Modular Theory, Non-Commutative Geometry and Quantum Gravity

TL;DR

This paper explores a novel approach to quantum gravity using modular theory and non-commutative geometry, aiming to reconstruct spectral geometries from an operational formalism of states and observables.

Contribution

It provides the first detailed exposition of a framework combining modular theory and non-commutative geometry for quantum gravity reconstruction.

Findings

Develops a covariant formalism for spectral geometry reconstruction.

Bridges modular theory with non-commutative geometry in quantum gravity.

Addresses foundational issues in the proposed approach.

Abstract

This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.