Gauge group and reality conditions in Ashtekar's complex formulation of canonical gravity

TL;DR

This paper analyzes the reality conditions and gauge transformations in Ashtekar's complex formulation of general relativity, providing a new stabilization framework and linking diffeomorphisms to reality conditions.

Contribution

It introduces a novel stabilization algorithm for reality conditions and establishes the full gauge structure, including time diffeomorphisms, within the canonical formalism.

Findings

Developed a general stabilization algorithm for reality conditions.

Linked spacetime diffeomorphisms to reality conditions.

Constructed the complete set of canonical gauge generators.

Abstract

We discuss reality conditions and the relation between spacetime diffeomorphisms and gauge transformations in Ashtekar's complex formulation of general relativity. We produce a general theoretical framework for the stabilization algorithm for the reality conditions, which is different from Dirac's method of stabilization of constraints. We solve the problem of the projectability of the diffeomorphism transformations from configuration-velocity space to phase space, linking them to the reality conditions. We construct the complete set of canonical generators of the gauge group in the phase space which includes all the gauge variables. This result proves that the canonical formalism has all the gauge structure of the Lagrangian theory, including the time diffeomorphisms.