Exploring the diffeomorphism invariant Hilbert space of a scalar field

TL;DR

This paper investigates the structure of the diffeomorphism invariant Hilbert space for a scalar field, providing explicit formulas and addressing quantization and ordering issues relevant to loop quantum gravity.

Contribution

It offers an explicit scalar product formula for the diffeomorphism invariant space and analyzes the quantization of invariant quantities, highlighting similarities with loop quantum gravity.

Findings

Explicit scalar product formula derived

Analysis of ordering ambiguities in quantization

Insights into the structure of diffeomorphism invariant states

Abstract

As a toy model for the implementation of the diffeomorphism constraint, the interpretation of the resulting states, and the treatment of ordering ambiguities in loop quantum gravity, we consider the Hilbert space of spatially diffeomorphism invariant states for a scalar field. We give a very explicit formula for the scalar product on this space, and discuss its structure. Then we turn to the quantization of a certain class of diffeomorphism invariant quantities on that space, and discuss in detail the ordering issues involved. On a technical level these issues bear some similarity to those encountered in full loop quantum gravity.