Pseudo-Riemannian spectral triples and the harmonic oscillator

TL;DR

This paper introduces pseudo-Riemannian spectral triples to extend spectral geometry to pseudo-Riemannian manifolds and their noncommutative analogues, establishing a link to genuine spectral triples and K-homology, with applications including the harmonic oscillator.

Contribution

It defines pseudo-Riemannian spectral triples and proves they can be associated with genuine spectral triples, broadening the scope of spectral geometry to noncommutative and pseudo-Riemannian contexts.

Findings

Main theorem links pseudo-Riemannian spectral triples to genuine spectral triples.

Application of the local index theorem under additional assumptions.

Harmonic oscillator example demonstrates broad applicability.

Abstract

We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a K-homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classical pseudo-Riemannian manifolds.