Editing Faddeev–Popov ghost (section)

===Faddeev–Popov procedure===
{{main|BRST quantization}}

It is possible, however, to modify the action, such that methods such as [[Feynman diagram]]s 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 particle]]s in Feynman diagrams – or as the ''absence'' of some gauge configurations. However, they are a necessary computational tool 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 must be obtained with all gauges since the gauge one chooses to carry out calculations is an arbitrary choice. The [[Feynman gauge|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 
:<math> 
\int \mathcal{D}[A] \exp i \int \mathrm d^4 x \left ( - \frac{1}{4} F^a_{\mu \nu} F^{a \mu \nu } \right ).
</math>
The integral needs to be constrained via gauge-fixing via <math> G(A) = 0 </math> to integrate only over physically distinct configurations. Following Faddeev and Popov, this constraint can be applied by inserting 
:<math> 
1 = \int \mathcal{D}[\alpha (x) ] \delta (G(A^{\alpha })) \mathrm{det} \frac{\delta G(A^{\alpha} )}{\delta \alpha }
</math>
into the integral. <math> A^{\alpha } </math> denotes the gauge-fixed field.<ref>{{Cite book |last1=Peskin |first1=Michael E. |author-link=Michael Peskin |url=https://books.google.com/books?id=_Q84DgAAQBAJ |title=An Introduction To Quantum Field Theory |last2=Schroeder |first2=Daniel V. |publisher=[[Avalon Publishing]] |year=1995 |isbn=978-0-8133-4543-7 |series=The advanced book program |location=Boulder, CO}}</ref> The determinant is then expressed as a [[Berezin integral]]. Indeed, for any square matrix <math>M</math>, one has the identity
:<math>\int \exp\left[-\theta^TM\eta\right] \,d\theta\,d\eta = \det M</math>
where the integration variables <math>\theta,\eta</math> are [[Grassmann variable]]s (aka [[supernumber]]s): 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 the [[Pauli 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 for [[spinor]]s passes through the [[Clifford algebra]], not the Grassmann algebra. The Clifford algebra has a natural [[filtration (abstract algebra)|filtration]] inherited from the [[tensor algebra]]; this induces a [[graded algebra|gradation]], the [[associated graded algebra]], which ''is'' naturally isomorphic to the Grassmann algebra. The details of this grading are presented at length in the article on [[Clifford algebra]]s.