Polymer representations and geometric quantization

TL;DR

This paper explores the connection between polymer representations of the Weyl algebra, used in loop quantum gravity, and geometric quantization, focusing on a two-dimensional phase space to understand their relationship.

Contribution

It provides a geometric quantization perspective on polymer representations, which are typically constructed algebraically, specifically for two-dimensional phase spaces.

Findings

Polymer representations can be understood via geometric quantization.

The approach offers new insights into the structure of loop quantum gravity.

A framework for extending to higher dimensions is suggested.

Abstract

Polymer representations of the Weyl algebra of linear systems provide the simplest analogues of the representation used in loop quantum gravity. The construction of these representations is algebraic, based on the Gelfand-Naimark-Segal construction. Is it possible to understand these representations from a Geometric Quantization point of view? We address this question for the case of a two dimensional phase space.