Representations of the Weyl Algebra in Quantum Geometry

TL;DR

This paper proves that, under certain conditions, all regular, cyclic, and diffeomorphism-invariant representations of the Weyl algebra in quantum geometry are equivalent to the fundamental one, highlighting its uniqueness.

Contribution

It establishes the uniqueness of the fundamental representation of the Weyl algebra in quantum geometry under broad assumptions, without domain restrictions or diffeomorphism pull-back requirements.

Findings

All such representations are unitarily equivalent to the fundamental representation.

The paper provides a direct proof of the irreducibility of the Weyl algebra.

It analyzes the behavior of C*-algebras generated by continuous functions and homeomorphisms.

Abstract

The Weyl algebra A of continuous functions and exponentiated fluxes, introduced by Ashtekar, Lewandowski and others, in quantum geometry is studied. It is shown that, in the piecewise analytic category, every regular representation of A having a cyclic and diffeomorphism invariant vector, is already unitarily equivalent to the fundamental representation. Additional assumptions concern the dimension of the underlying analytic manifold (at least three), the finite wide triangulizability of surfaces in it to be used for the fluxes and the naturality of the action of diffeomorphisms -- but neither any domain properties of the represented Weyl operators nor the requirement that the diffeomorphisms act by pull-backs. For this, the general behaviour of C*-algebras generated by continuous functions and pull-backs of homeomorphisms, as well as the properties of stratified analytic diffeomorphisms are studied. Additionally, the paper includes also a short and direct proof of the irreducibility of A.