Background independent quantizations: the scalar field II

TL;DR

This paper develops a background-independent quantization method for scalar fields inspired by Loop Quantum Gravity, focusing on polymer variables, algebraic structures, and invariant states, with detailed mathematical analysis and representation theory.

Contribution

It introduces a new polymer *-star algebra framework for scalar field quantization and characterizes all homeomorphism invariant states within this setting.

Findings

Derived the complete class of homeomorphism invariant states

Constructed GNS representations and analyzed their properties

Characterized invariant subspaces and self-adjoint extensions

Abstract

We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the `quantum' polymer *-star algebra and looking for positive linear functionals, called states. Assumed in our paper homeomorphism invariance allows to derive the complete class of the states. They are determined by the homeomorphism invariant states defined on the CW-complex *-algebra. The corresponding GNS representations of the polymer *-algebra and their self-adjoint extensions are derived, the equivalence classes are found and invariant subspaces characterized. In the preceding letter (the part I) we outlined those results. Here, we present the technical details.