielectricity of the vacuum (dielectric modelof confinement). Due to the so-called antiscreening mechanism, the effective dielectric constantis equal to zero. This means that the energy of an isolated charge goes to infinity. There arealso so-called classical-dielectric models of confinement (see Refs [24], [25]). The confinement isinduced by a special kind of dielectricity of the vacuum, such that −→ E = 0 and −→ D = 0 ( −→ E a = 0 , −→ D a = 0 ). In this case we do not have a distribution of a charge. This is similar to the electrictype of Meissner effect.It is easy to see that in our case (the Nonsymmetric Kaluza–Klein Theory) the dielectricityis induced by the nonsymmetric tensors g µν and ℓ ab . If g [ µν ] = 0 , −→ D = −→ E , −→ B = −→ H (in anelectromagnetic case see Ref. [9]). The gravitational field described by the nonsymmetric tensor g µν behaves as a medium for an electromagnetic field (or Yang–Mills’ field). In this way theskewon field g [ µν ] plays a double role:1) additional gravitational interaction from NGT,2) a strong interaction field connected to the confinement problem.In other words we can say that we get a confinement from higher dimensions due to a torsion inhigher dimensions.In Refs [26], [27] one can find some ideas of nonlocal field theory with an application toconfinement problem which can be connected to dielectric model of confinement.There is a body of works on classical models of confinement for Abelian and non-Abeliangauge fields (see Refs [28], [29], [30]) which are not directly connected to our approach. Moreover,it is worth to mention that an idea of a confinement in QCD for SU (3) group (see Ref. [31]) canbe applied for electrodynamics in order to get a confinement of plasma in thermonuclear fusion.We do not confuse here the “confinement” problem in strong interactions (i.e. the fact thatquarks are permanently bound in hadrons and never manifest themselves as free particles, unlikeleptons) and the “confinement” problem in thermonuclear fusion. We turn only an attentionof a reader that some ideas from strong interaction “confinement” problem can be applied to athermonuclear fusion problem. Moreover, the ideas are really far away from our idea of dielectricmodel of confinement (see Refs [28]–[31]).An important problem is to find an exact solution with axial symmetry for the NonsymmetricKaluza–Klein Theory with fermion sources for G = SU (3) . This could offer us a model of a hadronwith a confinement condition ( −→ D a = 0 , −→ E a = 0 ). The axially symmetric, stationary case seems tobe very interesting from more general point of view. We have in General Relativity very peculiarproperties of stationary, axially symmetric solutions of the Einstein–Maxwell equations. Thesesolutions describe the gravitational and electromagnetic fields of a rotating charged mass. Thuswe get the magnetic field component. Asymptotically (these solutions are asymptotically flat) themagnetic field behaves as a dipole field. We can calculate the gyromagnetic ratio at infinity, i.e.the ratio of the magnetic dipole moment and the angular momentum moment. It is worth noticingthat we get the anomalous gyromagnetic ratio, i.e. the gyromagnetic ratio for an electron (for a17harged Dirac particle). We cannot interpret the Kerr–Newman solution as a model of a fermionfor we have a singularity. In the Nonsymmetric Kaluza–Klein Theory we can expect completelynonsingular solutions. We can also expect the asymptotic behavior of Einstein–Maxwell theory.Thus it seems that we probably get the solutions with an anomalous gyromagnetic ratio. Such asolution could be treated as a (classical) model of -spin particle.In a non-Abelian case (for G = SU (3) c × U (1) em ) the solution of field equations could offer usa model of a charged barion (i.e. proton), where the skewon field g [ µν ] induces a confinement ofcolor. Such solutions should be considered also for a zero charge and without and with fermionsources. Let us mention that fermion fields (quarks fields) are coupled to the Riemannian part (aLevi-Civita connection induced by g ( αβ ) metric) of the connection ω αβ on E (i.e. e ω αβ ) .Let us come back to our presentation of the Nonsymmetric Kaluza–Klein Theory. One caneasily calculate e R ( e Γ ) (see Appendix A) getting e R ( e Γ ) = − ℓ ab h ab (2.55)(it is a cosmological constant). In the Nonsymmetric Kaluza–Klein Theory (in the non-Abeliancase) we consider a special nonholonomic frame. Moreover, we can consider a different framewhich is still nonholonomic, moreover, it looks more classical. Let us take a section e : E → P and attach to it a frame v a , a = 5 , , . . . , n + 4 , selecting x µ = const on a fiber in such a way that e is given by the condition e ∗ v a = 0 and the fundamental fields ζ a such that v a ( ζ b ) = δ ab satisfy [ ζ a , ζ b ] = λ C cab ζ c . Thus we have ω = 1 λ v a X a + π ∗ ( A aµ θ µ ) X a , (2.56)where e ∗ ω = A = A aµ θ µ X a . (2.57)In this frame a tensor γ takes a form γ AB = g αβ + λ ℓ ab A aα A bβ λℓ cb A cα λℓ ac A cβ ℓ ab ! . (2.58)This frame is also unholonomic. One gets dv a = − λ C abc v b ∧ v c . (2.59)In this way a non-Abelian gauge field four-potential is a part of our theory. We present herea model of a color confinement. This model is a dielectric model of a confinement. It is aclassical model of confinement. We know that a confinement of a color is a nonperturbativeeffect. Moreover, our theory is nonlinear and contains a gravity. Thus we should quantize thetheory using Ashtekar–Lewandowski method (see Ref. [32]) or using different methods describedin Conclusions, which we mentioned already above.In our theory test particles move along geodesic equations induced by Levi-Civita connectioninduced by a symmetric part of a metric γ , i.e. γ ( AB ) . This connection has a form e ω AB = π ∗ ( e ω αβ ) − h db e g ( µα ) H dµβ θ b H aβγ θ γ h bd e g ( αβ ) H dγβ θ γ ee ω ab ! (2.60)18here ee ω ab means a connection (Levi-Civita one) induced by a Killing–Cartan metric on G .One can write a geodetic equation on P : u A e ∇ A u B = 0 , (2.61)where u B ( τ ) is a tangent vector to geodetic and e ∇ A means a covariant derivative with respect toa Levi-Civita connection e ω AB (Eq. (2.60)). One gets e Du α dτ − u b h ba e g ( αβ ) H aβγ u γ = 0 du b dτ = 0 . (2.62)This equation is written on P . We have a normalization of a four-velocity u α , g ( αβ ) u α u β = 1 .The second equation gives us a constancy of a color charge of a test particle. We can identify q b = 2 mu b . (2.63)Moreover, if we take a section e : E → P we get e Du α dτ + Q c m u β g αδ h cd F dβδ = 0 dQ a dτ − C acb Q c A bν u ν = 0 (2.64)where e ∗ Ω = 12 F dβδ θ β ∧ θ δ X d e ∗ ( q b X b ) = Q b X b . (2.65)Eq. (2.64) is called a Wong equation in the case of G = SU (2) (see Ref. [33]). Moreover,for the first time the equation has been derived by R. Kerner (see Ref. [34]) in general case (anarbitrary group G ). In Non-Abelian Kaluza–Klein Theory W. Kopczyński derived this equationon a principal bundle P and afterwards projected it on E (see Ref. [35]), i.e. in the form (2.62).(This was of course Kaluza–Klein Theory with a symmetric metric.)In our theory an action is given by S = Z U d n +4 x R ( W ) p det γ AB (2.66)where U = V × G , V ⊂ E .The Palatini variational principle adopted here is along the main theoretical stream. Evenmore unconventional approach is advocated by J. Plebański, where we vary not only with respectto a metric and a connection, but also with respect to a curvature. We do not apply the mentionedformalism. The Palatini variational principle is really interesting if applied to Kaluza–KleinTheory in a nonsymmetric version.From the Palatini variational principle (with respect to W αβ , g µν , ω ) δS = vol( G ) δ Z V d x (cid:16) R ( W ) + R ( e Γ ) + ℓ cd (2 H c H d − L cµν H dµν ) (cid:17) √− g (2.67)19ne gets field equations R αβ ( W ) − g αβ R ( W ) = 8 π gauge T αβ + Λg αβ (2.68) ∼ g [ µν ] , = 0 (2.69) g µν,σ − g ζν Γ ζµσ − g µζ Γ ζσν = 0 (2.70) gauge ∇ µ (cid:0) ℓ ab ∼ L aαµ (cid:1) = 2 g [ αβ ]gauge ∇ β (cid:0) h ab ∼ g [ µν ] H aµν (cid:1) (2.71)where gauge T αβ = − ℓ ab π (cid:16) g γβ g τζ g εγ L aζα L bτε − g [ µν ] H ( aµν H b ) αβ − g αβ (cid:0) L aµν H bµν − H a H b (cid:1)(cid:17) . (2.72)The skew-symmetric part of the metric induces a current J αa = 12 π ∼ g [ αβ ]gauge ∇ β ( h ab g [ µν ] H bµν ) . (2.73)This current vanishes if g [ µν ] = 0 . One can easily see here that if D a ¯ a = 0 we have zero color charge distribution on the right-handside of Eq. (2.71). Moreover, a color charge is in general gauge-dependent.We have here the same problem to define a non-Abelian charge as to define an energy inGeneral Relativity. We cannot define an energy or a charge at a space-time point. However,in non-Abelian gauge field theory the situation is even more severe. According to Ref. [36],the most important difference between theories of Yang–Mills’ type and gravitation is that theunderlying bundle of the latter—the bundle of linear frames—is “concrete”, has more structurethan “abstract” bundles occurring in other gauge theories.A fundamental difference between these two theories (in NGT there is the same problem asin GR) happens if we consider asymptotic behavior (at large distances) of static fields. A gaugetransformation U of A → A , U : E → G , A µ ( x ) = U − ( x ) A µ ( x ) U ( x ) + U − ( x ) ∂ µ U ( x ) (2.74) F µν ( x ) = U − ( x ) F µν ( x ) U ( x ) (2.75)is compatible with a static A iff U ( r, ψ, ϕ ) = U ( ψ, ϕ ) (cid:16) u ( ψ, ϕ ) r + . . . (cid:17) (2.76)( U does not depend on time), where ψ and ϕ are defined as usual on S ( r, ψ, ϕ are spheri-cal coordinates). U : S → G , u ( ψ, ϕ ) is a real function. From Eqs (2.74)–(2.76) one gets( F µν θ µ ∧ θ ν = e ∗ ( H µν θ µ ∧ θ ν ) , H µν = H aµν X a ) F µν ( r, ψ, ϕ ) = U − F µν U + O ( r − ) . (2.77)In the case of gravitational fields we have g µν = η µν + O ( r − ) , (2.78)20 µν is a Minkowski tensor, g µν is nonsymmetric, Γ ≃ O ( r − ) , (2.79) Γ is a ω αβ connection in static configurations.One gets also for G aαµ G αµ = U − G αµ U + O ( r − ) (2.80)where G aαµ = g βα g µν G aβν , G aµν θ µ ∧ θ ν = e ∗ ( h αd ℓ db ∼ L dαµ θ α ∧ θ µ ) and G αµ = G aαµ X a .We define a Levi-Civita symbol and a dual Cartan basis η αβγδ , η = √− g (2.81) η αβγ = θ δ η αβγδ (2.82) η αβ = θ γ ∧ η αβγ (2.83) η α = θ β ∧ η αβ (2.84) η = θ α ∧ η α . (2.85)Eq. (2.71) can be rewritten after taking a section e∂ µ ( G aαµ ) = 4 πJ αa − C adc A cµ G dαµ (2.86)where we raise Latin indices by a Killing–Cartan tensor h ab .We rewrite Eq. (2.86) using dual forms b D ⋆ G = 4 π ⋆ J (2.87)or in the Gauss form d ⋆ G = 4 π ⋆ J − [ A, ⋆ G ] (2.88)where A = A µ θ µ , ⋆ J = J αa X a η α , ⋆ means a Hodge star and ⋆ J and ⋆ G mean dual forms for J α and G αµ ⋆ G = G αµ η αβ . b D means an exterior covariant derivative with respect to a gauge connection ω on a fiber bundle P (in a section e ), d is an ordinary exterior derivative of E . η α , η αβ mean a dual Cartan base.In this way a total non-Abelian charge π I ⋆ G (2.89)is ill-defined.A conservation law for a non-Abelian charge can be written d (cid:16) ⋆ J − π [ A, ⋆ G ] (cid:17) = 0 (2.90)(see Ref. [36]). 21ne gets J a = 12 π ∼ g [4¯ b ]gauge ∇ ¯ b ( h ab g [ µν ] H bµν ) (2.91)and ∂ ¯ m ( ℓ ab D a ¯ m ) + C dbe A e ¯ m D a ¯ m ℓ ab = J a . (2.92)Our confinement condition ( D a ¯ m = 0 ) is considered in a static (or stationary) limit. In this way J a = 0 and a fact that a total non-Abelian charge is ill-defined does not concern us. In the casewith some external sources, i.e. quark fields (fermion colored fields) J a will compensate a chargecaused by fermions and a total color charge distribution remains zero. In this situation we candevelop a color confinement program on the level of exact solutions with fermion sources whichwe mentioned before.In the case of gravitational field (see Ref. [36]) an analogue of a non-Abelian charge (2.89)is not ill-defined in static situation for an asymptotically flat space-time (also in the case ofnonsymmetric gravitation field).Let us consider a transformation of a connection ω αβ , ( Γ αβγ ), i.e. ω ′ = U − ( x ) ωU ( x ) + U − dU (2.93)where ω = ω αβ X βα (2.94)where X βα are generators of a GL (4 , R ) group and U ( x ) has the form (2.76). Now of course thegroup G is GL (4 , R ) group. In order to have the asymptotic behavior (2.79) in static configu-rations, U must be a constant Lorentz transformation, i.e. it belongs to SO(1 , ⊂ GL (4 , R ) .Introducing pseudotensor of an energy-momentum for a gravitational field we can define a con-served four-momentum from a conservation law d ( ⋆ T µ + ⋆ t µ ) = 0 (2.95)where T µ = T µν θ ν , t µ = t µν θ ν , (2.96) ⋆ T µ , ⋆ t µ are dual Hodge forms to T µ and t µ . T µν is an energy-momentum tensor for a matter (Yang–Mills’ field) which is nonsymmetricin general. In some future extensions we include also an energy-momentum tensor for fermion(quark) fields. t µν is a pseudotensor of an energy-momentum for a gravitational field, defined insuch a way that Eq. (2.95) is equivalent to gravitational field equations via Bianchi identity. Inthis way we can define superpotentials V µ such that dV µ = 4 π ( ⋆ T µ + ⋆ t µ ) (2.97)and a conserved 4-momentum P µ = 14 π I V µ . (2.98) P µ is well defined in static situation (under condition (2.78) and (2.79)).22sing (2.37) one easily gets ⋆ G = h ed g αµ g νω e ∗ (cid:16) ℓ dn (cid:16) H nωµ − µh na k ad H dωµ + (cid:0) H nαω e g ( αδ ) g [ δµ ] − H nαµ e g ( αδ ) g [ δω ] (cid:1) − µh na k ad e g ( δτ ) e g ( αβ ) H dδα g [ τω ] g [ βµ ] − µh na k ad e g ( δβ ) e g ( ατ ) H dβ [ ω g µ ] τ g [ δα ] + 2 µ h na h bc k ac k bd e g ( αβ ) H dα [ ω g | µ | β ] (cid:17) η µν X e (cid:17) , (2.99)where η µν = π ∗ ( η µν ) .One also derives V ′ µ = V ν U ν µ + O ( r − ) (2.100)where U = ( U ν µ ) ∈ SO(1 , .If we want to consider fermions (quarks) in the theory we should add a Lagrangian of fermions(quarks) L fermions = p − e g X f ( iψ jf ( γ µ D µ ) jk − m f δ jk ) ψ kf (2.101)where ( D µ ) jk = e ∇ µ δ jk + gA aµ ( X a ) jk (2.102)is a covariant derivative of a spinor field with respect to Riemannian part of a connection ω αβ ( e ω αβ ) generated by g ( αβ ) and a gauge field at once ( e g = det( g ( αβ ) ) ). e ∇ µ = ∂ µ + σ αβ e Γ αβµ , σ αβ = [ γ α , γ β ] , (2.103) g is a coupling constant, m f is the mass of a quark of flavour f . X a is a generator of the Liealgebra of a group G (equal to SU (3) ) in a fundamental representation and repeated indices aresummed over.One derives a color current for fermions (quarks) getting J qµa = ig π p − e g X f ψ jf γ µ ( X a ) jk ψ kf (2.104)and a color charge distribution J q a = ig π p − e g X f ψ jf γ ( X a ) jk ψ kf . (2.105)The color confinement condition reads now J q a + J a = 0 (2.106)or ig p − e g X f ψ jf γ ( X a ) jk ψ kf + ∼ g [4¯ b ] e ∗ (cid:16) gauge ∇ ¯ b ( h ab g [ µν ] H aµν ) (cid:17) = 0 . (2.107)Spinor fields (quark fields) are defined in the same gauge e . In this way we can write e ∗ (cid:16) ig p − e g X f Ψ jf γ ( X a ) jk Ψ kf + ∼ g [4¯ b ]gauge ∇ ¯ b ( h ab g [ µν ] H aµν ) (cid:17) = 0 , (2.108)23.e. in any “gauge”, Ψ jf , Ψ jf are spinor fields on P . This means that ig p − e g X f Ψ jf γ ( X a ) jk Ψ kf + ∼ g [4¯ b ]gauge ∇ ¯ b ( h ab g [ µν ] H aµν ) = 0 . (2.109)Eq. (2.109) means a dielectric color confinement condition in a presence of fermion (quark) sources.It is interesting to express −→ E a and −→ B a fields in terms of −→ D a and −→ H a fields. From Eq. (2.22)one gets ( ℓ ce ℓ dc g νω g µβ + δ ed g βµ g ων ) L dµν = H eβω . (2.110)From Eq. (2.110) we obtain E e ¯ w = (cid:16) ε ¯ n ¯ m ¯ l (cid:0) ℓ ce ℓ dc g ¯ n ¯ w g ¯ m + δ ed g n g ¯ w ¯ n (cid:1) H ¯ ld + (cid:0) ℓ ce ℓ dc ( g ¯ m ¯ w g − g w g ¯ m ) + δ ed ( g g ¯ w ¯ m − g m g ¯ w ) (cid:1) D d ¯ m (cid:17) (2.111) B e ¯ a = ε ¯ a ¯ w ¯ b h ε ¯ n ¯ m ¯ l (cid:0) ℓ ce ℓ dc g ¯ n ¯ w g ¯ m ¯ b + δ ed g ¯ b ¯ m g ¯ w ¯ n (cid:1) H ¯ ld + (cid:0) ℓ ce ℓ dc ( g ¯ n ¯ w g b − g w g ¯ n ¯ b ) + δ ed ( g ¯ b g ¯ w ¯ n − g ¯ b ¯ n g ¯ w ) (cid:1) D ¯ nd i (2.112)It is interesting to ask how to construct a Nonsymmetric Kaluza–Klein Theory with a group G = G ⊗ U (1) em with an obvious application G = SU (3) c .The simplest choice is to suppose that a nonsymmetric right-invariant tensor on G has theform ℓ ab − ! (2.113)where ℓ ab = h ab + µk ab is a nonsymmetric right-invariant tensor on G .In this case we can consider also a fermion sources (quarks) adding to the Lagrangian aLagrangian of fermion (quark) fields. We considered such a situation before. Moreover, now weshould add also a coupling with an electromagnetic field, i.e. iq p − e g X f ψ fj γ µ ψ fj A µ (2.114)where q is an elementary charge, q f is a charge of a quark measured in q , A µ is a four-potentialof an electromagnetic field and repeated indices are summed over.In this way we get an electric current as a source of Maxwell equations in our theory: J αn +5 = 12 π ∼ g [ αµ ] ∂ µ (cid:0) g [ νβ ] F νβ (cid:1) + q √− e g π X f q f X j ψ fj γ µ ψ fj (2.115)a density of a charge is J n +5 = 12 π ∼ g [4 ¯ m ] ∂ ¯ m ( g [ µν ] F νβ ) + q √− e g π X f q f X j ψ fj γ ψ fj . (2.116)If we want a confinement of a charge we should have −→ D = 0 (see Ref. [9]). This meansthat even −→ E = 0 , J n +5 = 0 , for G = SU (3) c n = 8 . Let us consider Eqs (2.110)–(2.112) for24 = U (1) , i.e. in an electromagnetic case. Let us notice that in the formulas below Latin smallcases a, b, c = 1 , , correspond to space indices as in Ref. [9]. One gets ( g νω g µβ + g βµ g ων ) H µν = F βω (2.117) E w = (cid:0) ε nme ( g nw g m + g m g wn ) H e + ( g nw g − g w g m − g g wm − g m g w ) D n (cid:1) (2.118) B a = ε awb (cid:0) ε nme ( g nw g mb + g bm g wn ) H e + ( g nw g b − g w g mb + g b g wn − g bn g w ) D n (cid:1) (2.119)Thus we get E a = K ae H e + L an D n (2.120) B a = K ae H e + L an D n (2.121)where K we = ε nme ( g nw g m + g m g wn ) (2.122) L an = ( g nw g − g w g m − g g wm − g m g w ) (2.123) K ae = ε wba ε mne ( g nw g mb + g bm g wm ) (2.124) L an = ε wba ( g nw g b − g w g mb − g b g wn − g bn g w ) (2.125)Let us give the following remark on confinement condition −→ D a = 0 . This condition should besatisfied outside a hadron, which in our model is a solution of field equation for the NonsymmetricKaluza–Klein Theory with or without fermion (quark) sources with G = SU (3) c or G = SU (3) c ⊗ U (1) em . In this way we consider a soliton model of hadrons. Inside a hadron, i.e. a solution offield equation this condition is not satisfied. The solution should be static (or stationary) withspherical symmetry or axial symmetry.Eq. (2.68) can be rewritten in a different shape R αβ ( W ) = 8 π gauge T αβ − Λg αβ . (2.126)From Eq. (2.126) one gets R ( αβ ) ( Γ ) = 8 π gauge T ( αβ ) − Λg ( αβ ) (2.127) R [ [ αβ ] ,γ ] ( Γ ) = 8 π gauge T [ [ αβ ] ,γ ] − Λg [ [ αβ ] ,γ ] (2.128)We use of course a fact that a trace of gauge T αβ is zero gauge T αβ g αβ = 0 . (2.129)Eq. (2.71) can be rewritten in the form gauge ∇ µ (cid:0) ℓ ab L aαµ (cid:1) = 2 g [ αβ ]gauge ∇ β (cid:0) h ab g [ µν ] H aµν (cid:1) , (2.130) gauge ∇ µ means a gauge derivative with respect to a connection ω , gauge ∇ µ means a covariant derivativewith respect to connection ω (a gauge derivative) and a connection ω αβ on E at once, ∼ L aµν = √− g L aµν , ∼ g [ µν ] = √− g g µν . (2.131)25et us consider a different approach to NGT coming from Einstein Unified Field Theory.This is a so called Hermitian-Nonsymmetric Theory (see Refs [18, 37]). In this theory we have afundamental tensor g µν as before. Moreover, now this tensor is complex and Hermitian g ∗ νµ = g µν . (2.132)(In this case the star ∗ means the complex conjugation. Do not mix up it with the Hodge star.)In such a way one gets g µν = g ( µν ) + g [ µν ] (2.133)and g [ µν ] is pure imaginary g [ µν ] = ip µν (2.134)where p µν is a real antisymmetric tensor p µν = − p νµ . (2.135)In the theory we have two connections as before (in the real version) ω αβ = Γ αβγ θ γ and W αβ = W αβγ θ . The first connection is an Hermitian connection (in holonomic system of coordinates) Γ ∗ αβγ = Γ αγβ (2.136)and Γ α [ βα ] = 0 (2.137)or Q αβα ( Γ ) = 0 (2.138)where Q αβγ ( Γ ) is a torsion of the connection ω αβ . The second connection is not Hermitian W αβ = ω αβ − δ αβ W (2.139)and the form W is pure imaginary.The Ricci tensor is defined as before (Moffat–Ricci) tensor. This tensor is Hermitian. Theinverse tensor of g µν , g µν is also Hermitian. It is easy to prove that in a nonholonomic system ofcoordinates we have in place of Eq. (2.136) Γ ∗ νµω = Γ νωµ + ( e Γ νµω − e Γ νωµ ) (2.140)where e Γ νωµ is a Levi-Civita connection generated by the symmetric part of g µν , g ( µν ) . Theconnection ω αβ satisfies Eqs (2.2)–(2.3).Now we construct a Nonsymmetric Kaluza–Klein Theory exactly as before (see Refs [3, 4]).In the 5-dimensional (electromagnetic) case we have in place of the nonsymmetric tensor g µν anHermitian tensor g ∗ µν = g νµ . Thus we get Nonsymmetric Hermitian Kaluza–Klein Theory. It isan Hermitian metrization of a fiber bundle. This is a natural Hermitian metrization of a fiberbundle (5-dimensional case).All the formulas are the same. Moreover, we should remember that g [ µν ] is pure imaginary.The connection ω AB on P is Hermitian in holonomic system of coordinates. Moreover, in ourlift-horizontal basis it satisfies a different condition Γ ∗ NMW = Γ NMW + ( e Γ NMW − e Γ NW M ) (2.141)26here e ω NM = e Γ NMW θ W is a Levi-Civita connection generated by γ ( AB ) , where N, W, M, A, B =1 , , , , . If a frame is holonomic, we get a condition to be Hermitian from Eq. (2.141).In the case of an exact solution from Ref. [9] we have g µν = − α ω − r − r sin θ − ω γ (2.142)where ω = iℓ r and γ = (cid:16) − ℓ r (cid:17) α − and all the remaining formulae are the same as in Ref. [9] (Eqs (4.1), (4.3), (4.5), (4.6) of Ref. [9]).The energy-momentum tensor for an electromagnetic field is also Hermitian em T ∗ αβ = em T βα . (2.143)Let us come to the Nonsymmetric Kaluza–Klein Theory in a general non-Abelian case. Inthis way we consider an Hermitian nonsymmetric tensor on P (as in this section) γ AB . But nowour constant µ is a pure imaginary constant, e.g. µ = iµ , where µ is a real number. Our tensoris right invariant with respect to the group action on P and Hermitian, γ ∗ AB = γ BA . (2.144)In all the formulae derived here it is enough to put iµ in place of µ . The connection on P , ω AB satisfies condition (2.141), but now A, B = 1 , , . . . , n + 4 , ℓ ab = h ab + µk ab = h ab + iµk ab , (2.145) ℓ ∗ ab = ℓ ba (2.146)In this way we consider a natural Hermitian metrization of a fiber bundle in a general non-Abeliancase (for a semisimple gauge group). All energy-momentum tensors are now Hermitian, e.g. gauge T αβ ∗ = gauge T βα . (2.147)Let us give the following remark. A lift horizontal basis in Kaluza–Klein Theory is nonholo-nomic. For this a Levi-Civita connection coefficients are not Christoffel symbols. They are notsymmetric in lower indices.One can consider in place of a complex (Hermitian) metric also a hypercomplex (Hermitian)metric (see Ref. [38]). Hypercomplex numbers (see Ref. [39]) are defined as x = x + Ix (2.148)where x , x are real numbers and I = − for complex numbers, +1 for hypercomplex numbers, for dual or parabolic numbers. (2.149)27ypercomplex numbers form a ring. They do not form a field. Addition and multiplication aredefined as usual taking into account the fact that I = 1 ( I = 1 ). An inverse of a number doesnot always exist. This ring contains also divisors of zero. One gets E , = ( I ± , E , = E , E · E = 0 . (2.150)In this way x = e x E + e x E = ( x + x ) E + ( x − x ) E (2.151)and the ring of hypercomplex numbers is isomorphic to a simple product of two copies of realnumbers field. In some sense it is a trivial structure in comparison with the complex numbersfield.Thus if we take e g µν = g ( µν ) + Ig [ µν ] = E g µν − E g νµ (2.152)where g µν = g ( µν ) + g [ µν ] (2.153)as in the real version of the theory, we get two disconnected real versions of the theory for g µν and transpose g νµ . The nonsymmetric natural metrization of a fiber bundle in Hermitian(hypercomplex) can be done analogously to the complex one in 5-dimensional case and in generalnon-Abelian (for a semisimple group) case.Moreover, all the calculations given by us in the case of a real version can be repeated remem-bering that g [ µν ] should be shifted to Ig [ µν ] and also k ab → Ik ab . In this way we get Nonsymmetric–Hermitian (Hypercomplex) Kaluza–Klein Theory. Moreover, we can write also in the case of atensor γ AB e γ AB = γ ( AB ) + Iγ [ AB ] = E γ AB − E γ BA (2.154) γ AB = γ ( AB ) + γ [ AB ] (2.155)and we have as before in a 4-dimensional case two disconnected real versions. Thus in the caseof Hermitian (Hypercomplex) Kaluza–Klein Theory we are reduced to a real version. Moreover,from the methodological point of view it is better to consider a Hermitian approach.The solution (2.142) will now look ω = Iℓ r and γ = (cid:16) ℓ r (cid:17) α − . (2.156)This solution can be written also in the form (2.153)–(2.154), i.e. e g µν = − α − r − r sin θ
00 0 0 γ + J l r − l r = E − α l r − r − r sin θ − l r γ − E − α − l r − r − r sin θ l r γ (2.157)where γ is given by the second formula of Eq. (2.156).28 Spontaneous symmetry breaking and Higgs’ mechanism in theNonsymmetric Kaluza–Klein Theory
In order to incorporate a spontaneous symmetry breaking and Higgs’ mechanism in our geomet-rical unification of gravitation and Yang–Mills’ fields we consider a fiber bundle P over a basemanifold E × G/G , where E is a space-time, G ⊂ G , G , G are semisimple Lie groups. Thuswe are going to combine a Kaluza–Klein theory with a dimensional reduction procedure.Let P be a principal fiber bundle over V = E × M with a structural group H and with aprojection π , where M = G/G is a homogeneous space, G is a semisimple Lie group and G itssemisimple Lie subgroup. Let us suppose that ( V, γ ) is a manifold with a nonsymmetric metrictensor γ AB = γ ( AB ) + γ [ AB ] . (3.1)The signature of the tensor γ is ( + − − − , − − − · · · − | {z } n ) . Let us introduce a natural frame on Pθ ˜ A = ( π ∗ ( θ A ) , θ = λω a ) , λ = const . (3.2)It is convenient to introduce the following notation. Capital Latin indices with tilde e A, e B, e C run , , , . . . , m + 4 , m = dim H + dim M = n + dim M = n + n , n = dim M , n = dim H .Lower Greek indices α, β, γ, δ = 1 , , , and lower Latin indices a, b, c, d = n + 5 , n + 5 , . . . ,n + 6 , . . . , m + 4 . Capital Latin indices A, B, C = 1 , , . . . , n + 4 . Lower Latin indices with tilde e a, e b, e c run , , . . . , n + 4 . The symbol over θ A and other quantities indicates that these quantitiesare defined on V . We have of course n = dim G − dim G = n − ( n − n ) , where dim G = n , dim G = n − n , m = n + n .On the group H we define a bi-invariant (symmetric) Killing–Cartan tensor h ( A, B ) = h ab A a B b . (3.3)We suppose H is semisimple, it means det( h ab ) = 0 . We define a skew-symmetric right-invarianttensor on H k ( A, B ) = k bc A b B c , k bc = − k cb . Let us turn to the nonsymmetric metrization of P . κ ( X, Y ) = γ ( X, Y ) + λ ℓ ab ω a ( X ) ω b ( Y ) (3.4)where ℓ ab = h ab + ξk ab (3.5)is a nonsymmetric right-invariant tensor on H . One gets in a matrix form (in the natural frame(3.2)) κ ˜ A ˜ B = γ AB ℓ ab ! , (3.6)29 et( ℓ ab ) = 0 , ξ = const and real, then ℓ ab ℓ ac = ℓ ba ℓ ca = δ cb . (3.7)The signature of the tensor κ is (+ , −−− , − · · · − | {z } n , − − · · · − | {z } n ) . As usual, we have commutationrelations for Lie algebra of H , h [ X a , X b ] = C cab X c . (3.8)This metrization of P is right-invariant with respect to an action of H on P .Now we should nonsymmetrically metrize M = G/G . M is a homogeneous space for G (withleft action of group G ). Let us suppose that the Lie algebra of G , g has the following reductivedecomposition g = g ˙+ m (3.9)where g is a Lie algebra of G (a subalgebra of g ) and m (the complement to the subalgebra g )is Ad G invariant, ˙+ means a direct sum. Such a decomposition might be not unique, but weassume that one has been chosen. Sometimes one assumes a stronger condition for m , the socalled symmetry requirement, [ m , m ] ⊂ g . (3.10)Let us introduce the following notation for generators of g : Y i ∈ g , Y ˜ ı ∈ m , Y ˆ a ∈ g . (3.11)This is a decomposition of a basis of g according to (3.9). We define a symmetric metric on M using a Killing–Cartan form on G in a classical way. We call this tensor h .Let us define a tensor field h ( x ) on G/G , x ∈ G/G , using tensor field h on G . Moreover,if we suppose that h is a biinvariant metric on G (a Killing–Cartan tensor) we have a simplerconstruction.The complement m is a tangent space to the point { εG } of M , ε is a unit element of G . Werestrict h to the space m only. Thus we have h ( { εG } ) at one point of M . Now we propagate h ( { f G } ) using a left action of the group Gh ( { f G } ) = ( L − f ) ∗ ( h ( { εG } )) .h ( { εG } ) is of course Ad G invariant tensor defined on m and L ∗ f h = h .We define on M a skew-symmetric 2-form k . Now we introduce a natural frame on M . Let f ijk be structure constants of the Lie algebra g , i.e. [ Y j , Y k ] = f ijk Y i . (3.12) Y j are generators of the Lie algebra g . Let us take a local section σ : V → G/G of a natural bundle G G/G where V ⊂ M = G/G . The local section σ can be considered as an introduction ofa coordinate system on M .Let ω MC be a left-invariant Maurer–Cartan form and let ω σMC = σ ∗ ω MC . (3.13)30sing decomposition (3.9) we have ω σMC = ω σ + ω σ m = θ b ı Y b ı + t ˜ a Y ˜ a . (3.14)It is easy to see that θ ˜ a is the natural (left-invariant) frame on M and we have h = h a ˜ b θ ˜ a ⊗ θ ˜ b (3.15) k = k a ˜ b θ ˜ a ∧ θ ˜ b . (3.16)According to our notation e a, e b = 5 , , . . . , n + 4 .Thus we have a nonsymmetric metric on Mγ ˜ a ˜ b = r (cid:0) h a ˜ b + ζk a ˜ b (cid:1) = r g ˜ a ˜ b . (3.17)Thus we are able to write down the nonsymmetric metric on V = E × M = E × G/G γ AB = g αβ r g ˜ a ˜ b ! (3.18)where g αβ = g ( αβ ) + g [ αβ ] g ˜ a ˜ b = h a ˜ b + ζk a ˜ b k a ˜ b = − k b ˜ a h a ˜ b = h b ˜ a ,α, β = 1 , , , , e a, e b = 5 , , . . . , n + 4 = dim M + 4 = dim G − dim G + 4 . The frame θ ˜ a isunholonomic: dθ ˜ a = 12 κ ˜ a ˜ b ˜ c θ ˜ b ∧ θ ˜ c (3.19)where κ ˜ a ˜ b ˜ c are coefficients of nonholonomicity and depend on the point of the manifold M = G/G (they are not constant in general). They depend on the section σ and on the constants f ˜ a ˜ b ˜ c .We have here three groups H, G, G . Let us suppose that there exists a homomorphism µ between G and H , µ : G → H (3.20)such that a centralizer of µ ( G ) in H , C µ is isomorphic to G . C µ , a centralizer of µ ( G ) in H ,is a set of all elements of H which commute with elements of µ ( G ) , which is a subgroup of H .This means that H has the following structure, C µ = G . µ ( G ) ⊗ G ⊂ H. (3.21)If µ is a isomorphism between G and µ ( G ) one gets G ⊗ G ⊂ H. (3.22)Let us denote by µ ′ a tangent map to µ at a unit element. Thus µ ′ is a differential of µ acting onthe Lie algebra elements. Let us suppose that the connection ω on the fiber bundle P is invariant31nder group action of G on the manifold V = E × G/G . According to Refs [13, 40, 41, 42] thismeans the following.Let e be a local section of P , e : V ⊂ U → P and A = e ∗ ω . Then for every g ∈ G there existsa gauge transformation ρ g such that f ∗ ( g ) A = Ad ρ − g A + ρ − g dg g , (3.23) f ∗ means a pull-back of the action f of the group G on the manifold V . According to Refs[40, 41, 42, 43, 44, 45] (see also Refs [46, 47, 48]) we are able to write a general form for suchan ω . Following Ref. [42] we have ω = e ω E + µ ′ ◦ ω σ + Φ ◦ ω σ m . (3.24)(An action of a group G on V = E × G/G means left multiplication on a homogeneous space M = G/G .) where ω σ + ω σ m = ω σMC are components of the pull-back of the Maurer–Cartanform from the decomposition (3.14), e ω E is a connection defined on a fiber bundle Q over a space-time E with structural group C µ and a projection π E . Moreover, C µ = G and e ω E is a 1-formwith values in the Lie algebra g . This connection describes an ordinary Yang–Mills’ field gaugegroup G = C µ on the space-time E . Φ is a function on E with values in the space e S of linearmaps Φ : m → h (3.25)satisfying Φ ([ X , X ]) = [ µ ′ X , Φ ( X )] , X ∈ g . (3.26)Thus e ω E = e ω iE Y i , Y i ∈ g ,ω σ = θ b ı Y b ı , Y b ı ∈ g ,ω σ m = θ ˜ a Y ˜ a , Y ˜ a ∈ m . (3.27)Let us write condition (3.24) in the base of left-invariant form θ b ı , θ ˜ a , which span respectivelydual spaces to g and m (see Refs [49, 50]). It is easy to see that Φ ◦ ω σ m = Φ a ˜ a ( x ) θ ˜ a X a , X a ∈ h (3.28)and µ ′ = µ a b ı θ b ı X a . (3.29)From (3.26) one gets Φ c ˜ b ( x ) f ˜ b b ı ˜ a = µ a b ı Φ b ˜ a ( x ) C cab (3.30)where f ˜ b b ı ˜ a are structure constants of the Lie algebra g and C cab are structure constants of the Liealgebra h . Eq. (3.30) is a constraint on the scalar field Φ a ˜ a ( x ) . For a curvature of ω one gets Ω = 12 H CAB θ A ∧ θ B X C = 12 e H iµν θ µ ∧ θ ν α ci X c + gauge ∇ µ Φ c ˜ a θ µ ∧ θ ˜ a X c + 12 C cab Φ a ˜ a Φ b ˜ b θ ˜ a ∧ θ ˜ b X c − Φ c ˜ d f ˜ d ˜ a ˜ b θ ˜ a ∧ θ ˜ b X c . (3.31)32hus we have H cµν = α ci e H iµν (3.32) H cµ ˜ a = gauge ∇ µ Φ c ˜ a = − H c ˜ aµ (3.33) H c ˜ a ˜ b = C cab · Φ a ˜ a Φ b ˜ b − µ c b ı f b ı ˜ a ˜ b − Φ c ˜ d f ˜ d ˜ a ˜ b (3.34)where gauge ∇ µ means gauge derivative with respect to the connection e ω E defined on a bundle Q overa space-time E with a structural group G Y i = α ci X c . (3.35) e H iµν is the curvature of the connection ω E in the base { Y i } , generators of the Lie algebra of theLie group G , g , α ci is the matrix which connects { Y i } with { X c } . Now we would like to remindthat indices a, b, c refer to the Lie algebra h , e a, e b, e c to the space m (tangent space to M ), b ı, b , b k tothe Lie algebra g and i, j, k to the Lie algebra of the group G , g . The matrix α ci establishes adirect relation between generators of the Lie algebra of the subgroup of the group H isomorphicto the group G .Let us come back to a construction of the Nonsymmetric Kaluza–Klein Theory on a mani-fold P . We should define connections. First of all, we should define a connection compatible witha nonsymmetric tensor γ AB , Eq. (3.18), ω AB = Γ ABC θ C (3.36) Dγ AB = γ AD Q DBC ( Γ ) θ C (3.37) Q DBD ( Γ ) = 0 where D is the exterior covariant derivative with respect to ω AB and Q DBC ( Γ ) its torsion.Using (3.18) one easily finds that the connection (3.36) has the following shape ω AB = π ∗ E ( ω αβ ) 00 b ¯ ω ˜ a ˜ b ! (3.38)where ω αβ = Γ αβγ θ γ is a connection on the space-time E and b ω ˜ a ˜ b = b Γ ˜ a ˜ b ˜ c θ ˜ c on the manifold M = G/G with the following properties Dg αβ = g αδ Q δβγ ( Γ ) θ γ = 0 (3.39) Q αβα ( Γ ) = 0 (3.40) b Dg ˜ a ˜ b = g ˜ a ˜ d b Q ˜ d ˜ b ˜ c ( b Γ ) . (3.41) b Q ˜ d ˜ b ˜ d ( b Γ ) = 0 D is an exterior covariant derivative with respect to a connection ω αβ . Q αβγ is a tensor oftorsion of a connection ω αβ . b D is an exterior covariant derivative of a connection b ω ˜ a ˜ b and b Q ˜ a ˜ b ˜ c ( b Γ ) its torsion. 33n a space-time E we also define the second affine connection W αβ such that W αβ = ω αβ − δ αβ W , (3.42)where W = W γ θ γ = ( W σγσ − W σγσ ) . We proceed a nonsymmetric metrization of a principal fiber bundle P according to (3.18). Thuswe define a right-invariant connection with respect to an action of the group H compatible witha tensor κ ˜ A ˜ B Dκ ˜ A ˜ B = κ ˜ A ˜ D Q ˜ D ˜ B ˜ C ( Γ ) θ ˜ C (3.43) Q ˜ D ˜ B ˜ D ( Γ ) = 0 where ω ˜ A ˜ B = Γ ˜ A ˜ B ˜ C e θ ˜ C . D is an exterior covariant derivative with respect to the connection ω ˜ A ˜ B and Q ˜ A ˜ B ˜ C its torsion. After some calculations one finds ω ˜ A ˜ B = π ∗ ( ω AB ) − ℓ db γ MA L dMB θ b L aBC θ C ℓ bd γ AB (2 H dCB − L dCB ) θ C e ω ab ! (3.44)where L dMB = − L dBM (3.45) ℓ dc γ MB γ CM L dCA + ℓ cd γ AM γ MC L dBC = 2 ℓ cd γ AM γ MC H dBC , (3.46) L dCA is Ad-type tensor with respect to H (Ad-covariant on P ) e ω ab = e Γ abc θ c (3.47) ℓ db e Γ dac + ℓ ad e Γ dcb = − ℓ db C dac (3.48) e Γ dac = − e Γ dca , e Γ dad = 0 . (3.49)We define on P a second connection W ˜ A ˜ B = ω ˜ A ˜ B − m + 2) δ ˜ A ˜ B W . (3.50)Thus we have on P all ( m + 4) -dimensional analogues of geometrical quantities from NGT, i.e. W ˜ A ˜ B , ω ˜ A ˜ B and κ ˜ A ˜ B . Let us calculate a Moffat–Ricci curvature scalar for the connection W ˜ A ˜ B R ( W ) = κ ˜ A ˜ B (cid:0) R ˜ C ˜ A ˜ B ˜ C ( W ) + R ˜ C ˜ C ˜ A ˜ B ( W ) (cid:1) (3.51)where R ˜ C ˜ C ˜ A ˜ B ( W ) is a curvature tensor for a connection W ˜ A ˜ B and κ ˜ A ˜ B is an inverse tensor for κ ˜ A ˜ B κ ˜ A ˜ C κ ˜ A ˜ B = κ ˜ C ˜ A κ ˜ B ˜ A = δ ˜ C ˜ B . (3.52)34sing results from Ref. [1] one gets (having in mind some analogies from a theory with a basespace E to the theory with the base space V = E × M = E × G/G ) R ( W ) = R ( W ) + 1 r R ( b Γ ) + 1 λ e R ( e Γ ) − λ ℓ ab (cid:0) H a H b − L aMN H bMN (cid:1) (3.53)where R ( W ) is a Moffat–Ricci curvature scalar on the space-time E for a connection W αβ , R ( b Γ ) is a Moffat–Ricci curvature scalar for a connection b ω ˜ a ˜ b on a homogeneous space M = G/G , e R ( e Γ ) is a Moffat–Ricci curvature scalar for a connection e ω ab , H a = γ [ AB ] H a [ AB ] = g [ αβ ] H aαβ + 1 r g [˜ a ˜ b ] H a ˜ a ˜ b (3.54) L aMN = γ AM γ BN L aAB = δ Mµ δ Nγ g αµ g βγ L aαβ + 1 r (cid:0) g αµ g ˜ b ˜ n L aα ˜ b + g ˜ a ˜ n g βγ L a ˜ aβ (cid:1) δ Mµ δ N ˜ n + 1 r g ˜ a ˜ m g ˜ b ˜ n L a ˜ a ˜ b δ M ˜ m δ N ˜ n . (3.55)One finds that − ℓ ab L aMN H bMN = − ℓ ab (cid:16) g αµ g βν L aαβ H bµν + 2 r g αµ g ˜ b ˜ n L aα ˜ b H bµ ˜ n + 1 r g ˜ a ˜ m g ˜ b ˜ n L a ˜ a ˜ b H b ˜ m ˜ n (cid:17) = − ℓ ab (cid:16) L aµν H bµν + 2 r g ˜ b ˜ n L αµ ˜ b H bµ ˜ n + 1 r g ˜ a ˜ m g ˜ b ˜ n L a ˜ a ˜ b H b ˜ m ˜ n (cid:17) . (3.56)We get conditions from Eq. (3.46) ℓ dc g µβ g γµ L dγα + ℓ cd g αµ g µγ L dβγ = 2 ℓ cd g αµ g µγ H dβγ (3.57) ℓ dc g ˜ m ˜ b g ˜ c ˜ m L d ˜ c ˜ a + ℓ cd g ˜ a ˜ m g ˜ m ˜ c L d ˜ b ˜ c = 2 ℓ cd g ˜ a ˜ m g ˜ m ˜ c H d ˜ b ˜ c (3.58) ℓ dc g µβ g γµ L dγ ˜ a + ℓ cd g ˜ a ˜ m g ˜ m ˜ c L dβ ˜ c = 2 ℓ cd g ˜ a ˜ m g ˜ m ˜ c H dβ ˜ c (3.59) L aµν = g αµ g βν L aαβ (3.60) L aµ ˜ b = g αµ L aµ ˜ b . (3.61)For ℓ ab H a H b = h ab H a H b we have the following: h ab H a H b = h ab H a H b + 2 r h ab H a H b + 1 r h ab H a H b (3.62)where H a = g αβ H aαβ , H a = g [˜ a ˜ b ] H a ˜ a ˜ b . (3.63)Finally, we have for a density of R ( W ) , i.e. q | κ | R ( W ) = √− g r n q | e g | q | ℓ | R ( W )= √− g r n q | e g | q | ℓ | (cid:18) R ( W ) + e R ( e Γ ) λ + 1 r R ( b Γ ) + λ ℓ ab (cid:0) H a H b − L aµν H bµν (cid:1) + λ r ℓ ab (cid:0) H ( a H b )1 − g ˜ b ˜ n L aµ ˜ b H bµ ˜ n (cid:1) + λ r ℓ ab (cid:0) H a H b − g ˜ a ˜ m g ˜ b ˜ n L a ˜ a ˜ b H b ˜ m ˜ n (cid:1)(cid:19) . (3.64)35e define an integral of action S ∼ Z U q | κ | R ( W ) d m +4 x, (3.65)where U = M × G × V, V ⊂ E, d m +4 x = d x dµ H ( h ) dm ( y ) ,dµ H ( h ) is a biinvariant measure on a group H and dm ( y ) is a measure on M induced by abiinvariant measure on G . R ( W ) is a Moffat–Ricci curvature scalar for a connection W αβ on E .Let us consider Eqs (3.57)–(3.59) modulo equations (3.32)–(3.34). One gets ℓ ij g µβ g γµ e L iγα + ℓ ji g αµ g µγ e L iβγ = 2 ℓ ji g αµ g µγ e H iβγ (3.66)where ℓ ij = ℓ cd α ci α dj is a right-invariant nonsymmetric metric on the group G and L cµν = α ci e L iµν . (3.67) e L iµν plays a role of an induction tensor for the Yang–Mills’ field with the gauge group G . e H iµν isof course the tensor of strength of this field. The polarization tensor is defined as usual e L iµν = e H iµν − π f M iµν . (3.68)We introduce two Ad G -type 2-forms with values in the Lie algebra g (of G ) e L = e L iµν θ µ ∧ θ ν Y i (3.69) f M = f M iµν θ µ ∧ θ ν Y i (3.70)and we easily write e L = e Ω E − π f M = e Ω E − Q (3.71)where e Q = e Q iµν θ µ ∧ θ ν Y i , e Q iµν = α ic Q cµν . e Ω E is a 2-form of a curvature of a connection e ω E (Eq. (3.27)) in Eq. (3.31) (the first term of this equation).In this way we get a geometrical interpretation of a Yang–Mills’ induction tensor in terms ofthe curvature tensor and torsion in additional dimensions (see Refs [1, 3]). Afterwards we get ℓ cd g ˜ m ˜ b g ˜ c ˜ m L d ˜ c ˜ a + ℓ cd g ˜ a ˜ m g ˜ m ˜ c L d ˜ b ˜ c = 2 ℓ cd g ˜ a ˜ m g ˜ m ˜ c (cid:0) C dab Φ a ˜ b Φ b ˜ b − µ d b ı f b ı ˜ b ˜ c − Φ d ˜ d f ˜ d ˜ b ˜ c (cid:1) , (3.72) ℓ cd g µβ g γµ L dγ ˜ a + ℓ cd g ˜ a ˜ m g ˜ m ˜ c L dβ ˜ c = 2 ℓ cd g ˜ a ˜ m g ˜ m ˜ c gauge ∇ β Φ d ˜ c . (3.73)Let us rewrite an action integral S = − V V r n Z U (cid:0) R ( W ) d n x (cid:1) d n x d x, U = V × M × H, V ⊂ E, (3.74) V = Z H q | ℓ | d n x (3.75) V = Z M q | e g | d n x. (3.76)Thus we get S = − Z V √− g d x L ( W , g, e A, Φ ) (3.77)36here L ( W , g, e A, Φ )= R ( W ) + λ (cid:16) π L YM ( e A ) + 2 r L kin ( gauge ∇ Φ ) + 1 r V ( Φ ) − r L int ( Φ, e A ) (cid:17) + λ c (3.78) L YM ( e A ) = − π ℓ ij (cid:0) e H i e H j − L iµν e H jµν (cid:1) (3.79)is the lagrangian for the Yang–Mills’ field with the gauge group G (see Eqs (2.33) and (2.39)), L kin ( gauge ∇ Φ ) = 1 V Z M q | e g | d n x (cid:0) ℓ ab g ˜ b ˜ n L aµ ˜ b gauge ∇ µ Φ b ˜ n (cid:1) = ℓ ab g αµ V Z M q | e g | d n x (cid:0) g ˜ b ˜ n L aα ˜ b gauge ∇ µ Φ b ˜ n (cid:1) (3.80)is a kinetic part of a lagrangian for a scalar field Φ a ˜ a . It is quadratic in gauge derivative of Φ a ˜ a andis invariant with respect to the action of groups H and G . V ( Φ ) = ℓ ab V Z M q | e g | d n x h g [ ˜ m ˜ n ] (cid:0) C acd Φ c ˜ m Φ d ˜ n − µ a b ı f b ı ˜ m ˜ n − Φ a ˜ e f ˜ e ˜ m ˜ n (cid:1) g [˜ a ˜ b ] (cid:0) C bef Φ e ˜ a Φ f ˜ b − µ b b f b ˜ a ˜ b − Φ b ˜ a f ˜ d ˜ a ˜ b (cid:1) − g ˜ a ˜ m g ˜ b ˜ n L a ˜ a ˜ b (cid:0) C bcd Φ c ˜ m Φ d ˜ n − µ b b ı f b ı ˜ m ˜ n − Φ b ˜ e f ˜ e ˜ m ˜ n (cid:1)i (3.81)is a self-interacting term for a field Φ . It is invariant with respect to the action of the groups H and G . This term is a polynomial of fourth order in Φ ’s (a Higgs’ field potential term) L int ( Φ, e A ) = h ab µ ai e H i g [˜ a ˜ b ] (cid:0) C bcd Φ c e a Φ d ˜ b − µ b b ı f b ı ˜ a ˜ b − Φ b ˜ d f ˜ d ˜ a ˜ b (cid:1) (3.82)where g [˜ a ˜ b ] = 1 V Z M q | e g | d n x g [˜ a ˜ b ] (3.83)is the term describing non-minimal coupling between the scalar field F and the Yang–Mills’ field.This term is also invariant with respect to the action of the groups H and G . λ c = 1 λ e R ( e Γ ) + 1 r V Z M q | e g | b R ( b Γ ) d n x = 1 λ e R ( e Γ ) + 1 r e P . (3.84)The condition (3.73) can be explicitly solved (see Appendix A). One gets L nω ˜ m = gauge ∇ ω Φ n ˜ m + ξk nd gauge ∇ ω Φ d ˜ m − (cid:0) ζ gauge ∇ ω Φ n ˜ a h a ˜ d k d ˜ m + e g ( αµ )gauge ∇ α Φ n ˜ m g [ µω ] (cid:1) − ξζk nd gauge ∇ ω Φ d ˜ d e g ( δα ) g [ αω ] h d ˜ a k a ˜ m + ξk nd (cid:0) ζ h ˜ d ˜ a gauge ∇ ω Φ d ˜ a k d ˜ b k m ˜ c h c ˜ b + gauge ∇ β Φ d ˜ m e g ( δβ ) g [ δα ] g [ ωµ ] e g ( αµ ) (cid:1) − ξ k nb k bd (cid:0) ζ gauge ∇ ω Φ d ˜ a h a ˜ b k m ˜ b + e g ( αβ )gauge ∇ a Φ d ˜ m g [ ωβ ] (cid:1) (3.85)where k nb = h na h bp k ap . (3.86)37he condition (3.72) can be also explicitly solved. One gets L n ˜ w ˜ m = H n ˜ w ˜ m + µk nd H d ˜ w ˜ m + ζ (cid:0) h a ˜ d H n ˜ a ˜ w k d ˜ m − h a ˜ d H n ˜ a ˜ m k a ˜ w (cid:1) − µζ h d ˜ c h a ˜ b H d ˜ d ˜ a k c ˜ w k b ˜ m − µζk nd h a ˜ p h d ˜ b H d ˜ b [ ˜ w k m ]˜ p k d ˜ a + 2 µ ζk nb k bd H d ˜ a [ ˜ w k m ]˜ p h p ˜ a . (3.87)In this case a kinetic term for a scalar field takes a form L kin ( gauge ∇ Φ ) = 1 V Z M q | e g | d n x h ℓ nk g ωµ g ˜ m ˜ p gauge ∇ µ Φ k ˜ p n gauge ∇ ω Φ n ˜ m + ζk nd gauge ∇ ω Φ d ˜ m − ζ gauge ∇ ω Φ d ˜ a h a ˜ q k q ˜ m − gauge ∇ α Φ a ˜ m e g ( αµ ) g [ ηω ] − ξζ gauge ∇ δ Φ d ˜ a k nd e g ( δα ) g [ αω ] h o ˜ d ˜ q k q ˜ m − ξ (cid:0) ζ k nd gauge ∇ ω Φ d ˜ a h b ˜ q h a ˜ w k q ˜ m k w ˜ b + k nd gauge ∇ β Φ d ˜ m e g ( αν ) e g ( βρ ) g [ νω ] g [ ρα ] (cid:1) + ξ (cid:0) ζk nb k bd gauge ∇ ω Φ d ˜ a h a ˜ q k q ˜ m + gauge ∇ α Φ d ˜ m e g ( αβ ) g [ βω ] (cid:1)oi . (3.88)In the case of g µν = η µν (a Minkowski space-time) one gets L kin ( gauge ∇ Φ ) = 1 V Z M q | e g | d n x h ℓ nk g ˜ m ˜ p gauge ∇ ω Φ k ˜ p n gauge ∇ ω Φ n ˜ m + ξk nd gauge ∇ ω Φ d ˜ m − ζ gauge ∇ ω Φ d ˜ a k a ˜ m − ξζ k nd k b ˜ m k a ˜ b gauge ∇ ω Φ d ˜ a + ξ ζk nb k bd k a ˜ m gauge ∇ ω Φ d ˜ a oi (3.89)where gauge ∇ ω Φ k ˜ p = η ωµ gauge ∇ µ Φ k ˜ p , k a ˜ b = h a ˜ c k c ˜ b .The Higgs potential is given by V ( Φ ) = 1 V Z M q | e g | d n x n g ˜ w ˜ p g ˜ m ˜ q h h nk H n ˜ w ˜ m + 2 ζh nk H n ˜ d ˜ w k d ˜ m + µζ (cid:16) k nk H n ˜ d ˜ w k d ˜ m + ζ (cid:0) − k kd H d ˜ d ˜ n k d ˜ w k a ˜ m − k kd H d e ı ˜ w k m ˜ a k e ı ˜ a + k kd H d e ı ˜ m k e ı ˜ a k w ˜ a (cid:1) + ζ (cid:0) k nk k nd H d e ı ˜ m k w ˜ r k e ı ˜ r − k nk k nd H d ˜ d ˜ a k d ˜ w k a ˜ m − k nk k nd H d e ı ˜ w k m ˜ r k e ı ˜ r (cid:1)(cid:17) + µ ζ (cid:0) k nk k nb k bd H d ˜ a ˜ w k m ˜ a − k nk k nb k bd H d ˜ a ˜ m k w ˜ a (cid:1)i H k ˜ p ˜ q − h cd (cid:0) H c ˜ p ˜ q g [˜ p ˜ q ] (cid:1)(cid:0) H d ˜ a ˜ b g [˜ a ˜ b ] (cid:1)o (3.90)or V ( Φ ) = 1 V Z M q | e g | d n x (cid:16) P kl [˜ p ˜ q ][˜ a ˜ b ] H k ˜ p ˜ q H l ˜ a ˜ b − h kl (cid:0) H k ˜ p ˜ q g [˜ p ˜ q ] (cid:1)(cid:0) H l ˜ a ˜ b g [˜ a ˜ b ] (cid:1)(cid:17) = 1 V Z M q | e g | d n x Q sk [˜ c ˜ d ][˜ p ˜ q ] H s ˜ c ˜ d H k ˜ p ˜ q . (3.91) Q [˜ c ˜ d ][˜ p ˜ q ] sk = Q [˜ p ˜ q ][˜ c ˜ d ] ks = − Q [ ˜ d ˜ c ][˜ p ˜ q ] sk = − Q [˜ c ˜ d ][˜ q ˜ p ] sk = Q [ ˜ d ˜ c ][˜ q ˜ p ] sk P sk [˜ c ˜ d ][˜ p ˜ q ] = g [˜ c [ ˜ p g ˜ d ]˜ q ] h sk − ζh sk k
0[ ˜ d | ˜ e | g ˜ c ][˜ p g | ˜ e | ˜ q ] + µζ (cid:16) − k sk k
0[ ˜ d | ˜ e | g ˜ c ][˜ p | ˜ e | ˜ q ] + ζ (cid:0) k sk k c | ˜ e | k d ]˜ f g ˜ e [˜ p g | ˜ f | ˜ q ] − k sk k e ˜ a k
0[ ˜ d | ˜ a | g ˜ c ][˜ p g | ˜ e | ˜ q ] − k sk k c | ˜ a | g ˜ d ][˜ q g | ˜ e | ˜ p ] k e ˜ a (cid:1)(cid:17) + µ ζ (cid:16) − k bs k kb k
0[ ˜ d | ˜ a | g ˜ c [˜ p g | ˜ a | ˜ q ] − k ns k nk k c | ˜ e | k d ]˜ f g ˜ e [˜ p g | ˜ e | ˜ q ] + ζ (cid:0) k ns k nk k a ˜ r k c | ˜ r | g ˜ a [ ˜ p g ˜ d ]˜ q ] + ζ (cid:0) k ns k nk k a ˜ r k c | ˜ r | g ˜ a [ ˜ p ˜ a ]˜ q ] − k ns k nk k c | ˜ e | k d ]˜ f g ˜ e [˜ p g | ˜ f | ˜ q ] (cid:1)(cid:1)(cid:17) + µ ζ (cid:0) − k bs k nb k nk k e [ ˜ d g ˜ c ][˜ p g | ˜ e | ˜ q ] − k bs k nb k nk k e [˜ c g | ˜ e | [ p g ˜ d ]˜ q ] (cid:1) + µ g [˜ c [ ˜ p g ˜ a ]˜ q ] k ps k pk , (3.92) Q sk [˜ c ˜ d ][˜ p ˜ q ] = P sk [˜ c ˜ d ][˜ p ˜ q ] − h sk g [˜ c ˜ d ] g [˜ p ˜ q ] . Let us do some manipulations concerning physical dimensions. The connection ω on the fiberbundle P has no correct physical dimensions. Let us pass in all formulas from ω to α s ~ c ω , ω α s ~ c ω, (3.93)where ~ is a Planck constant, c is the velocity of light in the vacuum and α s is a dimensionlesscoupling constant for the Yang–Mills’ field if this field couples to a matter. For example in theelectromagnetic case α s = √ . We use α g = α s = g ~ c where g is a coupling constant for a gaugefield. The redefinition of ω is equivalent to a usual treatment in local section e : V ⊃ U → P , e ∗ ω = g ~ c A .Let us notice that we do this redefinition for a connection ω , not only for ω E . This means thatwe treat Higgs’ field as a part of Yang–Mills’ field (gauge field). This is a part of our geometricalunification of fundamental interactions. One easily writes an integral of action S = − r Z √− g d x h R ( W )+ 8 πλ α s c ~ (cid:16) L YM + 14 πr L kin − πr V ( Φ ) − πr L int ( Φ, e A ) (cid:17) + λ c i . (3.94)If we want to be in line with an ordinary coupling between gravity and matter we should put πλ α s c ~ = 8 πG N c . (3.95)One gets λ = 2 α s ℓ pl = 2 √ α g ℓ pl (3.96)where ℓ pl is the Planck length ℓ pl = q G N ~ c ≃ − cm. In this case we have λ c = (cid:16) α s ℓ e R ( e Γ ) + e Pr (cid:17) . (3.97)Let us pass to spontaneous symmetry breaking and Higgs’ mechanism in our theory. In orderto do this we look for the critical points (the minima) of the potential V ( Φ ) . However, our fieldsatisfies the constraints Φ c ˜ b f ˜ b b ı ˜ a − µ a b ı Φ b ˜ a C cab = 0 . (3.98)39hus we must look for the critical points of V ′ = V + ψ b ı ˜ dc (cid:0) Φ c ˜ b f ˜ b b ı ˜ a − µ a b ı Φ b ˜ a C cab (cid:1) (3.99)where ψ b ı ˜ dc is a Lagrange multiplier. Moreover, we should change dimensions of the scalar field Φ a ˜ a in the potential. It is in the following exchange form H b ˜ a ˜ b = C dcd Φ c ˜ a Φ d ˜ b − µ b b ı f b ı ˜ a ˜ b − Φ b ˜ c f ˜ c ˜ a ˜ b (3.100)to H b ˜ a ˜ b = α s √ ~ c C dcd Φ c ˜ a Φ d ˜ b − α s √ ~ c µ b b ı f b ı e a e b − Φ b ˜ c f ˜ c ˜ a ˜ b . (3.101)It is easy to see that, if H a ˜ m ˜ n = 0 (3.102)then δV ′ δΦ = 0 (3.103)if (3.98) is satisfied.This was noticed in Refs [42], [1] and it is known in the symmetric theory. H a ˜ n ˜ m is a part ofthe curvature of ω over a manifold M . Thus it means that Φ crt satisfying Eq. (3.102) is a “puregauge”. If the potential V ( Φ ) is positively defined, then we have the absolute minimum of VV ( Φ ) = 0 . (3.104)But apart from this solution there are some others due to an influence of nonsymmetric metricon H and M . The details strongly depend on constants ξ , ζ and on groups G, G , H . There arealso some critical which are minima. Moreover, we expect the second critical point Φ = Φ such that V ( Φ ) = 0 and H a ˜ m ˜ n ( Φ ) = 0 (3.105) δV ′ δΦ ( Φ i crt ) = 0 , i = 0 , . (3.106)This means that Φ is not a “pure gauge” and a gauge configuration connected to Φ is nottrivial. This indicates that the local minimum is not a vacuum state. It is a “false vacuum” incontradiction to “true vacuum” for the absolute minimum Φ .Now we answer the question of what is a symmetry breaking if we choose one of the criticalvalues of Φ (we choose one of the degenerated vacuum states and the spontaneous breaking ofthe symmetry takes place). In Ref. [42] it was shown that if H a ˜ m ˜ n = 0 and Eq. (3.98) is satisfiedthen the symmetry is reduced to G . In the case of the second minimum (local minimum—falsevacuum) the unbroken symmetry will be in general different.Let us call it G ′ and its Lie algebra g ′ . This will be the symmetry which preserves Φ andthe constraint (3.98). It is easy to see that the Lie algebra of this unbroken group preserves Φ under Ad-action. For the symmetry group V is larger than G (it is H ) we expect some scalarswhich remain massless after the symmetry breaking in both cases (i.e., i = 0 , , “true” and falsevacuum case). They became massive only through radiative corrections. They are often referrredas the pseudo-Goldstone bosons. 40et us pass to the integral of action (3.94) in the two vacuum cases Φ , Φ . Let us expandthe Higgs’ field Φ ˜ aa in the neighbourhood of ( Φ k crt ) a ˜ b , k = 0 , , Φ a ˜ b = ( Φ k crt ) a ˜ b + ( ϕ k ) a ˜ b (3.107)and apply this formula for e ∗ ( gauge ∇ µ Φ b ˜ a ) : e ∗ ( gauge ∇ µ Φ b ˜ a ) = e ∗ ( gauge ∇ µ ( ϕ k ) b ˜ a ) + α s ~ c (cid:0) ( Φ k crt ) a ˜ a C bac α cj e A jµ + ( Φ k crt ) a ˜ b f ˜ b ˜ aj e A jµ (cid:1) (3.108)and for V ( Φ ) V ( Φ ) = V ( Φ k crt ) + e V k ( ϕ k ) , k = 0 , , (3.109)where V ( Φ k crt ) is the value for the critical value of Φ and e V k ( ϕ k ) is the polynomial of fourth orderin ϕ k . If we use Eq. (3.88) we get a mass matrix for vector bosons e A jµ which strongly dependson Φ k crt N µν M ij ( Φ k crt ) e A iµ e A jν . (3.110)The matrix N µν depends on g µν and in the case g µν = η µν (Minkowski tensor) we have N µν = η µν (3.111)and M ij ( Φ k crt ) = α s πr ~ c V Z M q | e g | d n x n ℓ np g ˜ m ˜ p ( k ) B pβ ( i (cid:16) ( k ) B d ˜ mj ) + ξk nd ( k ) B d ˜ mj ) − ζ ( k ) B d ˜ aj k a ˜ m − ξζ k nd k b ˜ m k a ˜ b ( k ) B d ˜ aj ) + ξ ζk nb k bd k a ˜ m ( k ) B d ˜ aj ) (cid:17)o , k = 0 , , (3.112)where ( k ) B b ˜ ni = (cid:2) δ ˜ m ˜ n C bms α si + δ bm f ˜ m ˜ ni (cid:3) [ Φ k crt ] m ˜ m ( M ij = M ji ) . (3.113)In the case of a symmetric theory ℓ ab = h ab , g ˜ a ˜ b = h a ˜ b one gets M ij = α s πr ~ c V Z M q | e g | d n x (cid:8) h bn h m ˜ p B b ˜ p ( i B n ˜ mj ) (cid:9) . (3.114)Let us consider an expression ( Φ k crt ) m ˜ n C bms α si + ( Φ k crt ) b ˜ m f ˜ m ˜ ni (3.115)in order to find its interpretation. One easily notices that it equals to (cid:0)(cid:2) Ad ′ H ( Y i ) + Ad ′ G ( Y i ) (cid:3) Φ k crt (cid:1) b ˜ m (3.116)( Ad ′ H and Ad ′ G mean the adjoint representation of Lie algebras of H ( h ) and G ( g ) , respectively).Thus if k = 0 (3.115) equals zero for Y i ∈ g and if k = 1 (3.115) equals zero for Y i ∈ g ′ . Thelatest statement comes from the invariancy of the vacuum state with respect to the action if thegroup G for k = 0 ( G ′ for k = 1 ). Generators of g ( g ′ ) should annihilate vacuum state. Thusthe matrix elements M ij ( Φ k crt ) are zero for i, j corresponding to g ( g ′ ).41rom the invariancy of the potential V with respect to the action of the group G one gets ∂ V∂Φ b ˜ n ∂Φ d ˜ d (cid:12)(cid:12)(cid:12) Φ = Φ k crt (cid:0) T b ˜ nc ˜ c (cid:1) i [ Φ k crt ] ˜ cc = 0 (3.117)where (cid:0) T b ˜ nc ˜ c (cid:1) i [ Φ k crt ] ˜ cc = [ Φ k crt ] m ˜ n C bms α si + [ Φ k crt ] b ˜ m f ˜ m ˜ ni . (3.118)Eigenvalues of M ij ( Φ k crt ) are the squares of the masses of the gauge bosons. The secular equation det( M − m I ) = 0 gives us a mass spectrum of massive vector bosons. Thus there is an orthogonalmatrix ( A ij ) = A such that A T = A − and A − M ( Φ k crt ) A = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( Φ k crt ) . . . ... . . . ... . . . m l k ( Φ k crt ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.119) l = n , l = dim G − dim G ′ . In this way we transform the broken vector fields into massive vector fields e B i ′ µ = l k X j =1 A i ′ j e A jµ (3.120)such that η µν l k X j =1 m j ( Φ k crt ) e B jµ e B jν = η µν M ij ( Φ k crt ) e A iµ e A jν . (3.121)Moreover, we should remember the formula (3.98) which is a constraint on Higgs’ field. The massmatrix of masses for Higgs’ bosons can be obtained in a similar way, V ( Φ ) = V ( Φ k crt ) + δV δΦ a ˜ a δΦ b ˜ b ϕ a ˜ a ϕ b ˜ b + . . . (3.122)The matrix m ( Φ k crt ) ˜ aa ˜ bb = − δ VδΦ a ˜ a δΦ b ˜ b (cid:12)(cid:12)(cid:12) Φ = Φ k crt (3.123)can be calculated for k = 0 . One gets m hf ˜ ea = − πr V Z M (cid:26) α s ~ c Q sk [˜ e ˜ a ][˜ n ˜ q ] C sac C kef ( Φ ) c ˜ a ( Φ ) e ˜ q − α s √ ~ c Q as [˜ p ˜ q ][˜ n ˜ a ] f ˜ e ˜ p ˜ q C sef ( Φ ) e ˜ a + 4 α s √ ~ c Q sf [˜ e ˜ a ][˜ p ˜ q ] f ˜ n ˜ p ˜ q C sea ( Φ ) a ˜ a + Q af [˜ c ˜ d ][˜ p ˜ q ] f e ˜ c ˜ d f ˜ n ˜ p ˜ q (cid:27)q | e g | d n x. (3.124)For k = 1 , H a ˜ p ˜ q ( Φ ) = 0 and Φ (if exists) satisfies the following equation α s √ ~ c Q sk [˜ e ˜ a ][˜ p ˜ q ] C sac ( Φ ) c ˜ a = Q ak [˜ c ˜ d ][˜ p ˜ q ] f ˜ e ˜ c ˜ d (3.125)42nd the supplementary condition (3.98).A mass matrix for Higgs’ bosons looks like m hf ˜ ea = − πr V Z M (cid:18) α s √ ~ c Q sk [˜ e ˜ h ][˜ p ˜ q ] H k ˜ p ˜ q ( Φ ) C saf (cid:19)q | e g | d n x. (3.126)We can diagonalize the mass matrix and we get m ( Φ k crt ) a ˜ ab ˜ b ϕ ˜ aa ϕ ˜ bb = l k X j =1 m j ( Φ k crt ) a ˜ a ψ a ˜ a ψ a ˜ a (3.127)where ψ a ˜ a = X b, ˜ b A ˜ bb ˜ aa ϕ ˜ bb . (3.128)For the mass matrix one has ( A − ) ˜ cca ˜ a m ( Φ k crt ) ˜ aa ˜ bb A ˜ ddb ˜ b = ( m ( Φ k crt ) ˜ cc ) δ ˜ d ˜ c δ cd . (3.129)The eigenvalue problem for m ( Φ k crt ) can be posed as follows m ( Φ k crt ) ˜ aa ˜ bb X a ˜ a = m ( Φ k crt ) X ˜ bb . (3.130)One gets the mass spectrum of Higgs’ particles from the secular equation det (cid:0) [ m ( Φ k crt )] ˜ aa ˜ bb − m ( Φ k crt ) I ˜ aa ˜ bb (cid:1) = 0 (3.131)where I ˜ aa ˜ bb = δ ab δ ˜ a ˜ b . (3.132)The diagonalization procedure of the matrix m ( Φ k crt ) ˜ aa ˜ bb can be achieved in the two follow-ing ways. The matrix defines a quadratic form on the representation space N for Higgs’ field.Moreover, the space N can be decomposed into Higgs’ multiplets m j and according to this de-composition the matrix can be written in a block diagonal form [ m ( Φ k crt )] = X j ⊕ m j ( Φ k crt ) . (3.133)We can diagonalize every matrix m j ( Φ k crt ) corresponding to the multiplet m j .Let us consider a problem of the Higgs’ multiplet Φ c ˜ a on E . One can find a representationspace N of Φ c ˜ a in the following way (see Ref. [45]). Let Ad G → X i ⊕ n i ⊕ Ad G (3.134)be the decomposition of the adjoint representation of G , where n i are irreducible representationsof G and let us consider the branching rule of Ad H Ad H → X j ⊕ ( n ′ j ⊗ m j ) (3.135)43here n ′ j are irreducible representations of G and m ′ j are irreducible representations of G . Thelatest formula comes from the known fact G ⊗ G ⊂ H . Thus for every pair ( n i , n ′ j ) where n i and n ′ j are identical irreducible representations of G there is an m j multiplet of Higgs’ field on E .In this way we can decompose Φ into a sum Φ = X ( n i ,n ′ j ) ⊕ Φ ( n i ,n j ) m j (3.136)or N = X ( n i ,n ′ j ) ⊕ m j . (3.137)Thus the multiplet of Higgs’ field is quite complicated in contradiction to the usual case wherethe Higgs’ field belongs to the adjoint representation of chosen group. Moreover, in our case wehave to do with smaller number of parameters in the theory. The theory is established by acoupling constant α s , a radius r , parameters coming from the nonsymmetricity of the theory ξ, ζ ,a homomorphism µ , an embedding of G in H , α ci ( g in h ) and an embedding of G in G (i.e. themanifold M ).The second way of diagonalization of the matrix [ m ( Φ k crt )] is based on the following observa-tion.The matrix m ( Φ k crt ) ˜ aa ˜ bb can be transformed into a different matrix ( n n ) × ( n n ) forming anindex from two indices e a and a a = αa + β e a + γ (3.138)where α, β, γ are integers. The new index a should be unambigous. Thus we must choose α, β, γ in such a way that for every a ∈ N n n the equation (3.138) has only one solution for a ∈ N n , e a ∈ N n .After this we diagonalize [ m ( Φ k crt )] as an ordinary matrix. What is a scale of masses in ourtheory? It is easy to see that m ˜ A = α s r (cid:16) ~ c (cid:17) (3.139)is this scale where m ˜ A is a typical vector boson mass obtained due to Higgs’ mechanism.Let us consider the following decomposition of the connection ω E defined on the principalfiber bundle Q : ω E = ω E + σ E , ω E ∈ g , σ E ∈ m , (3.140)corresponding to the decomposition of the Lie algebra g , g = g ˙+ m . In this way we consider a reduction of a bundle Q to Q induced by an embedding of G into G .The form ω E is a connection defined on Q and σ E is a tensorial form defined on Q ( E, G ) . Wesuppose that the reduction of the bundle Q to Q is possible.The form ω E corresponds to the Yang–Mills’ field (massless vector bosons) which remainsafter symmetry breaking. The tensorial form σ E corresponds to massive vector bosons.One gets for the curvature form Ω E = Ω E + D σ E + [ σ E , σ E ] , (3.141)44here Ω E is a curvature form for ω E and D means a covariant exterior derivative with respectto