Gauge fixing

Gauge fixing removes the degeneracy among gauge-equivalent field configurations so that the kinetic operator has an inverse propagator.

For the Abelian Higgs mechanism with scalar phase $\chi$ and vector mass $m$, take ${\mathcal{L}}_{\mathrm{gf}}=-(\partial \cdot A+\xi m\chi {)}^2/(2\xi )$ when the scalar kinetic mixing is $-m{A}_{\mu }{\partial }^{\mu }\chi$. The cross term cancels this mixing after integration by parts. The resulting vector propagator isAt fixed finite $\xi$ it decays as $1/k^2$. Would-be Goldstone bosons and the required Faddeev-Popov ghosts remain in intermediate calculations; BRST symmetry identifies the physical states and ensures cancellation of unphysical modes. The limit $\xi \to \infty$ gives the unitary gauge propagator and is not uniform in ultraviolet momentum.

A gauge condition $F(x)=0$ is regular at a representative $x_0$ if its derivative along the gauge orbit is an invertible matrix. The inverse function theorem then makes the gauge-condition values valid local coordinates along that orbit. A unique zero without this regularity is insufficient: the translation action on the line with $F(x)=x^3$ has a unique zero but a zero derivative there. Multiple regular intersections give a Gribov ambiguity rather than a globally unique slice.

In gauge field theory its standard form is $A^{+}=n_{\mu }{A}^{\mu }=0$ for a fixed null direction $n$. A light-cone gauge uses light-cone coordinates to fix a longitudinal variable, often choosing $x^{+}$ as the evolution parameter for a relativistic particle phase-space action. Related field gauges set selected minus components to zero when a field has the required gauge invariance. A massive field without that gauge freedom must instead eliminate dependent components through its equations of motion.

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.

If a real nonzero field $\xi (x)$ appears in the Lorentzian density $-F_{ab}{F}^{ab}/4+({\partial }_a{A}^a{)}^2/(2\xi )$ and is varied, its algebraic Euler-Lagrange equation is $-({\partial }_a{A}^a{)}^2/(2{\xi }^2)=0$. Hence it imposes the Lorenz gauge condition on real classical fields. Varying $A_b$ gives ${\partial }_a{F}^{ab}-{\partial }^b[(\partial \cdot A)/\xi ]=0$, including derivatives of $\xi$. This constrained field theory differs from holding the gauge parameter fixed in a Gaussian functional integral.

For an Abelian gauge theory with $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, a nonlinear gauge functional $F[A]$ has Faddeev-Popov determinant $\det (\delta F[A+\partial \alpha ]/\delta \alpha )$. This operator can depend on $A$ even though the Adjoint representation of $U(1)$ is trivial. Consequently Faddeev-Popov ghost fields need not decouple in a nonlinear Abelian gauge, unlike a field-independent linear gauge condition.

For $\delta A_{\mu }={\partial }_{\mu }\omega$, the real quadratic gauge functional varies by $(\square +2A\cdot \partial )\omega$. Its Faddeev-Popov operator depends on the gauge field, producing a ghost-gauge vertex even in an Abelian gauge theory. With $\Psi =\int \bar{c}(F[A]+\xi h/2)$, a consistent left BRST symmetry convention gives $s\Psi =hF+\xi h^2/2-\bar{c}(\square +2A\cdot \partial )c$. The auxiliary-field integral at $\xi =0$ imposes the exact condition; finite $\xi$ gives a weighted gauge condition. This real functional is distinct from a complex quadratic Abelian gauge condition.

The formal variation of this condition is $\delta F=({\partial }^2+2iA\cdot \partial )\alpha$. Thus its Faddeev-Popov ghost field operator can be chosen as ${\mathcal{O}}_A=-{\partial }^2-2iA\cdot \partial$. With $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$, one has $D^2={\partial }^2+2iA\cdot \partial +iF[A]$, so ${\mathcal{O}}_A=-D^2$ on the formal gauge slice. The reality obstruction for a complex Euclidean gauge condition prevents interpreting this slice as an ordinary real Euclidean gauge fixing without an additional complex-contour prescription.

A massless complex bosonic charged scalar field with $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$ contributes $[\det (-D^2){]}^{-1}$ to a Gaussian background-field integral. A Faddeev-Popov ghost field pair contributes $\det {\mathcal{O}}_A$. On the formal complex quadratic Abelian gauge condition, ${\mathcal{O}}_A=-D^2$, so the two determinants cancel and their effective-action contributions sum to zero. Use matching regulators, boundary conditions and removed zero modes. This is a formal cancellation of closed loops; the reality obstruction for a complex Euclidean gauge condition precludes deducing that physical scalar quantum electrodynamics has no quantum effects.

