On the Differential Geometry of Non-localized Field Theory: Poincaré Gravity

Abstract

In this paper, we study the differential structure of a spinor bundle in spaces where the metric tensor g _μν (x, ξ, $\overline \xi$ ) of the base manifold depends on the position variables x as well as on the spinor variable ξ and $\overline \xi$. Notions such as: gauge covariant derivatives of tensors, connections, curvatures, torsions and Bianchi identities are presented in the context of a modified gauge approach than the one proposed in [11], [13], due to introduction of a Poincaré group and the use of d — connections [6], [8] in the spinor bundle S ⁽²⁾ M. The introduction of basic 1 — form fields ρ _μ and spinors ζ _a , $\overline {{\zeta ^a}}$ with values in the Lie algebra of the Poincaré group is also essential in our study. The gauge field equations are derived by a Lagrangian density in an analogous to the Palatini method but a different situation than that developed by the authors [12].