Metric Gauge Fields in Deformed Special Relativity
aa r X i v : . [ phy s i c s . g e n - ph ] M a y Metric gauge fields in Deformed SpecialRelativity
Roberto Mignani − , Fabio Cardone , and Andrea Petrucci Dipartimento di Fisica ”E.Amaldi”, Universit`a degli Studi ”Roma Tre”Via della Vasca Navale, 84 - 00146 Roma, Italy GNFM, Istituto Nazionale di Alta Matematica ”F.Severi”Citt`a Universitaria, P.le A.Moro 2 - 00185 Roma, Italy I.N.F.N. - Sezione di Roma III Istituto per lo Studio dei Materiali Nanostrutturati (ISMN – CNR)Via dei Taurini - 00185 Roma, Italy ENEA, Italian National Agency for new Technologies,Energy and sustainable economic developmentVia Anguillarese 301 - 00123 Roma, ItalyJanuary 14, 2018
Abstract
We show that, in the framework of Deformed Special Relativity (DSR),namely a (four-dimensional) generalization of the (local) space-time struc-ture based on an energy-dependent ”deformation” of the usual Minkowskigeometry, two kinds of gauge symmetries arise, whose spaces either coin-cide with the deformed Minkowski space or are just internal spaces to it.This is why we named them ”metric gauge theories” . In the case of theinternal gauge fields, they are a consequence of the deformed Minkowskispace (DMS) possessing the structure of a generalized Lagrange space.Such a geometrical structure allows one to define curvature and torsionin the DMS. Introduction
It is well known that gauge theories play presently a basic role in describing allthe known interactions. In all cases, gauge symmetries are related to physicalfields directly arising from the symmetries ruling some given interaction; onone side, this leads to the rising of a new, dynamical gauge field; on the otherhand, if the gauge symmetry is broken, such a circumstance provides one withnew — often unforeseen — informations about the structural properties of theinteraction considered.Often, as known as well, in spite of the fact that the physical world is theusual Minkowski space-time, the gauge manifold is not the usual, Minkowskione. For instance, in the case of the usual Minkowski space, the gauge symmetryof electrodynamics does actually work in an auxiliary space (Weyl charge space).It is therefore worth to investigate when and where gauge symmetries can beintroduced in a Minkowski space, and to lead to significant physical results.It is just the purpose of the present paper to show that this circumstanceoccurs in the framework of
Deformed Special Relativity (DSR), namely a (four-dimensional) generalization of the (local) space-time structure based on anenergy-dependent ”deformation” of the usual Minkowski geometry [1, 2]. Aswe shall see, in DSR two kinds of gauge symmetries arise, whose spaces ei-ther coincide with the deformed Minkowski space (DMS) f M or are just internalspaces to it. This is why we named them ”metric gauge theories”. The paper is organized as follows. In Sect.2, we review the basic featuresof DSR that are relevant to our purposes. Sect.3 discuss DSR as a metricgauge theory. Metric gauge fields can be external (Subsect.3.1) or internal(Subsect.3.2). The last topic is related to the structure of DMS as generalizedLagrange space, whose main properties are summarized. Subsect. 3.2.2 dealswith the structure of f M as generalized Lagrange space. The internal gauge fieldsof f M are discussed in Subsect.3.2.4. In 3.3 we present a possible experimentalevidence for such metric gauge fields. Conclusions and perspectives are given inSect.4. The geometrical structure of the physical world — both at a large and a smallscale — has been debated since a long. After Einstein, the generally acceptedview considers the arena of physical phenomena as a four-dimensional space-time, endowed with a global , curved, Riemannian structure and a local , flat,Minkowskian geometry.However, an analysis of some experimental data concerning physical phe-nomena ruled by different fundamental interactions have provided evidence fora local departure from Minkowski metric [1, 2]: among them, the lifetime of the2weakly decaying) K s meson, the Bose-Einstein correlation in (strong) pion pro-duction and the superluminal propagation of electromagnetic waves in waveg-uides. These phenomena seemingly show a (local) breakdown of Lorentz in-variance, together with a plausible inadequacy of the Minkowski metric; on theother hand, they can be interpreted in terms of a deformed Minkowski space-time, with metric coefficients depending on the energy of the process considered[1, 2].All the above facts suggested to introduce a (four-dimensional) generaliza-tion of the (local) space-time structure based on an energy-dependent ”defor-mation” of the usual Minkowski geometry of M , whereby the correspondingdeformed metrics ensuing from the fit to the experimental data seem to providean effective dynamical description of the relevant interactions ( at the energyscale and in the energy range considered ).An analogous energy-dependent metric seems to hold for the gravitationalfield (at least locally, i.e. in a neighborhood of Earth) when analyzing someclassical experimental data concerning the slowing down of clocks.Let us shortly review the main ideas and results concerning the (four-dimensional) deformed Minkowski spacetime f M .The four-dimensional ”deformed” metric scheme is based on the assumptionthat spacetime, in a preferred frame which is fixed by the scale of energy E , isendowed with a metric of the form ds = b ( E ) c dt − b ( E ) dx − b ( E ) dy − b ( E ) dz = g DSRµν ( E ) dx µ dx ν ; g DSRµν ( E ) = (cid:0) b ( E ) , − b ( E ) , − b ( E ) , − b ( E ) (cid:1) , (1)with x µ = ( x , x , x , x ) = ( ct, x, y, z ) , c being the usual speed of light invacuum. We named ”Deformed Special Relativity” (DSR) the relativity theorybuilt up on metric (1).Metric (1) is supposed to hold locally, i.e. in the spacetime region wherethe process occurs. It is supposed moreover to play a dynamical role, and toprovide a geometric description of the interaction considered. In this sense,DSR realizes the so called ”Finzi Principle of Solidarity” between space-timeand phenomena occurring in it (see [3]). Futhermore, we stress that, from Let us recall that in 1955 the Italian mathematician Bruno Finzi stated his ”Princi-ple of Solidarity” (PS), that sounds ”
It’s (indeed) necessary to consider space-time TO BESOLIDLY CONNECTED with the physical phenomena occurring in it, so that its featuresand its very nature do change with the features and the nature of those. In this way notonly (as in classical and special-relativistic physics) space-time properties affect phenomena,but reciprocally phenomena do affect space-time properties. One thus recognizes in such anappealing ”Principle of Solidarity” between phenomena and space-time that characteristic ofmutual dependence between entities, which is peculiar to modern science. ” Moreover, referringto a generic N-dimensional space: ” It can, a priori , be pseudoeuclidean, Riemannian, non-Riemannian. But — he wonders — how is indeed the space-time where physical phenomenatake place? Pseudoeuclidean, Riemannian, non-Riemannian, according to their nature, asrequested by the principle of solidarity between space-time and phenomena occurring in it.”
