On a Lattice-Independent Formulation of Quantum Holonomy Theory

TL;DR

This paper develops a lattice-independent formulation of quantum holonomy theory, a promising approach to non-perturbative quantum gravity coupled with fermions, emphasizing background independence and algebraic structures.

Contribution

It introduces a flow-dependent QHD(M) algebra, constructs a kinematical Hilbert space, and demonstrates background independence of key operators.

Findings

Operators for Dirac and gravitational Hamiltonians are background independent.

A flow-dependent algebraic framework is established for quantum holonomy theory.

Necessary conditions for states in the new algebraic setting are formulated.

Abstract

Quantum holonomy theory is a candidate for a non-perturbative theory of quantum gravity coupled to fermions. The theory is based on the QHD(M) algebra, which essentially encodes how matter degrees of freedom are moved on a three-dimensional manifold. In this paper we commence the development of a lattice-independent formulation. We first introduce a flow-dependent version of the QHD(M) algebra and formulate necessary conditions for a state to exist hereon. We then use the GNS construction to build a kinematical Hilbert space. Finally we find that operators, that correspond to the Dirac and gravitational Hamiltonians in a semi-classical limit, are background independent.