# Deriving the mass of particles from Extended Theories of Gravity in LHC era

###### Abstract

We derive a geometrical approach to produce the mass of particles that could be suitably tested at LHC. Starting from a 5D unification scheme, we show that all the known interactions could be suitably deduced as an induced symmetry breaking of the non-unitary -group of diffeomorphisms. The deformations inducing such a breaking act as vector bosons that, depending on the gravitational mass states, can assume the role of interaction bosons like gluons, electroweak bosons or photon. The further gravitational degrees of freedom, emerging from the reduction mechanism in 4D, eliminate the hierarchy problem since generate a cut-off comparable with electroweak one at TeV scales. In this "economic" scheme, gravity should induce the other interactions in a non-perturbative way.

## I Introduction

The Standard Model of Particles can be considered a successful relativistic quantum field theory both from particle physics and group theory points of view. Technically, it is a non-Abelian gauge theory (a Yang-Mills theory) associated with the tensor product of the internal symmetry groups , where the color symmetry for quantum chromodynamics, is treated as exact, whereas the symmetry, responsible for generating the electro-weak gauge fields, is considered spontaneously broken.

So far, as we know, there are four fundamental forces in Nature; namely, electromagnetic, weak, strong and gravitational forces. The Standard Model well represents the first three, but not the gravitational interaction. On the other hand, General Relativity (GR) is a geometric theory of the gravitational field which is described by the metric tensor defined on pseudo-Riemannian space-times. The Einstein field equations are nonlinear and have to be satisfied by the metric tensor. This nonlinearity is indeed a source of difficulty in quantization of General Relativity. Since the Standard Model is a gauge theory where all the fields mediating the interactions are represented by gauge potentials, the question is why the fields mediating the gravitational interaction are different from those of the other fundamental forces. It is reasonable to expect that there may be a gauge theory in which the gravitational fields stand on the same footing as those of other fields oqr . As it is well-known, this expectation has prompted a re-examination of GR from the point of view of gauge theories.

While the gauge groups involved in the Standard Model are all internal symmetry groups, the gauge groups in GR must be associated with external space-time symmetries. Therefore, the gauge theory of gravity cannot be dealt under the standard of the usual Yang-Mills theories. It must be one in which gauge objects are not only gauge potentials but also tetrads that relate the symmetry group to the external space-time. For this reason we have to consider a more complex nonlinear gauge theory where all the interactions should be dealt under the same standard unification . In GR, Einstein took the space-time metric components as the basic set of variables representing gravity, whereas Ashtekar and collaborators employed the tetrad fields and the connection forms as the fundamental variables rovelli . We also will consider the tetrads and the connection forms as the fundamental fields but with the difference that this approach gives rise to a covariant symplectic formalism capable of achieving the result of dealing with physical fields under the same standard basini ; symplectism .

In order to frame historically our approach, let us sketch a quick summary of the various attempts where the Standard Model and GR have been considered under the same comprehensive picture. In 1956, Utiyama suggested that gravitation may be viewed as a gauge theory Utiyama in analogy to the Yang-Mills YangMills theory (1954). He identified the gauge potential due to the Lorentz group with the symmetric connection of the Riemann geometry, and reproduced the Einstein GR as a gauge theory of the Lorentz group , with the help of tetrad fields introduced in an ad hoc manner. Although the tetrads were necessary components of the theory (to relate the Lorentz group) adopted as an internal gauge group to the external space-time, they were not introduced as gauge fields. In 1961, Kibble Kibble constructed a gauge theory based on the Poincaré group , , , (the symbol represents the semi-direct product) which resulted in the Einstein-Cartan theory characterized by curvature and torsion. The translation group , is considered responsible for generating the tetrads as gauge fields. Cartan Cartan generalized the Riemann geometry in order to include torsion in addition to curvature. The torsion (tensor) arises from an asymmetric connection. Sciama Sciama , and others (Fikelstein Finkelstein , Hehl Hehl1 ; Hehl2 ) pointed out that intrinsic spin may be the source of torsion of the underlying space-time manifold.

Since the form and the role of tetrad fields are very different from those of gauge potentials, it has been thought that even Kibble’s attempt is not satisfactory as a full gauge theory. There have been a number of gauge theories of gravitation based on a variety of Lie groups Hehl1 ; Hehl2 ; Mansouri1 ; Mansouri2 ; Chang ; Grignani ; MAG . It was argued that a gauge theory of gravitation corresponding to GR can be constructed with the translation group alone, in the so-called teleparallel scheme teleparallelism .

