Complete quantization of a diffeomorphism invariant field theory

TL;DR

This paper introduces a simplified, yet comprehensive, model of a diffeomorphism invariant field theory based on Ashtekar variables, enabling explicit canonical quantization and analysis of its observables.

Contribution

It presents a reduced model for complexified Ashtekar gravity that allows complete canonical quantization and explicit expression of Dirac observables, advancing the understanding of quantum gravity.

Findings

Infinite-dimensional reduced phase space identified

Explicit form of Dirac observables derived

Scalar constraint is bilinear in momenta

Abstract

In order to test the canonical quantization programme for general relativity we introduce a reduced model for a real sector of complexified Ashtekar gravity which captures important properties of the full theory. While it does not correspond to a subset of Einstein's gravity it has the advantage that the programme of canonical quantization can be carried out completely and explicitly, both, via the reduced phase space approach or along the lines of the algebraic quantization programme. This model stands in close correspondence to the frequently treated cylindrically symmetric waves. In contrast to other models that have been looked at up to now in terms of the new variables the reduced phase space is infinite dimensional while the scalar constraint is genuinely bilinear in the momenta. The infinite number of Dirac observables can be expressed in compact and explicit form in terms of the original phase space variables. They turn out, as expected, to be non-local and form naturally a set of countable cardinality.