Mathematics

We study a noncommutative analogue of a space(time) foliated by (spacelike) hypersurfaces. First, in the classical (commutative) case, we show that the canonical Dirac operator on the total space(time) can be reconstructed from the family of Dirac operators on the (spacelike) hypersurfaces. Second, in the noncommutative case, the same reconstruction formula continues to make sense for an abstract family of spectral triples, and we prove that (in the case of Riemannian signature) the construction yields a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, the corresponding 'Lorentzian spectral triple' can also be viewed as the 'reverse Wick rotation' of such product spectral triples.