Inomata et al. Inomata proposed that Kibble’s gauge theory could be obtained, in a way closer to the Yang-Mills approach, by considering the de Sitter group , , which is reducible to the Poincaré group by a group-contraction. Unlike the Poincaré group, the de Sitter group is homogeneous and the associated gauge fields are all of gauge potential type and by the Wigner-Inönu group contraction procedure, one of the five vector potentials reduces to the tetrad.

It is standard to use the fiber-bundle formulation by which gauge theories can be constructed on the basis of any Lie group. Works by Hehl et al. MAG , on the so-called Metric Affine Gravity (MAG), adopted as a gauge group the affine group , , , which can be linearly realized. The tetrad has been identified by the nonlinearly realized translational part of the affine connection, on the tangent bundle. In MAG theory, the Lagrangian is quadratic in both curvature and torsion, in contrast to the Einstein-Hilbert Lagrangian of GR which is linear in the scalar curvature. The theory has the Einstein limit on one hand and leads to the Newtonian inverse distance potential plus the linear confinement potential (in the weak field approximation) on the other hand. In summary, as we have seen, there are many attempts to formulate gravitation as a gauge theory but currently no theory has been uniquely accepted as the gauge theory of gravity.

The nonlinear approach to group realizations was originally introduced by Coleman, Wess and Zumino CCWZ1 ; CCWZ2 in the context of internal symmetry groups (1969). It was later extended to the case of space-time symmetries by Isham, Salam, and Strathdee Isham considering the nonlinear action of , , modulus the Lorentz subgroup. In 1974, Borisov, Ivanov and Ogievetsky BorisovOgievetskii ; IvanovOgievetskii , considered the simultaneous nonlinear realization (NLR) of the affine and conformal groups. They stated that GR can be viewed as a consequence of spontaneous breakdown of the affine symmetry, in the same way that chiral dynamics, in quantum chromodynamics, is a result of spontaneous breakdown of chiral symmetry. In their model, gravitons are considered as Goldstone bosons associated with the affine symmetry breaking. As we will see below, this approach can be pursued in general.

In 1978, Chang and Mansouri ChangMansouri used the NLR scheme adopting , as the principal group. In 1980, Stelle and West StelleWest investigated the NLR induced by the spontaneous breakdown of , . In 1982 Ivanov and Niederle considered nonlinear gauge theories of the Poincaré, de Sitter, conformal and special conformal groups Ivanov1 ; Ivanov2 . In 1983, Ivanenko and Sardanashvily IvanenkoSardanashvily considered gravity to be a spontaneously broken , gauge theory. The tetrads fields arise, in their formulation, as a result of the reduction of the structure group of the tangent bundle from the general linear to Lorentz group. In 1987, Lord and Goswami Lord1 ; Lord2 developed the NLR in the fiber bundle formalism based on the bundle structure as suggested by Ne’eman and Regge NeemanRegge . In this approach, the quotient space is identified with physical space-time. Most recently, in a series of papers, Lopez-Pinto, Julve, Tiemblo, Tresguerres and Mielke discussed nonlinear gauge theories of gravity on the basis of the Poincaré, affine and conformal groups Julve ; Lopez-Pinto ; TresguerresMielke ; Tresguerres ; TiembloTresguerres1 ; TiembloTresguerres2 .

Now, following the prescriptions of GR, the physical space-time is assumed to be a four-dimensional differential manifold. In Special Relativity (SR), this manifold is the Minkwoski flat-space-time while, in GR, the underlying space-time is assumed to be curved in order to describe the effects of gravitation.

As we said, Utiyama Utiyama proposed that GR can be seen as a gauge theory based on the local Lorentz group in the same way that the Yang-Mills gauge theory YangMills is developed on the basis of the internal iso-spin gauge group. In this formulation, the Riemannian connection is the gravitational counterpart of the Yang-Mills gauge fields. While , in the Yang-Mills theory, is an internal symmetry group, the Lorentz symmetry represents the local nature of space-time rather than internal degrees of freedom. The Einstein Equivalence Principle, asserted for GR, requires that the local space-time structure can be identified with the Minkowski space-time possessing Lorentz symmetry.

