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Inphysics,Faddeev–Popov ghosts (also calledFaddeev–Popov gauge ghosts orFaddeev–Popov ghost fields) are extraneousfields which are introduced intogaugequantum field theories to maintain the consistency of thepath integral formulation. They are named afterLudvig Faddeev andVictor Popov.^[1][2]

A more general meaning of the word "ghost" intheoretical physics is discussed inGhost (physics).

The necessity for Faddeev–Popov ghosts follows from the requirement thatquantum field theories yield unambiguous, non-singular solutions. This is not possible in thepath integral formulation when agauge 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; themeasure of the path integrals contains a factor which does not allow obtaining various results directly from theaction.

It is possible, however, to modify the action, such that methods such asFeynman diagrams will be applicable by addingghost fields which break the gauge symmetry. The ghost fields do not correspond to any real particles in external states: they appear asvirtual particles in Feynman diagrams – or as theabsence of some gauge configurations. However, they are a necessary computational tool to preserveunitarity.

The exact form or formulation of ghosts is dependent on the particulargauge 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. TheFeynman–'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

${\displaystyle \int \mathcal{D}[A]\exp i\int {\mathrm{d}}^{4}x\left(-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a\mu \nu }\right).}$

The integral needs to be constrained via gauge-fixing via${\displaystyle G(A)=0}$ to integrate only over physically distinct configurations. Following Faddeev and Popov, this constraint can be applied by inserting

${\displaystyle 1=\int \mathcal{D}[\alpha (x)]\delta (G(A^{\alpha }))\mathrm{d}\mathrm{e}\mathrm{t}\frac{\delta G(A^{\alpha })}{\delta \alpha }}$

into the integral.${\displaystyle A^{\alpha }}$ denotes the gauge-fixed field.^[3] The determinant is then expressed as aBerezin integral. Indeed, for any square matrix${\displaystyle M}$, one has the identity

${\displaystyle \int \exp \left[-{\theta }^{T}M\eta \right]d\theta d\eta =detM}$

where the integration variables${\displaystyle \theta ,\eta }$ areGrassmann variables (akasupernumbers): they anti-commute and square to zero. In the present case, one introduces a field of Grassmann variables, one for every point in space-time (corresponding to the determinant at that point in space-time,i.e. one for each fiber of the gauge-field fiber bundle.) Used in the above identity for the determinant, these fields become the Fadeev-Popov ghost fields.

Because Grassmann numbers anti-commute, they resemble the anti-commutation property of thePauli exclusion principle, and thus are sometimes taken to be stand-ins for particles with spin 1/2. This identification is somewhat treacherous, as the correct construction forspinors passes through theClifford algebra, not the Grassmann algebra. The Clifford algebra has a naturalfiltration inherited from thetensor algebra; this induces agradation, theassociated graded algebra, whichis naturally isomorphic to the Grassmann algebra. The details of this grading are presented at length in the article onClifford algebras.

The Faddeev–Popov ghosts violate thespin–statistics relation, which is another reason why they are often regarded as "non-physical" particles.

For example, inYang–Mills theories (such asquantum chromodynamics) the ghosts arecomplexscalar fields (spin 0), but theyanti-commute (likefermions).

In general,anti-commuting ghosts are associated withbosonic symmetries, whilecommuting ghosts are associated withfermionic symmetries.

Every gauge field has an associated ghost, and where the gauge field acquires a mass via theHiggs mechanism, the associated ghost field acquires the same mass (in theFeynman–'t Hooft gauge only, not true for other gauges).

InFeynman 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 theS-matrix is exactly cancelled (in theFeynman–'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.)

The Lagrangian for the ghost fields${\displaystyle c^a(x)}$ inYang–Mills theories (where${\displaystyle a}$ is an index in the adjoint representation of thegauge group) is given by

${\displaystyle {\mathcal{L}}_{\text{ghost}}={\mathrm{\partial }}_{\mu }{\overline{c}}^a{\mathrm{\partial }}^{\mu }{c}^a+g{f}^{abc}\left({\mathrm{\partial }}^{\mu }{\overline{c}}^a\right)A_{\mu }^b{c}^c.}$

The first term is akinetic term like for regular complex scalar fields, and the second term describes the interaction with thegauge fields as well as theHiggs field. Note that inabelian gauge theories (such asquantum electrodynamics) the ghosts do not have any effect since thestructure constants${\displaystyle f^{abc}=0}$ vanish. Consequently, the ghost particles do not interact with abelian gauge fields.

• Faddeev, Ludwig (2009)."Faddeev-Popov ghosts".Scholarpedia.4 (4): 7389.Bibcode:2009SchpJ...4.7389F.doi:10.4249/scholarpedia.7389.ISSN 1941-6016.

• Gauge theories
• Quantum chromodynamics

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