The study of symmetries: some general techniques

TL;DR

This paper explores general techniques for studying symmetries in the context of self-adjoint operators relevant to quantum mechanics, focusing on automorphisms that preserve key relations and operations.

Contribution

It introduces several ideas and methods for analyzing the general form of symmetries in the mathematical foundations of quantum mechanics.

Findings

Identification of key automorphism properties

Development of techniques for symmetry classification

Application to operators in quantum mechanics

Abstract

Let $S(H)$ be the set of all self-adjoint bonded linear operators on $H$ and $\mathcal{V} \subset S(H)$ a subset that is pertinent in mathematical foundations of quantum mechanics. A symmetry is a bijective map $\phi :\mathcal{V} \to \mathcal{V}$ which is an automorphism with respect to one or more relations and/or operations on $\mathcal{V}$ that are relevant in mathematical physics. We will explain several ideas that can be used when studying the general form of symmetries.