View source for Zinn-Justin equation

===The quantized gauge action without matter===
Following Feynman (1948), one would naively expect the quantum evolution operator to be given by an integral over classical gauge fields of the form
:<math>\mathcal{U}=\int[\mathrm{d}\mathbf{A}_\mu]\exp\left({i\over \hbar}\mathcal{S}_{\mathrm{class.}}(\mathbf{A})\right),</math>
where <math>\mathcal{S}_{\mathrm{class.}}(\mathbf{A})</math> is the classical action \eqref{classicalaction}.
However, as a consequence of [[gauge invariance]], the integrand is constant along gauge orbits (the trajectories obtained by starting from one gauge field and acting on it with all gauge transformations) and thus the integral is not defined.
The idea is then to introduce a surface (section) that cuts all gauge orbits  once and to restrict the integral to this section with a measure that ensures that all choices of sections are equivalent. This surface is defined by a (Lie algebra valued) constraint  of the form <math>\mathbf{G}(\mathbf{A},x)=0\ ,</math> known as gauge fixing condition or simply '''gauge fixing'''. The appropriate integration measure has been identified by Faddeev and Popov and involves the determinant of a linear operator <math>\mathbf{M}(\mathbf{A})</math> defined by
: <math>[\mathbf{M}(\mathbf{A})\omega](x) =\Delta_\omega \mathbf{G}(\mathbf{A},x),</math>
where <math>\Delta_\omega \mathbf{G} </math> is the variation of <math>\mathbf{G}</math> at  first order in <math>\omega</math> corresponding to the variation of <math>\mathbf{A}_\mu</math> in \eqref{Agaugetrinf}:
: <math>\Delta_\omega \mathbf{G}(\mathbf{A},x)=\mathbf{G}(\mathbf{A}-\mathbf{D}\omega,x)-\mathbf{G}(\mathbf{A},x)+O(\|\omega\|)^2.</math>
In relativistic-covariant gauges  <math>\mathbf{M}(\mathbf{A})</math> is typically a differential operator. 
For example, in Landau's gauge, 
:<math>\label{Landaugauge}
\mathbf{G}(\mathbf{A},x)\equiv\sum_\mu \partial_\mu \mathbf{A}^\mu(x)=0 \ \Rightarrow \ [\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}](x)=-\sum_\mu\partial_\mu {\mathbf D}^\mu   \bar{\mathbf{C}}(x)\,.</math>

It follows from the rules of integration in Grassmann algebras that determinants can be represented by integrals over generators of [[Grassmann algebra]]s, which are anti-commuting variables (see the section [[#The origin of BRST symmetry]] and equation \eqref{detE} in particular). In gauge theories, this corresponds to introducing two ''spinless'' fermion fields <math>\mathbf{C}(x)</math> and <math>\bar{\mathbf{C}}(x)\ ,</math> often called  Faddeev-Popov ghosts, which are matrices belonging to the Lie algebra of <math>G\ .</math> Such spinless fermions are ''unphysical'' because they violate the [[Spin and statistics|spin-statistics theorem]]. In addition, one introduces a scalar field <math>\boldsymbol{\lambda}(x)\ ,</math> again belonging to the Lie algebra, which is used to enforce the gauge condition <math>\mathbf{G}(\mathbf{A},x)=0</math> in the field integral (''i.e''., it acts as a Lagrange multiplier). 

The evolution of the quantized gauge theory is then described in terms of an integral over four types of fields, <math>\mathbf{A}_\mu(x),\boldsymbol{\lambda}(x),\mathbf{C}(x),\bar{\mathbf{C}}(x),</math> of the form
:<math>\label{gaugefieldintegral}
\mathcal{U}=\int[\mathrm{d}\mathbf{A}_\mu][\mathrm{d}\boldsymbol{\lambda}][[\mathrm{d}\mathbf{C}\mathrm{d}\bar{\mathbf{C}}]\exp\left( {i\over \hbar}\mathcal{S}_{\mathrm{quant.}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})\right),</math>

where <math>\mathcal{S}_{\mathrm{quant.}}</math> is a ''local'' action that reads
:<math>\label{Squant}
\mathcal{S}_{\mathrm{quant.}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\mathcal{S}_{\mathrm{class.}}(\mathbf{A})+\hbar\mathcal{S}_{\mathrm{gauge}} (\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})</math>

with
:<math>\label{SLandau}
\mathcal{S}_{\mathrm{gauge}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\int\mathrm{d}^4 x\,\mathrm{tr} \left[\boldsymbol{\lambda}(x)\mathbf{G}(\mathbf{A},x)-\mathbf{C}(x)\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}(x)\right].</math>


Finally, instead of using a constraint of the form <math>\mathbf{G}(\mathbf{A},x)=0\ ,</math>  it is often convenient to impose <math>\mathbf{G}(\mathbf{A},x)=\mathbf{s}(x)</math> and to integrate over the field <math>\mathbf{s}(x)</math> with a Gaussian distribution of width <math>\xi\ .</math> This procedure corresponds to extending the field integral to a whole neighbourhood of the gauge section  and, after the integration over <math>\mathbf{s}(x)\ ,</math> it amounts to adding a term quadratic in <math>\boldsymbol{\lambda}</math> to the gauge action,
which becomes
:<math>\label{Sgauge}
\mathcal{S}_{\mathrm{gauge}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\int\mathrm{d}^4 x\,\mathrm{tr} \left[\boldsymbol{\lambda}(x)\mathbf{G}(\mathbf{A},x)-\mathbf{C}(x)\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}(x) +{\xi\over 2} \boldsymbol{\lambda}^2(x)\right].</math>

In the limit <math>\xi\rightarrow 0\ ,</math> Landau's gauge fixing and the correspondent gauge action \eqref{SLandau} are recovered.