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**Kalinowski MW ^{*}**

Bioinformatics Laboratory, Medical Research Centre, Polish Academy of Sciences, Poland

- *Corresponding Author:
- Kalinowski MW

Bioinformatics Laboratory, Medical Research Centre

Polish Academy of Sciences, Poland

**Tel:**61641643156

**E-mail:**[email protected]

**Received Date:** September 05, 2015; **Accepted Date:** January 13, 2016; **Published Date:** January 20, 2016

**Citation:** Kalinowski MW (2016) A Hierarchy of Symmetry Breaking in the Nonsymmetric Kaluza-Klein (Jordan-Thiry) Theory. J Phys Math 7:152. doi:10.4172/2090-0902.1000152

**Copyright: **© 2016 Kalinowski MW. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Physical Mathematics

The paper is devoted to the hierarchy of a symmetr y breaking in the Non symmetric Kaluza–Klein (Jordan–Thiry) Theory. The basic idea consists in a deformation of a vacuum states manifold to the cartesian product of vacuum states manifolds of every stage of a symmetry breaking .In the paper we consider a pattern of a spontaneous symmetry breaking including a hierarchy in the Non symmetr ic Kaluza–Klein (Jordan–Thiry) Theory.

In this paper we consider hierarchy of symmetry breaking in the Nonsymmetric Kaluza–Klein Theory and the Nonsymmetric Kaluza– Klein Theory with a spontaneous symmetry breaking and Higgs’ mechanism . In the second section we consider a Nonsymmetric Kaluza–Klein Theory and the Nonsymmetric Kaluza–Klein Theory with a spontaneous symmetry breaking and Higgs’ mechanism [1-6]. In the third section we develop a hierarchy of the symmetry breaking in our theory. For further development of the nonsymmetric Kaluza– Klein (Jordan–Thiry) Theory [7-10].

Elements of the Nonsymmetric Kaluza–Klein Theory in general non-Abelian case and with spontaneous symmetr y breaking and Higgs’ mechanism

Let be a principal fiber bundle over a space-time E with a structural group G which is a semisimple Lie group. On a space-time E we define a nonsymmetric tensor such that

g=det(gμv)≠0

(2.1)

g_{[μv]} is called as usual a skewon field (e.g., in NGT, [6,11-13] We define
on E a nonsymmetric connection compatible with g_{μv} such that

(2.2)

where is an exterior covariant derivative for a connection and is its torsion. We suppose also

(2.3)

We introduce on E a second connection

(2.4)

such that

(2.5)

(2.6)

Now we turn to nonsymmetric metrization of a bundle . We define a nonsymmetric tensor γ on a bundle manifold P such that

(2.7)

where π is a projection from P to E . On we define a connection ω (a 1-form with values in a Lie algebra g of G. In this way we can introduce on P (a bundle manifold) a frame such that

Thus our nonsymmetric tensor looks like

,A,B=1,2…..,n+4, (2.8)

(2.9)

where is a bi invarian t Killing–Cartan tensor on G and is a rightinvarian
t **skew- symmetric **tensor on G,

We have

(2.10)

Thus we can write (2.11)

(2.12)

(are structural constant s of the Lie algebra g).

is the symmetr ic part of γ and is the anti symmetr ic part of γ We have as usual

(2.13)

and

(2.14)

is a curvature of the connection ω

(2.15)

The frame θ^{A} on P is partially **nonholonomic**. We have

(2.16)

Even if the bundle is trivial, i.e. for Ω=0 This is different than in an electromagnetic case explanied by Kalinowski MW [3]. Our nonsymmetric metrization of a principal fiber bundle gives us a rightinvarian t structure on P with respect to an action of a group G on P [3]. Having P nonsymmetric ally metrized one defines two connection s on P right- invarian t with respect to an action of a group G on P. We have

(2.17)

In our left horizontal frame θ^{A}

(2.18)

(2.19)

where D is an exterior covariant derivative with respect to a connection on P and its **torsion**. One can solve Equation
(2.18)– (2.19) getting the following results

(2.20)S

where is an inverse tensor of g(α)

(2.21)

is an Ad-type tensor on P such that

(2.22)

is a connection on an internal space (typical fiber) compatible with a metric such that

(2.23)

(2.24)

and of course where is a torsion of the connection We also introduce an inverse tensor of g(α)

(2.25)

We introduce a second connection on P defined as

(2.26)

is a horizontal one form

(2.27)

(2.28)

In this way we define on P all analogues of four- dimension al
quantities from NGT [6,11]. It means, (*n+4*) dimension al analogues
from Moffat theory of gravitation, i.e. two connection s and a
nonsymmetric metric .Those quantities are right- invarian t with
respect to an action of a group G on P. One can calculate a scalar
curvature of a connection getting the following result [1-3].

