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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ G\rightrightarrows \Omega $\end{document}</tex-math></inline-formula> associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global". The microscopic point of view leads to the introduction of an additional layer over the grupoid <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, 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.</p>