The question addressed in this paper springs from the following observation. In his 1918 theory Weyl introduced local gauge transformations (transformations that depend on arbitrary functions of space and time), and it is local gauge symmetry that he connects with conservation of electric charge. According to the standard modern account, however, global gauge symmetry is invoked to deliver conservation of electric charge (see, for example, Leader and Predazzi, 1996; Ryder, 1985; Sakurai, 1964; Schweber, 1961; Sterman, 1993; Weinberg, 1995). Which is the correct symmetry to connect with charge conservation? This question might seem straightforward on the surface, but it turns out that a rather interesting story lies behind any satisfactory answer. The story involves a triangle of relationships, none of which has been adequately addressed in the literature to date. This triangle involves Weyl’s work, relativistic field theory, and Noether’s theorems.
note the “s” on “theorems”: a big part of this paper is Noether’s sometimes-underapprieciated second theorem, which addresses local symmetries.
particularly cute is the incorrect derivation showcased in the appendix, where we get charge conservation from Maxwell EM “for free"—except it wasn’t actually free, because assuming we could apply Noether’s theorems in the first place was equivalent to demanding charge conservation.
