A completely solvable model with an infinite number of Dirac observables for a real sector of complexified Ashtekar gravity

TL;DR

This paper introduces a solvable model of a real sector of complexified Ashtekar gravity with an infinite set of Dirac observables, enabling complete canonical quantization.

Contribution

It presents a new reduced model with an infinite-dimensional phase space and explicit Dirac observables, distinct from previous models, facilitating full quantization.

Findings

Model is completely solvable with explicit Dirac observables.

Infinite-dimensional reduced phase space with bilinear scalar constraint.

Quantization can be achieved via reduced phase space or algebraic methods.

Abstract

We introduce a reduced model for a real sector of complexified Ashtekar gravity that does not correspond to a subset of Einstein's gravity but for which the programme of canonical quantization can be carried out completely, both, via the reduced phase space approach or along the lines of the algebraic quantization programme.\\ This model stands in a certain 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.