In order to relate local Lorentz symmetry to the external space-time, we need to solder the local space to the external space. The soldering tools can be the tetrad fields. Utiyama regarded the tetrads as objects given a priori while they can be dynamically generated unification and the space-time has necessarily to be endowed with torsion in order to accommodate spinor fields classification . In other words, the gravitational interaction of spinning particles requires the modification of the Riemann space-time of GR to be a (non-Riemannian) curved space-time with torsion. Although Sciama used the tetrad formalism for his gauge-like handling of gravitation, his theory fell shortcomings in treating tetrad fields as gauge fields.

Following the Kibble approach Kibble , it can be demonstrated how gravitation can be formulated starting from a pure gauge viewpoint. In particular, gravity can be seen as a gauge theory which can be obtained starting from some local invariance, e.g. the local Poincaré symmetry, leading to a suitable unification scheme unification . This dynamical structure can be based on a nonlinear realization of the local conformal-affine group of symmetry transformations poincare .

Here, we start from a General Invariance Principle, as requested in the so called Open Quantum Relativity (OQR) oqr ; conservation and consider first the Global Poincaré Invariance and then the Local Poincaré Invariance. This approach leads to construct a given theory of gravity as a gauge theory. Such a viewpoint, if considered in detail, can avoid many shortcomings and could be useful to formulate self-consistent schemes for quantum gravity and then the unification of all interactions unification .

In particular, the idea of an unification theory, capable of describing all the
fundamental interactions of physics under the same standard, has
been one of the main issues of modern physics, starting from
the early efforts of Einstein, Weyl, Kaluza and Klein
kaluza until the most recent approaches
ross . Nevertheless, the large number of ideas, up to now
proposed, which we classify as unified theories, results
unsuccessful due to several reasons: the technical difficulties
connected with the lack of a unitary mathematical description of
all the interactions; the huge number of parameters introduced
to "build up" the unified theory and the fact that most of them
cannot be observed neither at laboratory nor at astrophysical (or
cosmological) conditions morselli ; the very wide (and several times
questionable since not-testable) number of extra-dimensions
requested by several approaches. Due to this situation, it seems
that unification is a useful (and aesthetic) paradigm, but far to
be achieved, if the trend is continuing to try to unify
interactions (i.e. to make something simple) by adding and adding
ingredients: new particles and new parameters (e.g. dark matter forest).
A different approach could be to consider the very essential
physical quantities and try to achieve unification without any ad hoc new ingredients. This approach
can be pursued starting from straightforward considerations which
lead to reconsider modern physics under a sort of economic issue: let us try to unifying forces approaching new schemes but without adding new parameters
^{2}^{2}2Following Occam’s Razor prescription: Entia non sunt multiplicanda praeter necessitatem..
A prominent role
deserves the conservation laws and the fact that each of them
brings out the existence of a symmetry unification .

As a general remark, the Noether Theorem states that, for every conservation law of Nature, a symmetry must exist. This leads to a fundamental result also from a mathematical point of view since the presence of symmetries technically reduces dynamics (i.e. gives rise to first integrals of motion) and, in several cases, allows to get the general solution. With these considerations in mind, we can try to change our point of view and investigate what will be the consequences of the absolute validity of conservation laws without introducing any arbitrary symmetry breaking.

In order to see what happens as soon as we ask for the absolute validity of conservation laws, we could take into account the Bianchi identities. Such geometrical identities work in every covariant field theory (e.g. Electromagnetism or GR) and can be read as equations of motion also in a fiber bundle approach bundle . We want to show that, the absolute validity of conservation laws, intrinsically contains symmetric dynamics; moreover, reducing dynamics from 5D to 4D, it gives rise to the physical quantities characterizing particles as the mass.

