Klein's programme and Quantum Mechanics

TL;DR

This paper explores the geometric structure of quantum mechanics, identifying the relevant transformation groups for closed and open systems, and suggests extending Klein's programme to include groupoids.

Contribution

It reveals that the general linear group, rather than just the unitary group, underpins the structure of open quantum systems, extending Klein's programme to groupoids.

Findings

Unitary group is the maximal compact subgroup for closed systems.

Open systems are associated with the general linear group.

The general linear group arises from the $C^{*}$-algebra as a Lie algebra.

Abstract

We review the geometrical formulation of Quantum Mechanics to identify, according to Klein's programme, the corresponding group of transformations. For closed systems, it is the unitary group. For open quantum systems, the semigroup of Kraus maps contains, as a maximal subgroup, the general linear group. The same group emerges as the exponentiation of the $C^{*}$--algebra associated with the quantum system, when thought of as a Lie algebra. Thus, open quantum systems seem to identify the general linear group as associated with quantum mechanics and moreover suggest to extend the Klein programme also to groupoids. The usual unitary group emerges as a maximal compact subgroup of the general linear group.