Symmetry and topology

Space-time symmetries, conservation laws and Nöther’s theorem are discussed. The Poincaré group, generators and Casimir invariants are outlined. Local charge conservation and the corresponding U(1) charge symmetry underlying electromagnetism are presented, showing the roles of minimal electromagnetic coupling and gauge transformations. Experimental demonstrations of the Aharonov–Bohm effect are described and the topological interpretation is recounted. How the Aharonov–Casher effect survives in the classical world is mentioned. Berry’s revelation of geometric phase is presented. The Bitter–Dubbers experiment confirming this analysis is presented. Some comments are given on a Hilbert space with a simple topology.