The geometrical structure of the physical world-both at a large and a small scale—has been debated since a long. After Einstein, the generally accepted view 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 phenomena ruled by different fundamental interactions have provided evidence for a local departure from Minkowski metric [3] [4] : among them, the lifetime of the (weakly decaying) meson, the Bose-Einstein correlation in (strong) pion production and the superluminal propagation of electromagnetic waves in waveguides. These phenomena seemingly show a (local) breakdown of Lorentz invariance, together with a plausible inadequacy of the Minkowski metric; on the other hand, they can be interpreted in terms of a deformed Minkowski spacetime, with metric coefficients depending on the energy of the process considered [3] [4] .

All the above facts suggested to introduce a (four-dimensional) generalization of the (local) space-time structure based on an energy-dependent “deformation” of the usual Minkowski geometry of M, whereby the corresponding deformed metrics ensuing from the fit to the experimental data seem to provide an effective dynamical description of the relevant interactions (at the energy scale and in the energy range considered).

An analogous energy-dependent metric seems to hold for the gravitational field (at least locally, i.e. in a neighborhood of Earth) when analyzing some classical experimental data concerning the slowing down of clocks.

Let us shortly review the main ideas and results concerning the (four-dimensional) deformed Minkowski spacetime.

The four-dimensional “ deformed” metric scheme is based on the assumption that spacetime, in a preferred frame which is fixed by the scale of energy , is endowed with a metric of the form

with being the usual speed of light in vacuum. We named “Deformed Special Relativity” (DSR) the relativity theory built up on metric (1), (2).

Metric (1), (2) is supposed to hold locally, i.e. in the spacetime region where the process occurs. It is supposed moreover to play a dynamical role, and to provide a geometric description of the interaction considered. In this sense, DSR realizes the so called “Finzi Principle of Solidarity” between space-time and phenomena occurring in it¹ (see [5] ). Futhermore, we stress that, from the physical point of view, is the measured energy of the system, and thus a merely phenomenological (non-metric) variable².

We notice explicitly that the spacetime described by (1), (2) is flat (it has zero four-dimensional curvature, at least at this level; but see below), so that the geometrical description of the fundamental interactions based on it differs from the general relativistic one (whence the name “deformation” used to characterize such a situation). Although for each interaction the corresponding metric reduces to the Minkowskian one for a suitable value of the energy (which is characteristic of the interaction considered), 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 usual Minkowski space by a suitable change of coordinates (e.g. by a rescaling), this would amount, in such a framework, to be a mere mathematical operation devoid of any physical meaning.

As far as phenomenology is concerned, it is important to recall that a local breakdown of Lorentz invariance may be envisaged for all the four fundamental interactions (electromagnetic, weak, strong and gravitational) whereby one gets evidence for a departure of the spacetime metric from the Minkowskian one (in the energy range examined). The explicit functional form of the metric (2) for all the four interactions can be found in [3] [4] . Here, we confine ourselves to recall the following basic features of these energy-dependent phenomenological metrics:

1) Both the electromagnetic and the weak metric show the same functional behavior, namely

with the only difference between them being the threshold energy, i.e. the energy value at which the metric parameters are constant, i.e. the metric becomes Minkowskian; the fits to the experimental data yield

2) for strong and gravitational interactions, the metrics read:

with

Let us stress that, in this case, contrarily to the electromagnetic and the weak ones, a deformation of the time coordinate occurs; moreover, the three-space is anisotropic³, with two spatial parameters constant (but different in value) and the third one variable with energy in an “over-Minkowskian” way (namely it reaches the limit of Minkowskian metric for decreasing values of, with) [3] [4] .

As a final remark, we stress that actually the four-dimensional energy-dependent spacetime is just a manifestation of a larger, five-dimensional space in which energy plays the role of a fifth dimension. Indeed, it can be shown that the physics of the interaction lies in the curvature of such a five-dimensional spacetime, in which the four-dimensional, deformed Minkowski space is embedded. Moreover, all the phenomenological metrics (2), (3) and (5), (6) can be obtained as solutions of the vacuum Einstein equations in this generalized Kaluza-Klein scheme [3] [4] .

