1. Multidimensional Geometrical Representation of Physical Reality

Since the beginning of the 20th century, a line of mathematical research on the multidimensional geometric formulation of physical phenomena has been active. The first to attempt this path was H. Poincaré ^[1], starting from the work of H. Lorentz ^[2], who, with his transformations, had removed the absolute nature of the time variable. These studies are part of the broader research theme related to non-Euclidean geometries that Poincaré conducted up to non-commutative geometries ^[3], consistent with the objective of the complete “geometrization” of the physical world according to the intentions later formulated by A. Einstein ^[4]. This also includes the research theme commonly called applied differential geometry, dedicated to the deepening of the properties characterizing some mathematical structures (manifold variety and more specifically bundles and fibers) and the related methods of representation (in particular in the Lagrangian space, see, e.g., ^[5]) and of transformation; the latter in the context of physics are called gauge calibration, see, e.g., ^[5][6], and are also the subject of group theory for the research of invariants and symmetries, see, e.g., ^[7][8], with reference to A. Einstein’s field equations. Geometrization should be regarded not as a replacement of more consolidated approaches based on forces and corresponding fields, but, in the spirit of Bohr’s Principle of complementarity ^[9], as an auxiliary formalism whose legitimacy may derive only from experimental verification, provided any of the specific predictions that such a representation yields are verified. In contrast, those stemming from other approaches are not verified with comparable extension. An illuminating presentation of the role of theoretical physics in connection with experimental results has been formulated by M. Gell-Mann ^[10].

2. Potentiality of the Generalization of Currently Used Representations

The approach based on the generalization of established representations is widely used and has often allowed an original contribution of mathematics to the advancement of knowledge on physics. As a classical example, we can cite the results obtained by Minkowski ^[11] that made possible the representation of both electron behavior and electrodynamics in general, through a metric formulation, continuing the attempt started by Poincaré ^[12] and abandoned by him.

More recent examples of results obtained by generalization are the theories like string theory ^[13]. In turn, these proposals led to a broad mathematical development whose main result is that the maximum number of dimensions to be used in such physical-mathematical theory is fixed in a deductive way, contrary to all other theoretical models where such number is not deducted but axiomatically established by choice ad libitum.

The adoption of the generalization tool can be considered to be of particular value when it allows an application of the ‘what if’ method in a deductive path, which is both interesting in itself from a logical-mathematical point of view and of potential value at the physics level, if the conjectures resulting from the generalized representation gave rise to forecasts, on a phenomenological ground, such as to be subject to experimental verification or refutation, in the wake of teaching by K. Popper ^[14] and R. Penrose ^[15]. It can be considered implicit in the constraint of experimental verification the selection of variables representing physical concepts in accordance with their definition by P. Bridgman ^[16], who states that in physics, “any concept is nothing more than a set of operations”.

4. Gauge Fields as an Intrinsic Consequence of Geometry

Another key and empowering aspect of the MPF proposal adopted here is to extend the gauge transformation ^{[26][27][32][33]}, commonly used for fields, to include the transformation of metrics. This inclusion can give a contribution to the ongoing investigations in view of the unification of forces, besides through an extension of the standard model to gravitation, also symmetrically intervening to describe strong and weak interactions, as well as the electromagnetic one, through the formalism centered on metrics, typically used for gravitational interactions.

From the modern point of view, we can advocate here that the expression “deformed Minkowski space” (DM), regardless of the number of dimensions, should be, more significantly and perhaps more correctly, denoted as “generalized Minkowski space”, endowed with generalized Lorentz transformations, defining the conditions for the Lorentz invariance, as we discuss in subsequent sections.

Regarding the energy definition problem, we are well aware that energy is not defined locally in general relativity. This is a consequence of the principle of equivalence, not specifically of Einstein’s equations. Correspondingly, and distinctly from Newtonian gravity (where the potential can be deduced from the Schwarzschild metric, when the curvature is set to zero in the exponential form of the latter ^[34][35]), in Einstein’s gravity, such an obstruction in defining the concept of energy persists. The MFP proposal suffers from the same difficulty. However, just as in the case of the experimental tests and validations of Einstein’s gravity and general relativity through the verifications of the phenomena based upon the notions of Riemann’s curvature and torsion tensor, we wish to propose, in the spirit of Popper and Penrose, to put to the “ordeal” of the experimental test, this approach, as well as the corresponding extension in the geometry, to include also the deflection. In this sense, a characteristic signature of the MFP approach ought to emerge in the possible dependence of certain experimental phenomena from the direction in space, i.e., in the appearance of anisotropic and asymmetric features.

Historically, attention was called upon the fact that the weak and null energy conditions are violated in solutions of Einstein’s theory with classical fields as material sources ^[36]. It was shown that the discussion is only meaningful when ambiguities in the definitions of stress–energy tensor and energy density of the matter fields are resolved, with emphasis on the positivity of the energy densities and covariant conservation laws and tracing the root of the ambiguities to the energy localization problem for the gravitational field ^[37]. Analogously, if we consider the response of a gravitational wave detector to scalar waves in connection to the possible choices of conformal frames for scalar–tensor theories, then a correction to the geodesic equation arising in the Einstein conformal frame yields a modification to the geodesic deviation equation, thus yielding a longitudinal mode that is absent in the Jordan conformal frame ^[38]. On the other hand, in the MFP approach, there are no problems of this kind with the energy density because the energy densities for each interaction giving rise to new phenomena descend from invariants of the theory that are experimentally determined threshold energies separating the flat metric from the non-flat one, for each interaction. A possible further development could be the use of the stress–energy tensor as an additional variable instead of scalar energy. Presently the investigation of the possibilities opened by the extension based on the introduction of scalar energy is worthwhile to be pursued.