ω E . Thus Ω E = 12 e H b ıµν θ µ ∧ θ ν β i b ı Y i (3.142) σ E = σ ˜ a Y ˜ a (3.143) e ∗ D σ E = gauge ∇ µ σ aν ] θ µ ∧ θ ν Y ˜ a (3.144) gauge ∇ µ σ aν ] = ∂ [ µ σ aν ] + g ~ c f ˜ a ˜ bi β i b ı σ ˜ b [ ν e A b ıµ ] (3.145) σ ∗ σ E = σ ˜ aν θ ν Y ˜ a = σ iν β ˜ ai Y ˜ a , (3.146) e ∗ ω E = e A iµ β i b ı θ µ Y i (3.147)where e is a local section of the principal bundle Q , matrices β ˜ ai , β i b ı define an embedding of g into g , gauge ∇ µ means a gauge derivative with respect to a connection ω E .If the symmetry is broken from G to G ′ we have a different decomposition ω E = ω ′ E + σ ′ E . (3.148)One can easily connect σ E or σ ′ E with e B (i.e. fields with defined non-zero rest mass, because e A have not defined masses) fields. One gets e ∗ σ E = g ~ c e B i ′ µ θ µ Y i ′ (3.149)where Y i ′ = ( A − ) ii ′ Y i = ( A − ) ii ′ β ˜ ai Y ˜ a . (3.150)The matrix A is defined by (3.119). The same holds for σ ′ E .Let us consider the following gauge transformation, i.e. a change of a local section of Q from e to f , e ( x ) = ( k ) U − ( x ) f ( x ) , (3.151)where ( k ) U ( x ) = exp (cid:16)X ˇ a ( k ) η ˇ a ( x ) Y ˇ a (cid:17) (3.152)for k = 0 , Y ˇ a ∈ m , i.e. ˇ a = e a ; for k = 1 , Y ˇ a ∈ m , and ( k ) η ˇ a ( x ) is a multiplet of scalar fields on E transforming according to the Ad G ( Ad G ′ ). Y ˇ a span m or m ′ ( k = 0 or k = 1 ). Such a gaugetransformation (a condition) is a “unitary gauge”.Let us consider the following parametrization of the Higgs’ field Φ c ˜ a = Ad G ( ( k ) U ( x )) ˜ a ′ a Ad H ( ( k ) U ( x )) ˜ cc ′ (cid:0) ( Φ k crt ) c ′ ˜ a ′ + ( k ) ϕ c ′ ˜ a ′ ( x ) (cid:1) . (3.153)We transform Higgs’ and gauge fields (cid:16) Φ ( k ) U (cid:17) c ˜ a = Ad H ( ( k ) U − ) cc ′ Ad G ( ( k ) U − ) ˜ a ′ ˜ a Φ c ′ ˜ a ′ = ( Φ k crt ) a ˜ a + ( k ) ϕ c ˜ a ( x ) (3.154) e B ′ iµ ( x ) Y i = Ad H ( ( k ) U ( x )) ij e B jµ Y i − ~ cg ∂ µ ( k ) U ( x ) ( k ) U − ( x ) . (3.155)45ne easily gets gauge ∇ µ Φ c ˜ a = Ad H ( ( k ) U ( x )) cc ′ Ad G ( ( k ) U − ( x )) ˜ a ′ ˜ a gauge ∇ µ (cid:0) Φ ( k ) U (cid:1) c ′ ˜ a ′ . (3.156)On the level of a tensorial form one gets f ∗ ( k ) σ E = Ad G ( ( k ) U − ( x )) (cid:0) e ∗ ( k ) σ E − d ( k ) U ( x ) ( k ) U − ( x ) (cid:1) (3.157)and L YM ( e A ) = L YM ( e B ) = L YM ( e B ′ ) . (3.158)It is important to notice that we should consider a local section for ℓ ij , i.e. a ij = e ∗ ℓ ij inthe lagrangian for the Yang–Mills’ field. The fields e B ′ are massive with the same masses as e B .The important point to notice is that the full lagrangian is still G -gauge invariant. Moreover,a choice of a particular value of Φ k crt (which is G invariant) reduces symmetry from G to G (spontaneously). The fields η ˇ a ( x ) disappear. They are eaten by the gauge transformation anddue to this the massive vector fields have three polarization degrees of freedom. Sometimes η ˇ a ( x ) are called “would-be Goldstone bosons”. In the matrix of masses they correspond to zero modes.Let us come back to field equations in our theory. From the Palatini variational principle forthe action S (see Eqs (3.77)–(3.78)) one gets (variation with respect to W λµ , g µν , ω E and Φ ) R µν ( W ) − g µν R ( W ) = 8 πG N c (cid:0) gauge T µν + T µν ( Φ ) + int T µν + g µν Λ (cid:1) (3.159) ∼ g [ µν ] ,ν = 0 (3.160) ∇ ν g [ µν ] = 0 (3.161) g µν,σ − g ξν Γ ξµσ − g µξ Γ ξσν = 0 (3.162) gauge ∇ ( e ℓ ij ∼ e L iαµ ) = 2 ∼ g [ αβ ]gauge ∇ β ( e h ij g [ µν ] e H iµν )+ 2 r √− g α s √ ~ c (cid:20) ℓ ab g ˜ b ˜ n g µα L aµ ˜ b (cid:0) Φ d ˜ c C bdc α cj + Φ b ˜ a f ˜ a ˜ nj (cid:1) + (cid:18) δL aβ ˜ b δ gauge ∇ α Φ w ˜ v (cid:19) ℓ ab g ˜ b ˜ n g βµ ( gauge ∇ µ Φ b ˜ n ) (cid:0) Φ d ˜ w C wdc α cj + Φ w ˜ a f ˜ anj (cid:1)(cid:21) av + 4 r √− g h ab µ ak e ℓ ij e ℓ ki e g [˜ a ˜ b ]gauge ∇ µ (cid:26) g [ µα ] (cid:20) √ ~ cα s C bcd Φ c ˜ a Φ d ˜ b − α s (cid:16) ~ c µ b b ı f b ı ˜ a ˜ b − α s ~ c Φ b ˜ d f d ˜ a ˜ b (cid:17)(cid:21)(cid:27) . (3.163) gauge ∇ µ ( ℓ ab ∼ L aµ ˜ b ) av = − √− g r n(cid:16) δV ′ δΦ b ˜ n (cid:17) g ˜ b ˜ n + 2 √− g µ ei ( e H iµν g [ µν ] ) h ed (cid:16) √ ~ cα s g [˜ a ˜ n ] C dcb Φ c ˜ a g ˜ b ˜ n − α s √ ~ c g [˜ c ˜ d ] f ˜ n ˜ c ˜ d g ˜ b ˜ n (cid:17)o (3.164)where gauge T αβ = − e ℓ ij π n g γβ g τρ g εγ e L iρα e L jτε − g [ µν ] e H ( iµν e H j ) αβ − g αβ (cid:16) e L iµν e H jµν − g [ µν ] e H iµν )( g [ γσ ] e H jγσ ) (cid:17)o (3.165)46s the energy momentum tensor for the gauge (Yang–Mills’) field with a zero trace gauge T αβ g αβ = 0 (3.166) T µν ( Φ ) = 14 πr (cid:0) ℓ ab g ˜ b ˜ n L aµ ˜ b gauge ∇ v Φ b ˜ n (cid:1) av (3.167) − g µν (cid:16) − πr V ( Φ ) + 14 πr ℓ ab ( g b ˜ n g αβ L aα ˜ b gauge ∇ β Φ b ˜ n ) av (cid:17) . (3.168)It is an energy-momentum tensor for a Higgs’ field int T µν = − πr h ab µ a b ı e H iµν (cid:18)e g [˜ a ˜ b ] (cid:16) √ ~ cα s C bcd Φ c ˜ a Φ d ˜ b − α s √ ~ c µ b b ı f b ı ˜ a ˜ b − α s √ ~ c Φ b ˜ d f ˜ d ˜ a ˜ b (cid:17)(cid:19) av + g µν πr h h ab µ ai e H iαβ g [ αβ ] e g [˜ a ˜ b ] (cid:16) √ ~ cα s C bcd Φ c ˜ a Φ d ˜ b − α s ~ c µ b b ı f b ı ˜ a ˜ b − α s √ ~ c Φ b ˜ d f ˜ d ˜ a ˜ b (cid:17)i av . (3.169)It is an energy-momentum tensor corresponding to the non-minimal interaction term L int ( e A, Φ ) . Λ = c πG N (cid:16) α s e R ( e Γ ) ℓ + e Pr (cid:17) = 16 πG N c λ c . (3.170)It plays a role of the “cosmological constant” ∼ e L iµν = √− g g βν g γν e L iβγ (3.171) ∼ g [ µν ] = √− g g [ µν ] (3.172) ( . . . . . . ) av = 1 V Z M q | e g | dx n ( . . . . . . ) . (3.173)We can write gauge ∇ µ ( e ℓ ij ∼ e L iαµ ) = √− g gauge ∇ µ ( e ℓ ij e L iαµ ) (3.174)where gauge ∇ µ means a covariant derivative with respect to a connection ω αβ on E and ω E at once.Let us come back to the equation of motion for test particles in our theory. According tothe usual interpretation we write down a geodetic equation on P with respect to a Levi-Civitaconnection induced by a symmetric part of κ ( ˜ A ˜ B ) .One writes u ˜ A e ∇ ˜ A u ˜ B = 0 (3.175)where e ∇ ˜ A means a covariant derivative with respect to a Levi-Civita connection induced by κ ( ˜ A ˜ B ) on P .One finds e Du α dτ + (cid:16) q c m (cid:17) u β h cd e g ( αδ ) H dβδ + (cid:16) q c m (cid:17) u ˜ b h cd e g ( αδ )gauge ∇ δ Φ d ˜ b = 0 (3.176) e Du ˜ a dτ + 1 r (cid:16) q c m (cid:17) u β h cd h a ˜ d gauge ∇ β Φ d ˜ d + 1 r (cid:16) q c m (cid:17) u ˜ b h cd h a ˜ d H d ˜ d ˜ b = 0 (3.177) ddτ (cid:16) q b m (cid:17) = 0 (3.178)47here e D means a covariant derivative along a line with respect to the connection e ω αβ on E . e D means a covariant derivative along a line with respect to the connection e ω ab on G/G ( r =const ), u ˜ A = ( u α , u ˜ a , u a ) (3.179) u a = q a m , (3.180) q a is a Yang–Mills’ charge known from the Non-Abelian Kaluza–Klein Theory (color (isotopic)charge), u α is a four-velocity of a test particle. u ˜ a is a charge associated with a Higgs’ field. This charge transforms according to the propertiesof a complement m with respect to G and G . Eq. (3.177) describes a movement of a test particlein a gravitational, gauge and Higgs’ field. Eq. (3.178) is an equation for a charge associatedwith Higgs’ field. This charge describes a coupling between a test particle and a Higgs’ field.Eq. (3.179) has a usual meaning (a constancy of a color (isotopic) charge). In this way we get ageneralization of Kerner–Wong–Kopczyński equation to the presence of a Higgs’ field. We have anormalization of a four-velocity u α , g ( αβ ) u α u β = 1 .Let us project the equation on a space-time E , i.e. we take a section e : E → P . One gets e Du α dτ + (cid:16) Q c m (cid:17) u β e g ( αδ ) F dβδ + (cid:16) Q c m (cid:17) u ˜ b h cd e g ( αδ ) e ∗ ( gauge ∇ δ Φ d ˜ b ) = 0 (3.181) e Du ˜ a dτ + 1 r (cid:16) Q c m (cid:17) u β h cd h a ˜ d e ∗ ( gauge ∇ β Φ d ˜ d ) + 1 r (cid:16) Q c m (cid:17) u ˜ b h cd h a ˜ d e ∗ ( H d ˜ d ˜ b ) = 0 (3.182) e ∗ ω = A aµ θ µ X a + Φ a ˜ b e θ b X a (3.183) e ∗ ( q c X c ) = Q c X c . (3.184)Equation (3.178) takes the form dQ a dτ − C acb Q c A bN u N = 0 , or dQ a dτ − C acb Q c A bν u ν − C acb Q c Φ b ˜ n u ˜ n = 0 , (3.185) e g ( αβ ) is defined by Eq. (2.25).Let us consider a Nonsymmetric Hermitian Kaluza–Klein Theory with spontaneous symmetrybreaking. We should introduce an Hermitian tensor on the manifold V = E × M = E × G/G .It is γ AB = g µν r g ˜ a ˜ b ! (3.186)but now g ˜ a ˜ b = h a ˜ b + iζk a ˜ b (3.187) g ∗ ˜ a ˜ b = g ˜ b ˜ a (3.188)and γ ∗ AB = γ BA . (3.189)48he tensor (in a nonholonomic frame) κ ˜ A ˜ B = γ AB ℓ ab ! (3.190)is such that ℓ ab = h ab + iξk ab (3.191) ℓ ∗ ab = ℓ ba (3.192)and κ ∗ ˜ A ˜ B = κ ˜ B ˜ A . (3.193)The connection Γ ˜ A ˜ B ˜ C is compatible with κ ˜ A ˜ B and we have Γ ∗ ˜ N ˜ M ˜ W = Γ ˜ N ˜ W ˜ M + ( e Γ ˜ N ˜ M ˜ W − e Γ ˜ N ˜ W ˜ M ) (3.194)where e N , f M , f W = 1 , , , . . . , ( m + 4) .All the formulae derived here (in this section) are the same but we should consider g [ µν ] as apure imaginary tensor and put iζ in place of ζ and iξ in place of ξ .The Ricci (Moffat–Ricci) tensor and all energy-momentum tensors are Hermitian. e Γ ˜ N ˜ M ˜ W is a connection generated by κ ( ˜ A ˜ B ) . In this theory we can consider Kähler structureson M = G/G .In the case of the Nonsymmetric Kaluza–Klein Theory with a spontaneous symmetry breakingand Higgs’ mechanism we have more possibilities. We can have a complex Hermitian structureas we describe above or a hypercomplex Hermitian structure on a P manifold. Moreover, we candefine on M = G/G a hypercomplex Hermitian metric tensor or a complex Hermitian metric.This means that we have ξ Iξ and ζ iζ (a pure imaginary). The last possibility seems tobe very interesting for we get Hermitian Theory with a mixture of hypercomplex and ordinarycomplex. In this way we get two disconnected real structures on E (a space-time) coupled toYang–Mills’ fields and to a Higgs’ field. For a base manifold V = E × M is a Cartesian productof E and M we have to do effectively with a real version and only on M a tensor is complex(Hermitian). In some cases the geometry of a whole space is effectively real and only on M wehave even Kählerian geometry. Let P be a principal fiber bundle P = ( P, V, π, H, H ) (4.1)over the base space V = E × S (where E is a space-time, S —a two-dimensional sphere) with aprojection π , a structural group H , a typical fiber H and a bundle manifold P . We suppose that H is semisimple. Let us define on P a connection ω which has values in a Lie algebra of H, h . Letus suppose that a group SO(3) is acting on S in a natural way. We suppose that ω is invariantwith respect to an action of the group SO(3) on V in such a way that this action is equivalent to49 O(3) action on S . This is equivalent to the condition (3.23). If we take a section e : E → P weget e ∗ ω = A aA θ A X a = A A θ A (4.2)where θ A is a frame on V and X a are generators of the Lie algebra h . [ X a , X b ] = C cab X c . (4.3)We define a curvature of the connection ωΩ = dω + 12 [ ω, ω ] . (4.4)Taking a section e e ∗ Ω = 12 F aAB θ A ∧ θ B X a = 12 F AB θ A ∧ θ B (4.5) F aAB = ∂ A A aB − ∂ B A aA − C acb A cA A cB . (4.6)Let us consider a local coordinate systems on V . One has x A = ( x µ , ψ, ϕ ) where x µ arecoordinate system on E , θ µ = dx µ , and ψ and ϕ are polar and azimuthal angles on S , θ = dψ , θ = dϕ . We have A, B, C = 1 , , . . . , , µ = 1 , , , . Let us introduce vector fields on V corresponding to the infinitesimal action of SO(3) on V (see Ref. [43]). These vector fields arecalled δ m = ( δ Am ) , m = 1 , , , A = 1 , , . . . , . Moreover, they are acting only on the last twodimensions ( A, B = 5 , , e a, e b = 5 , ). We get: δ µ ¯ m = 0 and δ ψ = cos ϕ, δ ϕ = − cot ψ sin ϕ,δ ψ = − sin ϕ, δ ϕ = − cot ψ cos ϕ,δ ψ = 0 , δ ϕ = 1 . (4.7)They satisfy commutation relation of the Lie algebra A of a group SO(3) , δ A ¯ m ∂ A δ B ¯ n − δ A ¯ n ∂ A δ B ¯ m = ε ¯ m ¯ n ¯ p δ B ¯ p . (4.8)The gauge field A A is spherically symmetric (invariant with respect to an action of a group SO(3) )iff for some V ¯ m —a field on V with values in the Lie algebra h — ∂ B δ A ¯ m A A + δ A ¯ m ∂ A A B = ∂ B V ¯ m − [ A B , V ¯ m ] . (4.9)It means that L δ ¯ m A A = ∂ B V ¯ m − [ A A , V ¯ m ] , (4.10)a Lie derivative of A A with respect to δ ¯ m results in a gauge transformation (see also Eq. (3.23)).Eq. (4.10) is satisfied if V = Φ sin ϕ sin ψ , V = Φ cos ϕ sin ψ , V = 0 (4.11)50nd A µ = A µ ( x ) , A ψ = − Φ ( x ) = A = Φ , A ϕ = Φ ( x ) sin ψ − Φ cos ψ = A = Φ (4.12)with the following constraints [ Φ , Φ ] = − Φ , [ Φ , Φ ] = Φ , [ Φ , A µ ] = 0 . (4.13) A µ , Φ , Φ are fields on E with values in the Lie algebra of H ( h ) , Φ is a constant element ofCartan subalgebra of h . Let us introduce some additional elements according to the NonsymmetricHermitian Kaluza–Klein Theory. According to Section 3 we have on E a nonsymmetric Hermitiantensor g µν , connections ω αβ and W αβ . On S we have a nonsymmetric metric tensor γ ˜ a ˜ b = r g ˜ a ˜ b = r (cid:0) h a ˜ b + ζk a ˜ b (cid:1) (4.14)where r is the radius of a sphere S and ζ is considered to be pure imaginary, h a ˜ b = − − sin ψ ! (4.15) k a ˜ b = ψ − sin ψ ! (4.16)and a connection compatible with this nonsymmetric metric g ˜ a ˜ b = − ζ sin ψ − ζ sin ψ − sin ψ ! (4.17) e g = det( g ˜ a ˜ b ) = sin ψ (1 + ζ ) (4.18) g ˜ a ˜ b = 1sin ψ (1 + ζ ) − sin ψ − ζ sin ψζ sin ψ − ! , (4.19) e a, e b = 5 , . In this way we have to do with Kählerian structure on S (Riemannian, symplectic andcomplex which are compatible). This seems to be very interesting in further research connectingunification of all fundamental interactions. On H we define a nonsymmetric metric ℓ ab = h ab + ξk ab (4.20)where k ab is a right-invariant skew-symmetric 2-form on H .One can rewrite the constraints (4.13) in the form [ Φ , Φ ] = iΦ [ Φ , e Φ ] = − i e Φ [ Φ , A µ ] = 0 (4.21)where Φ = Φ + iΦ , e Φ = Φ − iΦ (see Ref. [43]).51n this way our 6-dimensional gauge field (a connection on a fiber bundle) has been reduced toa 4-dimensional gauge one (a connection on a fiber bundle over a space-time E ) and a collection ofscalar fields defined on E satisfying some constraints. According to our approach there is definedon S a nonsymmetric connection compatible with a nonsymmetric tensor g ˜ a ˜ b , e a, e b = 5 , , b Dg ˜ a ˜ b = g ˜ a ˜ d Q ˜ d ˜ b ˜ c ( b Γ ) θ ˜ c Q ˜ d ˜ b ˜ d ( e Γ ) = 0 (4.22)where b D is an exterior covariant derivative with respect to a connection b ω ˜ a ˜ b = b Γ ˜ a ˜ b ˜ c θ ˜ c and Q ˜ d ˜ b ˜ c ( b Γ ) its torsion.Let us metrize a bundle P in a nonsymmetric way. On V we have nonsymmetric tensor (seeRef. [1]) γ AB = g µν r g ˜ a ˜ b ! (4.23)and a nonsymmetric connection ω AB = Γ ABC θ C compatible with this tensor Dγ AB = γ AD Q DBC ( Γ ) θ C Q DBD ( Γ ) = 0 . (4.24)The form of this connection is as follows ω AB = ω αβ b ω ˜ a ˜ b ! (4.25)where D is an exterior covariant derivative with respect to ω AB and Q DBC ( Γ ) its torsion.Afterwards we define on P a nonsymmetric tensor κ ˜ A ˜ B θ ˜ A ⊗ θ ˜ B = π ∗ ( γ AB θ A ⊗ θ B ) + ℓ ab θ a ⊗ θ b (4.26)where θ ˜ A = ( π ∗ ( θ A ) , λω a ) , (4.27) ω = ω X a is a connection defined on P ( ˜ A, ˜ B, ˜ C = 1 , , . . . , n + 6 ).We define on P two connections ω AB and W AB such that ω AB is compatible with a nonsym-metric tensor κ ˜ A ˜ B , Dκ ˜ A ˜ B = κ ˜ A ˜ D Q ˜ D ˜ B ˜ C ( Γ ) θ ˜ C Q ˜ D ˜ B ˜ D ( Γ ) = 0 , (4.28)where D is an exterior covariant derivative with respect to a connection ω ˜ A ˜ B and Q ˜ D ˜ B ˜ C ( Γ ) itstorsion.The second connection W ˜ A ˜ B = ω ˜ A ˜ B − n + 4) δ ˜ A ˜ B W ( n = dim H ) . (4.29)52n this way we have all quantities known from Section 3. We calculate a scalar of curvature(Moffat–Ricci) for a connection W ˜ A ˜ B and afterwards an action S = − V V Z U √− g d x Z H q | ℓ | d n x Z S q | e g | dΩ R ( W )= − r V V Z U √− g d x Z S q | e g | dΩ (cid:16) R ( W )+ 8 πG N c (cid:16) L YM + 14 πr L kin ( ∇ Φ ) − πr V ( Φ ) − πr L int ( Φ, e A ) (cid:17) + λ c (cid:17) (4.30)where V = R U p | ℓ | d n x , V = R S p | e g | dΩ , U ⊂ E , λ c = (cid:16) α s ℓ e R ( e Γ ) + 1 r e P (cid:17) (4.31)where e R ( e Γ ) is a Moffat–Ricci curvature scalar on a group H (see Section 3 for details). e P = 1 V Z S q | e g | dΩ b R ( b Γ ) (4.32)where b R ( b Γ ) is a Moffat–Ricci curvature scalar on S for a connection b ω ˜ a ˜ b . L YM = − π ℓ ij (cid:0) e H ( i e H j ) − e L iµν e H jµν (cid:1) (4.33)where ℓ ij g µβ g γµ e L iγα + ℓ ji g αµ g µγ e L iβγ = 2 ℓ ji g αµ g µγ e H iβγ (4.34)One gets from (3.45) L b ˜ a ˜ b = h bc ℓ cd H d ˜ b ˜ a , (4.35) V ( Φ ) = − V Z q | e g | dΩ (cid:0) h cd ( H c ˜ a ˜ b g ˜ a ˜ b )( H d ˜ c ˜ d g ˜ c ˜ d ) − ℓ cd g ˜ a ˜ m g ˜ b ˜ n L c ˜ a ˜ b H d ˜ m ˜ n (cid:1) = 1 V π p ζ κ (cid:0) ( ε ¯ r ¯ s ¯ t Φ ¯ t + [ Φ ¯ r , Φ ¯ s ]) , ( ε ¯ r ¯ s ¯ t Φ ¯ t + [ Φ ¯ r , Φ ¯ s ]) (cid:1) (4.36) κ de = (1 − ζ ) h de + ξ k cd k ce (4.37)where k cd = h cf k fd (4.38) V = Z S q | e g | dΩ = 4 π q ζ , (4.39) r, s, t = 1 , , , ε ¯ r ¯ s ¯ t is a usual antisymmetric symbol ε = 1 .We get also from (3.45) ℓ dc g µβ g γµ L dγ ˜ a + ℓ cd L dβ ˜ a = 2 ℓ cd F cβ ˜ a . (4.40)53sing Eq. (3.85) one gets L nω ˜ m = gauge ∇ ω Φ n ˜ m + ξk nd gauge ∇ ω Φ d ˜ m − e g ( αµ )gauge ∇ α Φ n ˜ m g [ µω ] + ξk nd gauge ∇ β Φ d ˜ m e g ( δβ ) g [ δα ] g [ ωµ ] e g ( αµ ) − ξ k nb k bd e g ( αβ )gauge ∇ Φ d ˜ m g [ ωβ ] . (4.41)Moreover, now we have to do with Minkowski space g µν = η µν and L nω ˜ m = H nω ˜ m + ξk nd H dω ˜ m . (4.42)We remember that e m = 5 , or ϕ, ψ and that H nµ ˜ m = gauge ∇ µ Φ n ˜ m . (4.43)We have L kin ( H nµ ˜ m ) = 1 V Z q | e g | dΩ ( ℓ ab η βµ L aβ ˜ b H bµ ˜ a g ˜ b ˜ a ) . (4.44)Finally we get L kin ( ∇ µ Φ ¯ m ) = 2 π V η µν p ζ ¯ κ (cid:0) gauge ∇ µ Φ ¯ m , gauge ∇ ν Φ ¯ m (cid:1) (4.45) κ ad = ( h ad + ξ k ab k bd ) (4.46)where gauge ∇ µ Φ ¯ m = ∂ µ Φ a ¯ m − [ A µ , Φ ¯ m ] . (4.47)Now we follow Ref. [43] and suppose rank H = 2 and afterwards H = G . In this way ourlagrangian can go to the GSW model where SU (2) × U (1) is a little group of Φ (see Appendix B).We get also a Higgs’ field complex doublet and spontaneous symmetry breaking and mass gen-eration for intermediate bosons. For simplicity we take ξ = 0 and also we do not consider aninfluence of the nonsymmetric gravity on a Higgs’ field. We get also a mixing angle θ W (Weinbergangle). If we choose H = G we get θ W = 30 ◦ . We get also some predictions of masses M H M W = 1cos θ W · q − ζ (4.48)where ζ is an arbitrary constant M H M W = 2 p − ζ √ . (4.49)We take M H ≃ GeV and M W ≃ GeV (see Refs [51, 52, 53, 54, 55]).One gets ζ = ± . i. (4.50)Thus ζ is pure imaginary. This means we can explain mass pattern in GSW model. r gives us ascale of mass and is an arbitrary parameter.Moreover, a scale of energy is equal to M = ~ cr √ π √ ζ which we equal to MEW (electro-weak) energy scale, i.e. to M W . One gets r ≃ . × − m. In the original Manton model Higgs’54oson is too light. We predict here masses for W, Z and Higgs bosons in the theory taking twoparameters, ζ (Eq. (4.50)) and r ≃ . × − m in order to get desired pattern of masses. Thevalue of the Weinberg angle derived here for H = G has nothing to do with “GUT driven” value for is a value of our sin θ W , not sin θ W . According to Ref. [43] a Lie group H should havea Lie algebra h with rank 2. We have only three possibilities: G2, SU (3) and SO(5) . The anglebetween two roots plays a role of a Weinberg angle. For
SO(5) θ = 45 ◦ and for SU (3) θ = 60 ◦ .Only for G , θ = θ W = 30 ◦ , which is close to the experimental value. In this way a unificationchooses H = G .Let us notice that dim G and for this dim P = 20 .Moreover, we have M Z = M W cos θ = M W cos θ W = 2 √ M W ≃ . (4.51)and we get from the theory sin θ W = 0 .
