Noncanonical quantization of gravity. II. Constraints and the physical Hilbert space

TL;DR

This paper advances a noncanonical approach to quantum gravity using affine variables, focusing on constraints and the structure of the physical Hilbert space, with a detailed phase space integral representation.

Contribution

It develops a phase space functional integral framework for quantizing gravity with affine variables, addressing constraints and the structure of the physical Hilbert space.

Findings

Constraints significantly alter operator representations

Ultralocal initial representations are replaced after imposing constraints

A well-defined phase space integral representation for the physical Hilbert space is constructed

Abstract

The program of quantizing the gravitational field with the help of affine field variables is continued. For completeness, a review of the selection criteria that singles out the affine fields, the alternative treatment of constraints, and the choice of the initial (before imposition of the constraints) ultralocal representation of the field operators is initially presented. As analogous examples demonstrate, the introduction and enforcement of the gravitational constraints will cause sufficient changes in the operator representations so that all vestiges of the initial ultralocal field operator representation disappear. To achieve this introduction and enforcement of the constraints, a well characterized phase space functional integral representation for the reproducing kernel of a suitably regularized physical Hilbert space is developed and extensively analyzed.