Of course, Finzi’s main purpose was to apply such a principle to Einstein’s Theory of GeneralRelativity, namely to the class of gravitational phenomena. However, its formulation is as E is the measured energy of the system , and thus amerely phenomenological (non-metric) variable .We notice explicitly that the spacetime f M described by (1) is flat (it has zerofour-dimensional curvature), so that the geometrical description of the funda-mental interactions based on it differs from the general relativistic one (whencethe name ”deformation” used to characterize such a situation). Although foreach interaction the corresponding metric reduces to the Minkowskian one fora suitable value of the energy E (which is characteristic of the interactionconsidered), the energy of the process is fixed and cannot be changed at will.Thus, in spite of the fact that formally it would be possible to recover the usualMinkowski space M by a suitable change of coordinates (e.g. by a rescaling),this would amount, in such a framework, to be a mere mathematical operationdevoid of any physical meaning.As far as phenomenology is concerned, it is important to recall that a localbreakdown of Lorentz invariance may be envisaged for all the four fundamentalinteractions (electromagnetic, weak, strong and gravitational) whereby one getsevidence for a departure of the spacetime metric from the Minkowskian one (in the energy range examined). The explicit functional form of the metric (1)for all the four interactions can be found in [1, 2]. Here, we confine ourselves torecall the following basic features of these energy-dependent phenomenologicalmetrics:1) Both the electromagnetic and the weak metric show the same functionalbehavior, namely g DSRµν ( E ) = diag (cid:0) , − b ( E ) , − b ( E ) , − b ( E ) (cid:1) ; (2) b ( E ) = (cid:26) ( E/E ) / , ≤ E ≤ E , E ≤ E (3)with the only difference between them being the threshold energy E , i.e.the energy value at which the metric parameters are constant, i.e. the metricbecomes Minkowskian; the fits to the experimental data yield E ,e.m. = 5 . ± . µeV ; E w = 80 . ± . GeV ; (4)2) for strong and gravitational interactions, the metrics read: g DSR ( E ) = diag (cid:0) b ( E ) , − b ( E ) , − b ( E ) , − b ( E ) (cid:1) ; (5) b ,strong ( E ) = b ,strong ( E ) = (cid:26) , ≤ E < E strong ( E/E strong ) , E strong < E ; b ,strong ( E ) = (cid:16) √ / (cid:17) ; b ,strong = (2 / ; (6) general as possible, so to apply in principle to all the known physical interactions. Therefore,Finzi’s PS is at the very ground of any attempt at geometrizing physics, i.e. describing physicalforces in terms of the geometrical structure of space-time. As is well known, all the present physically realizable detectors work via their electro-magnetic interaction in the usual space-time M . So, E is the energy of the system measuredin fully Minkowskian conditions. ,grav ( E ) = (cid:26) , ≤ E < E grav (1 + E/E grav ) , E grav < E (6’)with E s = 367 . ± . GeV ; E grav = 20 . ± . µeV. (7)Let us stress that, in this case, contrarily to the electromagnetic and the weakones, a deformation of the time coordinate occurs; moreover, the three-spaceis anisotropic , with two spatial parameters constant (but different in value)and the third one variable with energy in an ”over-Minkowskian” way (namelyit reaches the limit of Minkowskian metric for decreasing values of E , with E > E ) [1, 2].As a final remark, we stress that actually the four-dimensional energy-dependent spacetime f M is just a manifestation of a larger, five-dimensionalspace in which energy plays the role of a fifth dimension. Indeed, it can be shownthat the physics of the interaction lies in the curvature of such a five-dimensionalspacetime, in which the four-dimensional, deformed Minkowski space is em-bedded. Moreover, all the phenomenological metrics (2), (3) and (5), (6) canbe obtained as solutions of the vacuum Einstein equations in this generalizedKaluza-Klein scheme [1, 2].
We want now to show that the deformation of space-time, expressed by themetric g DSR (Eq.(1)), does affect also the external fields applied to the physicalsystem considered.Let us consider for instance the case of a physical process ruled by theelectromagnetic interaction. Therefore, the Minkowski space M is endowed withthe electromagnetic tensor F µν ( x ) (external e.m. field) acting on the system.Of course F µν ( x ) = g µρSR F ρν ( x ).In the deformed Minkowski space f M , the covariant components of the elec-tromagnetic tensor read e F µν = g DSRµρ F ρν = g DSRµρ g µσSR F σν , (8)where ( g DSRµρ g µσSR ) = diag ( b , b , b , b ) = ( b σ δ σρ ) . (9)We have therefore e F ν = b F ν ; e F ν = b F ν ; e F ν = b F ν ; e F ν = b F ν , (10)or e F µν = b µ F µν , µ, ν = 0 , , , At least for strong interaction; nothing can be said for the gravitational one.