(2.29)

Where

(2.30)

is a Moffat–Ricci curvature scalar for the connection is a is a Moffat–Ricci curvature scalar for the connection , and is a Moffat–Ricci curvature scalar for the connection

(2.31)

(2.32)

Usually in ordinary (symmetric) Kaluza–Klein Theory one has where GN is a Newtonian gravitational constant and c is the speed of light. In our system of units this is the same as in Non symmetric Kaluza–Klein Theory in an electromagnetic case [3,4] In the non- Abelian Kaluza –Klein Theory which unifies GR and Yang–Mills field theory we have a Yang–Mills lagrangian and a cosmological term. Here we have

(2.33)

and plays a role of a cosmolog ical term.In order to incorporate
a spontaneous symmetr y breaking and **Higgs’ mechanism **in our
geometrical unification of gravitation and Yang–Mills’ fields we
consider a fiber bundle over a base manifold where E
is a space-time, are semisimple Lie groups. Thus we are
going to combine a Kaluza–Klein theory with a dimension al reduction
procedure.

Let be a principal fiber bundle over *V=E × M* with a structural
group *H* and with a projection *π*, where

is a homogeneous space, G is a semisimple Lie group and
G_{o} its semisimple Lie subgroup. Let us suppose that (V, γ) is a manifold
with a **nonsymmetric **metric tensor

(2.34)

The signature of the tensor γ is Let us introduce a natural Phenomenon

(2.35)

It is convenient to introduce the following notation. Capital Latin
indices with tilde run *1,2,3…,m+4 ,*. Lower Greek indices *α,β,γ,δ=1,2,3,4* and lower Latin indices Capital
Latin indices. Lower Latin indices with tilde The symbol over θ^{A} and other quantities indicates that
these quantities are defined on V. We have of course where , .

On the group H we define a bi- invarian t ( symmetr ic) Killing– Cartan tensor

(2.36)

We suppose H is semisimple, it means .We define a
skew- symmetr ic right- invarian t tensor on *H*

Let us turn to the nonsymmetric metrization of .

(2.37)

where

(2.38)

is a nonsymmetric right-invarian t tensor on H.One gets in a matrix form (in the natural frame (2.35))

(2.39)

and real, then

(2.40)

The signature of the tensor k is . As usual, we have commutation relations for Lie algebra of H, h

(2.41)

This metrization of is right- invarian t with respect to an action
of H on P.Now we should nonsymmetric ally metrize M=G/G_{0}. M is a
homogeneous space for G (with left action of group G). Let us suppose
that the Lie algebra of G, g has the following reductive decomposition

(2.42)

where g_{0} is a Lie algebra of *G _{0}* (a subalgebra of g) and m (the complement
to the subalgebra g0) is

(2.43)

Let us introduce the following notation for generators of g:

(2.44)

This is a decomposition of a basis of g according to (2.42). We
define a symmetr ic metric on M using a Killing–Cartan form on G in
a classical way. We call this tensor h_{0}. Let us define a tensor field on *G/G _{0}*, using tensor field

We define on M a skew- symmetr ic 2-form *k ^{0}*

(2.45)

Yj are generators of the Lie algebra g. Let us take a local section s: of a natural bundle where. The local section s can be considered as an introduction of a coordinate system on M.

Let ω_{MC} be a left- invarian t Maurer–Cartan form and let

(2.46)

Using de composition (2.42) we have

(2.47)

It is easy to see that is the natural (left- invarian t) frame on M and we have

(2.48)

(2.49)

According to our notation

Thus we have a nonsymmetric metric on M

(2.50)

Thus we are able to write down the nonsymmetric metric on

(2.51)

where

The frame is unholonomic:

(2.52)

where are coefficient s of nonholonomicity and depend on the
point of the manifold *M=G/G _{0}* (they are not constant in general).
They depend on the section s and on the constants We have
here three groups H,G, G

(2.53)

such that a centralizer of *μ(G _{0})* in H,

(2.54)

If μ is a iso morphi sm between* G _{0}* and

(2.55)

Let us denote by μ ′ a tangent map to μ at a unit element. Thus μ ′ is a differential of μ acting on the Lie algebra element s. Let us suppose that the connection ω on the fiber bundle P is invarian t under group action of G on the manifold . According to Kobayashi [14-17] this means the following.