25 ( θ W = 30 ◦ ) . (4.52)However from the experiment we get sin θ W = 0 . ± . (4.53)which is not . .Moreover, from theoretical point of view the value . is a value without radiation correctionsand it is possible to tune it at Q = 91 . GeV/c in the MS scheme to get the desired value.Let us notice the following fact. In the electroweak theory we have a Lagrangian for neutralcurrent interaction L N = qJ em µ A µ + g cos θ W ( J µ − sin θ W J em µ ) Z µ = qJ em µ A µ + X f ψ f γ µ ( g fV − g fA γ ) ψ f Z µ (4.54)where g fV and g fA are coupling constants for vector and axial interactions for a fermion f . Onegets g fV = 2 q sin 2 θ W ( T f − q f sin θ W ) g fA = 2 q sin 2 θ W (4.55)where T f is the third component of a weak isospin of a fermion f and q f is its electric chargemeasured in elementary charge q , q f = T f + Y f (4.56)where Y f is a weak hypercharge for f . It is easy to see that for an electron we get g fV = 0 if θ W = 30 ◦ .Moreover, we know from the experiment that g fV = 0 (4.57)(see Ref. [51]).In the original GSW model a Weinberg angle θ W is a phenomenological parameter whichhas no geometrical interpretation in terms of Lie algebraic theory. Here this parameter has this55nterpretation. Moreover, this theory is still classical. How we can quantize it in a general caseincluding nonsymmetric gravity we describe in Conclusions and prospects of further research .Some quantum corrections can change many things going effectively to Eq. (4.57). Moreover,in Minkowski space g µν = η µν and with ξ = 0 the situation is much more simple and we canagree that radiative correction can go to Eq. (4.57) which is interpreted as a correction to sin θ W for example in MS scheme at Q = 91 . GeV/c. This means that even if g fV ( θ W = 30 ◦ ) = 0 thecorrections change g fV to be nonzero in such a way that θ W is not exactly equal to ◦ . Moreover,the unification scheme with H = G is still valid.Let us define a differential cross-section for f + f − → f ′ + f ′− scattering dσdt (cid:0) f − ( P ) f + → f ′− ( P ′ ) f + (cid:1) = 4 πα em s κ P P ′ (cid:12)(cid:12) M P P ′ ( − s ) (cid:12)(cid:12) (4.58)where κ P P ′ is a kinematic factor from the Dirac algebra equal to ( us ) for L (left) → L (left) and R (right) → R (right) and to ( ts ) for L (left) → R (right) and vice versa. At Z mass energies wecan ignore mass of fermion f and f ′ ( m Z > m f and m Z > m f ′ ).In this way the helicity is conserved. M P P ′ is an invariant amplitude which contains allnontrivial information about a coupling. It is defined in such a way that M P P ′ is equal to 1independently of P, P ′ for a simple s -channel photon exchange diagram of lowest order QED forelectrons. In GSW theory one gets M P P ′ ( Q ) = q f (cid:18) − sQ (cid:19) q f ′ + (cid:18) T f − q f sin θ W cos θ W sin θ W (cid:19)(cid:18) − sQ + M Z − Im(Π Z Z ( Q )) (cid:19)(cid:18) T f ′ − q f ′ sin θ W cos θ W sin θ W (cid:19) (4.59)where Im(Π Z Z ) = Γ Z M Z = α em θ W cos θ W X f (cid:20)(cid:20)(cid:18) T fL − q f sin θ W (cid:19)(cid:18) m f M Z (cid:19) + (cid:18) T fL (cid:19) (cid:18) − m f M Z (cid:19)(cid:21)(cid:18) − m f M Z (cid:19) C QCD ( f ) (cid:21) , (4.60)where T fL is a left-handed isospin component for a fermion f . The factor C QCD is C QCD =3 (cid:0) α s ( − M Z ) /π (cid:1) for quarks and for leptons. These formulas are very well known in alltextbooks and as we mention above θ W is an arbitrary parameter. If we evaluate the formulasfor θ W = π one gets M P P ′ ( Q ) = q f (cid:18) − sQ (cid:19) q f ′ (4.61)which simply means that g fV = 0 ( sin θ W = 0 . ).Moreover, we can introduce an effective Weinberg angle θ W = π + δ in such a way that allthe formulas are satisfied. In this way radiative corrections can be considered as corrections to ◦ Weinberg angle. The formula (4.59) can be evaluated in the following way: M P P ′ ( Q ) = q f (cid:18) − sQ + 4 δ (cid:18) − s ( Q + M Z Z − Im(Π Z Z ( Q ))) (cid:19)(cid:19) q f ′ (4.62)56 δ is a small correction to θ W = π ).One can use also some achievements from GSW model. Let us notice that M W = πα em (0) G F √ θ W (1 − ∆r ) (4.63)where ∆r is the 1-loop correction and its dominant contributions are ∆r = ∆r − − sin θ W sin θ W ∆ρ + ∆r rem (4.64) ∆r = 1 − α em (0) α em ( M Z ) (4.65) ∆ρ = 3 G F P f ( m f − m f )8 π √ ≃ G F ( m t − m b ) π √ (4.66) ∆ rem = √ G F M W π · (cid:16) ln (cid:16) M H M W (cid:17) − (cid:17) , (4.67) G F is a Fermi constant, m t and m b are top and bottom quark masses.The term ∆r corresponds to the running of α em from zero (it means, from Q = m e ≃ )to the electroweak scale Q = M Z . ∆ρ depends quadratically on the mass difference betweenthe members of the same fermion doublet. ∆r rem (the remainder) is dominated by Higgs’ bosoneffects and depends logarithmically on M H .We evaluate these formulas for θ W = π getting M W = 4 πα em G F √ − ∆r ) . (4.68)Now we proceed as before writing M W = 4 πα em G F √ θ W (4.69)where θ W = π δ (as above). δ is not a new phenomenological parameter. It is an effect of 1-loop corrections and a runningof α em . In Eq. (4.64) we can write θ W = π getting ∆r = ∆r − ∆ρ + ∆r rem . (4.70)In the formula (4.67) we can put the value of Higgs’ mass and bare value of M W obtained by us.In this way we get the desired value of sin θ W sin θ W = 4(1 − ∆r ) . (4.71)In terms of δ one gets δ = − √ ∆r (4.72)or δ = − √ (cid:16) − α em (0) α em ( M Z ) − ∆ρ + ∆r rem (cid:17) . (4.73)57e get exactly the same results if we use the MS definition of sin θ W = 1 − M W M Z which is alsoan effective value of sin θ W .Using results from Ref. [51] we can evaluate δ from the formula (4.73) getting δ = − . . (4.74)We have δ = − ′ ′′ and θ W = 29 ◦ ′ ′′ . This gives sin θ W = 0 . (4.75)which is not bad as compared with the experimental value for an effective Weinberg angle. Theformula (4.73) can be improved getting δ = − √ (cid:16) − α em (0) α em ( M Z ) − ∆ρ + ∆r rem (cid:17) / (1 − ∆ρ ) . (4.76)We get δ = − . (4.77) sin θ W = 0 . (4.78)More precise quantum field calculations can improve the result. The conclusion is as follows.The Weinberg angle is coming from the unification theory with G group. The value of thisparameter is equal to π . The δ correction is coming from radiative corrections.Now we have a physical relevance and correct description of the Nature. The results can beimproved starting from the formula M W (cid:16) − M W M Z (cid:17) = πα em (0) √ G F (1 + ∆r ) (4.79)using results from Refs [56, 57, 58] and references cited therein. The numerical results obtainedhere do not change significantly the full quantization scheme. Eventually we get some remarks.We have here to do with a finite renormalization of a parameter in the theory, i.e. with afinite renormalization of a Weinberg angle. According to the idea of a renormalization of anyparameter due to quantum interactions this is correct. We should renormalize not only masses orcharges (as in QED, an electron charge and its mass, which is an infinite renormalization), butreally any physical quantity as in solid state physics (an effective mass of an electron). An infiniterenormalization in QED showed us an impossibility to avoid a renormalization in general.The second remark is as follows. In classical field theory as our model for g µν = η µν we haveto do with parameters which have an interpretation as tree values. They should be renormalized.Only in a superrenormalizable theory they can remain the same in any order of perturbationcalculus. Our theory is not superrenormalizable.In our approach on a classical level we have the following parameters: r , ζ ( sin θ W is knownfrom the theory). In order to get a precise prediction we should translate them into G F and α em (0) , in particular G F = G µ . We take from Particle Data (see Ref. [51]) all the interactionconstants (coupling parameters) which can change running parameters.58et us use the above results to recalculate ζ and r in terms of M H and M Z . One gets M W = M Z cos θ W (4.80) M H M Z = q − ζ (4.81)and ζ = ± . i (4.82) r = ~ cM Z √ π p ζ = 2 . × − m . (4.83)In this way M W = M Z cos( π + δ ) (4.84)gives us for the value (4.75) M W = 79 . , (4.85)for the value (4.78) M W = 79 . . (4.86)In the above formulas we take for M H = 125 . GeV and for M Z = 91 . GeV. This means thatfor ζ given by (4.82) and r given by (4.83) we get desired values of M H and M Z . The predictedvalue for M W is a little smaller than the experimental value 80.385 GeV. Moreover, it seems thatconsideration of higher order corrections of perturbation calculus ( -loop corrections) can improvethe result to tune it to the experimental value. This has been done in Appendix E. It seems thateverything is self-consistent. The value of M W and sin θ W obtained in Appendix E indicates thathigher order corrections improve an agreement with an experiment. This means that our 20-di-mensional model works pretty well. The Weinberg angle is not here a phenomenological parameterand we have a confiance that a unification group H is G . This our future development is justifiedby an experiment. Appendix A
In this appendix we find formulae for L nων , L n ˜ w ˜ n , L nω ˜ n . We get these formulae using a generalformula from n -dimensional generalization of Einstein Unified Field Theory obtained by Hlavatýand Wrede (see Refs [19, 59]). One gets Γ NW M = e Γ NW M + (cid:0) K W MN − k [ M · A K W ] AB k NB (cid:1) + h NE (cid:8) K E ( W · A k M ) A + k C · B (cid:2) k ( M · C K W ) AB k E · A + K EAB k ( W · A k M ) · C (cid:3)(cid:9) (A.1) K ABC = − e ∇ A k BC − e ∇ B k CA + e ∇ C k AB , (A.2)where γ AB = h AB + k AB , (A.3) h AB = h BA , k AB = − k BA , (A.4)59 Γ NW M is the Levi-Civita connection generated by h AB = γ ( AB ) ( γ [ AB ] = k AB ), e ∇ A is a covariantderivative with respect to the connection e Γ NW M .The connection Γ NW M is the solution of the equation Dγ A + B − = Dγ AB − γ AD Q DBC ( Γ ) θ C = 0 , A, B, C, D, N, M = 1 , , . . . , N ,Q DBD ( Γ ) = 0 , (A.5)where D is an exterior covariant derivative with respect to the connection Γ . h AB h BC = δ AC (A.6)and all indices are raised by h AB . (E. Schrödinger was surprized that it was possible to find asolution to (A.5) in a covariant form.)Equation (A.1) is more general than that form Refs [19, 59] for in Eq. (A.1) e Γ NW M arecoefficients of the Levi-Civita connection. This connection can be nonsymmetric in indices
W, M for it can be considered in nonholonomic frame. In Refs [19, 59] e Γ NW M mean Christoffel symbols.Moreover, the proof is exactly the same as in Refs [19, 59]. The authors of Refs [19, 59] areusing the natural nonholonomic frame connected to the nonsymmetric metric γ AB in order to find(A.1). Moreover, this nonholonomic frame has nothing to do with the frame we consider.V. Hlavatý and C. R. Wrede were first to consider n -dimensional generalization of the geometryfrom Einstein Unified Field Theory with the nonsymmetric real tensor γ AB . Thus we can find L aµν from the nonsymmetric non-Abelian Kaluza–Klein theory where N = n + 4 (see Section 2).We can also consider non-Abelian theory with a spontaneous symmetry breaking where N =4 + n + n = 4 + m (see Section 3).In order to find L aµν we should calculate Γ nωµ = L nωµ . We should know a Levi-Civitaconnection generated by γ ( AB ) (and κ ( ˜ A ˜ B ) ) which is easy to find from Eqs (2.20) and (3.44) (e.g. e Γ nωµ = H nωµ ) in order to find covariant derivative of antisymmetric part of the metric. Thus oneeventually finds: L nων = H nων + ξH fνω k fe h ne + (cid:0) H nαω e g ( αδ ) g [ δν ] − H nαν e g ( αδ ) g [ δω ] (cid:1) − ξh na k ad e g ( δτ ) e g ( αβ ) H dδα g [ τω ] g [ βν ] − ξh na k ad e g ( βδ ) e g ( ατ ) H dβ [ ω g ν ] τ g [ δα ] + 2 ξ h na h bc k ac k bd e g ( αβ ) H dα [ ω g | µ | β ] (A.7)(One can try to get formula (A.7) using a different approach. It means to use an approximationformula from Ref. [60]. This is similar to our second approach from Ref. [9] in the case of anelectromagnetic field (see Appendix B of Ref. [9]). Moreover, this approach in the case of theNonsymmetric Kaluza–Klein Theory seems to be much more complex.) L n ˜ w ˜ n = H n ˜ w ˜ n + ξH f ˜ n ˜ w k fe h ne + ξ (cid:0) g [ ˜ w ˜ b ] e g (˜ b ˜ a ) H f ˜ n ˜ a − g [˜ n ˜ b ] ˜ g (˜ b ˜ a ) H f ˜ w ˜ a (cid:1) k fb k cd h cn h db = H n ˜ w ˜ n + ξH f ˜ n ˜ w k ef h ne + ξ ζ (cid:0) k w ˜ b e g (˜ b ˜ a ) H f ˜ n ˜ a − k n ˜ b e g (˜ b ˜ a ) H f ˜ w ˜ a (cid:1) k fb k cd h cn h db (A.8) L nω ˜ n = H nω ˜ n − ξH fω ˜ n k fe h ne − ξ (cid:0) g [ ωβ ] e g ( βα ) H fα ˜ n + ζk n ˜ b e g (˜ b ˜ a ) H fω ˜ a (cid:1) k fb k cd h cn h db = gauge ∇ ω Φ n ˜ n − ξ gauge ∇ ω Φ f ˜ n k fe h ne − ξ (cid:0) g [ ωβ ] e g ( βα )gauge ∇ a Φ f ˜ n + ζk n ˜ b e g (˜ b ˜ a )gauge ∇ ω Φ f ˜ a (cid:1) k fb k cd h cn h db (A.9)where gauge ∇ means a gauge derivative with respect to a connection ω on E , H c ˜ a ˜ b = C cab Φ a ˜ a Φ b ˜ b − µ c b ı f b ı ˜ a ˜ b − Φ c ˜ d f ˜ d ˜ a ˜ b . (A.10)60orking in the same way we get Einstein–Kaufmann connections on a group G and on a homo-geneous manifold M = G/G . This is important to find cosmological terms in the theory.We have Levi-Civita connections e ω AB = π ∗ (˜ ω αβ ) − h db e g ( µα ) H dµβ θ b H aβγ θ γ h bd e g ( αβ ) H dγβ θ γ e ω ab ( G ) ! (A.11)where e ω ab ( G ) is a Levi-Civita connection on G , e ω αβ is defined on E and e ω AB is defined on P , A, B = 1 , , . . . , n + 4 ; e ω ˜ A ˜ B = π ∗ (˜ ω AB ) − h db e γ ( MA ) H dMB θ b H aBC θ C h bd e γ ( AB ) H dCB θ C e ω ab ( H ) ! (A.12) e ω ab ( H ) is a Levi-Civita connection on a group H , e γ ( AB ) γ ( AC ) = δ BC . (A.13)Now e ω AB is defined on V = E × G/G and e ω ˜ A ˜ B on P , A, B = 1 , , . . . , n +4 , e A, e B = 1 , , . . . , n + n + 4 .Using (A.11) and (A.12) one can easily calculate covariant derivatives e ∇ A k BC or e ∇ ˜ A k ˜ B ˜ C andafterwards K ABC or K ˜ A ˜ B ˜ C in order to find desired connection coefficients of Γ NW M and Γ ˜ N ˜ W ˜ M .Let us do it for (A.11). One gets e Γ dβγ = H dβγ e Γ βγb = − h db h αβ H dαγ e Γ µaγ = h ad h αβ H dγβ e Γ aαc = e Γ δac = e Γ bcβ = 0 (A.14)(see also Ref. [61]).Using (A.1) one gets Γ nωµ = e Γ nωµ + 12 (cid:16) K ωµ · n − K [ µ · α K ω ] αb k nb − K [ µ · α K ω ] ab k nb + h ne (cid:16) k e ( ω · a k µ ) a + k γ · β (cid:2) k ( µ · γ K ω ) aβ k e · a − K eαβ k ( ω · α k µ ) · γ (cid:3)(cid:17)(cid:17) . (A.15)Moreover, we have K ωµe = − e ∇ ω k µe − e ∇ µ k eω − e ∇ e k ωµ = 2 H fµω k fe (A.16) K ωab = − e ∇ ω k ab − e ∇ a k bω − e ∇ b k ωa = 0 . (A.17)In all the formulae we keep original notation from Refs [19, 59] and after all calculations weswitch to our notation. h αβ → e g ( αβ ) , k ab → ξk ab . (A.18)In this way K ωµe = 2 ξH fµω k fe (A.19)61nd we get Eq. (A.7).Using (A.14) and switching according to (A.18) one gets e ∇ ω k µe = − h ed e g ( νβ ) H dωβ g [ µν ] − ξH mµω k me e ∇ µ k eω = − h ed e g ( εγ ) H dγµ g [ εω ] − ξH fµω k me e ∇ e k ωµ = − h ed e g ( νβ ) H dωβ g [ µν ] − h ed e g ( νβ ) H dµβ g [ ων ] e ∇ ω k ab = e ∇ a k bω = e ∇ b k ωa = 0 . We quote these formulae for a convenience of a reader.Working similarly we get (A.8) and (A.9).Let us come back to cosmological terms and calculate a connection (A.1) on G and G/G . Ona group G a right-invariant Einstein–Kaufmann connection reads Γ nwm = − C nwm + 12 (cid:0) K wm · n − µ k [ m · a K w ] ab k nb (cid:1) + h ne n µK e ( w · a k m ) a + µ k c · b (cid:2) k ( m · c K w ) ab k e · a − K eab k ( w · a k m ) · c (cid:3)o . (A.20)where K abc = − µ (cid:0) e ∇ a k bc − e ∇ b k ca + e ∇ c k ab (cid:1) ( ℓ ab = h ab + µk ab ) . (A.21) e ∇ a means a Riemannian covariant derivative on a semisimple Lie group G with respect to abiinvariant Killing tensor h ab .One gets e ∇ k bc = −
