Geometric approach to non-relativistic Quantum Dynamics of mixed states

TL;DR

This paper introduces a geometric framework for non-relativistic quantum mechanics of mixed states using Uhlmann's bundle, enabling a new perspective on quantum evolution, conserved observables, and recurrence phenomena.

Contribution

It develops a novel geometric approach to quantum dynamics of mixed states, incorporating a dynamic metric and geometric proofs of quantum recurrence.

Findings

Geometric description of quantum evolution via geodesics

Conserved observables as Killing vector fields

Proof of Poincare quantum recurrence in finite systems

Abstract

In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincare quantum recurrence in a physical system with finite energy levels.