Quantum Holonomy Theory and Hilbert Space Representations

TL;DR

This paper introduces a new formulation of quantum holonomy theory for quantum gravity, utilizing a Hilbert space representation of the QHD(M) algebra to achieve a background independent, non-perturbative framework.

Contribution

It proposes a novel Hilbert space representation of the QHD(M) algebra, advancing the mathematical foundation of quantum holonomy theory for quantum gravity.

Findings

New formulation of quantum holonomy theory presented

Hilbert space representation of QHD(M) algebra constructed

Existence of the generating state proved in future work

Abstract

We present a new formulation of quantum holonomy theory, which is a candidate for a non-perturbative and background independent theory of quantum gravity coupled to matter and gauge degrees of freedom. The new formulation is based on a Hilbert space representation of the QHD(M) algebra, which is generated by holonomy-diffeomorphisms on a 3-dimensional manifold and by canonical translation operators on the underlying configuration space over which the holonomy-diffeomorphisms form a non-commutative C*-algebra. A proof that the state that generates the representation exist is left for later publications.