# Pseudo-Riemannian spectral triples and the harmonic oscillator

July 06, 2012

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.

Keywords:

noncommutative geometry; pseudo-Riemannian manifold; spectral triple; K-homology; Harmonic oscillator