Abstract
Mathematically, gauge theories are extraordinarily rich --- so rich, in
fact, that it can become all too easy to lose track of the connections
between results, and become lost in a mass of beautiful theorems and
properties: indeterminism, constraints, Noether identities, local and
global symmetries, and so on.
One purpose of this short article is to provide some sort of a guide
through the mathematics, to the conceptual core of what is actually
going on. Its focus is on the Lagrangian, variational-problem
description of classical mechanics, from which the link between gauge
symmetry and the apparent violation of determinism is easy to
understand; only towards the end will the Hamiltonian description be
considered.
The other purpose is to warn against adopting too unified a perspective
on gauge theories. It will be argued that the meaning of the gauge
freedom in a theory like general relativity is (at least from the
Lagrangian viewpoint) significantly different from its meaning in
theories like electromagnetism. The Hamiltonian framework blurs this
distinction, and orthodox methods of quantization obliterate it; this
may, in fact, be genuine progress, but it is dangerous to be guided by
mathematics into conflating two conceptually distinct notions
without appreciating the physical consequences.