The minimal ingredient which we require to achieve these results is the fact that a 5-dimensional, singularity free space, where conservation laws are always and absolutely conserved, has to be defined. Specifically, in such a space, Bianchi identities are asked to be always valid and, moreover, the process of reduction to 4D-space generates the mass spectra of particles. In this sense, a dynamical unification scheme will be achieved where a fifth dimension has the physical meaning of inducing the mass of particles by deformations of space-time. In other words, we will show that deformations can be parameterized as "effective" scalar fields in a -group of diffeomorphisms. In this sense, we do not need any spontaneous symmetry breaking but just a self-consistent way to classify deformations as "gauge bosons". The layout of the paper is the following. In Sec.II, we discuss in detail the conformal-affine structure of gravitational field showing that the nonlinear realization of a group provides a way to determine the transformation properties of fields defined on a given quotient space . In other words, we show that gravitational field can be realized in many equivalent ways and we will use this feature to show that gravitational massive states are possible. Sec. III is devoted to the group structure. We show that the 4D-group of diffeomorphisms can be embedded in that in 5D. Furthermore, it is straightforward to show that contains all the generators of the Standard Model plus generators of the gravitational field. The space-time deformations as elements of -group are discussed in Sec.IV. The main result of this section is that deformations can be dealt as effective geometric scalar fields. In Sec.V, the 5D space-time structure and the reduction to 4D-dynamics is discussed. Such a reduction mechanism gives rise to effective theories of gravity (Extended Theories of Gravity odishap ; odirev ; capfra ; book ; odino ) where higher-order terms in curvature invariants or nonminimal couplings are naturally achieved. In Sec.VI, we discuss that these effective theories can be conformally related and the only singular theory (with null Hessian determinant) is GR. The straightforward consequence of such a result is that gravitational massive modes can be always generated. Sec.VII is devoted to the discussion of the mass generation while, in Sec.VIII, we derive massless and massive gravitational modes related to Extended Theories of Gravity. An interesting byproduct is the fact that 6 polarization states emerge and this result is perfectly in agreement with the fundamental Riemann theorem stating that in a -dimensional space, gravitational degrees of freedom are allowed. Sec.IX is devoted to the specific issue that massive gravitons could have observable effects between GeV-TeV scales and induce a symmetry breaking through a sort of regularization mechanism. Conclusions are drawn in Sec.X.

## Ii The Conformal-Affine Structure of Gravitational Field

### ii.1 Generalities on fiber bundle formalism

In this section, we shall take into account the fiber bundle formalism of gravitational field showing that it naturally exhibit a conformal-affine structure. This feature, in some sense, allow to compare all the theories of gravity, based on diffeomorphism invariance, under the same standard.

Before considering in details the conformal-affine structure of gravitational theories, let us briefly review the standard bundle approach to gauge theories. First, let us show that a usual gauge potential is the pullback of 1-form connection by the local sections of the bundle. Then, the transformation laws of the and under the action of the structure group are deduced.

Modern formulations of gauge field theories are geometrically expressible in the language of principal fiber bundles. A fiber bundle is a structure where (the total bundle space) and (the base space) are smooth manifolds, is the fiber space and the surjection (a canonical projection) is a smooth map of onto ,

(1) |

The inverse image is diffeomorphic to

(2) |

and it is called the fiber at . The partitioning is referred to as the fibration. Note that a smooth map is one whose coordinatization is differentiable; a smooth manifold is a space that can be covered with coordinate patches in such a manner that, a change from one patch to any overlapping patch is smooth Schwarz . Fiber bundles that admit a decomposition as a direct product, locally looking like , are called trivial. Given a set of open coverings of with satisfying , the diffeomorphism map is given by

(3) |

( represents the fiber product of elements defined over the space ) such that and and . Here, represents the identity element of the group . In order to obtain the global bundle structure, the local charts must be glued together continuously. Consider two patches and with a non-empty intersection . Let be the restriction of to defined by . Similarly let be the restriction of to . The composite diffeomorphism

(4) |

defined as

(5) |

constitutes the transition function between bundle charts and ( represents the group composition operation) where the diffeomorphism is written as and satisfies . The transition functions can be interpreted as passive gauge transformations. They satisfy some consistency conditions, i.e. the identity , the inverse and the cocycle . For trivial bundles, the transition function reduces to

(6) |

where is defined by , provided the local trivializations and it gives rise to the same fiber bundle.

A section is defined as a smooth map

(7) |

such that and satisfies

(8) |

where is the identity element of . It assigns to each point a point in the fiber over . Trivial bundles admit global sections.

