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

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.