Type of unphysical field in quantum field theory which provides mathematical consistency
This article is about a specific type of ghost field. For ghosts in the general physics sense, see
Ghosts (physics)
.
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
.
[1]
[2]
A more general meaning of the word "ghost" in
theoretical physics
is discussed in
Ghost (physics)
.
Overcounting in Feynman path integrals
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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
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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 some 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.
Consider for example non-Abelian gauge theory with
The integral needs to be constrained via gauge-fixing via
to integrate only over physically distinct configurations. Following Faddeev and Popov, this constraint can be applied by inserting
into the integral.
denotes the gauge-fixed field.
[3]
Spin?statistics relation violated
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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
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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
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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]
(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
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The Lagrangian for the ghost fields
in
Yang?Mills theories
(where
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
as well as the
Higgs field
. Note that in
abelian
gauge theories (such as
quantum electrodynamics
) the ghosts do not have any effect since the
structure constants
vanish. Consequently, the ghost particles do not interact with abelian gauge fields.
- ^
Feynman discovered empirically that "boxing" and simply dismissing these diagrams restored unitarity. "
Because, unfortunately, I also discovered in the process that the trouble is present in the Yang?Mills theory; and, secondly, I have incidentally discovered a tree?ring connection which is of very great interest and importance in the meson theories and so on. And so I'm stuck to have to continue this investigation, and of course you appreciate that this is the secret reason for doing any work, no matter how absurd and irrational and academic it looks: we all realize that no matter how small a thing is, if it has physical interest and is thought about carefully enough, you're bound to think of something that's good for something else.
"
[4]
References
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External links
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