Gauge symmetries in Ashtekar's formulation of general relativity

TL;DR

This paper explicitly constructs the gauge generators in Ashtekar's formulation of canonical gravity, clarifying how four-dimensional diffeomorphism invariance is preserved despite the Hamiltonian framework's apparent time-fixing.

Contribution

It provides an explicit construction of gauge generators in Ashtekar's formulation, demonstrating the preservation of four-dimensional diffeomorphism symmetry.

Findings

Complete set of gauge generators constructed

Symmetry group remains a gauge transformation group

Relation to the problem of time in quantum gravity

Abstract

It might seem that a choice of a time coordinate in Hamiltonian formulations of general relativity breaks the full four-dimensional diffeomorphism covariance of the theory. This is not the case. We construct explicitly the complete set of gauge generators for Ashtekar's formulation of canonical gravity. The requirement of projectability of the Legendre map from configuration-velocity space to phase space renders the symmetry group a gauge transformation group on configuration-velocity variables. Yet there is a sense in which the full four-dimensional diffeomorphism group survives. Symmetry generators serve as Hamiltonians on members of equivalence classes of solutions of Einstein's equations and are thus intimately related to the so-called "problem of time" in an eventual quantum theory of gravity.