Log-affine geodesics in the manifold of vector states on a von Neumann algebra

TL;DR

This paper introduces log-affine geodesics in the manifold of vector states on a von Neumann algebra, connecting concepts from quantum statistical physics and differential geometry to deepen understanding of quantum state spaces.

Contribution

It defines log-affine geodesics in von Neumann algebra state spaces, linking quantum statistical manifolds with geometric structures like exponential tangent spaces.

Findings

Defines log-affine geodesics in non-abelian von Neumann algebras

Connects quantum statistical states with geometric notions

Provides a framework linking physics and differential geometry

Abstract

This paper introduces the notion of a log-affine geodesic connecting two vector states on a von Neumann algebra. The definition is linked to the standard notion of Boltzmann-Gibbs states in statistical physics and the related notion of quantum statistical manifolds. In the abelian case it is linked to the notion of exponential tangent spaces.