Representations of the Quantum Holonomy-Diffeomorphism Algebra

TL;DR

This paper develops a framework for quantum gauge theories on curved backgrounds by constructing specific representations of the quantum holonomy-diffeomorphism algebra, enabling non-perturbative analysis of gauge operators.

Contribution

It introduces separable, strongly continuous representations of the quantum holonomy-diffeomorphism algebra, advancing the foundation of Quantum Holonomy Theory for quantum gauge fields.

Findings

Constructed physically relevant operators like the Yang-Mills Hamiltonian.

Established a non-perturbative framework for quantum gauge theories on curved backgrounds.

Provided a method for representing the algebraic structure of quantum gauge theories.

Abstract

In this paper we continue the development of Quantum Holonomy Theory, which is a candidate for a fundamental theory, by constructing separable strongly continuous representations of its algebraic foundation, the quantum holonomy-diffeomorphism algebra. Since the quantum holonomy-diffeomorphism algebra encodes the canonical commutation relations of a gauge theory these representations provide a possible framework for the kinematical sector of a quantum gauge theory. Furthermore, we device a method of constructing physically interesting operators such as the Yang-Mills Hamilton operator. This establishes the existence of a general non-perturbative framework of quantum gauge theories on a curved backgrounds. Questions concerning gauge-invariance are left open.