Poincaré gauge theory, gravitation, and transformation of matter fields

Abstract

AbstractIn the generalized Kaluza‐Klein theories a (4 + N)‐dimensional Riemannian space is used as a configuration space to unify gravitation with a nonabelien gauge symmetry characterized by a gauge group G of order N. The original metric space V4+N, however, is broken down in these theories to a local product structure M4 × VN (M4 being Minkowski space‐time) leading thereby to a description in terms of a fiber bundle over space‐time V4 with structural group G. In this paper we base the discussion from the very beginning on a bundle structure as the underlying geometric stratum and treat the quantum mechanical aspect of matter by representing it in the form of a generalized wave function, ϕ(x, X), defined as a section on a fiber bundle associated to a principal G‐bundle over space time having a homogeneous space of the group G as fiber. To make contact with gravitation we consider various soldered bundles possessing the Poincaré group, G = ISO(3, 1), as structural or gauge group and define Poincaré gauge fields as generalized matter fields on them. After reviewing the formulation of general relativity as a gauge theory of the Lorentz group we turn to the affine case and present a geometric formulation of a Poincaré gauge theory based on a Riemann‐Cartan space‐time U4 with axial vector torsion. Two sets of field equations are discussed relating the material source quantities to the geometric fields. Besides Einstein's equations coupling the metric to the classical distribution of energy and momentum of matter a second set of nonlinear field equations is set up relating a bilinear current in the Poincaré gauge fields to the axial vector torsion field of the underlying space‐time geometry. Finally, a breaking of the Poincaré symmetry is introduced in a manner involving the generalized matter field ϕ(x, X) freezing thereby the translational gauge degrees of freedom, however, leaving the Lorentz gauge symmetry unaffected.