Emergence of Time from a Twisted Spectral Triple in Almost-Commutative Geometry

TL;DR

This paper explores how Lorentzian structures can emerge from twisted spectral triples in noncommutative geometry, offering an algebraic approach as an alternative to Wick rotation for the Standard Model.

Contribution

It introduces a unified algebraic framework for deriving Lorentzian spectral triples from Riemannian ones within almost-commutative geometry, addressing the Lorentzian signature problem.

Findings

Demonstrates how twisted spectral triples can produce Lorentzian structures

Provides an algebraic mechanism connecting Riemannian and Lorentzian spectral triples

Offers an alternative to Wick rotation in noncommutative geometry

Abstract

This proceeding presents a synthesis of recent results on the emergence of pseudo-Riemannian structures from twisted spectral triples within the almostcommutative framework. It provides a unified algebraic mechanism for addressing the Lorentzian signature problem, demonstrating how the almost-commutative structure underlying the noncommutative Standard Model of particle physics may give rise to Lorentzian spectral triple from a purely Riemannian setting. This notably offers an alternative to Wick rotation, provided by a notion of morphism connecting twisted and pseudo-Riemannian spectral triples.