arXiv:1903.04928

On the parallel transport in quantum mechanics with an application to three-state systems

Raphaël Leone

Published Mar 11, 2019• 1 citation

Abstract

The aim of this article is to give a rigorous although simple treatment of the geometric notions around parallel transport in quantum mechanics. I start by defining the teleparallelism (or generalized Pancharatnam connection) between n-dimensional vector subspaces (or n-planes) of the whole Hilbert space. It forms the basis of the concepts of parallel transport and of both cyclic and non-cyclic holonomies in the Grassmann manifold of n-planes. They are introduced in the discrete case (broken lines) before being rendered 'continuous' (smooth curves) and the role of the geodesics is stressed. Then, I discuss the interest of such a construction to geometrize a part of the dynamics when a (quasi-)dynamical invariant is known, especially in the adiabatic limit. Finally, I illustrate the general theory with a three-state toy model allowing for non-Abelian adiabatic transports.

DOI: 10.48550/arxiv.1903.04928

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