The study of symmetries: some general techniques

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.