Gauge fixing

Gauge fixing is the procedure of removing redundant degrees of freedom from a gauge theory to make calculations tractable. Because gauge symmetry identifies physically equivalent field configurations as mathematically distinct, the gauge field contains more variables than the physics requires. Gauge fixing selects a representative from each equivalence class — a slice through the space of configurations — so that the equations of motion have unique solutions.

The choice of gauge is arbitrary, but some choices are more useful than others. The Lorenz gauge simplifies the electromagnetic wave equation. The Coulomb gauge separates transverse and longitudinal degrees of freedom. The Faddeev-Popov method provides a systematic way to fix gauges in non-abelian theories, introducing ghost fields that cancel unphysical contributions to loop diagrams.

But gauge fixing is not free. The Gribov ambiguity — the existence of multiple gauge-equivalent configurations that satisfy the same gauge condition — means that no global gauge fixing is possible in non-abelian theories. The Faddeev-Popov method works perturbatively but fails nonperturbatively. This is not a technical problem; it is a structural feature of gauge theories that tells us something about the topology of the configuration space.

The insistence that gauge fixing is merely a calculational convenience ignores the deeper point: the failure of global gauge fixing is the failure of a coordinate system to cover the space. It is the same phenomenon that produces coordinate singularities in general relativity and monopole solutions in electromagnetism. Gauge fixing is where the geometry of the theory makes contact with the topology of its space of states.