Let e be a local section of and Then for
every g∈G there exists a gauge transformation ρ_{g} such that

(2.56)

f *means a pull-back of the action f of the group G on the manifold V. According to Hlavaty [13-25] we are able to write a general form for such an ω. Following [17] we have

(2.57)

(An action of a group *G* on *V=E × G/G _{0}* means left multiplication
on a homogeneous space

This connection describes an ordinary Yang–Mills’ field gauge group *C ^{μ}*=G on the space-time E. Φ is a function on E with values in the space of linear maps

(2.58)

satisfying Φ

(2.59)

Thus

(2.60)

Let us write condition (2.57) in the base of left-invarian t form which span respectively dual spaces to *g _{0}* and m [24,25]. It is
easy to see that

(2.61)

and

(2.62)

From (2.59) one gets

(2.63)

where are structure constant s of the Lie algebra g and are structure constant s of the Lie algebra h Equation (2.63) is a constraint on the scalar field .For a curvature of ω one gets

(2.64)

Thus we have

(2.65)

(2.66)

(2.67)

where means gauge derivative with respect to the connection defined on a bundle q over a space-time E with a structur al group G

(2.68)

is the curvature of the connection in the base * {Y _{i}}*,
generators of the Lie algebra of the Lie group G g, is the matrix
which connects

Let us come back to a construction of the Nonsymmetric Kaluza–
Klein Theory on a manifold P. We should define connection s. First of
all, we should define a connection compatible with a nonsymmetric
tensor γ_{ΑΒ}, Equation (2.51)

(2.69)

(2.70)

where is the exterior covariant derivative with respect to and its torsion. Using (2.51) one easily finds that the connection (2.69) has the following shape

(2.71)

where γ is a connection on the space-time E and on the manifoldwith the following properties .

(2.72)

(2.73)

(2.74)

( is an exterior covariant derivative with respect to a connection is a tensor of torsion of a connection is an exterior covariant derivative of a connection and its torsion.On a space-time E we also define the second affine connection such that

(2.75)

We proceed a nonsymmetric metrization of a principal fiber bundle according to (2.51). Thus we define a right-invarian t connection with respect to an action of the group H compatible with a tensor

(2.76)

where D is an exterior covariant derivative with respect to the connection and its torsion. After some calculations one finds

(2.77)

Where

(2.78)

(2.79)

L^{d}_{c} is Ad-type tensor with respect to H (Ad- covariant on

(2.80)

(2.81)

(2.82)

We define on P a second connection

(2.83)

Thus we have on P all (m+4) dimension al analogues of geometrical quantities from NGT, i.e.

Let us calculate a Moffat–Ricci curvature scalar for the connection

Let us incorporate in our scheme a hierarchy of a symmetry breaking. In order to do this let us consider a case of the manifold

(3.1)

where

(3.2)

(3.3)

(3.4)

Every manifold Mi is a manifold of vacuum states if the symmetr y
is breaking from G_{i+1} to G_{i}, G_{k}=G.

Thus

(3.5)

We will consider the situation when

( 3.6)

This is a constraint in the theory. From the chain (3.5) one gets

(3.7)

and

(3.8)

The relation (3.6) means that there is a diffeomorphism g onto *G/
G _{0}* such that

(3.9)

This diffeomorphism is a deformation of a product (3.1) in *G/G _{0}*.
The theory has been constructed for the case considered before with

(3.10)

The fields Φ_{i}, i=0,…,k-1.

this way we get the following decomposition for a kinetic part of the field Φ and for a potential of this field:

(3.11)

(3.12)

where

(3.13)

(3.14)

(3.15)

Equation (3.5) is a non symmetr ic tensor on a manifold M_{i}.