5t follows that the tensor e F µν is not antisymmetric: e F µν = − e F νµ . (12)The result shown here for the electromagnetic interaction can be generalizedto other fundamental interactions described by tensor fields.On account of the well-known identification e F i = f E i , e F = − f B , e F = − f B , e F = − f B (13)(and analogously for F µν ), we can write, for the energy density e E of the deformedelectromagnetic field: e E = e E + e B π = b E + b B + b B + b B π , (14)to be compared with the standard expression for the e.m. field E , B : E = E + B π . (15) There is therefore a difference in the energy associated to the electromagneticfield in the deformed space-time region.
We have, for the energy density∆ E = E − e E . (16)We can state that the difference ∆ E represents the energy spent by the interac-tion in order to deform the space-time geometry. We can therefore conclude that the deformation of space-time does affect thefield itself that deforms the geometry of the space . There is therefore a feedbackbetween space and interaction which fully implements the Solidarity Principle.
It is clear from the discussion of the phenomenological metrics describing thefour fundamental interactions in DSR that the Minkowski space M is the space-time manifold of background of any experimental measurement and detection(namely, of any process of acquisition of information on physical reality). In par-ticular, we can consider this Minkowski space as that associated to the electro-magnetic interaction above the threshold energy E ,e.m. . Therefore, in modelingthe physical phenomena, one has to take into account this fact. The geometricalnature of interactions, i.e. assuming the validity of the Finzi principle, meansthat one has to suitably gauge (with reference to M ) the space-time metricswith respect to the interaction — and/or the phenomenon — under study. Inother words, one needs to ”adjust” suitably the local metric of space-time ac-cording to the interaction acting in the region considered. We can name such a6rocedure ”Metric Gaugement Process” (M.G.P.). Like in usual gauge theoriesa different phase is chosen in different space-time points , in DSR different met-rics are associated to different space-time manifolds according to the interactionacting therein. We have thus a gauge structure on the space of manifolds f M ≡ ∪ g DSR ∈P ( E ) f M ( g DSR ) , (17)where P ( E ) is the set of the energy-dependent pseudoeuclidean metrics of thetype (1). This is why it is possible to regard Deformed Special Relativity as a Metric Gauge Theory . In this case, we can consider the related fields as externalmetric gauge fields.
However, let us notice that DSR can be considered as a metric gauge theoryfrom another point of view, on account of the dependence of the metric coeffi-cients on the energy. Actually, once the MGP has been applied, by selecting thesuitable gauge (namely, the suitable functional form of the metric) accordingto the interaction considered (thus implementing the Finzi principle), the met-ric dependence on the energy implies another different gauge process. Namely,the metric is gauged according to the process under study, thus selecting the given metric, with the given values of the coefficients, suitable for the givenphenomenon.We have therefore a double metric gaugement, according, on one side, to theinteraction ruling the physical phenomenon examined, and on the other sideto its energy, in which the metric coefficients are the analogous of the gaugefunctions . We want now to show that the deformed Minkowski space f M of Deformed Spe-cial Relativity does possess another well-defined geometrical structure, besidesthe deformed metrical one. Precisely, we will show that f M is a generalizedLagrange space [6]. As we shall see, this implies that DSR admits a different, intrinsic gauge structure. Let us give the definition of generalized La-grange space [4], since usually one is not acquainted with it.Consider a N-dimensional, differentiable manifold M and its (N-dimensional)tangent space in a point, T M x ( x ∈ M ). As is well known, the union [ x ∈M T M x ≡ T M (18) The analogy of this second kind of metric gauge with the standard, non-abelian gaugetheories is more evident in the framework of the five-dimensional space-time ℜ (with energyas extra dimension) embedding f M , on which Deformed Relativity in Five Dimensions (DR5)is based (see [1, 2]). In ℜ , in fact, energy is no longer a parametric variable, like in DSR,but plays the role of fifth (metric) coordinate. The invariance under such a metric gauge,not manifest in four dimensions, is instead recovered in the form of the isometries of thefive-dimensional space-time-energy manifold ℜ . y the generic element of T M x ,namely a vector tangent to M in x . Then, an element u ∈ T M is a vectortangent to the manifold in some point x ∈ M . Local coordinates for T M areintroduced by considering a local coordinate system ( x , x , ..., x N ) on M andthe components of y in such a coordinate system ( y , y , ..., y N ). The 2 N num-bers ( x , x , ..., x N , y , y , ..., y N ) constitute a local coordinate system on T M .We can write synthetically u = ( x , y ). T M is a 2 N -dimensional, differentiablemanifold.Let π be the mapping ( natural projection ) π : u = ( x , y ) −→ x . ( x ∈ M , y ∈ T M x ). Then, the tern ( T M , π, M ) is the tangent bundle to the basemanifold M . The image of the inverse mapping π − ( x ) is of course the tangentspace T M x , which is called the fiber corresponding to the point x in the fiberbundle One considers also sometimes the manifold d T M = T M / { } , where 0is the zero section of the projection π . We do not dwell further on the theoryof the fiber bundles, and refer the reader to the wide and excellent literature onthe subject [5].The natural basis of the tangent space T u ( T M ) at a point u = ( x , y ) ∈ T M is (cid:26) ∂∂x i , ∂∂y j (cid:27) , i, j = 1 , , ..., N .A local coordinate transformation in the differentiable manifold T M reads x ′ i = x ′ i ( x ) , det (cid:18) ∂x ′ i ∂x j (cid:19) = 0 ,y ′ i = ∂x ′ i ∂x j y j . (19)Here, y i is the Liouville vector field on T M , i.e. y i ∂∂y i .On account of Eq.(19), the natural basis of T M x can be written as ∂∂x i = ∂x ′ k ∂x i ∂∂x ′ k + ∂y ′ k ∂x i ∂∂y ′ k ,∂∂y j = ∂y ′ k ∂y j ∂∂y ′ k . (20)Second Eq.(20) shows therefore that the vector basis (cid:18) ∂∂y j (cid:19) , j = 1 , , ..., N ,generates a distribution V defined everywhere on T M and integrable, too ( ver-tical distribution on T M ).If H is a distribution on T M supplementary to V , namely T u ( T M ) = H u ⊕ V u , ∀ u ∈ T M , (21)then H is called a horizontal distribution , or a nonlinear connection on T M .A basis for the distributions H and V are given respectively by (cid:26) δδx i (cid:27) and8 ∂∂y j (cid:27) , where the basis in H explicitly reads δδx i = ∂∂x i − H ji ( x , y ) ∂∂y j . (22)Here, H ji ( x , y ) are the coefficients of the nonlinear connection H . The basis (cid:26) δδx i , ∂∂y j (cid:27) = n δ i , ˙ ∂ j o is called the adapted basis. The dual basis to the adapted basis is (cid:8) dx i , δy j (cid:9) , with δy j = dy j + H ji ( x , y ) dx i . (23)A distinguished tensor (or d-tensor ) field of (r,s)-type is a quantity whosecomponents transform like a tensor under the first coordinate transformation(19) on T M (namely they change as tensor in M ). For instance, for a d-tensorof type (1,2): R ′ ijk = ∂x ′ i ∂x s ∂x r ∂x ′ j ∂x p ∂x ′ k R srp . (24)In particular, both (cid:26) δδx i (cid:27) and (cid:26) ∂∂y j (cid:27) are d-(covariant) vectors, whereas (cid:8) dx i (cid:9) , (cid:8) δy j (cid:9) are d-(contravariant) vectors.A generalized Lagrange space is a pair GL N =( M , g ij ( x , y )), with g ij ( x , y )being a d-tensor of type (0,2) (covariant) on the manifold T M , which is sym-metric, non-degenerate and of constant signature.A function L : ( x , y ) ∈ T M → L ( x , y ) ∈ R (25)differentiable on d T M and continuous on the null section of π is named a regularLagrangian if the Hessian of L with respect to the variables y i is non-singular.A generalized Lagrange space GL N =( M , g ij ( x , y )) is reducible to a Lagrangespace L N if there is a regular Lagrangian L satisfying g ij = 12 ∂ L∂y i ∂y j (26)on d T M . In order that GL N is reducible to a Lagrange space, a necessarycondition is the total symmetry of the d-tensor ∂g ij ∂y k . If such a condition issatisfied, and g ij are 0-homogeneous in the variables y i , then the function L = g ij ( x , y ) y i y j is a solution of the system (26). In this case, the pair ( M , L ) isa Finsler space ( M , Φ), with Φ = L . One says that GL N is reducible to aFinsler space. Namely it must be rank k g ij ( x , y ) k = N . Let us recall that a Finsler space is a couple ( M , Φ), where M is be an N-dimensionaldifferential manifold and Φ : T M ⇒ R a function Φ( x , ξ ) defined for x ∈ M and ξ ∈ T x M such that Φ( x , · ) is a possibly non symmetric norm on T x M .
9f course, GL N reduces to a pseudo-Riemannian (or Riemannian) space( M , g ij ( x )) if the d-tensor g ij ( x , y ) does not depend on y . On the contrary,if g ij ( x , y ) depends only on y (at least in preferred charts), it is a generalizedLagrange space which is locally Minkowskian.Since, in general, a generalized Lagrange space is not reducible to a Lagrangeone, it cannot be studied by means of the methods of symplectic geometry, onwhich — as is well known — analytical mechanics is based.A linear H− connection on T M (or on d T M ) is defined by a couple ofgeometrical objects C Γ( H ) = ( L ijk , C ijk ) on T M with different transforma-tion properties under the coordinate transformation (19). Precisely, L ijk ( x , y )transform like the coefficients of a linear connection on M , whereas C ijk ( x , y )transform like a d-tensor of type (1,2). C Γ( H ) is called the metrical canonical H− connection of the generalized Lagrange space GL N .In terms of L ijk and C ijk one can define two kinds of covariant derivatives:a covariant horizontal (h-) derivative , denoted by ” p ”, and a covariant vertical(v-) derivative , denoted by ” | ”. For instance, for the d-tensor g ij ( x , y ) one has g ij p k = δg ij δx k − g sj L sik − g is L sjk ; g ij | k = ∂g ij ∂x k − g sj C sik − g is C sjk . (27)The two derivatives g ij p k and g ij | k are both d-tensors of type (0,3).The coefficients of C Γ( H ) can be expressed in terms of the following gener-alized Christoffel symbols : L ijk = g is (cid:18) δg sj δx k + δg ks δx j + δg jk δx s (cid:19) ; C ijk = g is (cid:18) ∂g sj ∂x k + ∂g ks ∂x j + ∂g jk ∂x s (cid:19) . (28) Curvature and torsion in a generalized Lagrange space
