Families of spectral triples and foliations of space(time)

Abstract

We study a noncommutative analogue of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a 'Lorentzian spectral triple', which can also be viewed as the 'reverse Wick rotation' of a product spectral triple. This construction of 'Lorentzian spectral triples' fits well into the Krein space approach to noncommutative Lorentzian geometry.

22 pages

References in corpus (8)

• A Lorentzian version of the non-commutative geometry of the standard model of particle physics
• Causality theory for closed cone structures with applications
• Equivariant Lorentzian Spectral Triples
• The Lorentzian distance formula in noncommutative geometry
• Causality in noncommutative two-sheeted space-times
• The index of generalised Dirac-Schrödinger operators
• The disappearance of causality at small scale in almost-commutative manifolds
• On the definition of spacetimes in Noncommutative Geometry, part II

Cited by in corpus (4)

• Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
• An index theorem on asymptotically static spacetimes with compact Cauchy surface
• The APS-index and the spectral flow
• Index Theory for Globally Hyperbolic Spacetimes