Faddeev-Popov ghost

In physics, Faddeev-Popov ghosts are additional fields which need to be introduced in the realization of gauge theories as consistent quantum field theories.

Overview

The necessity for Faddeev-Popov ghosts follows from the requirement that in the path integral formulation of quantum field theories, 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 in external states: they may only appear as virtual particles in Feynman diagrams. The exact form or formulation of ghosts is dependent on the particular gauge chosen. The Feynman-'t Hooft gauge is usually the simplest gauge, and is assumed for the rest of this article. The Faddeev-Popov ghosts violate the spin-statistics relation. For example, in Yang-Mills theories (such as 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. Every gauge field has an associated ghost, and where the gauge field acquires a mass via the Higgs mechanism, the associated ghost field acquires the same mass (in the Feynman-'t Hooft gauge only, not true for other gauges). In Feynman diagrams the ghosts appear as closed loops, attached to the rest of the diagram via a gauge particle at each vertex. Their contribution to the S-matrix is exactly cancelled by a contribution from a similar loop of gauge particles with only 3-vertex couplings or gauge attachments to the rest of the diagram. (The loop of gauge particles makes other contributions that survive the ghost cancellation.) The opposite sign of the contribution of the ghost and gauge loops is due to them having opposite fermionic/bosonic natures. (Closed fermion loops have an extra -1 associated with them; bosonic loops don't.) The Lagrangian for the ghost fields <math>c^a(x)\,</math> in Yang-Mills theories (where <math>a</math> is an index in the adjoint representation of the gauge group) is given by

<math>\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 </math>

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 (such as quantum electrodynamics) the ghosts do not have any effect since <math>f^{abc} = 0</math> and, consequently, the ghost particles do not interact with the gauge fields. 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, such as Pauli-Villars ghosts—whose existence allows the probabilities to be negative.