By means ofthe connection C Γ( H ) it is possible to define a d-curvature in T M by means ofthe tensors R ij kh , S ij kh and P ij kh given by R ij kh = δL ijk δx h − δL ijh δx k + L rjk L irh − L rjh L irk + C ijr R rkh ; S ij kh = ∂C ijk ∂y h − ∂C ijh ∂y k + C rjk C irh − C rjh C irk ; P ij kh = ∂L ijk ∂y h − C ij p h + C ijr P rkh . (29) Notice that every Riemann manifold ( M , g ) is also a Finsler space, the norm Φ( x , ξ ) beingthe norm induced by the scalar product g ( x ).A finite-dimensional Banach space is another simple example of Finsler space, whereΦ( x , ξ ) ≡ k ξ k . R ijk is related to the bracket of the basis (cid:26) δδx i (cid:27) : (cid:20) δδx i , δδxj (cid:21) = R sij ∂∂y s (30)and is explicitly given by R ijk = δH ij δx k − δH ik δx j . (31)The tensor P ijk , together with T ijk , S ijk , defined by P ijk = ∂H ij ∂y k − L ijk ; T ijk = L ijk − L ikj ; S ijk = C ijk − C ikj (32)are the d-tensors of torsion of the metrical connection C Γ( H ).¿From the curvature tensors one can get the corresponding Ricci tensors of C Γ( H ): R ij = R si js ; S ij = S si js ; P ij = P si js P ij = P si sj , (33)and the scalar curvatures R = g ij R ij ; S = g ij S ij . (34)Finally, the deflection d-tensors associated to the connection C Γ( H ) are D ij = y i p j = − H ij + y s L isj ; d ij = y i | j = δ ij + y s C isj , (35)namely the h- and v-covariant derivatives of the Liouville vector fields.In the generalized Lagrange space GL N it is possible to write the Einsteinequations with respect to the canonical connection C Γ( H ) as follows: R ij − Rg ij = κ H T ij ; P ij = κ T ij ; S ij − Sg ij = κ V T ij ; P ij = κ T ij , (36) R ijk plays the role of a curvature tensor of the nonlinear connection H . The correspondingtensor of torsion is instead t ijk = ∂H ij ∂y k − ∂H ik ∂y j . κ is a constant and H T ij , V T ij , T ij , T ij are the components of the energy-momentum tensor. f M On the basis of the previous considerations, let us analyze the geometrical struc-ture of the deformed Minkowski space of DSR f M , endowed with the by nowfamiliar metric g µν,DSR ( E ) . As said in Sect.2, E is the energy of the processmeasured by the detectors in Minkowskian conditions. Therefore, E is a func-tion of the velocity components, u µ = dx µ /dτ , where τ is the (Minkowskian)proper time : E = E (cid:18) dx µ dτ (cid:19) . (37)The derivatives dx µ /dτ define a contravariant vector tangent to M at x ,namely they belong to T M x . We shall denote this vector (according to thenotation of the previous Subsubsection) by y = ( y µ ). Then, ( x , y ) is a point ofthe tangent bundle to M . We can therefore consider the generalized Lagrangespace GL = ( M, g µν ( x , y )), with g µν ( x , y ) = g µνDSR ( E ( x , y )) ,E ( x , y ) = E ( y ) . (38)Then, it is possible to prove the following theorem [6]: The pair GL = ( M, g
DSR,µν ( x , y )) ≡ f M is a generalized Lagrange spacewhich is not reducible to a Riemann space, or to a Finsler space, or to a Lagrangespace. Notice that such a result is strictly related to the fact that the deformedmetric tensor of DSR is diagonal.If an external electromagnetic field F µν is present in the Minkowski space M , in f M the deformed electromagnetic field is given by e F µν ( x , y ) = g µρDSR F ρν ( x )(see Eq.(8)). Such a field is a d-tensor and is called the electromagnetic tensorof the generalized Lagrange space . Then, the nonlinear connection H is given by H µν = (cid:26) µνρ (cid:27) y ρ − e F µν ( x , y ) , (39)where (cid:26) µνρ (cid:27) , the Christoffel symbols of the Minkowski metric g µν , are zero,so that H µν = − e F µν ( x , y ) , (40)namely, the connection coincides with the deformed field. Contrarily to ref.[6], we shall not consider the restrictive case of a classical (non-relativistic) expression of the energy, but assume a general dependence of E on the velocity(eq.(38)). H reads therefore δδx µ = ∂∂x µ + e F νµ ( x , y ) ∂∂y ν . (41)The local covector field of the dual basis (cfr. Eq.(23)) is given by δy µ = dy µ − e F µν ( x , y ) dx ν . (42) f M The derivation operators applied to the deformed metric tensor of the space GL = f M yield δg DSRµν δx ρ = ∂g DSRµν ∂x ρ + e F σρ ∂g DSRµν ∂y σ = e F σρ ∂g DSRµν ∂E ∂E∂y σ , (43) ∂g DSRµν ∂y σ = ∂g DSRµν ∂E ∂E∂y σ . (44)Then, the coefficients of the canonical metric connection C Γ( H ) in f M (seeEq.(28)) are given by L µνρ = g µσDSR ∂E∂y α (cid:18) ∂g DSRσν ∂E e F αρ + ∂g DSRσρ ∂E e F αν − ∂g DSRνρ ∂E e F ασ (cid:19) ,C µνρ = g µσDSR ∂E∂y α (cid:18) ∂g DSRσν ∂E δ αρ + ∂g DSRσρ ∂E δ αν − ∂g DSRνρ ∂E δ ασ (cid:19) . (45)The vanishing of the electromagnetic field tensor, F αρ = 0, implies L µνρ = 0.One can define the deflection tensors associated to the metric connection C Γ( H ) as follows (cfr. Eq.