On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories

TL;DR

This paper investigates the uniqueness of the Ashtekar-Lewandowski representation in loop quantum gravity, demonstrating its uniqueness under mild assumptions using a new C*-algebra similar to Weyl algebras in quantum field theory.

Contribution

It introduces a new C*-algebra approach and proves the uniqueness of the diffeomorphism invariant representation in quantum gauge theories.

Findings

The Ashtekar-Lewandowski representation is unique under mild assumptions.

A new C*-algebra analogous to Weyl algebra is constructed.

The approach advances understanding of the superselection sectors in quantum gravity.

Abstract

Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Lewandowski representation, has been constructed. This representation is singled out by its mathematical elegance, and up to now, no other diffeomorphism invariant representation has been constructed. This raises the question whether it is unique in a precise sense. In the present article we take steps towards answering this question. Our main result is that upon imposing relatively mild additional assumptions, the AL-representation is indeed unique. As an important tool which is also interesting in its own right, we introduce a C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories.