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Faddeev-Popov ghost and anti-ghost.

They are generated for any gauge vector particle, in other words, for those particles which have the mark 'g' in the 'Aux' column of the Particles Table described in the previous section. The names of Faddeev-Popov ghosts and anti-ghosts are constructed by means of suffixes 'c' and 'C' respectively. For example, G.c, G.C are gluon ghosts and W+.c, W+.C, W-.c, W-.C are W-boson ghosts.

The operation of Hermitian conjugation transforms a Faddeev-Popov ghost to itself whereas an anti-ghost is transformed to itself with the opposite sign. Thus the rules of Hermitian conjugation of gluon and W-boson ghosts are

$$\begin{eqnarray*} (G.c)^+&=&G.c\\ (G.C)^+&=&-G.C\\ (\mbox{W+.c})^+&=& \mbox{W-.c} \\ (\mbox{W+.C})^+&=&-\mbox{W-.C} \end{eqnarray*}$$

Faddeev-Popov (anti)ghosts are scalar, anti-commutative fields ¹. The nonzero propagators for these fields are:

$$<0\vert T[\mbox{A+.c}(p_1), \; A.C(p_2)]\vert> = <0\vert T[\mbox{A+.C}(p_1), \; A.c(p_2)]\vert> = ScPr(p_1,p_2,M),$$

where $M$ is a mass of the prototype particle. This equality of masses is a consequence of the choice of the t'Hooft-Feynman gauge.

The appearance of the Faddeev-Popov ghosts in gauge theories can be explained in the following way. In the case of the t'Hooft-Feynman gauge the quantum field of massless gauge particle has two unphysical components, because the four-dimensional vector field describes a particle with two polarization states. The contributions of Faddeev-Popov ghost and anti-ghost compensate the contribution of these unphysical polarizations. In the case of massive gauge boson there are three physical polarizations and the compensation by means of Faddeev-Popov ghosts looks wrong from the viewpoint of naive arguments based on counting degrees of freedom. Indeed, in this case one additional Goldstone ghost appears [Bjorken&Drell].

Footnote

1  The well-known spin-statistic relation is not valid for unphysical fields.