(36)): D µν = y µ p ν = δy µ δx ν + y α L µαν = f F µν + y α L µαν ; d µν = y µ | ν = δ µν + y α C µαν . (46)The covariant components of these tensors read D µν = g µσ,DSR D σν = g µσ,DSR (cid:16) f F σν + y α L σαν (cid:17) == F µν ( x ) + 12 y σ ∂E∂y α (cid:18) ∂g DSRµσ ∂E e F αν + ∂g DSRµν ∂E e F ασ − ∂g DSRσν ∂E e F αµ (cid:19) ; d µν = g µσ,DSR d σν == g DSR,µν + 12 y σ ∂E∂y α (cid:18) ∂g DSRµσ ∂E δ αν + ∂g DSRµν ∂E δ ασ − ∂g DSRσν ∂E δ αµ (cid:19) . (47)13t is important to stress explicitly that, on the basis of the results of 3.2.1, the deformed Minkowski space f M does possess curvature and torsion, namely itis endowed with a very rich geometrical structure. This permits to understandthe variety of new physical phenomena that occur in it (as compared to thestandard Minkowski space) [1, 2].Following ref.[6], let us show how the formalism of the generalized La-grange space allows one to recover some results on the phenomenological energy-dependent metrics discussed in Sect.2.Consider the following metric ( c = 1): ds = a ( E ) dt + ( dx + dy + dz ) (48)where a ( E ) is an arbitrary function of the energy and spatial isotropy ( b = 1)has been assumed. In absence of an external electromagnetic field ( F µν = 0),the non-vanishing components C µνρ of the canonical metric connection C Γ( H )(see Eq.(46)) are C = a ′ a y , C = − a ′ a y , C = − a ′ a y , C = a ′ a y ,C = − a ′ y , C = − a ′ y , C = − a ′ y , (49)where the prime denotes derivative with respect to E : a ′ = dadE . According to the formalism of generalized Lagrange spaces, we can write theEinstein equations in vacuum corresponding to the metrical connection of thedeformed Minkowski space (see Eqs.(37)). It is easy to see that the independentequations are given by a ′ = 0; (50)2 aa ′′ − ( a ′ ) = 0 . (51)The first equation has the solution a = const. , namely we get the Minkowskimetric. Eq.(52) has the solution a ( E ) = 14 (cid:18) a + EE (cid:19) , (52)where a and E are two integration constants.This solution represents the time coefficient of an over-Minkowskian metric.For a = 0 it coincides with (the time coefficient of) the phenomenological metricof the strong interaction, Eq.(6). On the other hand, by choosing a = 1, onegets the time coefficient of the metric for gravitational interaction, Eq.(6’).In other words, considering f M as a generalized Lagrange space permits torecover (at least partially) the metrics of two interactions (strong and gravita-tional) derived on a phenomenological basis.
14t is also worth noticing that this result shows that a spacetime deformation(of over-Minkowskian type) exists even in absence of an external electromagneticfield (remember that Eqs.(51),(52) have been derived by assuming F µν = 0). As we have seen, the deformed Minkowski space f M , considered as a generalizedLagrange space, is endowed with a rich geometrical structure. But the importantpoint, to our purposes, is the presence of a physical richness, intrinsic to f M .Indeed, let us introduce the following internal electromagnetic field tensors on GL = f M , defined in terms of the deflection tensors : F µν ≡
12 ( D µν − D νµ ) == F µν ( x ) + 12 y σ ∂E∂y α (cid:18) ∂g DSRµσ ∂E e F αν − ∂g DSRνσ ∂E e F αµ (cid:19) (53)( horizontal electromagnetic internal tensor ) and f µν ≡
12 ( d µν − d νµ ) == 12 y σ ∂E∂y α (cid:18) ∂g DSRµσ ∂E δ αν − ∂g DSRνσ ∂E δ αµ (cid:19) (54)( vertical electromagnetic internal tensor ).The internal electromagnetic h- and v-fields F µν and f µν satisfy the following generalized Maxwell equations F µν p ρ + F νρ p µ + F ρµ p ν ) = y α (cid:0) R βµν C βαρ + R βνρ C βαµ + R βρµ C βαν (cid:1) ,R βµν = g βσ ∂F µν ∂x σ ; (55) F µν | ρ + F νρ | µ + F ρµ | ν = f µν p ρ + f νρ p µ + f ρµ p ν ; (56) f µν | ρ + f νρ | µ + f ρµ | ν = 0 . (57)Let us stress explicitly the different nature of the two internal electromag-netic fields. In fact, the horizontal field F µν is strictly related to the presenceof the external electromagnetic field F µν , and vanishes if F µν = 0. On the con-trary, the vertical field f µν has a geometrical origin, and depends only on thedeformed metric tensor g DSRµν ( E ( y )) of GL = f M and on E ( y ) . Therefore,it is present also in space-time regions where no external electromagnetic fieldoccurs. As we shall see, this fact has deep physical implications.A few remarks are in order. First, the main results obtained for the (abelian)electromagnetic field can be probably generalized (with suitable changes) to non-abelian gauge fields. Second, the presence of the internal electromagnetic h- andv-fields F µν and f µν , intrinsic to the geometrical structure of f M as a generalized15agrange space, is the cornerstone to build up a dynamics (of merely geometricalorigin) internal to the deformed Minkowski space. The important point worth emphasizing is that such an intrinsic dynamicssprings from gauge fields.
