Faddeev–Popov ghost

A Grassmann-valued auxiliary field introduced when gauge fixing a path integral. Its functional integral reproduces the Faddeev–Popov determinant, accounting for gauge redundancy in quantum calculations.

Faddeev–Popov procedure: A path-integral method that fixes gauge redundancy by inserting a gauge condition and its associated determinant. This procedure derives the determinant that the ghost field represents.
Gauge symmetry: A redundancy in a theory's mathematical description that leaves physical observables unchanged. Ghosts arise because this redundancy makes the unfixed path integral overcount configurations.
Quantum chromodynamics: The quantum field theory of quarks and gluons based on the non-Abelian gauge group SU(3). Its gluon self-interactions make ghost loops and vertices essential in covariant gauges.
Faddeev–Popov ghost (particle physics): A hypothetical spin-zero particle proposed in early theories of weak interactions, distinct from gauge-fixing ghost fields. The shared name masks a different historical concept with no role as a gauge-fixing field.
Faddeev–Popov determinant: The functional determinant of the variation of a gauge-fixing condition along gauge orbits. Integrating over ghost fields reproduces this determinant.
Gauge fixing: A choice of condition selecting representative field configurations from each gauge-equivalence class. The chosen condition determines the determinant represented by the ghosts.
Yang–Mills theory: A gauge field theory whose dynamics are determined by a Lie-group-valued connection and its curvature. Faddeev–Popov ghosts are standard fields in gauge-fixed quantization of non-Abelian Yang–Mills theories.
Physical polarization: A propagating polarization state that contributes to the physical particle spectrum. Ghost states are not physical polarizations, even though their propagators appear in diagrams.
Grassmann variable: An algebraic variable that anticommutes with other Grassmann variables and squares to zero. Anticommuting integration makes the ghost field produce a determinant rather than its inverse.
Path integral formulation: A quantum formulation that computes amplitudes by summing over possible field histories with weighted phases. The Faddeev–Popov determinant enters when the path integral is gauge fixed.

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