Representations of the Weyl Algebra in Quantum Geometry

TL;DR

The work proves a Stone–von Neumann-type uniqueness for the Weyl algebra of quantum geometry by showing that any regular representation with a cyclic diffeomorphism-invariant vector is unitarily equivalent to the fundamental representation, under mild hypotheses such as dim M ≥ 3 and stratified-analytic diffeomorphisms. It achieves this by employing exponentiated fluxes (Weyl operators) and a stratified-diffeomorphism framework to avoid domain issues and to extend locality beyond analyticity. A detailed quantum-geometric background is developed, including the Ashtekar–Lewandowski measure μ0, the decomposition of paths, quasi-surfaces and quasi-flux actions, and the construction of Weyl operators, followed by proofs of irreducibility and a thorough treatment of stratified diffeomorphisms. The results have implications for the foundations of loop quantum gravity by clarifying representation-uniqueness under diffeomorphism symmetry and providing a robust, geometry-informed operator framework that generalizes previous holonomy-flux approaches. These insights enhance the mathematical coherence and physical plausibility of quantum geometry models by tightly constraining the allowable representations of the Weyl algebra.

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.

Figures (1)

• Figure 1: Stratified Diffeomorphism in Lemma \ref{['lem:winddiffeo-bumps']}