Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics

TL;DR

This paper demonstrates how spectral triplets naturally emerge in non-commutative quantum mechanics formulated on quantum Hilbert space, enabling the computation of a Connes-inspired distance function that links quantum statistics and geometry.

Contribution

It introduces a general algorithm to compute Connes' distance in non-commutative quantum mechanics, revealing a connection between quantum statistics and emergent geometry.

Findings

Spectral triplets arise naturally in non-commutative quantum mechanics.

A simple algorithm for computing Connes' distance is developed.

Distance between states reflects a link between statistics and geometry.

Abstract

We show that when non-commutative quantum mechanics is formulated on the Hilbert space of Hilbert-Schmidt operators (referred to as quantum Hilbert space) acting on a classical configuration space, spectral triplets as introduced by Connes in the context of non-commutative geometry arise naturally. A distance function as defined by Connes can therefore also be introduced. We proceed to give a simple and general algorithm to compute this function. Using this we compute the distance between pure and mixed states on quantum Hilbert space and demonstrate a tantalizing link between statistics and geometry.