For real $A_{\mu }$ in positive-definite Euclidean signature, $\partial \cdot A$ is real and $A_{\mu }{A}_{\mu }\ge 0$. Therefore $\partial \cdot A+i{A}^2=0$ forces $A^2=0$, and hence $A=0$. Its only real gauge orbit is the zero-field orbit; a field with nonzero electromagnetic curvature cannot be gauge-transformed to zero. Consequently this complex equation is not an admissible gauge fixing for general Hermitian Euclidean U(1) gauge symmetry configurations. A formal complexified treatment needs a specified contour and cannot be justified by a real delta-functional insertion.

The Faddeev-Popov ghost field couples to both the gauge field and the adjoint scalar field. In a convention with $D_{\mu }={\partial }_{\mu }+[A_{\mu },\cdot ]$, the ghost action after integration by parts contains $f^{abc}({\partial }_{\mu }{\bar{c}}^a)A_{\mu }^b{c}^c$ and $f^{abc}(n\cdot \partial {\bar{c}}^a){\varphi }^b{c}^c/\sqrt{2}$.

For gauge fixing with $F^a=\partial \cdot A^a+(n\cdot \partial {\varphi }^a)/\sqrt{2}$, integrating out the Nakanishi-Lautrup field produces gauge-scalar mixing. The Schur complement of the free gauge-field kernel cancels the extra $(n\cdot p{)}^2/2$ in the scalar kernel, leaving $⟨{\varphi }^a(p){\varphi }^b(-p)⟩={\delta }^{ab}/p^2$. This is independent of the gauge vector $n$, but still depends on the full momentum $p$.

A zero mode of a quadratic field operator is a nonzero field variation that the operator maps to zero. Infinitesimal pure-gauge variations are zero modes of an unfixed gauge kinetic operator, preventing it from having an inverse propagator.

For a minimally coupled massless scalar field in a flat radiation-dominated universe, conformal time gives $a=\alpha \tau$ and $(a\varphi {)}^{\prime \prime }-{\nabla }^2(a\varphi )=0$. Every nonzero Fourier mode is therefore a free oscillator divided by $a$. The zero-wavenumber equation instead gives $a{\varphi }_0=A+B\tau$, hence the displayed constant-plus-decaying solution. Substituting $k=0$ into a fixed-amplitude oscillatory formula retains only the $1/a$ term and loses the constant field solution. A truly general solution must include this zero mode or explicitly exclude a homogeneous background.

The Gribov ambiguity is the nonperturbative failure of a local gauge condition to intersect every gauge orbit exactly once. Distinct gauge-equivalent fields satisfying the same condition are Gribov copies.

Adding $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\alpha )$ gives a family of Lorentz-covariant gauges; $\alpha =1$ is Feynman gauge.

The Fourier-space quadratic Maxwell kernel after adding $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$. For nonzero $\xi$ and non-null four-momentum, its transverse and longitudinal momentum projectors give $K=-k^2(P_T+{\xi }^{-1}{P}_L)$ and its inverse is $-(P_T+\xi P_L)/k^2$. The inversion of the gauge-fixed Maxwell kinetic operator supplies the photon propagator with its vacuum boundary prescription.

The $\xi \to 0$ limit of covariant gauge fixing by $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$. It imposes the divergence condition sharply and gives a transverse photon propagator. This is distinct from the vector-potential choice called Landau gauge for a uniform magnetic field when computing Landau levels.

The covariant Landau gauge propagator is the transverse limit of the inversion of the gauge-fixed Maxwell kinetic operator. Contracting with $k^{\mu }$ gives zero away from the pole. The Lorenz gauge condition still has residual transformations satisfying $\square \chi =0$, so boundary conditions and the pole prescription complete the choice.

Use $F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$ and $(D_{\mu }c{)}^a={\partial }_{\mu }{c}^a+g{f}^{abc}{A}_{\mu }^b{c}^c$. The Faddeev-Popov gauge-orbit identity with Lorenz gauge function $G^a={\partial }^{\mu }{A}_{\mu }^a$, followed by Gaussian averaging of $G^a=f^a$, supplies the quadratic gauge-fixing term. Represent the determinant of $-{\partial }^{\mu }{D}_{\mu }$ using a Grassmann Gaussian integral over a Faddeev-Popov ghost field and antighost field to obtain the ghost term. The minus sign in this operator differs by a field-independent determinant factor from the direct gauge-condition Jacobian; it is a consistent antighost convention.

