Gauge fixing

Gauge fixing removes the degeneracy among gauge-equivalent field configurations so that the kinetic operator has an inverse propagator.

Adding

- (\partial_{μ} A^{μ})^{2} / (2 α)

gives a family of Lorentz-covariant gauges;

α = 1

Faddeev-Popov determinant ( $Δ_{FP}$ )

Δ_{FP} [A] = det (\frac{δ G ( A ^{α} )}{δ α})_{G = 0}

that compensates for the change from integration along a gauge orbit to a gauge-fixing condition

G (A) = 0

A Faddeev-Popov ghost field is a Grassmann-valued scalar field whose Gaussian functional integral represents the Faddeev-Popov determinant. Ghosts occur only on internal lines and cancel unphysical gauge-field contributions.

An axial gauge imposes

n^{μ} A_{μ} = 0

for a fixed vector

n^{μ}

. Its Faddeev-Popov operator is

n^{μ} D_{μ}

; on the strict gauge slice its gauge-field-dependent part vanishes, so its ghosts decouple.

BRST symmetry is a nilpotent fermionic symmetry of a gauge-fixed action. Its differential

s

replaces an infinitesimal gauge parameter by the ghost field and satisfies

s^{2} = 0

s Ψ

supplies the gauge-fixing and ghost terms. Nilpotence gives

s (s Ψ) = 0

immediately.

BRST cohomology identifies physical states and observables with BRST-closed objects modulo BRST-exact ones. A change of gauge-fixing fermion changes the action by a BRST-exact term and therefore leaves BRST-cohomology classes unchanged when the measure has no BRST anomaly.

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Gauge fixing is a procedure used in theoretical physics, particularly in the context of gauge theories, to eliminate the redundancy caused by gauge symmetries. Gauge symmetries are transformations that can be applied to the fields in a theory without changing the physical content of the theory. Because of these symmetries, multiple field configurations can describe the same physical situation, leading to an overcounting of degrees of freedom.