Indeed, the two internal fields F µν and f µν (inparticular the latter one) do satisfy equations of the gauge type (cfr. Eqs.(57)-(58)). Then, we can conclude that the (energy-dependent) deformation of themetric of f M , which induces its geometrical structure as generalized Lagrangespace, leads in turn to the appearance of (internal) gauge fields .Such a fundamental result can be schematized as follows: f M = ( M, g
DSRµν ( E )) = ⇒ GL = ( M, g µν ( x , y )) = ⇒ (cid:16) f M , F µν , f µν (cid:17) (58)(with self-explanatory meaning of the notation).We want also to stress explicitly that this result follows by the fact that, indeforming the metric of the space-time, we assumed the energy as the physical(non-metric) observable on which letting the metric coefficients depend . Thisis crucial in stating the generalized Lagrangian structure of f M , as shown above. We want now to discuss some results on anomalous interference effects, whichadmit a quite straightforward interpretation in terms of the intrinsic gauge fieldsof DSR.In double-slit-like experiments in the infrared range, we collected evidencesof an anomalous behaviour of photon systems under particular (energy andspace) constraints [7, 8, 9, 10]. The experimental set-up is reported in Fig.1.This layout shows the horizontal view of the interior of a closed box dividedinto different rooms by panels. The box was 20 cm long, 12 cm large and 7 cm high. It contained two infrared LEDs S and S , three detectors A, B andC (either photodiodes or phototransistors) and three apertures F , F and F .The source S was aligned with the detector A through the aperture F , thesource S was aligned with the detector C which was right on the apertureF . The detector B was in front of the aperture F and did not receive anyphoton directly. The position of the detectors, the sources and the apertureswas designed so that the detector A was not influenced by the lighting state ofthe source S according to the laws of physics governing photons propagation. Inother words, A did not have to distinguish whether S was on or off. Besides, inorder to prevent reflections of photons, the internal surfaces of the box had beencoated by an absorbing material. While the detectors B and C were controllingdetectors, A was devoted to perform the actual experiment. In particular, wecompared the signal, measured on A when S was on and S was off, with thesignal on A when both sources S and S were on. As to what it has beensaid about the incapability of A to distinguish between S off or on, these twocompared conditions were expected to produce compatible results. However, itturned out that the sampling of the signal on A with S on and S on and the16igure 1: Schematic layout of the box used to detect the anomalous interferenceeffect.sampling of the signal on A when only S was on do not belong to the samepopulation and are represented by two different gaussian distributions whosemean values are significantly different. Besides, the difference between the twomean values was less than 4.5 µeV , as predicted by the theory of DeformedSpace-time [1, 2]. Since it was experimentally verified that no photons passedthrough the aperture F , this result shows an anomalous behaviour of the photonsystem. The same experiment was carried out by different sources, detectors, bytwo different boxes and different measuring systems. Nevertheless, every timewe obtained the same anomalous result [8, 9, 10]. Moreover, the same kind ofgeometrical structure and the same spatial distances were used in other kindof experiments carried out in the microwave region of the spectrum and by alaser system [11, 12, 1, 2]. Although these experiments had completely differentexperimental set-ups from our initial one, they succeeded in finding out the samekind of anomalous behaviour that we had found out by the box experiments.The anomalous effect in photon systems, at least in those experimental set-ups that were used, disagrees both with standard quantum mechanics (Copen-hagen interpretation) and with classical and quantum electrodynamics. Somepossible interpretations can be given in terms of either the existence of deBroglie–Bohm pilot waves associated to photons, and/or the breakdown of localLorentz invariance (LLI) [7, 8, 9, 10]. Besides, it turns out that it is also possibleto move a step forward and hypothesise the existence of an intriguing connec-tion between the pilot wave interpretation and that involving LLI breakdown.One might assume that the pilot wave is, in the framework of LLI breakdown,a local deformation of the flat Minkowskian spacetime.The interpretation in terms of DSR is quite straightforward. Under theenergy threshold E ,em =4.5 µeV , the metric of the electromagnetic interaction17s no longer Minkowskian. The corresponding space-time is deformed. Such aspace-time deformation shows up as the hollow wave accompanying the photon,and is able to affect the motion of other photons. This is the origin of theanomalous interference observed ( shadow of light ). The difference of signalmeasured by the detector A in all the double-slit experiments can be regardedas the energy spent to deforme space-time. In space regions where the externalelectromagnetic field is present (regions of ”standard” photon behavior), we canassociate such energy to the difference ∆ E , Eq.(16), between the energy densitycorresponding to the external e.m. field F µν and that of the deformed one e F µν given by Eq.(8).But it is known from the experimental results that the anomalous interfer-ence effects observed can be explained in terms of the shadow of light, namelyin terms of the hollow waves present in space regions where no external e.m.field occurs. How to account for this anomalous photon behavior within DSR?The answer is provided by the internal structure of the deformed Minkowskispace discussed above. In fact, we have seen that the structure of the deformedMinkowski space f M as Generalized Lagrange Space implies the presence of twointernal e.m. fields, the horizontal field F µν and the vertical one, f µν . Whereas F µν is strictly related to the presence of the external electromagnetic field F µν ,vanishing if F µν = 0, the vertical field f µν is geometrical in nature, dependingonly on the deformed metric tensor g DSR , µν ( E ) of GL = f M and on E . There-fore, it is present also in space-time regions where no external electromagneticfield occurs. In our opinion, the arising of the internal electromagnetic fieldsassociated to the deformed metric of f M as Generalized Lagrange space is atthe very physical, dynamic interpretation of the experimental results on theanomalous photon behavior. Namely, the dynamic effects of the hollow waveof the photon, associated to the deformation of space-time — which manifestthemselves in the photon behavior contradicting both classical and quantumelectrodynamics — , arise from the presence of the internal v-electromagneticfield f µν (in turn strictly connected to the geometrical structure of f M ).Moreover, as is well known, in relativistic theories, the vacuum is nothingbut Minkowski geometry. A LLI breaking connected to a deformation of theMinkowski space is therefore associated to a lack of Lorentz invariance of thevacuum. Then, the view by Kostelecky [13] that the breakdown of LLI is relatedto the lack of Lorentz symmetry of the vacuum accords with our results inthe framework of DSR, provided that the quantum vacuum is replaced by thegeometric vacuum. As is well known, successfully embodying gauge fields in a space-time struc-ture is one of the basic goals of the research in theoretical physics startingfrom the beginning of the XX century. The almost unique tool to achieve suchobjective is increasing the number of space-time dimensions. In such a kindof theories (whose prototype is the celebrated Kaluza-Klein formalism), one18reserves the usual (special-relativistic or general-relativistic) structure of thefour-dimensional space-time, and gets rid of the non-observable extra dimen-sions by compactifying them (for example to circles). Then the motions of theextra metric components over the standard Minkowski space satisfy identicalequations to gauge fields. The gauge invariance of these fields is simply a con-sequence of the Lorentz invariance in the enlarged space. In this framework,gauge fields are external to the space-time, because they are added to it by thehypothesis of extra dimensions.In the case of the DSR theory, gauge fields arise from the very geometrical,basic structure of f M , namely they are a consequence of the metric deformation.The arising gauge fields are intrinsic and internal to the deformed space-time , and do not need to be added from the outside. As a matter of fact,
DSR is thefirst theory based on a four-dimensional space-time able to embody gauge fieldsin a natural way.
