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For $L[A^a]=n^{\mu }{A}_{\mu }^a$, eliminating $B^a$ gives the axial gauge term $(n\cdot A^a{)}^2/(2\xi )$ and ghost operatorIn the strict $\xi \to 0$ gauge, $n\cdot A=0$. The gauge-field-dependent part of $n\cdot D$ then vanishes, the Faddeev-Popov determinant becomes field independent, and the ghosts decouple. The one-loop gluon propagator therefore receives the gluon bubble but no ghost bubble. As in the covariant gauge, the four-gluon tadpole can occur with a hard cutoff and vanishes as a scaleless integral in dimensional regularization. Gauge-invariant observables agree between the two gauges even though their individual propagators and diagrammatic decompositions differ.

For the convention in the question, a finite Yang-Mills gauge transformation acts covariantly on the field strength:or with $U$ and $U^{†}$ exchanged if the opposite convention is used for $D_{\mu }$. Cyclicity of the matrix trace givesso the Yang-Mills theory Lagrangian is gauge invariant.

The quadratic gauge-field operator has zero directions $A_{\mu }\sim A_{\mu }+D_{\mu }\alpha$ along each gauge orbit. It therefore has no inverse on the full field space. Gauge fixing removes this degeneracy and produces a propagator, while the Faddeev-Popov determinant accounts for the corresponding Jacobian.

For axial gauge,so the Faddeev-Popov determinant is $\det (n^{\mu }{D}_{\mu })$. Representing it with a Faddeev-Popov ghost field pair givesThe delta functional sets $\omega =n\cdot A$ in its Gaussian weight, and henceSubstitution yieldsIn the strict axial-gauge limit $n\cdot A=0$, $n\cdot D=n\cdot \partial$, so the ghost determinant is independent of $A$ and can be absorbed into $N$.

Choose a reference metric in each conformal class. An infinitesimal worldsheet diffeomorphism generated by $c^{\alpha }$ and a Weyl transformation decompose the metric variation into its trace and the traceless operatorInserting the gauge condition and its Faddeev-Popov determinant cancels the formal gauge-orbit volume. Representing $\det P_1$ by anticommuting worldsheet ghost fields giveswhere $b_{\alpha \beta }$ is symmetric and traceless and $c^{\alpha }$ is a vector. The gauge-fixed integral is consequentlyOn higher-genus worldsheets one must additionally integrate over moduli and treat conformal-Killing and ghost zero modes separately; these finite-dimensional factors are suppressed in the displayed formal expression.

For a relativistic particle, the Hamiltonian gauge parameter changes the worldline einbein by its time derivative. Fixing the einbein to a constant therefore gives a Faddeev-Popov determinant of ${\partial }_t$, represented by anticommuting FP ghosts. Gauge zero modes and the proper-time modulus are treated separately. The displayed phase convention gives the odd canonical bracket $\{b,c\}=-i$.

In two dimensions the metric variation splits into a scalar trace and the trace-free diffeomorphism operator $(P_{1}v{)}_{ab}={\nabla }_a{v}_b+{\nabla }_b{v}_a-g_{ab}\nabla \cdot v$. Integrating the Weyl parameter in the inverse Faddeev-Popov determinant removes the trace delta function. The remaining determinant is exponentiated by an anticommuting vector ghost and symmetric trace-free antighost, yielding the two chiral $b\bar{\partial }c$ terms. Conformal Killing vectors and moduli are zero modes treated separately from this local determinant.

On a unit interval with gauge parameters vanishing at its ends, $e=s+\dot{ϵ}$ separates the worldline einbein into its invariant average $s$ and a canonical gauge transformation. The Jacobian for nonconstant modes is $\det {\partial }_t$ between the appropriate boundary-condition spaces. On a circle its prime excludes the constant gauge-parameter zero mode. This Faddeev-Popov determinant can be represented by anticommuting $b,c$ fields with an action proportional to $\int b\dot{c}dt$.

The trace-free metric variation is $(P_{1}c{)}_{ab}={\nabla }_a{c}_b+{\nabla }_b{c}_a-{\gamma }_{ab}{\nabla }_d{c}^d$. A symmetric trace-free antighost exponentiates the Faddeev-Popov determinant. In light-cone worldsheet coordinates its two chiral kinetic terms are $b_{++}{\partial }_{-}{c}^{+}$ and $b_{--}{\partial }_{+}{c}^{-}$.