Gauge Fixing - Gauge Freedom

Gauge Freedom

The archetypical gauge theory is the Heaviside-Gibbs formulation of continuum electrodynamics in terms of an electromagnetic four-potential, which is presented here in space/time asymmetric Heaviside notation. The electric field E and magnetic field B of Maxwell's equations contain only "physical" degrees of freedom, in the sense that every mathematical degree of freedom in an electromagnetic field configuration has a separately measurable effect on the motions of test charges in the vicinity. These "field strength" variables can be expressed in terms of the scalar potential and the vector potential A through the relations:

If the transformation

(1)

is made, then B remains unchanged, since

.

However, this transformation changes E according to

.

If another change

(2)

is made then E also remains the same.

Hence, the E and B fields are unchanged if we take any function ψ(r, t) and simultaneously transform A and φ via the transformations (1) and (2).

A particular choice of the scalar and vector potentials is a gauge (more precisely, gauge potential) and a scalar function ψ used to change the gauge is called a gauge function. The existence of arbitrary numbers of gauge functions ψ(r, t) corresponds to the U(1) gauge freedom of this theory. Gauge fixing can be done in many ways, some of which we exhibit below.

Although classical electromagnetism is now often spoken of as a gauge theory, it was not originally conceived in these terms. The motion of a classical point charge is affected only by the electric and magnetic field strengths at that point, and the potentials can be treated as a mere mathematical device for simplifying some proofs and calculations. Not until the advent of quantum field theory could it be said that the potentials themselves are part of the physical configuration of a system. The earliest consequence to be accurately predicted and experimentally verified was the Aharonov-Bohm effect, which has no classical counterpart. Nevertheless, gauge freedom is still true in these theories. For example, the Aharonov-Bohm effect depends on a line integral of A around a closed loop, and this integral is not changed by

Gauge fixing in non-abelian gauge theories, such as Yang-Mills theory and general relativity, is a rather more complicated topic; for details see Gribov ambiguity, Faddeev–Popov ghost, and frame bundle.