Let us remark the mathematically self-evident, but physically basic, point that the generalized metric (2) (and the corresponding interval (1)) is clearly not preserved by the usual Lorentz transformations. If is the matrix representing a standard Lorentz transformation, this amounts to say that the similarity transformation generated by does not leave the deformed metric tensor invariant:

(where denotes transpose) namely, standard Lorentz invariance is violated.

However, in DSR it is possible to introduce generalized Lorentz transformations which are the isometries of the deformed Minkowski space [3] [4] . They are also referred to as deformed Lorentz transformations (DLT). If denotes a column four-vector, a DLT is therefore a matrix connecting two inertial frames,

and leaving the deformed interval (1) invariant, namely

Therefore, unlike the case of a standard LT, a deformed Lorentz transformation generates a similarity transformation which preserves the deformed metric tensor. Let us also notice the explicit dependence of on the energy. This means that in DSR Lorentz invariance is recovered, although in a generalized sense.

The explicit form of the deformed Lorentz transformations can be found in [3] [4] .

It is clear from the discussion of the phenomenological metrics describing the four fundamental interactions in DSR that the Minkowski space 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 particular, we can consider this Minkowski space as that associated to the electromagnetic interaction above the threshold energy. Therefore, in modeling the physical phenomena, one has to take into account this fact. The geometrical nature of interactions, i.e. assuming the validity of the Finzi principle, means that one has to suitably gauge (with reference to M) the space-time metrics with respect to the interaction-and/or the phenomenon-under study. In other words, one needs to “adjust” suitably the local metric of space-time according to the interaction acting in the region considered. We can name such a procedure “Metric Gaugement Process” (M.G.P.). Like in usual gauge theories a different phase is chosen in different space-time points, in DSR different metrics are associated to different space-time manifolds according to the interaction acting therein. We have thus a gauge structure on the space of manifolds

where is the set of the energy-dependent pseudoeuclidean metrics of the type (2). This is why it is possible to regard Deformed Special Relativity as a Metric Gauge Theory [6] . In this case, we can consider the related fields as external metric gauge fields.

However, let us notice that DSR can be considered as a metric gauge theory from another point of view, on account of the dependence of the metric coefficients on the energy. Actually, once the MGP has been applied, by selecting the suitable gauge (namely, the suitable functional form of the metric) according to the interaction considered (thus implementing the Finzi principle), the metric 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 given phenomenon.

We have therefore a double metric gaugement, according, on one side, to the interaction ruling the physical phenomenon examined, and on the other side to its energy, in which the metric coefficients are the analogous of the gauge functions⁴.

Let us give the definition of generalized Lagrange space [8] [9] , since usually one is not acquainted with it.

Consider a N-dimensional, differentiable manifold and its (N-dimensional) tangent space in a point,. As is well known, the union

has a fibre bundle structure. Let us denote by the generic element of, namely a vector tangent to in. Then, an element is a vector tangent to the manifold in some point. Local coordinates for are introduced by considering a local coordinate system on and the components of in such a coordinate system. The numbers constitute a local coordinate system on. We can write synthetically. is a -dimensional, differentiable manifold.

Let be the mapping (natural projection). (,). Then, the tern is the tangent bundle to the base manifold. The image of the inverse mapping is of course the tangent space, which is called the fiber corresponding to the point in the fiber bundle One considers also sometimes the manifold, where 0 is the zero section of the projection. We do not dwell further on the theory of the fiber bundles, and refer the reader to the wide and excellent literature on the subject [10] .

The natural basis of the tangent space at a point is,

.

A local coordinate transformation in the differentiable manifold reads

Here, is the Liouville vector field on, i.e..

