Faddeev-Popov ghost

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In physics, Faddeev-Popov ghosts are auxiliary fields which need to be introduced in the realization of gauge theories as consistent quantum field theories. In the path integral formulation, the path integrals should not overcount field configurations related by gauge symmetries since those correspond to the same physical state. Consequently, the measure of the path integrals contains an additional factor, which does not allow obtaining various results directly from the action using the regular methods (e.g., Feynman diagrams). It is possible, however, to modify the action such that the regular methods will be applicable. This often requires adding some additional fields, which are called the ghost fields. This technique is called the Faddeev-Popov procedure (see also BRST quantization). The ghost fields are a computational tool, and they do not correspond to any real particles: they may only appear as virtual particles in Feynman diagrams.

The Faddeev-Popov ghosts violate the spin-statistics relation. For example, in Yang-Mills theories (like the quantum chromodynamics) the ghosts are complex scalar fields (spin 0), but they anticommute (like fermions). In general, anticommuting ghosts are associated with bosonic symmetries, while commuting ghosts are associated with fermionic symmetries.

The Lagrangian for the ghost fields $c^a(x)\,$ in Yang-Mills theories (where $a$ is an index in the adjoint representation of the gauge group) is given by

$\mathcal{L}_\mathrm{ghost} = \partial_\mu \overline{c}^a\partial^\mu c^a + g f^{abc}(\partial^\mu\overline{c}^a) A_\mu^b c^c$

The first term is a kinetic term like for regular complex scalar fields, and the second term describes the interaction with the gauge fields. Note that in abelian gauge theories (like the quantum electrodynamics) the ghosts do not have any effect since $f a b c = 0$ .

The Faddeev-Popov ghosts are sometimes referred to as "good ghosts". The "bad ghosts" represent another, more general meaning of the word "ghost" in theoretical physics: states of negative norm - or fields with the wrong sign of the kinetic term - whose existence allows the probabilities to be negative.