Emergence of Lorentz symmetry from an almost-commutative twisted spectral triple

TL;DR

This paper explores how Lorentzian geometry naturally emerges from almost-commutative twisted spectral triples, offering a new perspective on the Lorentzian signature problem without relying on Wick rotation.

Contribution

It demonstrates the transition from Riemannian to pseudo-Riemannian spectral triples within an almost-commutative framework, suggesting a geometric origin of Lorentzian signature in noncommutative geometry.

Findings

Lorentzian spectral triples can arise from twisted spectral triples.

An alternative to Wick rotation is proposed that avoids complex numbers.

The approach links noncommutative geometry to the Lorentzian signature problem.

Abstract

This article demonstrates how the transition from a (Riemannian) twisted spectral triple to a pseudo-Riemannian spectral triple arises within an almost-commutative spectral triple. This opens a new perspective on the Lorentzian signature problem, showing that the almost-commutative structure at the heart of the noncommutative standard model of particle physics could be the origin of the emergence of a Lorentzian spectral triple, starting from self-adjoint Dirac operators and conventional inner product structures, in the framework of twisted spectral triples. We present an alternative to Wick rotation, acting on the metric and the Christoffel symbols in a way that does not introduce any complex numbers.