Gauge theory is widely utilized in particle physics, accounting for the Standard Theory by which all other competing theories are judged. A good technical summary article on gauge theories is John Taylor, "Gauge theories in particle physics," in Paul Davies, Ed., The New Physics, p. 458-480.
Elsewhere we find that classical electromagnetics was the first gauge theory. Also, J.D. Jackson, Classical Electrodynamics, 2nd Edition, 1975, p. 220-223 covers this succinctly, although in the system of Gaussian units rather than today's more familiar rationalized MKSA units. He shows that Maxwell's four equations can first be reduced to a set of two coupled equations in the A &
representation. He shows that the potentials A and
therein are arbitrary in a specific sense, since the A vector can be replaced with A' = A +
, where
is a scalar function and
is its gradient. The B field is given by B =
x A, so that the new B' field becomes