A bundle is a principal fiber bundle provided that the Lie group acts freely (i.e. if then ) on to the right , , preserves fibers on (), and finally is transitive on fibers. Furthermore, there must exist local trivializations compatible with the action. Hence, is homeomorphic to and the fibers of are diffeomorphic to . The trivialization or inverse diffeomorphism map is given by

(9) |

such that , , where we see from the above definition that is a local mapping of into satisfying for any and any . Let us observe that the elements of which are projected onto the same are transformed into one another by the elements of . In other words, the fibers of are the orbits of and at the same time, they are the set of elements which are projected onto the same . This observation motivates calling the action of the group vertical and the base manifold horizontal. The diffeomorphism map is called the local gauge since maps onto the direct (Cartesian) product . The action of the structure group on defines an isomorphism of the Lie algebra of onto the Lie algebra of vertical vector fields on , tangent to the fiber at each called fundamental vector fields

(10) |

where is the space of tangents at , i.e. . The map is a linear isomorphism for every and is invariant with respect to the action of , that is, , where is the differential push forward map induced by defined by .

Since the principal bundle is a differentiable manifold, we can define tangent and cotangent bundles. The tangent space defined at each point may be decomposed into a vertical and horizontal subspace as (where represents the direct sum). The space is a subspace of consisting of all tangent vectors to the fiber passing through , and is the subspace complementary to at . The vertical subspace is uniquely determined by the structure of , whereas the horizontal subspace cannot be uniquely specified. This result is very important because it makes possible to fix the Cauchy conditions on the dynamics. Thus we require the following condition: when transforms as , transforms as Nakahara ,

(11) |

Let the local coordinates of be where and . Let denote the generators of the Lie algebra corresponding to group satisfying the commutators , where are the structure constants of . Let be a connection form defined by . Let be a connection 1-form defined by

(12) |

( represents the differential pullback map) belonging to where is the dual space to . In such a case, the differential pullback map, applied to a test function and -forms and , satisfies , and. If is represented by a -dimensional matrix, then , , where , , , ,. Thus, assumes the form

(13) |

If is -dimensional, the tangent space is -dimensional. Since the vertical subspace is tangential to the fiber , it is -dimensional. Accordingly, is -dimensional. The basis of can be taken to be . Now, let the basis of be denoted by

(14) |

where . The connection 1-form projects onto . In order for to belong to , it has to be , . In other words,

(15) |

from which can be determined. The inner product appearing in is a map defined by , where the 1-form and vector are given by and . Observe also that, .

We parameterize an arbitrary group element as , ,. The right action on , i.e. , defines a curve through in . Define a vector by Nakahara

(16) |

where is an arbitrary smooth function. Since the vector is tangent to at , , the components of the vector are the fundamental vector fields at which constitute . We have to stress that the components of may also be viewed as a basis element of the Lie algebra . Given , ,

(17) | |||||

where use was made of . Hence, . An arbitrary vector may be expanded in a basis spanning as . By direct computation, one can show

(18) |

Equation (18) yields

(19) |

from which we obtain

(20) |

In this manner, the horizontal component is completely determined. An arbitrary tangent vector defined at takes the form

(21) |

where and are constants. The vector field is comprised of horizontal and vertical components.

Let and , then

Observing that and the first term on the RHS of (LABEL:Rightw) reduces to while the second term gives . We therefore conclude

(23) |

where the adjoint map is defined by

(24) |

The potential can be obtained from as . To demonstrate this, let and be specified by the inverse diffeomorphism or trivialization map (9) with for . We find , where we have used , and at implying Nakahara . Hence,

(25) |

To determine the gauge transformation of the connection 1-form , we use the fact that for and the transition functions defined between neighboring bundle charts (6). By direct computation we get

(26) | |||||

where is a curve in with boundary values and . Thus, we obtain the useful result

(27) |

Applying to Eq.(27), we get

(28) |

Hence, the gauge transformation of the local gauge potential reads,

(29) |

Since we obtain, from Eq.(29), the gauge transformation law of

(30) |

We have now all the ingredients to investigate the bundle structure of the gravitation field.

### ii.2 The Bundle Structure for Gravitation

Let us recall the definition of gauge transformations in the context of ordinary fiber bundles. Given a principal fiber bundle , ; with base space and standard -diffeomorphic fiber, gauge transformations are characterized by bundle isomorphisms exhausting all diffeomorphisms on Giachetta . This mapping is called an automorphism of provided it is equivariant with respect to the action of . This amounts to restrict the action of along local fibers leaving the base space unaffected. Indeed, with regard to gauge theories of internal symmetry groups, a gauge transformation is a fiber preserving bundle automorphism, i.e. diffeomorphisms with . The automorphisms form a group called the automorphism group of . The gauge transformations form a subgroup of called the gauge group (or in short) of .

