Gauge Fixing - Lorenz Gauge

See also: Covariant formulation of classical electromagnetism

The Lorenz gauge is given, in SI units, by:

and in Gaussian units by:

This may be rewritten as:

where A_μ = (φ/c, −A) is the electromagnetic four-potential, ∂μ the 4-gradient .

It is unique among the constraint gauges in retaining manifest Lorentz invariance. Note, however, that this gauge was originally named after the Danish physicist Ludvig Lorenz and not after Hendrik Lorentz; it is often misspelled "Lorentz gauge". (Neither was the first to use it in calculations; it was introduced in 1888 by George F. FitzGerald.)

The Lorenz gauge leads to the following inhomogeneous wave equations for the potentials:

It can be seen from these equations that, in the absence of current and charge, the solutions are potentials which propagate at the speed of light.

The Lorenz gauge is incomplete in the sense that there remains a subspace of gauge transformations which preserve the constraint. These remaining degrees of freedom correspond to gauge functions which satisfy the wave equation

These remaining gauge degrees of freedom propagate at the speed of light. To obtain a fully fixed gauge, one must add boundary conditions along the light cone of the experimental region.

Maxwell's equations in the Lorenz gauge simplify to

where jν = (ρc, −j) is the four-current.

Two solutions of these equations for the same current configuration differ by a solution of the vacuum wave equation

.

In this form it is clear that the components of the potential separately satisfy the Klein-Gordon equation, and hence that the Lorenz gauge condition allows transversely, longitudinally, and "time-like" polarized waves in the four-potential. The transverse polarizations correspond to classical radiation, i. e., transversely polarized waves in the field strength. To suppress the "unphysical" longitudinal and time-like polarization states, which are not observed in experiments at classical distance scales, one must also employ auxiliary constraints known as Ward identities. Classically, these identities are equivalent to the continuity equation

.

Many of the differences between classical and quantum electrodynamics can be accounted for by the role that the longitudinal and time-like polarizations play in interactions between charged particles at microscopic distances.