Such a conventional, intrinsic gauge structure is related to a given deformedMinkowski space f M , in which the deformed metric is fixed: f M = ( M, ¯ g DSRµν ( E )) . (59)On the contrary, with varying g DSR , we have another gauge-like structure — asalready stressed in Sect.3 — namely what we called an external metric gauge.In the latter case, the gauge freedom amounts to choosing the metric accordingto the interaction considered.The circumstance that the deformed Minkowski space f M is endowed withthe geometry of a generalized Lagrange space testifies the richness of non-trivialmathematical properties present in the seemingly so simple structure of thedeformation of the Minkowski metric. In this connection, let us recall that f M (contrarily to the usual Minkowski space) is not flat, but does possess curvatureand torsion (see 3.2.1).Let us stress that — as already mentioned — the deformed Minkowski space f M can be naturally embedded in a five-dimensional Riemannian space ℜ (see[1, 2]). We denoted by DR5 the generalized theory based on this five-dimensionalspace.In embedding the deformed Minkowski space f M in ℜ , energy does lose itscharacter of dynamic parameter (the role it plays in DSR), by taking insteadthat of a true metrical coordinate, E = x , on the same footing of the space-timeones. This has a number of basic implications. In such a change of role of energy,with the consequent passage from f M to ℜ , some of the geometrical and dynamicfeatures of DSR are lost, whereas others are still present and new propertiesappear. The first one is of geometrical nature, and is just the passage from a(flat) pseudoeuclidean metric to a genuine (curved) Riemannian one. The otherconsequences pertain to both symmetries and dynamics. Among the former, werecall the basic one — valid at the slicing level x = const. ( dx = 0) —, relatedto the Generalized Lagrange Space structure of f M , which implies the naturalarising of gauge fields , intimately related to the inner geometry of the deformedMinkowski space. Let us also stress that, in the framework of ℜ , the dependence19f the metric coefficients on a true metric coordinate make them fully analogousto the gauge functions of non-abelian gauge theories, thus implementing DR5as a metric gauge theory (in the sense specified in Subsect.3.1). Let us recallthat the metric homomorphisms of ℜ are strictly connected to the invarianceunder what we called the Metric Gaugement Process of DSR (see Subsect.3.1).Concerning the influence of the extra dimension on the physics in the four-dimensional deformed space-time, points worth investigating are the possibleconnection between Lorentz invariance in DR5 and the usual gauge invariance,and the occurrence of parity violation as consequence of space anisotropy whenviewed from the standpoint of the space-time-energy manifold ℜ .A further basic topic deserving study in DSR is the extension to the non-abelian case of the results obtained for the abelian gauge fields (like the e.m.one), based on the structure of the deformed Minkowski space f M as General-ized Lagrange Space (see Subsubsect.3.2.1). In other words, it would be worthverifying if also non-abelian internal gauge fields can exist in absence of externalfields, due to the intrinsic geometry of f M .20 eferences [1] F. Cardone and R. Mignani: Energy and Geometry – An Introduction toDeformed Special Relativity (World Scientific, Singapore, 2004).[2] F. Cardone and R. Mignani:
Deformed Spacetime – Geometrizing Inter-actions in Four and Five Dimensions (Springer, Heidelberg, Dordrecht,2007); and references therein.[3] F. Cardone, R. Mignani and A. Petrucci: “The principle of solidarity:Geometrizing interactions” , in
Einstein and Hilbert: Dark matter , V.V.Dvoeglazov ed. (Nova Science, Commack, N.Y., 2011), p.19.[4] See e.g.
R. Miron and M. Anastasiei:
The Geometry of Lagrange Spaces:Theory and Applications (Kluwer, 1994); R. Miron, D. Hrimiuc, H. Shi-mada and S.V. Sabau:
The Geometry of Hamilton and Lagrange Spaces (Kluwer, 2002); and references therein.[5] See e.g.
N. Steenrod:
The Topology of Fibre Bundles (Princeton Univ.Press, 1951).[6] R. Miron, A. Jannussis and G. Zet: in
Proc. Conf. Applied DifferentialGeometry - Gen. Rel. and The Workshop on Global Analysis, DifferentialGeometry and Lie Algebra, 2001.,
Gr. Tsagas ed. (Geometry Balkan Press,2004), p.101, and refs. therein.[7] F. Cardone, R. Mignani, W. Perconti and R. Scrimaglio,
Phys. Lett. A ,1 (2004).[8] F. Cardone, R. Mignani, W. Perconti, A. Petrucci and R. Scrimaglio,
Int.J. Mod. Phys. B , 85 (2006).[9] F. Cardone, R. Mignani, W. Perconti, A. Petrucci and R. Scrimaglio, Int.J. Mod. Phys. B , 1107 (2006).[10] F. Cardone, R. Mignani, W. Perconti, A. Petrucci and R. Scrimaglio, An-nales Fond. L. de Broglie , 319, (2008).[11] A. Ranfagni, D. Mugnai and R. Ruggeri, Phys. Rev. E , 027601 (2004).[12] A. Ranfagni and D. Mugnai, Phys. Lett. A , 146 (2004).[13] See e.g. CPT and Lorentz Symmetry I, II, IIIe.g. CPT and Lorentz Symmetry I, II, III