The map is required to satisfy two conditions, namely its commutability with the right action of the equivariance condition

(31) |

according to which fibers are mapped into fibers, and the verticality condition

(32) |

where and belong to the same fiber. The last condition ensures that no diffeomorphisms given by

(33) |

be allowed on the base space . In a gauge description of gravitation, one is interested in gauging external transformation groups. This means that the group action on space-time coordinates cannot be neglected. The spaces of internal fiber and external base must be interlocked in the sense that transformations in one space must induce corresponding transformations in the other. The usual definition of a gauge transformation, i.e. as a displacement along local fibers not affecting the base space, must be generalized to reflect this interlocking. One possible way of framing this interlocking is to employ a nonlinear realization of the gauge group , provided a closed subgroup exists. The interlocking requirement is then transformed into the interplay between groups and one of its closed subgroups .

Let us denote by a Lie group with elements . Let be a closed subgroup of specified by

(34) |

with elements and known linear representations . Here is a projection map and is the right group action. Let be a differentiable manifold with points to which and may be referred, i.e. and . Being that and are Lie groups, they are also manifolds. The right action of on induces a complete partition of into mutually disjoint orbits . Since , all elements of are defined over the same . Thus, each orbit constitutes an equivalence class of point , with equivalence relation where .

By projecting each equivalence class onto a single element of the quotient space , the group becomes organized as a fiber bundle in the sense that . In this manner the manifold is viewed as a fiber bundle with -diffeomorphic fibers and base space . A composite principal fiber bundle , ; is one whose -diffeomorphic fibers possess the fibered structure described above. The bundle is then locally isomorphic to . Moreover, since an element is locally homeomorphic to the elements of are - by transitivity - also locally homeomorphic to where (locally) . Thus, an alternative view of , ; is provided by the -associated -bundle , ; Tresguerres . The total space may be regarded as -bundles over the base space or equivalently as -fibers attached to the manifold .

The nonlinear realization (NLR) technique CCWZ1 ; CCWZ2 provides a way to determine the transformation properties of fields defined on the quotient space . The NLR of Diff becomes tractable due to a theorem by Ogievetsky. According to this theorem BorisovOgievetskii , the algebra of the infinite dimensional group Diff can be taken as the closure of the finite dimensional algebras of , and , . Remind that the Lorentz group generates transformations that preserve the quadratic form on Minkowski space-time built from the metric tensor, while the special conformal group generates infinitesimal angle-preserving transformations on Minkowski space-time.

The affine group is a generalization of the Poincaré group where the Lorentz group is replaced by the group of general linear transformations. As such, the affine group generates translations, Lorentz transformations, volume preserving shear and volume changing dilation transformations. As a consequence, the NLR of Diff, can be constructed by taking a simultaneous realization of the conformal group , and the affine group , , on the coset spaces , , and , , . One possible interpretation of this theorem is that the conformal-affine group CA (defined below) may be the largest subgroup of Diff, whose transformations may be put into the form of a generalized coordinate transformation. We remark that a NLR can be made linear by embedding the representation in a sufficiently higher dimensional space. Alternatively, a linear group realization becomes nonlinear when subject to constraints. One type of relevant constraints may be those responsible for symmetry reduction from Diff to , for instance.

We take the group , as the basic symmetry group . The CA group consists of the groups , and , . In particular, CA is proportional to the union , , . We know however that the affine and special conformal groups have several group generators in common. These common generators reside in the intersection , , of the two groups, within which there are two copies of , , where is the group of scale transformations (dilations) and , , is the Poincaré group.

Finally we define the CA group as the union of the affine and conformal groups minus one copy of the overlap , i.e. , , , . Being defined in this way we recognize that , is a parameter Lie group representing the action of Lorentz transformations , translations , special conformal transformations , space-time shears and scale transformations . All these transformations can be adopted to define any conformally-affine theory of gravity.

In this paper, we obtain the NLR of , modulo , as 4D realizations starting from 5D-manifold unification . This procedure has been recently adopted also in holographic approaches to Quantum Chromodynamics cappiello .

