A Topos Model for Loop Quantum Gravity

TL;DR

This paper explores how loop quantum gravity can be formulated within a topos-theoretical framework using Bohrification, revealing structural properties like non-sobriety of the state space while maintaining key invariances.

Contribution

It applies the topos approach to LQG via Bohrification, providing a novel perspective on its mathematical structure and invariance properties.

Findings

Proves the non-sobriety of the external state space in Bohrified LQG.

Demonstrates that the construction respects diffeomorphism invariance.

Shows gauge invariance is maintained in the topos-theoretical formulation.

Abstract

One of the main motivations behind so-called topos physics, as developed by Chris Isham and Andreas Doering [4-7], is to provide a framework for new theories of quantum gravity. In this article we do not search for such theories, but ask instead how one of the known candidates for a final theory, loop quantum gravity (LQG), fits into the topos-theoretical approach. In the construction to follow, we apply the 'Bohrification' method developed by Heunen, Landsman and Spitters [10, 11] to the C*-algebra version of LQG introduced by Christian Fleischhack [9]. We then bring together LQG results and methods from topos physics in a proof of the non-sobriety of the external state space S of the Bohrified LQG theory, and show that the construction obeys the standard requirements of diffeomorphism and gauge invariance.