Spectral geometry with a cut-off: Topological and metric aspects

Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutative geometry, and focus on the Connes distance associated to a spectral triple (A,H,D)(A,H,D). A high momentum (short distance) cut-off is implemented by the action of a projection PP on the Dirac operator DD and/or on the algebra AA. This action induces two new distances. We individuate conditions making them equivalent to the original distance. We also study the Gromov–Hausdorff limit of the set of truncated states, first for compact quantum metric spaces in the sense of Rieffel, then for arbitrary spectral triples. To this aim, we introduce a notion of “state with finite moment of order 11” for noncommutative algebras. We then focus on the commutative case, and show that the cut-off induces a minimal length between points, which is infinite if PP has finite rank. When PP is a spectral projection of DD, we work out an approximation of points by non-pure states that are at finite distance from each other. On the circle, such approximations are given by Fejér probability distributions. Finally we apply the results to the Moyal plane and the fuzzy sphere, obtained as Berezin quantization of the plane and the sphere respectively.