Direct gauging of the Poincaré group. II

Abstract

This paper continues the study of direct gauge theory of the Poincaré groupP ₁₀. The meanings and implications of transformations induced by the local action ofP ₁₀ are studied, and transformation rules for all field quantities are derived for the local action ofP ₁₀ in a sufficiently small neighborhood of the identity. These results lead directly to a system of fundamental partial differential equations that are both necessary and sufficient for invariance of the “free field” Lagrangian density. Homogeneity arguments and the classical theory of invariants are used to obtain the most general “free field” Lagrangian density. Gauge conditions are shown to imply coordinate conditions, and an algebraic system of antiexact gauge conditions is implemented. The underlying Minkowski space,M ₄, and the resulting Riemann-Cartan space,U ₄, become attached at their “centers,” as do their respective frame and coframe bundles. Weak constraints of vanishing torsion are studied. All field quantities are shown to be determined in terms of the compensating l-forms for the Lorentz sector alone provided an explicit system of integrability conditions is satisfied. Field equations of the Einstein type are shown to result.