Okay I can't resist - this stuff is just too cool. Here is my take on some of this stuff; assume grains of salt throughout as I am an engineer of sorts. (And s

Okay I can't resist - this stuff is just too cool. Here is my take on some of this stuff; assume grains of salt throughout as I am an engineer of sorts. (And sorry if this is old hat - and that the direction I go in strays pretty far, at times, from the original question about gauge theory actually having to do with physical things) It's funny that charge conservation, noted below as a consequence of gauge symmetry, is not exact. See CP symmetry violation and "all that". Furthermore there are various notions of internal symmetries... in engineer talk, perhaps gauge symmetry is a symmetry of parameterization - in math talk the base space of a fibre bundle "parameterizes" (in a coordinate free way) a larger space. --Gauge symmetry is a freedom in how exactly this is done (shout out to your comment about equivalence classes - totally agree). (For example the space time manifold showing how to put a vector space at every location in the manifold via tangent space constructions, or similar) I get lost sometimes.. Anyway, looking at part of the Wikipedia charge conservation article we read that: "...an electromagnetic field is unchanged when the scalar and vector potential are shifted by the gradient of an arbitrary scalar field." In quantum mechanics the scalar field is equivalent to a phase shift in the wavefunction of the charged particle..." This shift in the Lagrangian (to call it a different way) is gauge symmetry (U(1)) again. Interestingly, the phase shift can occur even if the space is "E&M flat" (no detectable field) and this shift IS observable: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect -A purely topological effect due to the fact that loops (around the solenoid in the experiment) in the space cannot be shrunk to a point! (not without leaving "the space" that is) Any phase shift is in fact a manifestation of Weyl's "path dependent length" which he'd speculated about to Einstein. (Einstein shot it down, but it was reborn as phase) But wait, path dependent lengths (generalized path length shift == phase shift) mean we have some, admittedly abstract, notion of curvature! This is the connection (ahem) between E&M theories and General Relativity. (and even closer connection with E&M-like, but non-abelian (Yang Mills) ) The potential, A, in electrodynamics plays a role analogous to that of the connection coefficient (Christoffel) in differential geometry. The tensor field strength in electrodynamics is analogous to curvature! (luckily for those who like their minds un-exploded E&M "happens" in flat space.) Once you get to Yang Mills, the commutator of the potential has to be added to the derivative of the Faraday tensor to get the total field strength. This is just the same as in General Relativity, where the curvature is the derivative of the connection plus its commutator. I guess what I am saying is, depending on how you look at it, the field theories are sort of shot-through with gauge symmetry, to both quantifiable effect and mathematical purpose, and it is neat.