12 ( C fbc k fe + C fec k bf ) (A.22)and K abc = µ ( C fba k fc + C fac k fb − C fbc k fa ) . (A.23)If we write a connection on Γ in the form Γ nwm = − C nwm + u nwm (A.24)one gets for a Moffat–Ricci tensor on GR bd = e R bd + e ∇ a u abd − e ∇ d u aba + 12 (cid:0) e ∇ b u aad − e ∇ d u aab (cid:1) (A.25)where e ∇ a u ced = − (cid:0) C fea u cfd + C fda u cef − C cfa u fed (cid:1) (A.26) u nwm = − µ (cid:0) L wmn − µk [ m · a L w ] ab k nb (cid:1) − µ h ne (cid:0) L e ( w · a k m ) a + µ k c · b (cid:2) k ( m · e L w ) ab k ea − L eb k ( w · a k m ) · c (cid:3)(cid:1) (A.27)where L abc = C fba k fc + C fac k fb − C fbc k fa . (A.28)62ventually we find R bd = e R bd − C fdb ( u aaf + u afa ) − C afd (2 u fba + u fab ) + 14 ( C afb u fad − C fbd u afa ) . (A.29) e R bd is a Moffat–Ricci (equals to Ricci tensor) for a Levi-Civita connection on G generated by h ab e R bd = − h bd . (A.30)Moreover, if k ab = C fab V f (A.31)where e ∇ k V f = − C efk V e (A.32)we get u nwm = µ C snw C pms V p − µ C pas V p C rnb V r C q [ m · a V | q | C s | b | w ] + µ C fcb V f (cid:2) C p ( m · c V | p | C s | b | w ) C qas V q C rna V r − C sbn C pas V p C q ( wa V q C rm ) V r (cid:3) . (A.33) u nwm can be calculated explicitly in a general form. One gets u nwm = 12 µ (cid:0) C fmw k fn + C f · w · n k fm − C fmn k fw (cid:1) − µ h C fbw k fa ( k na k mb − k nb k ma ) + C fmb k fa ( k na k wb − k nb k wa ) − k nb k ma C fab k fw − (cid:0) k fa k mc C fwn + 2 k fm C anf k wa − C fwn k fa k ma + C fmn k fa k wa (cid:1)i + 12 µ h k cb C fab k fn k wa k mb + C fbw k fa k mc ( k ca k nb − k cb k na ) + C fbm k fa k wc ( k ca k nb − k cb k na )+ C fab k cb k wc k am k nf + C fnb k fa k mc ( k ca k wb − k bc k wa ) + C anf k bf ( k cb k ma k wc − k wb k mc k ab ) i (A.34)and R G = ℓ ab R ab . (A.35)In the case of the Einstein–Kaufmann connection on M = G/G manifold one gets b Γ ˜ n ˜ w ˜ m = ( e n e w e m ) + u ˜ n ˜ w ˜ m (A.36)where (cid:8) e n e w e m (cid:9) is the Christoffel symbol built from h a ˜ b . In this way a cosmological term reads P = 1 V Z M q | e g | b R ( b Γ ) d n x (A.37) b R ( b Γ ) = g ˜ a ˜ b b R ˜ b ˜ d ( b Γ ) (A.38) b R ˜ b ˜ d = e R ˜ b ˜ d + e ∇ ˜ a u ˜ a ˜ b ˜ d − e ∇ ˜ d u ˜ a ˜ b ˜ a + 12 (cid:0) e ∇ ˜ a u ˜ a ˜ a ˜ d − e ∇ ˜ d u ˜ a ˜ a ˜ b (cid:1) (A.39)63here b R ˜ b ˜ d is a Moffat–Ricci tensor for a connection b Γ ˜ a ˜ b ˜ c and e R ˜ b ˜ d is a Ricci tensor of a Levi-Civitaconnection formed for a metric tensor h a ˜ b , where u ˜ n ˜ w ˜ m = 12 (cid:0) K ˜ w ˜ m ˜ n − e g [ m · ˜ a ] K ˜ w ]˜ a ˜ b e g [˜ n ˜ b ] (cid:1) + h n ˜ e n K ˜ e ˜ a ( ˜ w e g | ˜ m | ˜ a ) + e g [˜ c · ˜ b ] he g [( | ˜ m |· ˜ c ] K ˜ w )˜ a ˜ b e g [˜ e · ˜ a ] − K ˜ c ˜ a ˜ b e g [( ˜ w · ˜ a ] e g [ ˜ m · ˜ c ] io . (A.40)During the calculations in Section 4 we used the following identities: g ˜ a ˜ m g ˜ m ˜ c = g ˜ m ˜ a g ˜ m ˜ c = δ ˜ c ˜ a (A.41) g ˜ m ˜ a g ˜ c ˜ m = g ˜ a ˜ m g ˜ c ˜ m = δ ˜ c ˜ a (A.42)where e m, e a, e b = 5 , , ( ϕ, ψ ) and F µψ = F µ = − F µ = − F ψµ = − gauge ∇ µ Φ ( x ) = e ∗ ( H µψ ) (A.43) F µϕ = sin ψ gauge ∇ µ Φ ( x ) = − F ϕµ = F µ = − F µ = e ∗ ( H µϕ ) (A.44)We have also F ψϕ = F = cos ψ (cid:0) Φ ( x ) − [ Φ , Φ ( x )] (cid:1) + sin ψ (cid:0) Φ + [ Φ ( x ) , Φ ( x )] (cid:1) = e ∗ ( H ) = − F ϕψ = − e ∗ ( H ) . (A.45) Appendix B
Following Ref. [43] we use the following formulae Φ = 12 ( ϕ ∗ x − α + ϕ ∗ x − β − ϕ x α − ϕ x β ) (B.1) Φ = sin ψ i ( ϕ x α + ϕ x β + ϕ ∗ x − α + ϕ ∗ x − β ) − Φ cos ψ. (B.2) Φ is constant and commutes with a reduced connection. SU (2) × U (1) is a little group of Φ , Φ = 12 i (2 − h γ, α i ) − ( h α + h β ) , (B.3) x α , x − α , x β , x − β are elements of a Lie algebra h of H (see Ref. [62]) corresponding to roots α, − α, β, − β , h α and h β are elements of Cartan subalgebra of h such that h α = 2 α i α · α H i = [ x α , x − α ] , (B.4)where α = ( α , . . . , α k ) , k = rank( h ) , γ = α − β , [ H i , x ω ] = ω i x ω , H i form Cartan subalgebraof h , [ x ω , x τ ] = C ω,τ x ω + τ if ω + τ is a root, if ω + τ is not a root x ω and x τ commute. We take k = 2 . h γ, α i = 2 γ · αα · α = 2 | γ || α | cos θ. (B.5)64n this way we get a Higgs’ doublet (cid:0) ϕ ϕ (cid:1) = e ϕ .The SU (2) × U (1) generators are given by t = 12 i ( x γ + x − γ ) t = 12 ( x γ − x − γ ) t = 12 ih γ y = 12 ih. (B.6) h is an element of Cartan subalgebra orthogonal to h γ with the same norm. Now everything isexactly the same as in Ref. [43] except the fact that ¯ k ad = h ad − ξ k ab k bd (B.7) k ad = (1 − ζ ) h ad − ξ k ab k bd . (B.8)In Ref. [43] ¯ k ad = k ad = h ad . (B.9)A four-potential of Yang–Mills’ field (a connection ω E ) can be written as A µ = X i =1 A µ t i + B µ y (B.10)or A µ = 12 i ( A − µ x γ + A + µ x − γ + A µ h γ + B µ h ) (B.11) A ± µ = A µ ± iA µ . (B.12)We have (see Ref. [43]) h ( t i , t j ) = − γ · γ δ ij h ( y, y ) = − γ · γh ( t i , y ) = 0 F µν = (cid:0) ∂ µ A aν − ∂ ν A aµ + ε abc A bµ A cν (cid:1) t a + ( ∂ µ B ν − ∂ ν B µ ) y = F aµν t a + B µν y (B.13) h ( F µν , F µν ) = − δ ab γ · γ F aµν F bµν − γ · γ B µν B µν (B.14) gauge ∇ µ Φ = (cid:16) ∂ µ ϕ − iA − µ ϕ − iA µ ϕ − i tan θB µ ϕ (cid:17) x α + (cid:16) ∂ µ ϕ − iA + µ ϕ + 12 iA µ ϕ − i tan θB µ ϕ (cid:17) x β (B.15) gauge ∇ µ e Φ = − (cid:16) ∂ µ ϕ ∗ + 12 iA + µ ϕ ∗ + 12 iA µ ϕ ∗ + 12 i tan θB µ ϕ ∗ (cid:17) x − α − (cid:16) ∂ µ ϕ ∗ + 12 iA − µ ϕ ∗ − iA µ ϕ ∗ + 12 i tan θB µ ϕ ∗ (cid:17) x − β (B.16)65e redefine the fields A aµ , B µ and e ϕ with some rescaling ( g is a coupling constant) A ′ aµ = L A aµ , B ′ µ = L B µ , e ϕ ′ = L e ϕ (B.17)where L = 1 g γ · γ ) / (B.18) L = 1 g (cid:16) γ · γα · α (cid:17) / (B.19)We proceed the following transformation Z µ A µ ! = cos θ − sin θ sin θ cos θ ! A µ B µ ! . (B.20)According to the classical results we also have g ′ g = tan θ , assuming q = g sin θ , where q isan elementary charge and g and g ′ are coupling constants of A aµ and B µ fields. The spontaneoussymmetry breaking and Higgs’ mechanism in the Manton model works classical if we take forminimum of the potential e ϕ = v √ ! e iα , α arbitrary phase, (B.21)and we parametrize e ϕ = (cid:0) ϕ ϕ (cid:1) in the following way e ϕ ( x ) = exp (cid:16) i v σ a t a ( x ) (cid:17) v + H ( x ) √ ! . (B.22)For a vacuum state we take e ϕ = v √ ! , (B.23) t a ( x ) and H ( x ) are real fields on E . t a ( x ) has been “eaten” by A aµ , a = 1 , , and Z µ fields makingthem massive. H ( x ) is our Higgs’ field. σ a are Pauli matrices.In the formulae (B.7)–(B.8) we take ξ = 0 . One gets in the Lagrangian mass terms: M W W + µ W − µ + 12 M Z Z µ Z µ − M H H , where W + µ = A + µ , W − µ = A − µ , getting masses for W ± , Z bosons and a Higgs boson (see Eqs(4.48)–(4.52)). For G2 h γ, α i = 3 and θ = 30 ◦ , θ is identified with the Weinberg angle θ W .In order to proceed a Higgs’ mechanism and spontaneous symmetry breaking in this modelwe use the following gauge transformation e ϕ ( x ) U ( x ) e ϕ ( x ) = 1 √ v + H ( x ) ! , (B.24)where v = 2 √ rg cos θ (B.25)66 vacuum value of a Higgs field U ( x ) = exp (cid:16) − v t a ( x ) σ a (cid:17) . (B.26) H ( x ) is the remaining scalar field after a symmetry breaking and a Higgs’ mechanism. One gets A µ A uµ = ad ′ U − ( x ) A µ + U − ( x ) ∂ µ U ( x ) (B.27) F µν F uµν = ad ′ U − ( x ) F µν . (B.28) Appendix C
In this appendix we derive a Kerner–Wong–Kopczyński equation in GWS-model. One gets (seeEqs (3.176)–(3.178)) e Du α dτ + u β m h ( q, H βδ ) + 1 m e g ( αδ ) (cid:16) u · h ( q, gauge ∇ δ Φ ) + u h ( q, gauge ∇ δ Φ ) (cid:17) = 0 (C.1) e Du dτ − r u β m h ( q, gauge ∇ β Φ ) − r u h ( q, H ) = 0 (C.2) e Du dτ − r u b m sin ψ h ( q, gauge ∇ β Φ ) − r u m sin ψ h ( q, H ) = 0 (C.3) ddτ (cid:16) qm (cid:17) = 0 . (C.4) q is an isotopic charge belonging to a Lie algebra of H ( h ), u ˜ a = ( u , u ) is a charge which couplesa test particle to Higgs’ field, H βδ is a strength of SU (2) × U (1) Yang–Mills’ field, Φ , Φ arescalar fields before a spontaneous symmetry breaking (see Eq. (4.12)). e Ddτ is a covariant derivativealong a line with respect to a connection e ω αβ on E , e Ddτ is a covariant derivative with respect to aLevi-Civita connection on S . We have of course g ( αβ ) u α u β = 1 .Using some additional fields Φ , Φ , Φ and also Φ and e Φ , we can write gauge ∇ µ Φ and gauge ∇ µ Φ interms of Higgs’ fields ϕ and ϕ (see Appendix B), m is the mass of a test particle. gauge ∇ µ Φ = 12 gauge ∇ µ ( Φ + e Φ ) = 12 h(cid:16) ∂ µ ϕ − iA − µ ϕ − iA µ ϕ − i tan θB µ ϕ (cid:17) x α + (cid:16) ∂ µ ϕ − iA + µ ϕ + 12 iA + µ ϕ − iB µ ϕ tan θ (cid:17) x β − (cid:16) ∂ µ ϕ ∗ + 12 iA + µ ϕ ∗ + 12 iA µ ϕ ∗ + 12 iB µ ϕ ∗ tan θ (cid:17) x − α − (cid:16) ∂ µ ϕ ∗ + 12 iA − µ ϕ ∗ − iA µ ϕ ∗ + 12 i tan θB µ ϕ ∗ (cid:17) x − β i (C.5) gauge ∇ µ Φ = sin ψ i gauge ∇ µ ( Φ − e Φ ) = sin ψ i h(cid:16) ∂ µ ϕ − iA − µ ϕ − iA µ ϕ − i tan θB µ ϕ (cid:17) x α + (cid:16) ∂ µ ϕ − iA + µ ϕ + 12 iA + µ ϕ − iB µ ϕ tan θ (cid:17) x β (cid:16) ∂ µ ϕ ∗ + 12 iA + µ ϕ ∗ + 12 iA µ ϕ ∗ + 12 iB µ ϕ ∗ tan θ (cid:17) x − α − (cid:16) ∂ µ ϕ ∗ + 12 iA − µ ϕ ∗ − iA µ ϕ ∗ + 12 i tan θB µ ϕ ∗ (cid:17) x − β i (C.6)Let q = q γ x γ + q − γ x − γ + qh + e qh γ + q α x α + q − α x − α + q β x β + q − β x − β . (C.7)It is easy to see that the first part of q , q = q + q , (C.8) q = q γ x γ + q − γ x − γ + qh + e qh γ (C.9)couples to Yang–Mills’ field and the second part q = q α x α + q − α x − α + q β x β + q − β x − β (C.10)to scalar fields Φ and Φ .In this way in a GSW model a test particle has a weak isotopic charge, weak hyperchargewhich are equivalent to weak charge and an electric charge. It has also an additional weak chargewhich couples it to Higgs’ field, i.e. q . Moreover, we have also u ˜ a = ( u , u ) charge. It would bevery interesting to observe this additional charges in an experiment. H = ∂ Φ − ∂ Φ + [ Φ , Φ ] (C.11) H = ∂ Φ − ∂ Φ + [ Φ , Φ ] (C.12)We get e Du α dτ − u β m e g ( αδ ) e Q i δ ij F iβδ − u b m e g ( αδ ) · Q · B βδ + 1 m e g ( αδ ) (cid:16) u · h (cid:0) q, e ∗ ( gauge ∇ δ Φ ) (cid:1) + u h (cid:0) q, e ∗ ( gauge ∇ δ Φ ) (cid:1)(cid:17) = 0 (C.13) e Du dτ − r u β m h (cid:0) Q, e ∗ ( gauge ∇ β ) Φ (cid:1) − r u h ( Q, e ∗ ( H )) = 0 (C.14) e Du dτ − r u b m sin ψ h (cid:0) Q, e ∗ ( gauge ∇ β Φ ) (cid:1) − r u m sin ψ h ( Q, e ∗ ( H )) = 0 (C.15) dQ a dτ − C ccb Q c A bM u M = 0 (C.16)where e ∗ ω E = A iµ θ µ t i + B µ θ µ y (C.17) e ∗ ( q c X c ) = Q c X c (C.18) e ∗ ω = α ci A iµ θ µ e t i + Φ a ˜ a θ ˜ a X a , (C.19) e Q i = Q i γ · γ is an isotopic charge, e Q = Qγ · γ is a weak hypercharge, e t i = t i , i = 1 , , , e t = y, (C.20)68 ( x, y ) = h ab x a y b . (C.21)Let us consider the transformation (B.20) and the following transformation Q q ! = cos θ sin θ − sin θ cos θ ! e Q Q ! . (C.22) θ plays of course a role of the Weinberg angle θ W . Q is a neutral weak charge and q an electriccharge. In this way we get in Eq. (C.16) a very familiar term − u β m e g ( αδ ) qF βδ (C.23)where F βδ = ∂ β A δ − ∂ δ A β (C.24)is a strength of an electromagnetic field and q an electric charge, i.e. a Lorentz force term.One gets for H H = − H = − i sin ψ (cid:16) ϕ ϕ ∗ α i α · α + ϕ ϕ ∗ β i β · β (cid:17) H i − i cos ψϕ x α − i cos ψϕ x b − i cos ψϕ ∗ x − α − i cos ψϕ ∗ x − β + i ψϕ ϕ ∗ C α, − β x γ + i ψϕ ϕ ∗ C β, − α x − γ . (C.25)Let us proceed a spontaneous symmetry breaking and Higgs’ mechanism in our Kerner–Wong–Kopczyński equation. In this way we transform gauge ∇ µ Φ ˜ a ad ′ U − ( x )gauge ∇ µ Φ ˜ a = gauge ∇ µ Φ u ˜ a , e a = 5 , , (C.26)where gauge ∇ µ Φ u = 12 √ h ∂ µ H ( x )( x β − x − β )+ i v + H ( x )) (cid:0) A uµ ( x β + x − β ) + B µ tan θ ( x − β − x β ) − A + uµ x − α + A − uµ x α (cid:1)i (C.27) gauge ∇ µ Φ u = sin ψ i h ∂ µ H ( x )( x β + x − β )+ i v + H ( x )) (cid:0) A + uµ x − α − A − uµ x α + A uµ ( x β − x − β ) + B µ tan θ ( x − β − x β ) (cid:1)i (C.28) H ad ′ U − ( x ) H u , (C.29)where H u = − sin ψ ( v + H ( x ))2 (cid:16) ( v + H ( x )) β i β · β H i + √ ψ ( x β + x − β ) (cid:17) (C.30) H u = − H u (C.31)where A + µ A + uµ = (cid:0) ad ′ U − ( x ) A µ (cid:1) + + i v ∂ µ t + ( x ) (C.32)69 − µ A − uµ = (cid:0) ad ′ U − ( x ) A µ (cid:1) − + i v ∂ µ t − ( x ) (C.33) A µ A uµ = (cid:0) ad ′ U − ( x ) A µ (cid:1) + i v ∂ µ t ( x ) . (C.34)Simultaneously we proceed a transformation on charges Q Q u = ad ′ U − ( x ) Q. (C.35)In this way we have from Eqs (C.13)–(C.16) e Du α dτ − u β m e g ( αδ ) e Q iu δ ij W iβδ − u β m e g ( αδ ) Q u Z βδ − u β m e g ( αδ ) qF βδ + 1 m e g ( αδ ) (cid:16) u h ( Q u , gauge ∇ δ Φ u ) + u h ( Q u , gauge ∇ δ Φ u ) (cid:17) = 0 (C.36) e Du dτ − r u β m h ( Q u , gauge ∇ β Φ u ) − r u h ( Q u , H u ) = 0 (C.37) e Du dτ − r u β m sin ψ h ( Q u , gauge ∇ β Φ u ) − r u m sin ψ h ( Q u , H u ) = 0 . (C.38)In Eqs (C.36)–(C.38) a test particle is coupled to physical fields only, i.e. W iµν , F µν and H .One derives a final form of h ( Q u , e ∗ ( gauge ∇ µ Φ )) , h ( Q u , e ∗ ( gauge ∇ µ Φ )) , h ( Q u , e ∗ ( H )) , getting h ( Q u , e ∗ ( gauge ∇ µ Φ )) = 12 √ (cid:16) α · α (cid:0) q − α W − uµ − q α W + uµ (cid:1) + 2 β · β (cid:16) ∂ µ H ( q − β − q β )+ i v + H ( x ))( Z uµ cos θ + A µ sin θ )( q − β + q β ) + ( − Z uµ sin θ tan θ + A µ sin θ )( q − β − q β ) (cid:17) + h ( x α , x α ) q α W − uµ + h ( x β , x β ) q β (cid:16) ∂ µ H + i v + H ( x ))( Z uµ cos θ + A µ sin θ )+ ( Z uµ sin θ tan θ − A µ sin θ ) (cid:17) + h ( x − α , x − a ) q − α W + uµ + h ( x − β, − β ) q − β (cid:16) − ∂ µ H + i v + H ( x ))( Z uµ cos θ + A µ sin θ )+ ( − Z uµ sin θ tan θ + A µ sin θ ) (cid:17)(cid:17) (C.39) h ( Q u , e ∗ ( gauge ∇ µ Φ )) = sin ψ √ i (cid:18) i ( v + H ( x )) α · α ( q α W + uµ − q − α W − uµ )+ 2 β · β (cid:16) ∂ µ H ( q β + q − β ) + ( q − β − q β ) Z uµ cos θ (cid:17) − i h ( x α , x α ) q α W − uµ + i h ( x − α , x − α ) q − α ( v + H ( x )) W + uµ + h ( x β , x β ) q β (cid:16) ∂ µ H + Z uµ cos θ (cid:17) + h ( x − β , x − β ) q β (cid:16) ∂ µ H − Z uµ cos θ (cid:17)(cid:19) (C.40)70 ( Q u , e ∗ ( H )) = − h ( Q u , e ∗ ( H ))= − sin 2 ψ √ (cid:18) ( q β + q − β ) (cid:16) β · β + h ( x β , x β ) + h ( x − β , x − β ) (cid:17)(cid:19) (C.41)Here the superscript u means that all quantities are in a gauge U . In this way all couplings of atest particle are expressed by physical fields after a spontaneous symmetry breaking and Higgs’mechanism.Let us consider Eq. (C.16) in more details using Eq. (C.6) and let us change a gauge using agauge changing function U ( x ) . One finds dQ uγ dτ − i (cid:0) ( Z uµ cos θ − sin θA µ ) Q uγ − W − uµ e q (cid:1) u µ + v + H ( x ) √ h γ, α i (cid:16) u + i sin ψ u (cid:17) q α − i u cos ψ (cid:0) h γ, β i + h γ, α i (cid:1) Q uγ = 0 (C.42) dQ u − γ dτ − i (cid:0) W + uµ e Q u − ( Z uµ cos θ − sin θA µ ) Q u − γ (cid:1) u µ + 1 √ v + H ( x )) h γ, α i q − α (cid:16) u − i sin ψ u (cid:17) + i u cos ψ ( h γ, α i + h γ, β i ) Q uγ = 0 (C.43) dqdτ = 0 (C.44) d e Q u dτ −