(3.16)

are structure constant s of the Lie algebra gi . The scheme of
the symmetr y breaking acts as follows from the group *G _{i+1}* to

(3.17)

(3.18)

The shape of g is a true indicator of a reality of the symmetr y breaking pattern. If

g = Id +δ g (3.19)

where δg is in some sense small and Id is an identity, the sums (3.11)- (3.12) are close to the analogous formulae from the expanation of Kalinowski [5,10]. The smallness of g is a criterion of a practical application of the symmetr y breaking pattern (3.5) . It seems that there are a lot of possibilities for the condition (3.9). Moreover, a smallness of δg plus some natural conditions for groups Gi can narrow looking for grand unified models. Let us notice that the decomposition of M results in decomposition of cosmolog ical terms

(3.20)

where

(3.21)

where is a non symmetr ic connection on Mi compatible with
the non symmetr ic tensor and its curvature scalar. The
truncation procedure can be proceeded in several ways. Finally let us
notice that the energy scale of broken gauge bosons is fixed by a radius *r _{i}* at any stage of the symmetr y breaking in our scheme.

Let us consider Equation (3.10) in more details. One gets

(3.22)

where

(3.23)

(3.24)

is a matrix of Higgs’ fields transformation.

According to our assumptions one gets also:

(3.25)

For g is an invertible map we have det g* ( y) ≠ 0 .

We have also

(3.26)

and

(3.27)

(3.28)

(3.29)

such that

(3.30)

(3.31)

For an inverse tensor one easily gets

(3.32)

We have

(3.33)

In this way we have for the measure

(3.34)

where

(3.35)

(3.36)

In the case of one gets

(3.37)

where

(3.38)

where

(3.39)

Moreover, to be in line in the full theory we should consider a chain
of groups *H _{i}* i = 0,1, 2,..., k −1 , in such a way that

(3.40)

For every group *H _{i}* we have the following assumptions

(3.41)

and *G _{i+1}* is a centralizer of

(3.42)

We know from elementary particles physics theory that

and that G2 is a group which plays the role of H in the case of a
symmetr y breaking from to *U _{el}* (1) . However, in this
case because of a factor

It seems that in a reality we have to do with two more stages of a
symmetr y breaking. Thus* k=3*. We have

(3.43)

(3.44)

(3.45)

and

(3.46)

(3.47)

(3.48)

and

(3.49)

(3.50)

We can take for *G, SU(5), SU(10), E6 or SU(6)*. Thus there are a lot
of choices for *G _{2}*,

(3.51)

(3.50)

(3.52)

Thus

(3.53)

We can try with F4= *H _{0}*

In the case of H

Thus we can try with *E _{7}, E_{8}*

(3.55)

In this way we have

Thus we can try with

(3.57)

But in this case

This seems to be nonrealistic. For instance, if G= SO(10), E6,

In this case we get

And H could be SO(10), SO(18), SO(20).In this approach we try to consider additional dimensions connecting to the manifold M more seriously, i.e. as physical dimensions, additional space-like dimensions. We remind to the reader that gauge-dimensions connecting to the group H have different meaning. They are dimensions connected to local gauge symmetr ies (or global) and they cannot be directly observed. Simply saying we cannot travel along them. In the case of a manifold M this possibility still exists. However, the manifold M is diffeomorphically equivalent to the product of some manifolds Mi, i = 0,1, 2,..., k −1, with some characteristic sizes ri The radii ri represent energy scales of symmetr y breaking. The lowest energy scale is a scale of weak interactions (Weinberg–Glashow–Salam model) cm. In this case this is a radius of a sphere S2 The possibility of this “travel” will be considered in the concept explanied by Kalinowski [26]. In this case a metric on a manifold M can be dependent on a point x∈E (parametrically).It is interesting to ask on a stability of a symmetr y breaking pattern with respect to quantum fluctuations. This difficult problem strongly depends on the details of the model. Especially on the Higgs sector of the practical model. In order to preserve this stability on every stage of the symmetr y breaking we should consider remaining Higgs’ fields (after symmetry breaking ) with zero mass. According to S. Weinberg, they can stabilize the symmetry breaking in the range of energy

(3.58)

i.e. for a symmetry breaking from *G _{i+1}* to

It seems that in order to create a realistic grand unified model based on non symmetr ic Kaluza–Klein (Jordan–Thiry) theory it is necessary to nivel cosmolog ical terms. This could be achieved in some models due to choosing constant s ξ and ζ and μ . After this we can control the value of those terms, which are considered as a selfinteraction potential of a scalar field Y. The scalar field Y can play in this context a role of a quintessence .