On account of Equation (14), the natural basis of can be written as

Second Equation (15) shows therefore that the vector basis, , generates a distribution

defined everywhere on and integrable, too (vertical distribution on).

If is a distribution on supplementary to, namely

then is called a horizontal distribution, or a nonlinear connection on. A basis for the distributions

and are given respectively by and, where the basis in explicitly reads

Here, are the coefficients of the nonlinear connection. The basis is called the adapted basis.

The dual basis to the adapted basis is, with

A distinguished tensor (or d-tensor) field of (r,s)-type is a quantity whose components transform like a tensor under the first coordinate transformation (19) on (namely they change as tensor in). For instance, for a d-tensor of type (1,2):

In particular, both and are d-(covariant) vectors, whereas, are d-(contravariant) vectors.

A generalized Lagrange space is a pair, with being a d-tensor of type (0,2) (covariant) on the manifold, which is symmetric, non-degenerate⁵ and of constant signature.

A function

differentiable on and continuous on the null section of is named a regular Lagrangian if the Hessian of with respect to the variables is non-singular.

A generalized Lagrange space is reducible to a Lagrange space if there is a regular Lagrangian satisfying

on. In order that is reducible to a Lagrange space, a necessary condition is the total symmetry of the d-tensor. If such a condition is satisfied, and are 0-homogeneous in the variables, then the function is a solution of system (21). In this case, the pair is a Finsler space⁶, with. One says that is reducible to a Finsler space.

Of course, reduces to a pseudo-Riemannian (or Riemannian) space if the d-tensor does not depend on. On the contrary, if depends only on (at least in preferred charts), it is a generalized Lagrange space which is locally Minkowskian.

Since, in general, a generalized Lagrange space is not reducible to a Lagrange one, it cannot be studied by means of the methods of symplectic geometry, on which—as is well known—analytical mechanics is based.

A linear -connection on (or on) is defined by a couple of geometrical objects on with different transformation properties under the coordinate transformation (14). Precisely, transform like the coefficients of a linear connection on, whereas transform like a d-tensor of type (1,2). is called the metrical canonical -connection of the generalized Lagrange space.

In terms of and one can define two kinds of covariant derivatives: a covariant horizontal (h-) derivative, denoted by “” , and a covariant vertical (v-) derivative, denoted by “”. For instance, for the d-tensor one has

The two derivatives and are both d-tensors of type (0,3).

The coefficients of can be expressed in terms of the following generalized Christoffel symbols:

By means of the connection it is possible to define a d-curvature in by means of the tensors, and given by

Here, the d-tensor is related to the bracket of the basis:

and is explicitly given by⁷

The tensor, together with, , defined by

are the d-tensors of torsion of the metrical connection.

From the curvature tensors one can get the corresponding Ricci tensors of:

and the scalar curvatures

Finally, the deflection d-tensors associated to the connection are

namely the hand v-covariant derivatives of the Liouville vector fields.

In the generalized Lagrange space it is possible to write the Einstein equations with respect to the canonical connection as follows:

where is a constant and, , , are the components of the energy-momentum tensor.

On the basis of the previous considerations, let us analyze the geometrical structure of the deformed Minkowski space of DSR, endowed with the by now familiar metric As said in Section 2, is the energy of the process measured by the detectors in Minkowskian conditions. Therefore, is a function of the velocity components, , where is the (Minkowskian) proper time:⁸

The derivatives define a contravariant vector tangent to at, namely they belong to. We shall denote this vector (according to the notation of the previous Subsubsection) by. Then, is a point of the tangent bundle to. We can therefore consider the generalized Lagrange space, with

Then, it is possible to prove the following theorem [7] :

The pair is a generalized Lagrange space which is not reducible to a Riemann space, or to a Finsler space, or to a Lagrange space.

Notice that such a result is strictly related to the fact that the deformed metric tensor of DSR is diagonal.