12 ( W + uµ Q uγ − W − uµ Q u − γ ) = 0 (C.45) dq α dτ + 12 i (cid:16) (cos θA µ − Z uµ sin θ ) h α, γ i q α − Z uµ cos θ − sin θA µ ) q α γ α − γ α γ · γ (cid:17) u µ + u ( v + H ( x )) √ Q uγ (cid:16) u + i sin ψ √ u (cid:17) − i u cos ψq α (2 + h α, β i ) = 0 (C.46) dq − α dτ + 12 i (cid:16) θA µ − Z uµ sin θ ) q − α γ α − γ α γ · γ + ( Z uµ cos θ − sin θA µ ) q − α h α, γ i (cid:17) u µ + v + H ( x ) √ Q u − γ (cid:16) u − i sin ψu (cid:17) + i ψq − α (2 + h α, β i ) = 0 (C.47)Simultaneously we get q β = q − β = 0 . (C.48)In this way our equations are simpler, e.g. h ( Q u , e ∗ ( H )) = h ( Q u , e ∗ ( H )) = 0 . Let usnotice that q α and q − α charges are not influenced by the gauge transformation U ( x ) . The electriccharge q does not feel any movement of additional charges.Thus one gets eventually e Du µ dτ − e Q iu m e g ( µδ ) δ ij u β W iuβδ − Q u m e g ( µδ ) u β Z uβδ − qm e g ( µδ ) u β F βδ + 1 √ m e g ( µδ ) (cid:18) u (cid:16) α · α ( q − α W − uδ − q α W + uδ ) + 1 β · β (cid:0) h ( x α , x α ) q α W − uδ + h ( x − α , x − α ) q − α W + uδ (cid:1)(cid:17) + u sin ψ (cid:16) v + H ( x ) α · α ( q α W + uδ − q − α W − uδ ) − β · β ) (cid:0) h ( x α , x α ) q − α ( v + H ( x )) (cid:1) W + uδ (cid:17)(cid:19) = 0 (C.49)71 Du dτ − r u β √ m (cid:16) α · α ( q − α W − uβ − q α W + uβ )+ 1 β · β (cid:0) h ( x α , x α ) q α W − uβ + h ( x − α , x − α ) q − α W + uβ (cid:1)(cid:17) = 0 (C.50) e Du dτ − r u β √ m sin ψ (cid:16) v + H ( x ) α · α ( q α W + uβ − q − α W − uβ )+ 12( β · β ) (cid:0) − h ( x α , x α ) q α W − uβ + h ( x − α , x − α )( v + H ( x )) W + uβ (cid:1)(cid:17) = 0 . (C.51)Let us suppose that H = G . In this case one gets | β | = | α | = √ , | γ | = √ ,α · α = β · β = 2 , γ · γ = 6 , h γ, α i = 3 , h γ, β i = h α, β i = − ,γ α − γ α γ · γ = √ ,θ = 30 ◦ , cos θ = √ , sin θ = 12 . (C.52)Thus one gets dQ uγ dτ − i (cid:16)
12 ( Z uγ √ − A µ ) Q uγ − W − uγ e q (cid:17) u µ − v + H ( x )) √ q α (cid:16) u + i sin ψ u (cid:17) − i cos ψQ uγ = 0 (C.53) dQ u − γ dτ − iu µ (cid:16) W + uµ e Q u −
12 ( √ Z uµ − A µ ) Q u − γ (cid:17) + 3 u √ v + H ( x )) q − α (cid:16) u − i sin ψ u (cid:17) + iu cos ψQ u − γ = 0 (C.54) dqdτ = 0 (C.55) d e Q u dτ −
12 ( W + uµ Q uγ − W − uµ Q u − γ ) = 0 (C.56) dq α dτ + 12 i (cid:16)
32 ( √ A µ − Z uµ ) − √
36 ( √ Z uµ − A µ ) (cid:17) q α u µ + u ( v + H ( x )) √ Q uγ (cid:16) u + i sin ψ √ u (cid:17) − i u cos ψq α = 0 (C.57) dq − α dτ + i (cid:16) √
36 ( √ A µ − Z uµ ) + 32 ( √ Z uµ − A µ ) (cid:17) u µ q − α + v + H ( x ) √ Q u − γ (cid:16) u − i sin ψ u (cid:17) + i ψq − α = 0 . (C.58)Eqs (C.49)–(C.51) and (C.53)–(C.58) are generalized Kerner–Wong–Kopczyński equations inGSW model. 72t the end of this appendix we consider a cosmological constant in GSW model. In this casewe have from Eq. (4.31) λ c = (cid:16) α s ℓ e R ( e Γ ) + 1 r e P (cid:17) . Moreover, now we have also an additional term for a Higgs’ potential V (0) = 0 . One gets λ ′ c = λ c − π (1 − ζ ) p ζ r (C.59)and eventually λ ′ c = α s ℓ e R ( e Γ ) + 1 r (cid:16) e P − π (1 − ζ ) p ζ r (cid:17) (C.60)where e P is given by the formula (4.32).Moreover, we should add to cosmological constant term also V (0) (see Appendix D). Theterm e P has been calculated in Refs [1, 5] for S . e R ( e Γ ) is equal to e R G ( G = G , see Eq. (A.35)).One gets λ ′ c = α s ℓ e R G + 1 r (cid:18)(cid:16) | ζ | ζ + 1)(1 + ζ ) / (cid:16) ζ E (cid:16) | ζ | p ζ + 1 (cid:17) − ζ + 1) K (cid:16) | ζ | p ζ (cid:17)(cid:17) + 8 ln (cid:16) | ζ | q ζ + 1 + 4(1 + 9 ζ − ζ ) | ζ | ζ ) / (cid:17)(cid:17) | ζ | p ζ − π p ζ (cid:16) − ζ ) + ξ p ζ K ( h α + h β , h α + h β ) (cid:17)(cid:19) (C.61)where K ( k ) = Z π/ dθ √ − k sin θ (C.62) E ( k ) = Z π/ p − k sin θ dθ (C.63)are elliptic integrals of the first and second order. e R G strongly depends on k ab and ξ . It seems that we can tune λ ′ c to the desired value knownfrom observational data. K ( · , · ) is defined by Eq. (D.15). Appendix D
In this appendix we give details of an interaction of the Higgs’ field and nonsymmetric (Hermitian)gravity. One gets from Eq. (3.88) L kin ( gauge ∇ Φ ) = 12 πr (1 + ζ ) ( gauge ∇ µ Φ k ∇ ω Φ d (cid:16) ξζ ( ζ − − − ξ ) k kd + (cid:0) ξζ (2 ξ − ζ − ξζ − π − ξ (cid:1) k bd k bk − h kd + ξ ζ (2 ξ − π ) k nb k nk k bd (cid:17) g ωµ + gauge ∇ µ Φ d ∇ γ Φ n (cid:16) − ξ h nd e g ( γβ ) g [ βω ] ξ e g ( γβ ) g [ βω ] k nd + 2 e g ( αν ) e g ( γρ ) g [ νω ] g [ ρα ] k nd + 2 ξ k nd k nd e g ( αν ) e g ( γρ ) g [ νω ] g [ ρα ] (cid:17) g ωµ + (cid:16) − ζ gauge ∇ µ Φ k ∇ ω b Φ n + 2 ζ gauge ∇ µ b Φ k ∇ ω Φ n + 2 ζ ( ζ + 1) k nd gauge ∇ µ Φ k ∇ ω b Φ d + 2 ξ ( ζξ + 1) k nd gauge ∇ µ Φ k ∇ ω b Φ d + 2 ξ ζk nb k bd gauge ∇ µ b Φ k ∇ ω Φ d (cid:17) g ωµ ℓ nk + ℓ nk g ωµ k nd (cid:0) π + 4 ζ + 4 ζξ + 2 πξζ + 4 ζξ (cid:1) g αη g [ ηω ]gauge ∇ µ Φ n ∇ α b Φ d + 2 ξζℓ nk g ωµ e g ( αν ) e g ( γρ ) g [ νω ] g [ ρα ] (cid:0) gauge ∇ µ Φ k ∇ γ b Φ d + ζ ∇ µ b Φ k ∇ γ Φ d + 2 gauge ∇ µ b Φ k ∇ γ Φ d (cid:1) + ℓ nk g ωµ h(cid:16) − gauge ∇ µ b Φ k ∇ ω b Φ n − ξ ( ζ − k nd gauge ∇ µ b Φ k ∇ ω b Φ d + 2 ξ ζk nb k bd gauge ∇ µ b Φ k ∇ ω b Φ d (cid:17) + e g ( αη ) g [ ηω ] (cid:0) gauge ∇ α b Φ d ∇ µ b Φ k − ξ ∇ µ b Φ k ∇ α b Φ d − ζ ξk nd gauge ∇ µ b Φ k ∇ α b Φ d (cid:1)i) (D.1)where gauge ∇ µ Φ is given by the formula (C.5) before an electro-weak symmetry breaking and by theformula (C.27) after an electro-weak symmetry breaking (in a gauge U ), gauge ∇ µ b Φ = 1sin ψ gauge ∇ µ Φ (D.2)and gauge ∇ µ Φ is given by the formula (C.6) before an electro-weak symmetry breaking and by theformula (C.28) after an electro-weak symmetry breaking (in a gauge U ). ζ is pure imaginary. Weshould do a rescaling (B.17) in (C.5) and (C.27).One can derive L int (see Eq. (3.82)) and gets L int = iζ √ πr (1 + ζ ) (cid:0) F µν g µν ( ϕ ϕ ∗ − ϕ ϕ ∗ ) − ϕ ϕ ∗ ( F − µν g µν + F + µν g µν ) (cid:1) . (D.3)One eventually gets in a gauge U L int = − iζ √ v + H ( x )) πr (1 + ζ ) ( Z uµν + √ F µν ) g µν . (D.4)Let us notice the following fact. In the formulas (D.1) and (D.3) an interaction between Higgs’and “gauge” fields is covariant, however non-minimal. The interaction between gravity and Higgs’fields has a non-classical kinetic term. The real significance of this interaction demands moreinvestigations.They are “interference effects” between Higgs’ field from GSW model and a nonsymmetricgravity being an effect of a unification. L int is an effect of a unification as well.Let us consider L kin ( gauge ∇ Φ ) in a Minkowski space g µν = η m and let us suppose that electroweaksymmetry breaking took place. One gets L kin ( gauge ∇ Φ ) = 12 πr (1 + ζ ) η ωµ h(cid:0) ( ξζ (2 ξ − ζ − ξζ −
2) + ( π − ζ ) k bd k bk − h dk (cid:1) gauge ∇ ω Φ d ∇ µ Φ k + (cid:0) ζ (2( ζ + 1) + ξ ) h nk + 2 ξ ( ξ ( ζξ + 1) − ζ ) k nd k nk (cid:1) gauge ∇ ω Φ k ∇ ω b Φ d ( h nk − ξ (2 ζ − k pn k pk ) gauge ∇ µ b Φ k ∇ ω b Φ n i (D.5)where gauge ∇ µ Φ = 12 √ h −
12 ( v + H ( x ))( W − uµ x α + W + uµ x − α )+ (cid:16) ∂ µ H + 12 i ( v + H ( x )) (cid:16) W + uµ + 12 ( √ Z uµ − A µ ) (cid:17) x β (cid:17) (D.6) − (cid:16) ∂ µ H + i √ v + H ( x )) Z uµ (cid:17) x − β i (D.7) gauge ∇ µ b Φ = 12 √ (cid:16) i ( v + H ( x ))( W − uµ x α + W + uµ x − α ) (cid:17) − (cid:16) ∂ µ H + 12 i ( v + H ( x )) (cid:16) W + uµ + 12 ( √ Z uµ − A µ ) (cid:17) x β (cid:17) + (cid:16) ∂ µ H + i √ v + H ( x )) Z uµ (cid:17) x − β . (D.8)Let us come back to the Higgs’ potential. One can write V ( ϕ , ϕ ) = 1 V π p ζ κ (cid:16) i (2 − | ϕ | ) h α + 12 i (2 − | ϕ | ) h β − iϕ ϕ ∗ x γ − iϕ ϕ ∗ x − γ , i (2 − | ϕ | ) h α + 12 i (2 − | ϕ | ) h β − iϕ ϕ ∗ x γ − iϕ ϕ ∗ x − γ (cid:17) (D.9)where κ ( x, y ) = κ ad x a y d (D.10) κ ad = (1 − ζ ) h ad + ξ k cd k ce (D.11)and ζ is pure imaginary.This is of course in the case of Hermitian Kaluza–Klein Theory. This generalized potential ismuch more complicated than in the GWS model and can go to some complicated Higgs’ sectorstructure. Moreover, in the simplest case for ξ = 0 it can predict a good agreement with anexperiment for a pattern of masses for W, Z bosons and Higgs’ boson.The Higgs’ potential can be written in the following form V ( ϕ , ϕ ) = V ( ϕ , ϕ ) + V ( ϕ , ϕ ) (D.12) V ( ϕ , ϕ ) = − π (1 − ζ )2(1 + ζ ) r h γ · γ | ϕ | | ϕ | − α · α (2 − | ϕ | ) − β · β (2 − | ϕ | ) − α · β )( α · α )( β · β ) (2 − | ϕ | )(2 − | ϕ | ) i (D.13) V ( ϕ , ϕ ) = πξ ζ ) r (cid:20) −
14 (2 − | ϕ | ) K ( h α , h α ) −
12 (2 − | ϕ | )(2 − | ϕ | ) K ( h α , h β )+ 32 (2 − | ϕ | ) ϕ ϕ ∗ K ( h α , x γ ) + 32 (2 − | ϕ | ) ϕ ϕ ∗ K ( h α , x − γ ) −
14 (2 − | ϕ | ) K ( h β , h β ) + 32 (2 − | ϕ | ) ϕ ϕ ∗ K ( h β , x γ ) + 32 (2 − | ϕ | ) ϕ ϕ ∗ K ( h β , x − γ )
94 ( ϕ ϕ ∗ ) K ( x γ , x γ ) −
94 ( ϕ ∗ ϕ ) K ( x − γ , x − γ ) − | ϕ | | ϕ | K ( x − γ , x − γ ) (cid:21) (D.14)where K ( x, y ) = k de x d y e = k cd k ce x d y e ,k cd = h cb k bd , k dc = k cd , k ab = − k ba (D.15)(a right invariant tensor on H = G ). V ( ϕ , ϕ ) is a part of Higgs’ potential known in Manton model corrected by a constant fromNonsymmetric Kaluza–Klein Theory (Hermitian version). V ( ϕ , ϕ ) is an additional term fromHermitian Kaluza–Klein Theory and can give additional Higgs’ phenomena. Moreover, accordingto an experiment we do not see new phenomena. Moreover, we should do a rescaling (B.17) anduse primed fields.One gets V ′ ( ϕ ′ , ϕ ′ ) = V (cid:16) g (cid:16) α · αγ · γ (cid:17) / ϕ ′ , g (cid:16) α · αγ · γ (cid:17) / ϕ ′ (cid:17) = − π (1 − ζ )(1 + ζ ) r (cid:20) g ( α · α )( γ · γ ) | ϕ ′ | | ϕ ′ | − α · α ) (cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17) − β · β ) (cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17) − α · β )( α · α )( β · β ) (cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17)(cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17)(cid:21) (D.16) V ′ ( ϕ ′ , ϕ ′ ) = V (cid:16) g (cid:16) α · αγ · γ (cid:17) / ϕ ′ , g (cid:16) α · αγ · γ (cid:17) / ϕ ′ (cid:17) = πξ ζ ) r (cid:20) − (cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17) K ( h α , h α ) − (cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17)(cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17) · K ( h α , h β )+ 3 g ( α · α )2( γ · γ ) (cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17) ϕ ′ ϕ ′ ∗ (cid:0) K ( h α , x γ ) + K ( h α , x − γ ) (cid:1) − (cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17) K ( h β , h β ) + 3 g ( α · α )2( γ · γ ) (cid:16) − g ( α · α )( γ · γ ) | ϕ ′ | (cid:17)(cid:0) K ( h β , x γ ) + K ( h β , x − γ ) (cid:1) − g ( α · α ) γ · γ ) (cid:0) ( ϕ ′ ϕ ′ ∗ ) K ( x γ , x γ ) + ( ϕ ′ ∗ ϕ ′ ) K ( x − γ , x − γ ) (cid:1) − g ( α · α ) ( γ · γ ) | ϕ ′ | | ϕ ′ | K ( x − γ , x − γ ) (cid:21) (D.17)76q. (4.33) can be rewritten in the form below (see Eq. (2.39)): L YM = 18 π (cid:20) − γ · γ h δ ¯ n ¯ k H ¯ kωµ H ¯ nωµ − δ ¯ c ¯ d H ¯ c H ¯ d + 2 δ ¯ n ¯ k H ¯ kωµ H ¯ nδω g [ αµ ] e g ( αδ ) i + ξ h k ¯ n ¯ k H ¯ kωµ H ¯ nδω e g ( δα ) g [ αµ ] − k ¯ k ¯ d H ¯ kωµ H ¯ dδα e g ( δβ ) e g ( αρ ) g [ βω ] g [ ρµ ] − k ¯ k ¯ d H ¯ kωµ H ¯ dηω e g ( ηβ ) e g ( αρ ) g [ µα ] g [ βρ ] + k ¯ k ¯ d H ¯ kωµ H ¯ dηω e g ( ηδ ) e g ( αρ ) g [ δβ ] r [ ωδ ] i + ξ h k n ¯ k k n ¯ d H ¯ kωµ H ¯ dηµ e g ( ρβ ) e g ( ηα ) g [ ωβ ] g [ αρ ] − k n ¯ k k n ¯ d H ¯ kωµ H ¯ dδα e g ( δη ) e g ( αρ ) g [ ηω ] g [ ρµ ] − k n ¯ k k n ¯ d H ¯ kωµ H ¯ dηω e g ( ρα ) e g ( ηβ ) g [ µα ] g [ βρ ] + k ¯ kb k b ¯ d H ¯ kωµ H ¯ dαω e g ( αβ ) g [ µα ] − k ¯ kb k b ¯ d H ¯ kωµ H ¯ dαµ e g ( αβ ) g [ ωβ ] + k p ¯ n k p ¯ k H ¯ kωµ H ¯ nωµ i + ξ h k n ¯ k k nb k b ¯ d H ¯ kωµ H ¯ dαω e g ( αβ ) g [ µβ ] − k n ¯ k k nb k b ¯ d H ¯ kωµ H ¯ dαµ e g ( αβ ) g [ ωβ ] i(cid:21) (D.18)where H ¯ c = H ¯ cµν g [ µν ] , ¯ n, ¯ k, ¯ c, ¯ d = 1 , , , in such a way that e ∗ ω E = A µ θ µ + B µ θ µ = ( A µa t a + B µ y ) θ µ (D.19)( a = 1 , , ), t = i ( x γ + x − γ ) , t = ( x γ − x − γ ) , t = ih γ = iγ · γ ( γ H + γ H ) , t = y = ih = iγ · γ ( γ H − γ H ) , e ∗ Ω E = e ∗ (cid:16) H ¯ k θ µ ∧ θ µ (cid:17) = 12 ( F aµν t a + B µν y ) θ µ ∧ θ µ , (D.20) a, b, c = 7 , , . . . , ( h = G ) in such a way that c = α, − α, β, − β, γ, − γ, α ′ , − α ′ , β ′ , − β ′ , γ ′ , − γ ′ , , , where α, β etc. correspond to 12 roots of the algebra G and to generators x α , x β etc., , correspond to the generators H , H —elements of Cartan subalgebra of G .One gets ℓ ¯ n ¯ d = − γ · γ δ ¯ n ¯ d + ξα c ¯ n α d ¯ d k cd (D.21)where α c ¯ n x c = t ¯ n , (D.22) ¯ n = 1 , , , t ¯ n = t a , ¯ n = 4 , t = y .In this way we have H ¯ kωµ → F aωµ , ¯ k = 1 , , ,H ωµ → B ωµ . (D.23)In a gauge U one gets F aωµ → W auωµ , a = 1 , ,F ωµ → F uωµ = 12 ( Z uωµ + √ F ωµ ) B µν = 12 ( √ Z uµν − F µν ) . (D.24)One should remember that we have to do with Hermitian version of Nonsymmetric Kaluza–Klein Theory (Hermitian Kaluza–Klein Theory). Moreover, we consider Hypercomplex–Hermitianversion which is effectively equivalent to a real version of a theory.77 ppendix E In this appendix we consider our δ -deviation from θ W = π in a deeper level. It means we considera ∆r theory up to the second order known in the literature (see Ref. [56] and references therein,see also [63], [64]). We have ∆r = (cid:16) − α em (0) α em ( M Z ) (cid:17)(cid:16) − C W S W ∆ρ (cid:17) + ∆r rem ,C W = cos θ W , S W = sin θ W , (E.1)where ∆ρ = 3 x t (cid:0) x t ρ (2) ( z ) + δρ QCD (cid:1) (E.2) x t = G F m t π √ (E.3) δρ QCD = − α s ( µ ) π c + (cid:16) α s ( µ ) π (cid:17) c ( µ ) (E.4) c = 23 (cid:16) π (cid:17) (E.5) c = − . (E.6) ρ (2) ( z ) = 494 + π + 272 log z + 32 log z + z (cid:0) − π + 12 log z −
27 log z (cid:1) + z (cid:0) − π − z −
720 log z (cid:1) (E.7) z = m t m H (E.8) ∆r rem = √ G F M W π · (cid:16) ln (cid:16) M H M W (cid:17) − (cid:17) . (E.9)From the equation sin θ W = sin ( π + δ ) = 4(1 − ∆r ) (E.10)one gets δ = − √ (cid:0) − α em (0) α em ( M Z ) − ∆ρ (cid:0) α em (0) α em ( M Z ) (cid:1) + ∆r rem (cid:1) (cid:0) − ∆ρ (cid:0) − α em (0) α em ( M Z ) (cid:1)(cid:1) . (E.11)Taking α s ( M Z ) = 0 . (E.12) G F = G µ = 1 . × − (GeV) − (E.13) m t = 173 .