Let us notice that using the equation

(3.59)

and (3.27) one gets

(3.60)

In this way we get constraints for Higgs’ fields, Φ_{0}, Φ_{1}, Φ_{k-1}

Solving these constraints we obtain some of Higgs’ fields as functions of in dependent components [26]. This could result in some cross terms in the potential (3.12) between Φ’s with different i. For example a term

where Φ′ means in dependent fields. This can cause some problems in a stability of symmetry breaking pattern against radiative corrections. This can be easily seen from Equation (3.59) solved by in dependent Φ′ ,

Φ = BΦ′ (3.61)

(3.62)

Where B is a linear operator transforming in dependent Φ′ into Φ.

We can suppose for a trial a condition similar to (3.59) for every i = 0,..., k −1 ,

(3.63)

where are structure constant s for the Lie algebra hi of the group *H _{i}* are structure constant s of the Lie algebra g

In this way

(3.64)

In this case we should have a consistency between (3.63) and (3.60)
which impose constraints on *C,f μ* and where refer to *H _{i}, G_{i}+1*. Solving (3.63) via introducing in dependent fields Φ

(3.65)

Combining (3.62) , (3.64) , (3.65) one gets

(3.66)

Equation (3.66) gives a relation between in dependent Higgs’ fields
Φ′and Φ ′_{i}. Simultaneously it is a consistency condition between
Equation (3.59) and Equation (3.63). However, the condition (3.63)
seems to be too strong and probably it is necessary to solve a weaker
condition (3.60) which goes to the mentioned terms V( Φ_{i}′, Φ′_{j}) . The
conditions (3.63) plus a consistency (3.66) avoid those terms in the
Higgs potential. This problem demands more investigation. *ϕ (g)=
{gG _{1}}*

It seems that the condition (3.9) could be too strong. In order to
find a more general condition we consider a simple example of (3.5).
Let *G _{0}= {e}* and K=2 In this case we have

(3.67)

(3.68)

(3.69)

In this way *G _{1}× G/G_{1}* is diffeomorphically equivalent to G. Moreover,
we can consider a fibre bundle with base space

(3.70)

where *U⊂ G/G _{1}* is an open set. Thus in a place of (3.9) we consider a
local diffeomorphism

(3.71)

where

*U _{i}* ⊂

where

(3.72)

i.e.

(3.73

in a unique way. This could give us a fibration of *G /G _{0}* in

For gG_{i+1} we simply define

(3.74)

If we define

(3.75)

Thus in general

(3.76)

where

(3.77)

Thus in a place of (3.9) we have to do with a structure

(3.78)

such that

(3.79)

where

(3.80)

This generalizes (3.9) to the local conditions (3.71). Now we can
repeat all the considerations concerning a decomposition of Higgs’
fields using local diffeomorphisms *g _{U}* (

(3.81)

suggested by Salam and Pati, where SU(4) unifies This will be helpful in our future consideration concerning extension
to super symmetr ic groups, i.e. *U(2,2)* which unifies to the super Lie group *U(2,2)* considered by Mohapatra. Such models
on the phenomenological level incorporate fermions with a possible
extension to the super symmetric SO(10) model. They give a natural
framework for lepton flavour mixing going to the neutrino oscillations
incorporating see-saw mechanism for mass generations of neutrinos.
In such approaches the see-saw mechanism is coming from the grand
unified models. Our approach after incorporating manifolds with
anticommuting parameters, super Lie groups, super Lie algebras and
in general supermanifolds (superfibrebundles) can be able to obtain
this. However, it is necessary to develop a formalism (in the language
of supermanifolds, superfibrebundles, super Lie groups, super Lie
algebras) for non symmetr ic connections, non symmetr ic Kaluza–
Klein (Jordan–Thiry) theory. In particular we should construct an
analogue of Einstein–Kaufmann connection for supermanifold, a non
symmetr ic Kaluza–Klein (Jordan–Thiry) theory for superfibrebundle
with super Lie group. In this way we should define first of all a non
symmetr ic tensor on a super Lie group and afterwards a non symmetr
ic metrization of a superfibrebundle. Let us notice that on every stage
of symmetr y breaking, i.e. from *G _{i+1}*to Gi, we have to do with group

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