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{{Quantum field theory|cTopic=Some models}}
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In [[physics]], '''Faddeev–Popov ghosts''' (also called '''ghost fields''') are additional [[field (physics)|fields]] which are introduced into [[gauge theories|gauge]] [[quantum field theory|quantum field theories]] to maintain the consistency of the [[path integral formulation]]. They are named after [[Ludvig Faddeev]] and [[Victor Popov]].<ref>W. F. Chen. [http://arxiv.org/abs/0803.1340v2 Quantum Field Theory and Differential Geometry]</ref>
 
There is also a more general meaning of the word "'''ghost'''" in [[theoretical physics]], which is discussed below (see ''[[#General ghosts in theoretical physics|general ghosts in theoretical physics]]'').
 
==Overcounting in Feynman path integrals==
The necessity for Faddeev–Popov ghosts follows from the requirement that in the [[path integral formulation]], [[quantum field theory|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 (mathematics)|measure]] of the path integrals contains a factor which does not allow obtaining various results directly from the original [[Action (physics)|action]] using the regular methods (e.g., [[Feynman diagram]]s). 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 particle]]s in Feynman diagrams – or as the ''absence'' of some gauge configurations.  However they are necessary to preserve [[unitarity (physics)|unitarity]].
 
The exact form or formulation of ghosts is dependent on the particular [[Gauge fixing|gauge]] chosen, although the same physical results are obtained with all the gauges. The [[Gauge fixing|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 theory|Yang–Mills theories]] (such as [[quantum chromodynamics]]) the ghosts are [[complex number|complex]] [[scalar field theory|scalar fields]] ([[spin (physics)|spin]] 0), but they [[anticommutativity|anti-commute]] (like [[fermions]]).
 
In general, [[anticommutativity|anti-commuting]] ghosts are associated with [[fermion]]ic symmetries, while [[commutativity|commuting]] ghosts are associated with [[boson]]ic 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 diagram]]s 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|S-matrix]] is exactly cancelled (in the [[Feynman gauge|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&nbsp;−1 associated with them; bosonic loops don't.)
 
==Ghost field Lagrangian==
The Lagrangian for the ghost fields <math>c^a(x)\,</math> in [[Yang–Mills theory|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 field]]s. 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.
 
==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 regularization|Pauli–Villars ghosts]]—whose existence allows [[Negative probability|the probabilities to be negative]] thus violating [[Unitarity (physics)|unitarity]].
 
== Changing the symmetry ==
Ghost particles could obtain the symmetry or break it in gauge fields. The "good ghost" particles actually obtain the symmetry by unchanging the "gauge fixing lagrangian" in a gauge transformation, while bad ghost particles break the symmetry by bringing in the non-abelian G-matrix which does change the symmetry, and this was the main reason to introduce the Gauge covariant and contravariant derivatives.
 
==References==
{{reflist}}
* L.D. Faddeev and V.N. Popov, Feynman Diagrams for the Yang-Mills Field, Phys. Lett. B25 (1967) 29.
 
==External links==
* [http://www.scholarpedia.org/article/Faddeev-Popov_ghosts Scholarpedia]
* {{cite web|last=Copeland|first=Ed|title=Ghost Particles|url=http://www.sixtysymbols.com/videos/ghost_particles.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|coauthors=Padilla, Antonio (Tony)}}
 
{{particles}}
 
{{DEFAULTSORT:Faddeev-Popov ghost}}
[[Category:Quantum field theory]]
[[Category:Quantum chromodynamics]]

Latest revision as of 22:37, 10 January 2015

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