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

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 ⇉ Ω 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.