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A gauge-fixed action supplements a gauge-invariant action with a term imposing the gauge condition and with Faddeev-Popov ghost fields representing the Faddeev-Popov determinant. A local gauge condition gives local differential operators and a local action.

Use the Minkowski metric, path-integral weight $e^{iS}$ and the Abelian gauge theory transformation $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\omega$. The gauge functional $F[A]=\partial \cdot A+A^2$ varies asThus the Faddeev-Popov operator is ${\mathcal{M}}_A=\square +2A\cdot \partial$. It depends on the gauge field despite the gauge group being Abelian: the ghosts interact because this gauge condition is nonlinear.

Choose the gauge-fixing fermion $\Psi =\int d^{4}x\bar{c}(F[A]+\xi h/2)$. The gauge-fixed action $S=S_{\mathrm{Maxwell}}+s\Psi$ isHere the tensor $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is distinct from the scalar gauge functional $F[A]$. With $\xi =0$, integrating over $h$ imposes the exact printed constraint. For nonzero $\xi$, eliminating $h=-F[A]/\xi$ instead givesThis version displays the additional gauge-dependent cubic and quartic gauge-field vertices as well as the ghost interaction; the strict condition is its $\xi \to 0$ limit.

For the Fourier transform convention $c(x)=\int d^{4}p(2\pi {)}^{-4}{e}^{-ipx}c(p)$, the quadratic ghost kernel is $p^2$. With the ordering $⟨c(p)\bar{c}(q)⟩$,The term $-2\bar{c}{A}^{\mu }{\partial }_{\mu }c$ has Fourier coefficient $2i{p}_{\mu }$, where $p$ is the incoming ghost momentum. Multiplication by $i$ in the Feynman rule givesThese signs refer to the displayed action, Fourier convention and ghost ordering. Reversing the ghost/antighost convention changes corresponding signs consistently. A closed ghost loop has the additional minus sign from Grassmann variables.

Use Lie algebra generators with $[T_b,T_c]=f^a{}_{bc}{T}_a$, and define the adjoint gauge covariant derivative by $(D_{\mu }X{)}^a={\partial }_{\mu }{X}^a+f^a{}_{bc}{A}_{\mu }^b{X}^c$. Varying the gauge field strength gives $\delta F_{\mu \nu }=D_{\mu }\delta A_{\nu }-D_{\nu }\delta A_{\mu }$. Antisymmetry followed by integration by parts therefore givesFor variations vanishing at the boundary, the Yang-Mills equations areThe Jacobi identity for the commutators of gauge covariant derivatives, using $[D_{\mu },D_{\nu }]X=[F_{\mu \nu },X]$, gives the gauge-theory Bianchi identity:In the second form of the gauge-theory Bianchi identity ${\tilde{F}}^{\mu \nu }=\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{F}_{\rho \sigma }$. The identity is a geometric consequence of the definition of curvature, not a second dynamical field equation.

For the usual non-Abelian Yang-Mills theory, the quantum running coupling becomes strong at low energies. Dimensional transmutation generates a scale absent from the classically scale-invariant equations. The expected confining dynamics and massive colour-singlet spectrum involve confinement and a mass gap, rather than freely propagating weakly coupled coloured waves. These nonperturbative effects are not captured by simply solving the classical equations; this is not a claim of a mathematical proof of the Yang-Mills mass gap.

Write the gauge condition as ${\chi }^a[A]=f^a[A]$ to distinguish it from the structure constants. Under an infinitesimal gauge transformation, ${\delta }_{\omega }{A}_{\mu }=D_{\mu }\omega$. Define the Faddeev-Popov operator by ${\mathcal{M}}^{ab}=\delta {\chi }^a[A+D\omega ]/\delta {\omega }^b{|}_{\omega =0}$. In a Euclidean convention, the gauge-fixing and ghost additions can be chosen asA nonlocal kernel would require a double integral; a local functional $\chi$ instead makes $\mathcal{M}$ a local differential operator. Choosing a local gauge functional thus retains a local gauge-fixed action and the ordinary local interaction structure of perturbative quantum field theory.

The fields $c^a$ and ${\overline{c}}^a$ are independent Grassmann-valued fields in the Adjoint representation, Lorentz scalars with ghost numbers $+1$ and $-1$. They are the Faddeev-Popov ghost fields: their functional integral produces $\det \mathcal{M}$, the Faddeev-Popov determinant correcting the volume element along a gauge orbit. They are internal fields, not physical asymptotic particles. Their statistics supply a minus sign for every closed ghost loop. No auxiliary field is needed in the displayed gauge-fixing representation.

In axial gauge, ${\chi }^a=n^{\mu }{A}_{\mu }^a$, soWith the Fourier transform convention $e^{ik\cdot x}$, the ghost Feynman rules in the original normalization of $A$ areMomentum conservation accompanies the vertex; each ghost loop has the extra minus sign. With a canonically normalized field $A=g{A}_{\mathrm{can}}$, the vertex is instead $-g{f}^a{}_{bc}{n}_{\mu }$. The gauge fixing term is quadratic and introduces no further interaction vertex.

The Euclidean quadratic kernel of $A$ is $g^{-2}[k^2{\delta }_{\mu \nu }-k_{\mu }{k}_{\nu }+{\xi }^{-1}{n}_{\mu }{n}_{\nu }]$. For $n\cdot k\ne 0$, its inverse isThis follows by multiplication with the kernel: the $k^2\delta -kk$ piece gives $I-n{k}^T/(n\cdot k)$, and the ${\xi }^{-1}nn$ piece supplies the missing $n{k}^T/(n\cdot k)$. The strict axial-gauge propagator is thereforeIt obeys $n^{\mu }{D}_{\mu \nu }=0$. Every ghost attachment to an internal gauge propagator therefore vanishes in the strict limit; equivalently, $\mathcal{M}=n\cdot \partial$ on the gauge slice. Thus the interacting Faddeev-Popov ghosts decouple. Residual gauge transformations and the poles at $n\cdot k=0$ require compatible boundary conditions and an axial-gauge pole prescription.

After Wick rotation, the corresponding Minkowski propagator has ${\delta }_{\mu \nu }$ replaced by ${\eta }_{\mu \nu }$ and the usual overall factor $-i$, with a compatible causal prescription. The fixed $n$ makes the gauge-fixed propagator nonmanifestly Lorentz covariant. Nevertheless, gauge invariance makes physical observables independent of this gauge-choice vector, so their perturbative predictions are Lorentz invariant. Already at tree level, contraction with conserved external currents removes all terms containing $k_{\mu }$ or $k_{\nu }$, leaving $g^{2}j\cdot j^{\prime }/k^2$. The quantum Ward identities give the analogous cancellation in complete physical amplitudes; arbitrary gauge-dependent Green functions need not be independent of $n$.