In physics, Faddeev-Popov ghosts (also called ghost fields) are additional fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulation. There is also a more general meaning of the word "ghost" in theoretical physics, which is discussed in the section on General ghosts in theoretical physics. Named after Ludvig Faddeev and Victor Popov.[1]
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Overcounting in Feynman path integrals
The necessity for Faddeev-Popov ghosts follows from the requirement that in the path integral formulation of quantum field theories should yield unambiguous, non-singular solutions. This is not possible when a gauge symmetry is present since there is no procedure for selecting any one solution from a range of physically-equivalent solutions, all related by a gauge transformation. The problem stems from the path integrals overcounting field configurations related by gauge symmetries since those correspond to the same physical state; the measure of the path integrals contains a factor, which does not allow obtaining various results directly from the original 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 by adding some additional fields, which break the gauge symmetry, 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 in that they do not correspond to any real particles in external states: they only appear as virtual particles in Feynman diagrams. However they are necessary to preserve overall-all unitarity.
The exact form or formulation of ghosts is dependent on the particular gauge chosen, although the same physical results are obtained with all the gauges. The Feynman-'t Hooft gauge is usually the simplest gauge for this purpose, and is assumed for the rest of this article.
Spin-statistics relation violated
The Faddeev-Popov ghosts violate the spin-statistics relation, which is another reason why they are often regarded as "non-physical" particles.
For example, in Yang-Mills theories (such as quantum chromodynamics) the ghosts are complex scalar fields (spin 0), but they anti-commute (like fermions).
In general, anti-commuting ghosts are associated with bosonic symmetries, while commuting ghosts are associated with fermionic symmetries.
Gauge fields and associated ghost fields
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).
Appearance in Feynman diagrams
In Feynman diagrams the ghosts appear as closed loops wholly composed of 3-vertices, attached to the rest of the diagram via a gauge particle at each 3-vertex. Their contribution to the S-matrix is exactly cancelled (in the Feynman-'t Hooft gauge) by a contribution from a similar loop of gauge particles with only 3-vertex couplings or gauge attachments to the rest of the diagram. (A loop of gauge particles not wholly composed of 3-vertex couplings is not cancelled by ghosts.) 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.)
Ghost field Lagrangian
The Lagrangian for the ghost fields 
in Yang-Mills theories (where a is an index in the adjoint representation of the gauge group) is given by
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 fabc = 0 and, consequently, the ghost particles do not interact with the gauge fields.
General ghosts in theoretical physics
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 thus violating unitarity.
Footnotes
- ^ W. F. Chen. Quantum Field Theory and Differential Geometry
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