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

TL;DR

This paper investigates how introducing a spectral cut-off in noncommutative geometry affects topological and metric properties, including distances and limits, with applications to quantum spaces like the Moyal plane and fuzzy sphere.

Contribution

It develops a framework for analyzing truncated spectral triples, compares induced distances to original ones, and studies the Gromov-Hausdorff limits in noncommutative settings.

Findings

Cut-off induces a minimal length between points in commutative cases.

Conditions identified for equivalence of truncated and original distances.

Approximation of points by finite-distance states on the circle using Fejer distributions.

Abstract

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 Connes distance associated to a spectral triple (A, H, D). A high momentum (short distance) cut-off is implemented by the action of a projection P on the Dirac operator D and/or on the algebra A. 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 1" 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 P has finite rank. When P is a spectral projection of $D$, 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 Fejer probability distributions. Finally we apply the results to Moyal plane and the fuzzy sphere, obtained as Berezin quantization of the plane and the sphere respectively.