21 GeV (E.14) M H = 125 . (E.15) M W = 80 .
385 GeV (E.16) M Z = 91 .
18 GeV (E.17)78 em (0) ≃ α em ( m e ) = 1137 . (E.18) α em ( M Z ) ≃ α ( M W ) = 1128 (E.19) ∆ρ = − . (E.20)and eventually δ = − . (E.21)or δ = − ′ . ′′ (E.22) sin θ W = sin (cid:0) π + δ (cid:1) = 0 . (E.23)and M W = 79 . (E.24)which is almost correct value of a mass of W ± bosons, θ W = 29 ◦ ′ . ′′ . (E.25)It seems that this is self-consistent. Conclusions and prospects for further research
In the paper we consider a color confinement in the nonsymmetric non-Abelian Kaluza–KleinTheory. We derive a condition for a dielectric confinement in the theory. We remind to the readersome notions of the Nonsymmetric Kaluza–Klein Theory and a new version of the Kerner–Wong–Kopczyński equation in this theory. We solve constraints in Nonsymmetric (Non-Abelian) Kaluza–Klein Theory and also constraints in the Nonsymmetric Kaluza–Klein Theory with spontaneoussymmetry breaking and Higgs’ mechanism.In our geometrical unification we consider all interactions unified by one connection defined onmany dimensional manifold (see Refs [1, 65]). We consider also spontaneous symmetry breakingand Higgs’ mechanism in the Nonsymmetric Kaluza–Klein Theory in a general scheme applicablein a general version for a unification of the nonsymmetric gravity (NGT) with a grand unifiedmodel of gauge field interactions in a bosonic sector. We combine in this case a dimensionalreduction model with Kaluza–Klein Theory. In this approach Higgs’ field is a part of a Yang–Mills’ field on an extended space-time with a symmetry. A base manifold V = E × M , where M = G/G is a vacuum state manifold (classical vacuum). This approach has been suggested forthe first time in Ref. [66]. We derive a generalization of the Kerner–Wong–Kopczyński equationfor a case with a Higgs’ field presence.Due to a geometrical origin of these equations we get a new kind of a “charge” which couplesto Higgs’ field as an electric charge couples to an electromagnetic field in a Lorentz force term.This charge is a generalization of a color (isotopic) charge which couples to Yang–Mills’ field.We consider also a Manton model of electro-weak interactions in the framework of Nonsymmet-ric Kaluza–Klein Theory. In this way we unify electromagnetic and weak interaction (a bosonicsector) with a nonsymmetric gravity (NGT). (This is a 6-dimensional Manton model with G group.) We get a possibility to obtain a realistic mass spectrum for W ± , Z bosons and a re-cently discovered Higgs’ boson. Our unification is justified by the fact that a small correction to79 W = π (Weinberg angle) obtained in the theory can be got by renormalization procedure knownin the literature ( ∆r theory).Let us give the following remark. The classical 5-dimensional Kaluza–Klein Theory (formu-lated as a metrized electromagnetic fiber bundle) gives the exact results of Maxwell electrody-namics with a Lorentz force term and Einstein General Relativity, on a unified geometrical basis.This theory can be considered as a quintessence of classical physics, even it does not give any“interference effects” between gravity and electromagnetic theory. Our 20-dimensional unificationof Hermitian gravity and GSW model (a bosonic part) in a framework of Hermitian Kaluza–KleinTheory with spontaneous symmetry breaking can be treated as a prequantum geometrical unifica-tion of gravity and electro-weak interactions. Our 12-dimensional unification of Hermitian gravityand Nonabelian Yang–Mills’ field for G = SU (3) into Hermitian Nonabelian Kaluza–Klein The-ory can be treated as prequantum geometrical unification of gravity and strong interactions (abosonic part of QCD). Both unifications give “interference effects” between gravity, electro-weakinteractions and strong interactions.There are some further prospects for a research. First of all it is necessary to incorporatefermions in the theory.The beautiful theories as Kaluza–Klein theory (a Kaluza miracle) and its descendents shouldpass the following test if they are treated as real unified theories. They should incorporate chiralfermions. Since the fundamental scale in the theory is a Planck’s mass, fermions should be masslessup to the moment of spontaneous symmetry breaking. Thus they should be zero modes. In ourapproach they can obtain masses on a dimensional reduction scale. Thus they are zero modes in (4 + n ) -dimensional case. In this way ( n + 4) -dimensional fermions are not chiral (according tothe very well known Witten’s argument on an index of a Dirac operator). Moreover, they are notzero modes after a dimensional reduction, i.e. in 4-dimensional case. It means we can get chiralfermions under some assumptions.We should look for some possibilities of Grand Unified Models (see Ref. [67]). First of all weshould look for a group G such thatSU (3) c ⊗ SU (2) L ⊗ U (1) Y ⊂ G. There are a lot of possibilities. One of most promising is G = SO(10) . Moreover, we need alsoa group G such that M = G/G (see Refs [42, 43]). In our world G = SU (3) c ⊗ U (1) em . Thegroup H , G = SO(10) and G = U (1) em ⊗ SU (3) c should be such that SO(10) ⊗ ( SU (3) c ⊗ U (1) em ) ⊂ H. The simplest choice is H = SO(16) . Why? First of all G ⊂ SO(16) and
SO(10) ⊗ SO(6) ⊂ SO(16) .Moreover,
SO(6) ≃ SU (4) and SU (3) ⊗ U (1) ⊂ SU (4) . Thus if we identify U (1) with U (1) em andSU (3) with SU (3) c we get what we want. In this way M = SO(10) / SU (3) ⊗ U (1) ,S ⊂ M , dim SO(16) = 120 , dim SO(10) = 45 , n = dim M = 36 . There is also a possibility toconsider a different possibility M ′ = SO(10) / SU (3) and SO(10) ⊗ SU (3) ⊂ H for a U (1) is an Abelian factor, which is a little in a spirit of theManton approach for GSW model (see Ref. [43]). Coming back to the problem of fermions we80an try to couple multidimensional spinors in a minimal coupling scheme to multidimensionalconnections describing a unified field theory. In this way we can get chiral fermions coupled togravity, Yang–Mills’ and Higgs’ fields.Thus a Yukawa mechanism is possible in our approach. The Yukawa sector in the theory canbe obtained due to a minimal coupling to a total covariant derivative (“gauge” and with respectto a Levi-Civita connection generated by a symmetric part on a multidimensional metric on atotal (4 + n + n ) -dimensional manifold at one) to many-dimensional spinor in N -dimensionalspace, where N = Ent( n + n ) . In this way we can write a Lagrangian of the spinor field inthe form i ~ c ( Ψ Γ M gauge e ∇ M Ψ + gauge e ∇ M Ψ Γ M Ψ ) , where gauge e ∇ M is a derivative mentioned above. Γ M are N -dimensional generalization of Dirac matrices and Ψ = Γ Ψ + . Due to a dimensional reductionprocedure, taking only zero-modes for N -dimensional spinor we can get 4-dimensional spinorsdefined on a space-time E . We can try to get chiral spinors (a Witten argument of an index ofa Dirac operator does not work on a 4-dimensional space) and also to arrange many-dimensional( N -dimensional) spinor as a collection of 4-dimensional spinors to get fermions (known froman experiment). Due to a coupling to many-dimensional Yang–Mills’ field (after a dimensionalreduction decomposed into 4-dimensional Yang–Mills’ field and a multiple of scalar (four-dimen-sional) fields—Higgs’ fields) we get a Yukawa-type terms for 4-dimensional spinors. Thus dueto a Higgs’ mechanism (geometrized in our theory) we get a pattern of masses for 4-dimensionalfermions. The scale of mass for such fermions is given by a parameter r —a radius of a manifold M = G/G . Very heavy fermions with masses of order of a Planck’s mass are removed from thetheory by conditions of zero-mode for Ψ . The bare masses obtained here can interact accordingto the Newton law.Generalized Dirac matrices are defined by the relations { Γ A , Γ B } = 2 η AB or { Γ ˜ A , Γ ˜ B } = 2 η ˜ A ˜ B where η AB = diag {− , − , − , , − , . . . , − | {z } n } η ˜ A ˜ B = diag {− , − , − , , − , . . . , − | {z } n , − , . . . , − | {z } n } . For ( n + 4) or ( n + n + 4) equal to l + 2 (the even case) we define Γ ± = ( ± Γ + Γ ) ,Γ ¯ A ± = ( Γ A ± iΓ A +1 ) , A = 1 , . . . , l. It is easy to show { Γ ¯ A ± , Γ ¯ B ± } = − δ ¯ A ¯ B { Γ ¯ A + , Γ ¯ B + } = { Γ ¯ A − , Γ ¯ B − } = 0 . In particular ( Γ ¯ A + ) = ( Γ ¯ A − ) = 0 . In this way we always have a spinor Ψ such that Γ ¯ A − Ψ = 0 A . We get all possible spinors acting on Ψ by Γ ¯ A + . We get l +1 such spinors (a fullrepresentation). Γ A or Γ ˜ A can be derived in such a base by using iterative method.In the case of l + 3 (an odd case) we should have Γ l +3 = i − ( l +1) Γ · · · Γ l +2 such that ( Γ l +3 ) = − , { Γ l +3 , Γ ¯¯ A } = 0 , A = 1 , . . . , l + 2 . It is easy to define a basis of spinors for both cases. Let ζ = ( ζ , . . . , ζ l ) , ζ ¯ A = ± , Ψ ζ = (cid:16) l Y ¯ A =0 ( Γ ( l + ¯ A ) ) ζ ( l + ¯ A ) +1 / (cid:17) Ψ .Γ l +3 in the even case distinguishes between two classes of spinors Γ l +3 Ψ ζ = + Ψ ζ ( l -dimension—first representation) Γ l +3 Ψ ζ = − Ψ ζ ( l -dimension—second representation)In the odd case we have only one representation of l +1 -dimension.We can introduce also generators of SO(1 , n ) or SO(1 , n + n ) algebra b σ AB or b σ ˜ A ˜ B b σ AB = i Γ A , Γ B ] b σ ˜ A ˜ B = i Γ ˜ A , Γ ˜ B ] . We have of course [ b σ MN , b σ RS ] = − i [ η NS b σ MR + η RN b σ SM + η MR b σ NS + η SM b σ RS ][ b σ ˜ M ˜ N , b σ ˜ R ˜ S ] = − i [ η ˜ N ˜ S b σ ˜ M ˜ R + η ˜ R ˜ N b σ ˜ S ˜ M + η ˜ M ˜ R b σ ˜ N ˜ S + η ˜ S ˜ M b σ ˜ R ˜ S ] . Our spinors transform as Ψ → exp (cid:0) α AB b σ AB ) Ψ or Ψ → exp (cid:0) α ˜ A ˜ B b σ ˜ A ˜ B ) Ψα AB = − α BA α ˜ A ˜ B = − α ˜ B ˜ A . We also have ( b σ AB ) + Γ = Γ b σ AB ( b σ ˜ A ˜ B ) + Γ = Γ b σ ˜ A ˜ B . In our particular cases with or without spontaneous symmetry breaking we get our matricesusing ordinary Dirac matrices and their tensor products with some special matrices. One gets forcovariant derivatives e DΨ = dΨ + e ω AB b σ AB Ψ e DΨ = dΨ + e ω ˜ A ˜ B b σ ˜ A ˜ B Ψ. gauge e D Ψ = hor e DΨ = gauge d Ψ + hor( e ω AB ) b σ AB Ψ gauge e D Ψ = hor e DΨ = gauge d Ψ + hor( e ω ˜ A ˜ B ) b σ ˜ A ˜ B Ψ and also e DΨ = dΨ − e ω AB Ψ b σ AB or e DΨ = dΨ − e ω ˜ A ˜ B Ψ b σ ˜ A ˜ B where Ψ = Ψ + Γ and similarly gauge e D Ψ = gauge d Ψ − hor( e ω AB ) Ψ b σ AB or gauge e D Ψ = gauge d Ψ − hor( e ω ˜ A ˜ B ) Ψ b σ ˜ A ˜ B . e ω AB and e ω ˜ A ˜ B are Levi-Civita connections defined on P with respect to a symmetric part ofmetrics γ ( AB ) and γ ( ˜ A ˜ B ) .How does an iterative method for a construction of Γ matrices work? Let us suppose we haveordinary Dirac matrices γ µ and let us define Γ µ = γ µ ⊗ − ! , µ = 1 , , , ,Γ = I ⊗ ! ,Γ = I ⊗ − ii ! , I an identity matrix, × . Next step Γ A = Γ A ⊗ − ! , A = 1 , , , , , ,Γ = I ⊗ I ⊗ ! ,Γ = I ⊗ I ⊗ − ii ! , I an identity matrix, × . The Lagrangian for out spinor field (multidimensional) looks like L ( Ψ, Ψ , gauge e D ) = i ~ c (cid:0) Ψ ℓ ∧ gauge e D Ψ + gauge e D Ψ ∧ ℓΨ ) where ℓ = Γ µ η µ and η µ is a dual Cartan basis on E . 83e also write new type of covariant derivative gauge e D as D Ψ and D Ψ (see Ref. [9]).The interesting problem is to find exact solutions of field equations in the case of GSW-model of Hermitian Kaluza–Klein Theory with spontaneous symmetry breaking and also for theNonsymmetric Nonabelian (real or Hermitian) Kaluza–Klein Theory with G = SU (3) . We expectsome nonsingular, particle-like stationary solutions in the case of spherical symmetry. Axiallysymmetric stationary solutions in both cases seem to be very interesting from more general pointof view and we will seek for them. These solutions can be considered with and without fermionsources. We also look for some wave-like solutions: a non-Abelian plane wave, spherical andcylindrical waves. The waves can be considered as gravito-Yang–Mills’ waves.Let us give some comments. There are two versions of the Nonsymmetric Kaluza–KleinTheory: real and Hermitian. Both versions work very well in the case of 5-dimensional (electro-magnetic) and in the case of Non-Abelian Yang–Mills’ field. We get charge and color confinementand nonsingular solutions. However, if we want to apply the theory for GSW model (a bosonicpart of this model), only Hermitian version works getting pattern of masses of Z and W ± bosonsand Higgs’ boson agreed with an experiment. It seems that an experiment chooses the Hermitianversion. In this way an idea of deriving a unified field theory from higher-dimensional gravity ismaintained, together with much of the appealing simplicity and unity of the theory.Hermitian Kaluza–Klein Theory seems to be closer to quantum theory even it is a classical fieldtheory. According to A. Einstein Hermitian version of Unified Field Theory would be prequantumgravity.Let us express E c and H e in terms of D n and B d in the Nonsymmetric Kaluza–Klein Theory(electromagnetic case): E c = ∆ rc · L rn D n + ∆ rc K re C ed B d where L rn , K re and C ed are given resp. by Eqs (2.123), (2.122), and by Eq. (1.73) of Ref. [9] and ∆ rc is an inverse tensor of δ cr − K re A ec , i.e. ( δ cr − K re A ec ) ∆ rd = δ cd where A ec is given by Eq. (1.70) of Ref. [9] and det( δ cr − K re A ec ) = 0 H e = Ξ ea A ac L cn D n + Ξ ea C ad B d and Ξ ea is an inverse tensor of δ ae − A ec K ce , i.e. ( δ ae − A ec K ce )Ξ ef = δ af , det( δ ae − A ec K ce ) = 0 . In the case of Hermitian Kaluza–Klein Theory we define vectors F c = 1 √ D c + iB c ) and G c = 1 √ E c + iH c ) . One gets G c = 1 √ (cid:2) ( ∆ rc L rn + i Ξ ca A af L fn ) D n + ( ∆ rc K re C ed + i Ξ ca C ad ) B d (cid:3) .
84e remind to the reader that Latin indices (3-dimensional space indices) are keeping in up ordown position only for convenience. Thus we exchange Ξ ca → Ξ ca D n → D n L fn → L fn L rn → L rn H c → H c . In this way we describe the Riemann–Silberstein vector in the Hermitian Kaluza–Klein Theoryin an electromagnetic case (see Refs [68], [69], [70], [71]) F c and the second vector G c .For a vector F c is considered as a wave function of a photon we are closer to the quantumtheory. This really is a prequantum theory.We can do the same in the case of Yang–Mills’ field getting F e ¯ c = 1 √ D e ¯ c + iB e ¯ c ) and G e ¯ c = 1 √ E e ¯ c + iH e ¯ c ) , which can be calculated in the Hermitian-Nonabelian Kaluza–Klein Theory using formulas (2.45)–(2.50) and (2.111)–(2.112).How to quantize the Nonsymmetric Kaluza–Klein Theory? First of all we can quantize itusing Ashtekar–Lewandowski formalism considering it as GR with additional sources, i.e. g [ µν ] ,gauge fields, Higgs’ fields (see Ref. [32]). This will be done elsewhere. The second approach is toconsider our theory also as GR with additional geometrized sources (see Appendix E of Ref. [9])and develop it into a nonlocal theory. There are several approaches of quantization of nonlocaltheories (see Refs [72], [73]). In this case we can avoid infinities appearing in perturbation calculus,getting a theory which is renormalizable, super-renormalizable and even finite.Nonlocal theories, roughly speaking, are equivalent to theories with higher derivatives up toan infinite order. An integral transformation is equivalent to a differential operator of an infiniteorder.Moreover, introducing nonlocality or a differential operator of an infinite order can be consid-ered as a special type of a regularization procedure to remove infinities from Feynman diagramscalculations in perturbation calculus. It is possible to consider such a procedure as a general-ized (covariant and gauge invariant) Pauli–Villars regularization procedure. Simultaneously wecan quantize the theory (as an ordinary field theory) using Faddeev–Popov prescription in path-integral formalism for gravity and Yang–Mills’ field. A divergence of a one loop in the case ofgravity can be removed using dimensional regularization-renormalization procedure. In order toavoid massive ghosts we should carefully design higher derivative corrections to gravity, Yang–Mills’s fields, Higgs’ fields using differential operator of infinite order. According to Ref. [74, 75]we should add ℓ ab L aµν h (cid:16) − ∆Λ (cid:17) H bµν ∆ is a Laplace operator, a gauge covariant and a covariant with respect to a Levi-Civitaconnection generated by g ( αβ ) , Λ is a scale: ∆ = e g ( αβ )gauge e ∇ α gauge e ∇ β .h is an entire function (non-polynomial) which should be carefully chosen.We should also add h ab H a h (cid:16) − ∆Λ (cid:17) H b R ( W ) h (cid:16) − ∇ Λ (cid:17) R ( W ) R µν ( W ) h (cid:16) − ∇ Λ (cid:17) R µν ( W ) ∇ = e g ( αβ ) e ∇ α e ∇ β .R ( W ) and R µν ( W ) should be expressed by e R , e R µν and additional fields, i.e. g [ µν ] and W µ .In the case where we have to do with Higgs’ fields and spontaneous symmetry breaking weadd also the terms V Z M q | e g | d m x (cid:16) ℓ ab e g ˜ b ˜ n L aµb h (cid:16) − ∆Λ (cid:17) gauge ∇ µ Φ b ˜ n (cid:17) and h ab µ ai e H i h (cid:16) − ∆Λ (cid:17) g [˜ a ˜ b ] (cid:0) C bcd Φ c ˜ a Φ d ˜ b − µ b ˆ ı f ˆ ı ˜ a ˜ b − Φ b ˜ d f ˜ d ˜ a ˜ b (cid:1) where h , h , h and h are entire transcendental functions of a complex variable.The problem which arises now is as follows: Is it possible to choose h , h , h , h , h , h in sucha way that no physical poles are introduced while the theory will be (super-) renormalizable andunitary. It seems that such entire transcendental functions can be defined (see also Refs [76, 77]).This will be examined elsewhere.Let us consider the following Lagrangian in the theory L = R ( W ) + R ( W ) h (cid:16) − ∇ Λ (cid:17) R ( W ) + R µν ( W ) h (cid:16) − ∇ Λ (cid:17) R µν ( W )+ ℓ ab L aµν h (cid:16) − ∆Λ (cid:17) H bµν + h ab H a h (cid:16) − ∆Λ (cid:17) H b + R ( Γ ) , i.e. a Lagrangian (2.67) plus higher order in derivatives terms (this Lagrangian can be extendedto the case with Higgs’ fields).Let us apply a path-integral method to quantize gravitational and Yang–Mills’ field. We writerather formally (see Ref. [78]) Z = Z e iS [ A,g, ¯ W ] D A D g D W , where S is a classical action D A = Y µ D A µ = Y x,µ dA µ ( x ) D g = Y α,βα ≤ β Dg ( αβ ) Y α,βα<β Dg [ αβ ] = Y x,α,βα ≤ β dg ( αβ ) ( x ) Y x,α,βα<β dg [ αβ ] ( x ) W = Y µ D W µ = Y x,µ dW µ ( x ) mean functional (nonexiting) measure for gauge field and gravity.According to Ref. [74] we add gauge-fixing terms L g = − η µν β f ν [ g ] W g (cid:16) − (cid:3) Λ (cid:17) f µ [ g ] − β f a [ A ] W YM (cid:16) − (cid:3) Λ (cid:17) h ab f b [ A ] − β h [ W ] f W (cid:16) − (cid:3) Λ (cid:17) h [ W ] , where β , β , β are constants and (cid:3) is an ordinary d’Alembert operator in a Minkowski space. f µ [ g ] = f µ [ g αβ ] is a gauge fixing function for a gravitational field, f a [ A ] for a gauge field, W g is agravity gauge-fixing weighting function, W YM is a gauge-term weight for a Yang–Mills’ field. Weadd also a gauge-fixing function for W µ field with its weight e w .We can also add gauge-fixing terms for Higgs’ fields. Sometimes it is possible to consider agauge condition which involves gauge and Higgs’ fields together. In this way we can get also ad-ditional Faddeev–Popov ghosts. But this does not threaten us. These ghosts are easily exorcized.The most important problems in this theory are possible massive ghosts which could appear iffunctions h i are not properly chosen.The FP (Faddeev–Popov) ghosts are not dangerous as we mention above from quantum fieldtheory point of view. They are also exorcized from geometrical point of view, i.e. they can begeometrized (see Ref. [79]). According to Refs [79, 80] a gauge field (in specific fixed gauge, i.e. ina section of a principal bundle) plus a ghost field is a globally defined connection on the principalbundle (see also Eq. (2.56)). The anticommuting property of a ghost field can be easily derivedand a nilpotent BRST charge obtained as a differential operator. In order to proceed a functionalintegration we apply a well-known Faddeev–Popov trick in order to do an integration over thoseconfigurations which satisfy a gauge fixing conditions. In this approach ghosts fields appear inthe Lagrangian. We have two kinds of ghosts—gauge field ghosts and gravity field ghosts. Thuswe have a ghost field Lagrangian L gh = c a M ab c b + c µ N µν C ν + cM c coming from an exponentiation of a Faddeev–Popov determinant (an infinite analogue of a “Ja-cobian”) such that M ab c b = δ c f a [ A, x ] (scalars) N µν c µ = δ α f v [ g, x ] (vector) M c = δh [ W , x ] (scalar)where δ c f a is the infinitesimal transformation of f a with gauge parameters c b , δ α f v is the infinites-imal transformation of f v with a changing of a frame with parameter c µ and δh is an analogueof a gauge changing of W ν . They do not depend on weighting functions. The Faddeev–Popovghosts are ghost fields in this sense that they do not have a right statistics. In order to get ghostLagrangian we should integrate using anticommuting fields. In this way, they are anticommutingbosons. Thus one gets Z = Z e iS [ A,g, ¯ W ,c a ,c µ ,c ] DF Dh = V · Z e iS [ A,g, ¯ W ,c a ,c µ ,c ] DF DF = Y x dA fix ( x ) Y x dg fix ( x ) Y x dW fix ( x ) Y x ( dc a ( x ) dc a ( x )) Y x ( dc µ ( x ) dc µ ( x )) Y x ( dc ( x ) dc ( x )) Dh = Y x dh ( x ) (integration over a gauge group), c a , c µ , c mean antighost fields. V is an “infinite volume” of a local gauge group, A fix ( x ) meansthat the gauge has been fixed. The same for g fix ( x ) and W fix ( x ) . (In a more geometrical languagewe say that an integration is over an orbits space of a local gauge group.) S [ A, g, W , c a , c µ , c ] = Z d x √− g ( L + L g + L gh ) . The above formulae are starting points for a path-integral quantization of the NonsymmetricKaluza–Klein Theory after a careful choice of entire functions h i , i = 1 , , , . We hope to findthem to get (super-) renormalizable or even finite theory unifying nonsymmetric gravity and otherfundamental interactions (in a bosonic section) with “interference effects” obtained on a level of aclassical field theory. After an inclusion of fermion fields this program accomplishes the Einsteinidea of a Unified Field Theory of all interactions, which is geometrical (geometrization of physicalinteractions), nonlinear (nonlinear field equations) and also non-local. This nonlocality shouldof course be causal and this depends on functions h i . Such functions are entire transcendentalfunctions. They are not polynomials, i.e. h ( z ) = ∞ X n =0 a n z n and lim n →∞ n q | a n | = 0 . It means they are defined on a whole open complex plane and according to Liouville theorem theyhave a pole or an essential singularity at infinity.The construction of such functions can be done according to Refs [74, 75, 76, 77]. In any caseswe can write ( e g is a coupling constant) h ( z ) = 1 + e g exp (cid:16)Z p γ ( z )0 − ζ ( w ) w dw − (cid:17) where p γ ( z ) is a real polynomial of degree γ and p γ (0) = 0 , ζ ( z ) is an entire function and real onthe real axis and ζ (0) = 1 , | ζ ( z ) | → ∞ for | z | → ∞ , z ∈ C . There are several propositions for such functions in some applications for GR and Yang–Mills’fields. We can perform a perturbation calculus using Feynman diagrams for S-matrix which isunitary. The full program will be developed elsewhere.We should look for Kähler structure on M = G/G (on a homogeneous space) and also oncompact Lie groups G and H in order to get a more sound mathematical constructions.Let us notice that all the conclusions from Appendix D of Ref. [9] can be applied also here.The important problem in this theory is a problem of ghosts and tachions. This problemshould be solved on the level of a classical field theory before a quantization, on a prequantumlevel. The problem is connected with the existence of the skew-symmetric tensor g [ µν ] —a skewon.88his particle in a linear approximation is massive getting mass term from a cosmological constantwhich appears in the theory. According to modern ideas a cosmological constant is not zero andwe are (in principle) able to tune our cosmological term to the observation data. In this way wecan predict a value of a mass of a skewon. Skewon has a spin zero and has a positive energy inthe case of a pure real or hypercomplex Hermitian Theory (see Refs [81], [82], [83], [84]).Thus our 20-dimensional unification should be considered in the case of hypercomplex Hermi-tian gravity with Kählerian structure on S combined to Hermitian (complex or hypercomplex)Kaluza–Klein Theory.Our 12-dimensional unification should be considered in the case of hypercomplex Hermitiangravity combined to Hermitian (complex or hypercomplex) Kaluza–Klein Theory.Our 5-dimensional unification from Ref. [9] should be considered in the case of hypercomplexHermitian gravity combined to Hermitian (complex or hypercomplex) Kaluza–Klein Theory.We do not exhaust a possible research in this direction. First of all we can consider a Quater-nionic (or Split-Quaternion) Hermitian gravity (if we find some applications of additional degreesof freedom) to extend to Hermitian (quaternionic or split-quaternion) Hermitian Kaluza–KleinTheory. The second direction is to consider in place of a Cartesian product V = E × M = E × G/G or V = E × S a nontrivial principal fiber bundle over E with a fiber M = G/G or S . Moreover,we need application of additional degrees of freedom, i.e. a connection on the bundle.In Ref. [85] the authors write in a very pessimistic way. We quote: “ Unfortunately, althoughour understanding of gauge theories has continued to develop, we have made very little progressin understanding the origin of spontaneous symmetry breakdown. For the most part, the Higgs’mechanism continues to be described by the ad hoc introduction into the Lagrangian of elementary,weakly self-coupled scalar fields. In the minimal model, a complex SU (2) doublet is used, providingthree Goldstone bosons (longitudinal W and Z bosons) and one physical massive scalar. ”According to our research this pessimistic view of Higgs’ sector is not longer true. Higgs’ fieldsare part of gauge fields (dimensional reduction procedure). A full bosonic sector of GSW modelcan be incorporated as a part of the Hermitian (Nonsymmetric) Kaluza–Klein Theory gettingmasses of W and Z bosons, Higgs’ boson and Weinberg angle agreed with an experiment. Allmentioned particles have been discovered and some additional phenomena can be predicted dueto the existence of antisymmetric tensors in the theory.Let us give some historical remarks. The dimensional reduction and invariant connectionswhich lead to the interpretation of the Higgs-like scalar multiplets as a part of Yang–Mills’ fieldin higher dimensional bundle over a quotient space has been also introduced in Refs [86, 87, 88].Moreover, in our approach we follow Refs [40]–[48]. They are by any means published earlier. Letus notice the following fact. In 1977 A. Trautman communicated to me that K. A. Olive founda possibility to get a kinetic term for a scalar field from the fifth dimension which was similar tomy observation.The idea to interpret the non-Abelian gauge field as a torsion appeared in Refs [89, 90]. Thequartic potential of generalized Higgs’ field can be obtained in the framework of non-commutativegeometry, see Refs [91, 92]. Actually we do not follow this approach.We mention above on our plans to consider chiral fermions in our approach. Probably we usesome ideas from Ref. [93].In Refs [94, 95] an idea was considered to use supergroups in order to unify physical interac-tions. In this way a nonsymmetric tensor on a supergroup appears naturally as a part of gen-eralized Killing–Cartan tensor connected anticommuting generators. We mention this possibility89n Ref. [1]. Moreover, in Refs [94, 95] this idea is not connected to any nonsymmetric geometryas in Ref. [1]. Let us notice the idea to geometrized BRST symmetry has been developed inRef. [96]. Let us notice that in Refs [94, 95] supergroups are considered as global symmetries.Our suggestions from Ref. [1] considered a supergroup as a local symmetry combined via Einsteingeometry with nonsymmetric gravity or even supergravity. This idea can be considered as futureprospects for further research. Let us notice that we give some historical remarks on EinsteinUnified Field Theory in the last section of Ref. [9]. It is interesting that A. Einstein in Ref. [17]came back to his first ideas in Unified Field Theory (from 1920–30) and we develop them fur-ther in a new context. Let us mention on Ref. [97] where A. Crumeyrolle developed a programof geometrization and unification using a manifold with hypercomplex coordinates close to ourprospects with quaternionic metric.Finally, let us give some remarks. We do not consider in our (even do not touch) approachesin which weak interactions and gravity are cross-correlated. In particular, the possibility tosee gravitational interactions as emerging from the long-range behavior of the Higgs’ field (seeRef. [98]), the possibility that gravitation morphs into weak interactions at the Fermi scale (seeRef. [99]) and the relationship between weak interactions chirality and gravity (see Refs [100, 101]).All mentioned approaches are very interesting in principle. However, they have not any applicationin our approach—geometrization and unification of fundamental interactions. Only in Ref. [99] wesee some possibility to extend it to an Einstein–Cartan-like theory in order to get current-currentinteraction known in an old weak interaction theory. However, this approach even interestingfrom conceptual point of view cannot be maintained because we have now GSW-model employingYang–Mills’ and Higgs’ fields. The relationship between weak interactions chirality and gravity(see Refs [100, 101]) is not applicable in our Nonsymmetric Kaluza–Klein Theory because we canget chiral fermions in a completely different setting (mentioned above).We should look for a flavor-chiral fermion representation in our approach as we describedabove. However this is a still unresolved problem to be considered in future works. Moreover,we can claim that we deal with GSW-model and QCD-model in the Nonsymmetric Kaluza–KleinTheory. Our theory is a real candidate of TOE. Moreover, it does not make any developmentto “modern” Kaluza–Klein Theory for the reasons given in Conclusions of Ref. [9]. Now we arewaiting for results from new LHC and future accelerators.The nonsymmetric metric considered here is a crucial point and it has real physical motivationdescribed in the paper (dielectric model of a confinement and a correct pattern of W ± , Z andHiggs bosons in GSW-model). The gravitational influence on GSW-model (Higgs kinetic energy)can be of course testable in an experiment as a skewon-Higgs’ interaction which may be discoveredin LHC even before graviton-Higgs’ interaction. Acknowledgement
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