If an external electromagnetic field is present in the Minkowski space, in the deformed electromagnetic field is given by. Such a field is a d-tensor and is called the electromagnetic tensor of the generalized Lagrange space. Then, the nonlinear connection is given by

where, the Christoffel symbols of the Minkowski metric, are zero, so that

namely, the connection coincides with the deformed field.

The adapted basis of the distribution reads therefore

The local covector field of the dual basis (cfr. Equation (18)) is given by

The derivation operators applied to the deformed metric tensor of the space yield

Then, the coefficients of the canonical metric connection in (see Equation (23 )) are given by

The vanishing of the electromagnetic field tensor, , implies.

One can define the deflection tensors associated to the metric connection as follows (cfr. Equation (30)):

The covariant components of these tensors read

It is important to stress explicitly that, on the basis of the results of 3.2.1, the deformed Minkowski space does possess curvature and torsion, namely it is endowed with a very rich geometrical structure. This permits to understand the variety of new physical phenomena that occur in it (as compared to the standard Minkowski space) [3] [4] . Notice that this result follows by the fact that, in deforming the metric of the space-time, we assumed the energy as the physical (non-metric) observable on which letting the metric coefficients depend. This is crucial in stating the generalized Lagrangian structure of, as shown above.

Following ref. [7] , let us show how the formalism of the generalized Lagrange space allows one to recover some results on the phenomenological energy-dependent metrics discussed in Section 2.

Consider the following metric:

where is an arbitrary function of the energy and spatial isotropy has been assumed. In absence of an external electromagnetic field, the non-vanishing components of the canonical metric connection (see Equation (40)) are

where the prime denotes derivative with respect to

According to the formalism of generalized Lagrange spaces, we can write the Einstein equations in vacuum corresponding to the metrical connection of the deformed Minkowski space (see Equations (31)). It is easy to see that the independent equations are given by

The first equation has the solution, namely we get the Minkowski metric. Equation (46) has the solution

where and are two integration constants.

This solution represents the time coefficient of an over-Minkowskian metric. For it coincides with (the time coefficient of) the phenomenological metric of the strong interaction, Equation (7). On the other hand, by choosing, one gets the time coefficient of the metric for gravitational interaction, Equation (7’).

In other words, considering as a generalized Lagrange space permits to recover (at least partially) the metrics of two interactions (strong and gravitational) derived on a phenomenological basis.

It is also worth noticing that this result shows that a spacetime deformation (of over-Minkowskian type) exists even in absence of an external electromagnetic field (remember that Equations (45) and (46) have been derived by assuming).

As we have seen, the deformed Minkowski space, considered as a generalized Lagrange space, is endowed with a rich geometrical structure. But the important point, to our purposes, is the presence of a physical richness, intrinsic to. Indeed, let us introduce the following internal electromagnetic field tensors on, defined in terms of the deflection tensors:

(horizontal electromagnetic internal tensor) and

(vertical electromagnetic internal tensor).

The internal electromagnetic hand v-fields and satisfy the following generalized Maxwell equations

Let us stress explicitly the different nature of the two internal electromagnetic fields. In fact, the horizontal field is strictly related to the presence of the external electromagnetic field, and vanishes if. On the contrary, the vertical field has a geometrical origin, and depends only on the deformed metric tensor of and on. Therefore, it is present also in space-time regions where no external electromagnetic field occurs. 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 hand v-fields and, intrinsic to the geometrical structure of as a generalized Lagrange space, is the cornerstone to build up a dynamics (of merely geometrical origin) internal to the deformed Minkowski space.

The important point worth emphasizing is that such an intrinsic dynamics springs from gauge fields. Indeed, the two internal fields and (in particular the latter one) do satisfy equations of the gauge type (cfr. Equations (51) and (52)). Then, we can conclude that the (energy-dependent) deformation of the metric of, which induces its geometrical structure as generalized Lagrange space, leads in turn to the appearance of (internal) gauge fields [6] .

Such a fundamental result can be schematized as follows:

(with self-explanatory meaning of the notation).