Quantum Systems as Lie Algebroids

TL;DR

This paper explores the geometric structure of quantum systems using Lie algebroids, connecting classical geometric frameworks to quantum representations like Schrödinger and Heisenberg.

Contribution

It introduces a novel geometric perspective on quantum systems through Lie algebroids, unifying Schrödinger and Heisenberg representations.

Findings

Quantum systems are modeled as collections of self-adjoint operators on Kähler manifolds.

The geometry of the Heisenberg representation is described by Poisson structures on co-adjoint orbits.

Schrödinger and Heisenberg representations are shown to be equivalent.

Abstract

Lie algebroids provide a natural medium to discuss classical systems, however, quantum systems have not been considered. In aim of this paper is to attempt to rectify this situation. Lie algebroids are reviewed and their use in classical systems is described. The geometric structure of the Schr\"{o}dinger and Heisenberg representations of quantum systems is examined and their relationship to Lie algebroids is explored. Geometrically, a quantum system is seen to be a collection of bounded, linear, self-adjoint operators on a Hilbert, or more precisely, a K\"{a}hler manifold. The geometry of the Heisenberg representation is given by the Poisson structure of the co-adjoint orbits on the dual of the Lie algebra. Finally, it is shown that the Schr\"{o}dinger and Heisenberg representations are equivalent.