GaugeFixing

Gauge fixing is a procedure in gauge theories to remove redundant degrees of freedom associated with gauge invariance. In many theories, fields are defined up to a gauge transformation, so many field configurations describe the same physical situation. To perform calculations, one imposes a gauge condition that selects a unique representative from each equivalence class of gauge-related configurations.

Common gauges include the Lorenz gauge ∂^μ A_μ = 0, the Coulomb gauge ∇·A = 0, and the temporal gauge A^0 = 0. In non-Abelian gauge theories, gauge fixing is more subtle because the space of gauge orbits is more intricate.

In the quantization of gauge theories, gauge fixing is essential for defining the path integral and propagators. The Faddeev-Popov procedure inserts a delta function enforcing the gauge condition and a determinant to compensate for overcounting. This leads to the introduction of ghost fields, which are needed to preserve unitarity and gauge invariance of physical observables, at least perturbatively. BRST symmetry provides a formal framework for these structures. Observables that are gauge invariant do not depend on the choice of gauge, but gauge-dependent quantities such as propagators do.

A significant complication is the Gribov problem: in non-Abelian theories, a given gauge condition can intersect the same gauge orbit multiple times, making global gauge fixing ambiguous and affecting nonperturbative results.

In lattice gauge theory, gauge invariance is maintained exactly, and gauge fixing is optional; it is commonly used to study gauge-variant quantities and to define certain correlators, with Landau or Coulomb gauges among popular choices.

See also: Faddeev–Popov procedure, BRST symmetry, Gribov ambiguity, lattice gauge theory, Dirac's constrained quantization.