Unified Field Theoretical Models from Generalized Affine Geometries II

Abstract

The space-time structure of the new Unified Field Theory presented in previous reference (Int. J. Theor. Phys. 49:1288-1301, 2010 is analyzed from its SL(2C) underlying structure in order to make precise the notion of minimal coupling. To this end, the framework is the language of tensors and particularly differential forms and the condition a priory of the existence of a potential for the torsion is relaxed. We shown trough exact cosmological solutions from this model, where the geometry is Euclidean (2), the relation between the space-time geometry and the structure of the gauge group. Precisely this relation is directly connected with the relation of the spin and torsion fields. The solution of this model is explicitly compared with our previous ones and we find that: (i) the torsion is not identified directly with the Yang Mills type strength field, (ii) there exists a compatibility condition connected with the identification of the gauge group with the geometric structure of the space-time: this fact lead the identification between derivatives of the scale factor a(τ ) with the components of the torsion in order to allows the Hosoya-Ogura ansatz (namely, the alignment of the isospin with the frame geometry of the space-time), (iii) this compatibility condition precisely mark the fact that local gauge covariance, coordinate independence and arbitrary space time geometries are harmonious concepts and (iv) of two possible structures of the torsion the "tratorial" form (the only one studied here) forbids wormhole configurations, leading only, cosmological instanton space-time in eternal expansion.

Unified Field Theoretical Models from Generalized Affine Geometries III

“…As 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.…”

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…”

“…µC (see [19], [20] for details) to the SO(2, 2) group. The absolute value of the determinant in (1) is assumed.…”