## Iii The group structure in 5D and 4D-spaces

In this section, we will discuss the group structure of a 5D-Riemannian manifold (in particular the Lorentz group) and its reduction to 4D-manifold. Such an approach gives a useful tool to deal with the realization of effective theories of gravity in 4D and the problem of mass generation. Let us start with the necessary definition of the Minkowski space-time endowed with the metric

(35) |

where is a four-vector, is the time coordinate and is an ordinary vector in . The Lorentz transformations are those linear transformations of Minkowski-type space that leave , the scalar product of four-vectors, invariant:

(36) |

being . If is the Minkowski metric with signature , is a Lorentz transformation when . The set of such transformations is the orthogonal group , namely considering the time-like and space-like components, group characterized, as well known, by the properties that det and the number of generators is 6. The coset decomposition of such a group is

(37) |

where is the proper orthochronus Lorentz group with det, whose elements preserve parity (spatial orientation) and the direction of time; is the group of spatial inversion (parity inversion); are the time reversal transformations and are the total space-time inversion, where we have taken into account all the components without arbitrarily discarding any part of them. The covering group of is the simply connected complex group whose physical meaning is that particles (or in general fields) transform according to its representations.

Now we want to extend this scheme to a 5D-space (which we, initially, consider a flat manifold), where we do not define a priori a signature for the metric and which, after a 4D-reduction procedure, must be capable of reproducing all the features of Lorentz group. For the sake of generality, we do not specify the signature and the number of dimensions. Below we will assume .

Let be a manifold where are integers and such that with the flat metric and . A general signature is

(38) |

where are the time-like directions and are the space-like directions. As particular cases, we have

It is important to stress that the other flat (pseudo)-Riemannian spaces have more than one equivalent (independent) time-like directions and hence have no distinction between future and past time-like directions as they have in Minkowski space. This fact means that the space-like pseudo-spheres are connected hypersurfaces, rather than having two disjoint components as in Minkowski space. The metric can be written as

(39) |

where time-like and space-like components are clearly separated. Some considerations are necessary at this point. The metric (39) is invariant under rotations of the time-like directions among themselves (except for and which are degenerate particular cases, since in the first case there are no time arrows and in the second case, only one time arrow exists by definition) and of the space-like directions among themselves. The remaining independent pseudo-rotations are all boosts each involving a time-like and a space-like direction. The physical meaning of such a result is that close time-like paths are an usual feature in pseudo-Riemannian manifolds, moreover a definite time arrow distinguishing the past from the future is only a particular characteristic of Minkowski spaces where Lorentz transformations work.

Let us now take into account the possible linear transformations on this -manifold. A pseudo-orthogonal group can be defined on this pseudo-Riemannian manifold. This group consists of all the linear transformations such that the metric (39) is invariant, i.e.

(40) |

more precisely we can say that

(41) |

where are non-singular matrices in dimensions. Note that

(42) |

where the determinant is for rotations and for inversions, inversions which do not constitute a sub-group (the product of two inversions gives a rotation). In the first case, we have

(43) |

which is a special pseudo-orthogonal group. An important feature of such a group is that it consists of two disconnected pieces when both and are odd (see gilmore for the general demonstration). Special examples of are

The group can be decomposed as follows

(44) |

where are square matrices which rotate the time-like directions among themselves, are square matrices which rotate the space-like directions among themselves, and the boosts are, in general, or rectangular matrices which rotate time-like and space-like directions.

The number of generators of the group, i.e. the number of independent elements or the dimension of the group, can be easily calculated being, in general,

An important remark is in order at this point. It is well-known, since an old result by Riemann riemann , that a N-dimensional metric has degrees of freedom, that is, it is locally equivalent to give independent functions. This feature is related to the choice of local charts but it is also related to the number of degrees of freedom of gravitational field. As we will discuss below, this is a key ingredient of our discussion. In our case, we have

(45) |

The result is

(46) |

where is the number of independent pairs of one space-like and one time-like direction. For , we have 10 independent elements.

Clearly the rotations form a sub-group of but the boosts do not; boosts along different directions combine to give a boost plus a rotation.

Let us now add the translations to the pseudo-orthogonal group , consisting of rotations and inversions. This fact yields the full group of motions of , which can be classified in the most general inhomogeneous pseudo-orthogonal group . Not taking into account the inversions, a remarkable sub-group is , the inhomogeneous special pseudo-orthogonal group, of dimension