The study of symmetries: some general techniques

2025/07/30 by Semrl, Peter
#47B15 #47B49 #81R15 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2507.22572

Abstract

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

Citations

• A geometric approach to Wigner-type theorems
• Loewner's theorem for maps on operator domains
• When is a quantum measurement?

Related

• Symmetries in nuclear, atomic and molecular spectroscopy
• Symmetries in QFT
• Studies of symmetries that give special quantum states the "right to exist"
• Symmetry Groups in Quantum Mechanics and the Theorem of Wigner on the Symmetry Transformations
• Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond