Spectral noncommutative geometry and quantization: a simple example

TL;DR

This paper investigates the quantization of a simple noncommutative geometric model using spectral triples, revealing a discrete spectrum for the Connes distance and constructing a quantum Hilbert space.

Contribution

It introduces a covariant canonical quantization method for finite spectral triples, demonstrating the discretization of geometric distances in quantum noncommutative geometry.

Findings

Connes distance becomes discrete upon quantization

Constructed a quantum Hilbert space for noncommutative geometry

Quantized the Dirac operator within the spectral triple framework

Abstract

We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its quantization. In particular, we consider a simple model based on a finite dimensional spectral triple (A, H, D), which mimics certain aspects of the spectral formulation of general relativity. We find the physical phase space, which is the space of the onshell Dirac operators compatible with A and H. We define a natural symplectic structure over this phase space and construct the corresponding quantum theory using a covariant canonical quantization approach. We show that the Connes distance between certain two states over the algebra A (two ``spacetime points''), which is an arbitrary positive number in the classical noncommutative geometry, turns out to be discrete in the quantum theory, and we compute its spectrum. The quantum states of the noncommutative geometry form a Hilbert space K. D is promoted to an operator *D on the direct product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization of the family of the triples (A, H, D).