Schwinger's picture of quantum mechanics: 2-groupoids and symmetries

TL;DR

This paper extends Schwinger's quantum mechanics framework by introducing 2-groupoids to describe microscopic symmetries and groups of bisections for global symmetries, offering a new perspective on quantum symmetries.

Contribution

It proposes a novel algebraic structure of 2-groupoids to model microscopic symmetries and constructs a group of symmetries from 2-groupoids for the global perspective in the groupoid approach.

Findings

Introduction of 2-groupoid structures for microscopic symmetries.

Construction of symmetry groups from 2-groupoids.

Analog of Wigner's theorem for quantum symmetries in the groupoid framework.

Abstract

Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced.It is shown that, given a groupoid $G\rightrightarrows \Omega$ associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global".The microscopic point of view leads to the introduction of an additional layer over the grupoid $G$, giving rise to a suitable algebraic structure of 2-groupoid.On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid.The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.