Unified Field Theoretical Models from Generalized Affine Geometries III

Abstract

In this work the underlying structure of new type of Unified Field Theoretical model introduced in by the authors is elucidated and analyzed from the geometrical and group theoretical point of view. Our goal is to take advantage of the geometrical and topological properties of this theory in order to determine the minimal group structure of the resultant spacetime manifold able to support a fermionic structure. From this fact, the relation between antisymmetric torsion and Dirac structure of the spacetime is determined and important physical consequences enumerated. In the case of spacetime with torsion the real meaning of the spin-frame alignment is find and the question of the minimal coupling is discussed based in the important cases of tratorial, totally antisymmetric and general torsion fields. Motivation and Summary of the ResultsAs was pointed out by us in later works [1,2], the cornerstone of the problem of the Unification is where to start conceptually to reformulate the theoretical arena where the Fundamental Theory will be placed, and where the geometry is the unifying essence. According to Mach spacetime doesn't exists without matter. Then, two basic ideas immediately arise to fulfill the observation given by Mach: the concept of dualistic or non-dualistic theories. In the first one the simplest and economical description can be formulated in terms of the gravitational field without torsion plus the energy momentum tensor that, however, is added "by hand" in order to cover the lack of knowledgement of a fundamental structure of the space

Non-Riemannian geometry, Born–Infeld models and trace free gravitational equations

“…Similar thing happens in the non-Riemannian case, as pointed out in [12] [13] [14][15] [16], where the corresponding affine geometrical structure induces naturally the following constraint: K g = constant. This natural constraint impose a condition (ratio) between both basic tensors through their determinants: the metric determinant g and the fundamental one K (in the sense of a nonsymmetric theory that contains the antisymmetric structures),…”

Fermion Interactions, Cosmological Constant and Space-Time Dimensionality in a Unified Approach Based on Affine Geometry

“…As it was shown in [19][20][21][22][23][24] for this model of gravity (see Ref. [25], for astrophysical neutrino applications), the Dirac equation is derived from the same space-time manifold and acquires a coupling modification of the form…”

“…As pointed out in references [19][20][21][22][23][24], the torsion vector h = h α dx α (the 4-dimensional dual of the torsion field T βγδ ) plays multiple roles and can be constrained in several different physical situations. Mathematically, it is defined by the Hodge-de Rham decomposition given by the 4-dimensional Helmholtz theorem which states:…”

“…It is possible to show [19][20][21][22][23][24] that to derive the Dirac equation, one needs, as starting point, the symmetric part of the gravitational field equations derived from δ g L g = 0, that is…”

Dark matter candidates, helicity effects and new affine gravity with torsion

“…Notice that f a µ , in a sharp contrast with the tetrad field e a µ , carries the symmetry e aµ f a ν = f µν = −f νµ . See [5,6] for more mathematical and geometrical details of the theory. Consequently, the generalized Ricci tensor splits into a symmetric and antisymmetric part, namely:…”