Background independent quantizations: the scalar field I

TL;DR

This paper develops a background-independent quantization method for scalar fields inspired by Loop Quantum Gravity, introducing a new algebraic framework and classifying the resulting quantum states and representations.

Contribution

It introduces a novel polymer *-star algebra for scalar fields and classifies the associated states and representations under diffeomorphism invariance.

Findings

Complete class of states determined by homeomorphism invariance

Identification of GNS representations and irreducible classes

Characterization of states leading to self-adjoint momentum operators

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. The assumed in our paper homeomorphism invariance allows to determine a complete class of the states. Except one, all of them are new. In this letter we outline the main steps and conclusions, and present the results: the GNS representations, characterization of those states which lead to essentially self adjoint momentum operators (unbounded), identification of the equivalence classes of the representations as well as of the irreducible ones. The algebra and topology of the problem, the derivation, all the technical details and more are contained in the paper-part II.