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

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.
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https://hal.univ-lorraine.fr/hal-02062626
Contributor : Raphaël Leone <>
Submitted on : Saturday, March 9, 2019 - 2:51:18 PM
Last modification on : Friday, March 15, 2019 - 1:25:06 AM
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  • HAL Id : hal-02062626, version 1
  • ARXIV : 1903.04928

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Raphaël Leone. On the parallel transport in quantum mechanics with an application to three-state systems. 2019. ⟨hal-02062626⟩

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