Irreducibility of the Ashtekar-Isham-Lewandowski representation

TL;DR

This paper proves that the Ashtekar-Isham-Lewandowski representation in loop quantum gravity is irreducible when viewed as a representation of a specific C*-algebra similar to the Weyl algebra, reinforcing its fundamental uniqueness.

Contribution

It demonstrates the irreducibility of the Ashtekar-Isham-Lewandowski representation as a C*-algebra representation, advancing the understanding of its mathematical structure.

Findings

The AIL-representation is irreducible as a C*-algebra representation.

Supports the uniqueness and fundamental role of the AIL-representation in loop quantum gravity.

Links the AIL-representation to the Weyl algebra framework used in quantum field theory.

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-Isham-Lewandowski representation, has been constructed. Recently, several uniqueness results for this representation have been worked out. In the present article, we contribute to these efforts by showing that the AIL-representation is irreducible, provided it is viewed as the representation of a certain C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories.