Connes duality in Lorentzian geometry

TL;DR

This paper extends Connes' duality formula from Riemannian to Lorentzian geometry, highlighting the importance of spacetime structure and proposing an algebraic framework for potential quantization.

Contribution

It introduces a Lorentzian version of Connes' duality, emphasizing the role of global spacetime structure and proposing an algebraic approach for quantization.

Findings

Duality depends on global spacetime structure

A classification principle for spacetimes is proposed

An algebraic framework for quantization is suggested

Abstract

The Connes formula giving the dual description for the distance between points of a Riemannian manifold is extended to the Lorentzian case. It resulted that its validity essentially depends on the global structure of spacetime. The duality principle classifying spacetimes is introduced. The algebraic account of the theory is suggested as a framework for quantization along the lines proposed by Connes. The physical interpretation of the obtained results is discussed.