Emergence of Time from a Twisted Spectral Triple in Almost-Commutative Geometry
Gaston Nieuviarts
TL;DR
The paper addresses the long-standing issue of incorporating time and Lorentzian signature into noncommutative geometry (NCG) by developing a framework of twisted spectral triples and a $K$-morphism that connects twisted and pseudo-Riemannian (Krein) triples. It demonstrates how an almost-commutative, Standard-Model–like geometry, when endowed with a twist $\rho$ and a finite-part operator $K$, yields a Dirac operator of the form $D_p = D \otimes 1_F + K \otimes D_F$ whose fluctuations respect a twisted first-order condition and culminate in a Krein-space geometry, effectively realizing a pseudo-Riemannian metric without complex Wick rotations. A central result is that signature change and the emergence of time arise as purely algebraic consequences of these structures, with the twisted metric $g_R(u,v) = g(u, rv)$ linked to a spacelike reflection $r$ and the Krein product encoding Lorentzian inner products. This provides a conceptually new, algebraic route to a Lorentzian formulation of the noncommutative Standard Model and clarifies how twisted and almost-commutative spectral triples underpin the appearance of time in this geometric framework.
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.