In physics, Faddeev–Popov ghosts (also called
Faddeev–Popov gauge ghosts or Faddeev–Popov
ghost fields) are extraneous fields which
are introduced into gauge quantum field theories
to maintain the consistency of the path integral
formulation. They are named after Ludvig Faddeev
and Victor Popov.A more general meaning of
the word ghost in theoretical physics is discussed
in Ghosts (physics).
== Overcounting in Feynman path integrals
==
The necessity for Faddeev–Popov ghosts follows
from the requirement that quantum field theories
yield unambiguous, non-singular solutions.
This is not possible in the path integral
formulation when a gauge symmetry is present
since there is no procedure for selecting
among physically equivalent solutions related
by gauge transformation. The path integrals
overcount field configurations corresponding
to the same physical state; the measure of
the path integrals contains a factor which
does not allow obtaining various results directly
from the action.
=== Faddeev–Popov procedure ===
It is possible, however, to modify the action,
such that methods such as Feynman diagrams
will be applicable by adding ghost fields
which break the gauge symmetry. The ghost
fields do not correspond to any real particles
in external states: they appear as virtual
particles in Feynman diagrams – or as the
absence of gauge configurations. However,
they are a necessary computational tool to
preserve unitarity.
The exact form or formulation of ghosts is
dependent on the particular gauge chosen,
although the same physical results must be
obtained with all gauges since the gauge one
chooses to carry out calculations is an arbitrary
choice. 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 fermionic symmetries, while commuting
ghosts are associated with bosonic 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
c
a
(
x
)
{\displaystyle c^{a}(x)\,}
in Yang–Mills theories (where
a
{\displaystyle a}
is an index in the adjoint representation
of the gauge group) is given by
L
ghost
=
∂
μ
c
¯
a
∂
μ
c
a
+
g
f
a
b
c
(
∂
μ
c
¯
a
)
A
μ
b
c
c
.
{\displaystyle {\mathcal {L}}_{\text{ghost}}=\partial
_{\mu }{\bar {c}}^{a}\partial ^{\mu }c^{a}+gf^{abc}\left(\partial
^{\mu }{\bar {c}}^{a}\right)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 as well as the Higgs field. Note that
in abelian gauge theories (such as quantum
electrodynamics) the ghosts do not have any
effect since
f
a
b
c
=
0
{\displaystyle f^{abc}=0}
and, consequently, the ghost particles do
not interact with the gauge fields
