Novel Analysis of Spinor Interactions and non-Riemannian Geometry

Abstract

A novel analysis of the gauge theory of the local Lorentz group is implemented both in flat and in curved space-time, and the resulting dynamics is analyzed in view of the geometrical interpretation of the gauge potential. The Yang-Mills picture of local Lorentz transformations is first approached in a second-order formalism. For the Lagrangian approach to reproduce the second Cartan structure equation as soon as the Lorentz gauge connections are identified with the contortion tensor, an interaction term between the Lorentz gauge fields and the spin connections has to be postulated. The full picture involving gravity, torsion and spinors is described by a coupled set of field equations, which allows one to interpret both gravitational spin connections and matter spin density as the source term for the Yang-Mills equations. The contortion tensor acquires a propagating character, because of its non-Abelian feature, and the pure contact interaction is restored in the limit of vanishing Lorentz connections.