The ghost term equals $({\partial }^{\mu }{\bar{c}}^a)(D_{\mu }c{)}^a$ up to a boundary term. Thus it contains a ghost-gluon vertex with the sign matching this kinetic convention. An equivalent auxiliary-field form replaces the gauge-fixing term by $B^a{G}^a+\xi B^a{B}^a/2$. The Nakanishi-Lautrup field equation gives $B^a=-G^a/\xi$ and recovers the displayed Lagrangian. With left-acting BRST transformations $sA=Dc$, $sc=-g[c,c]/2$, $s\bar{c}=B$, and $sB=0$, the gauge-fixing and ghost terms together are $s[{\bar{c}}^a(G^a+\xi B^a/2)]$.

Gauge fixing removes the noninvertible longitudinal direction of the Maxwell quadratic action. In a gauge with parameter $\xi$, the Fourier kernel is $-k^2{\eta }^{\mu \nu }+(1-{\xi }^{-1})k^{\mu }{k}^{\nu }$. Multiplying its inverse by $i$ gives the photon propagator, with the Feynman i-epsilon prescription. At $\xi =1$ the answer is $-i{\eta }_{\mu \nu }/(k^2+i0)$.

For fixed $\xi \ne 0$, the Lorentzian quadratic density $-F^2/4+(\partial \cdot A{)}^2/(2\xi )$ has momentum kernel $K^{ab}=-p^2{\eta }^{ab}+(1+1/\xi )p^a{p}^b$. At $p^2\ne 0$ its inverse, defined by $K^{ac}{G}_{cb}={\delta }_b^a$, is $G_{ab}=-{\eta }_{ab}/p^2+(1+\xi )p_a{p}_b/(p^2{)}^2$. A vacuum photon propagator is $iG$ with the appropriate Feynman i-epsilon prescription. The usual parameter in the negative-sign covariant gauge term is $\alpha =-\xi$, so Feynman gauge is $\xi =-1$.

For the positive-sign covariant gauge-fixing inverse, $p^a{G}_{ab}=\xi p_b/p^2$. It is not identically zero for fixed $\xi \ne 0$ and is a nonzero vector at a nonzero non-null four-momentum. Null momenta require the chosen pole prescription, and individual zero components are possible. The covariant Landau gauge limit suppresses this longitudinal part. The Ward identity removes longitudinal contributions from physical conserved-current amplitudes.

With $D={\partial }_{\mu }{A}^{\mu }$, expand the Maxwell Lagrangian and its Feynman gauge term to findCommuting partial derivatives verifies the divergence identity. Both densities give $\square A^{\mu }=0$ under the usual variational boundary conditions. Their canonical momentum components differ: the original density gives $-F^{0\nu }-{\eta }^{0\nu }D$, whereas the subtracted density gives $-{\dot{A}}^{\nu }$. Thus a free-photon momentum expansion using the latter requires a stated boundary-term convention.

In mostly-minus signature, adding the boundary term ${\partial }_{\mu }{K}^{\mu }$ with $K^{\mu }=\frac{1}{2}(A_{\nu }{\partial }^{\nu }{A}^{\mu }-A^{\mu }{\partial }_{\nu }{A}^{\nu })$ to the subtracted Feynman gauge density changes ${\tilde{\pi }}^{\mu }=-{\dot{A}}^{\mu }$ to the displayed canonical momenta. For fields with vanishing spatial boundary terms, $\int K^0{d}^{3}x=\int A_0{\partial }_i{A}_i{d}^{3}x$, whose functional derivatives are these momentum shifts. They depend only on coordinates, so $[A_{\mu },{\pi }^{\nu }]=i{\delta }_{\mu }{}^{\nu }{\delta }^3$ is unchanged. The only potentially nonzero mixed momentum bracket is $[{\pi }^0(x),{\pi }^i(y)]=i({\partial }_{y^i}+{\partial }_{x^i}){\delta }^3(x-y)=0$. Thus both density representatives define canonical commutation relations, although their momenta must not be equated componentwise.

Gupta-Bleuler quantization keeps Lorentz covariance by first quantizing all four photon polarizations in an indefinite inner-product space. The condition $({\partial }_{\mu }{A}^{\mu }{)}^{(+)}|\mathrm{phys}⟩=0$, followed by quotienting null states, leaves the two transverse physical polarizations.

Choose $C_{\mathbf{p}}=a_{\mathbf{p}}^0-a_{\mathbf{p}}^3$ with the standard temporal and longitudinal polarization vectors. The physical pre-space is the common kernel of all $C_{\mathbf{p}}$. In each regulated mode, $a^0$ acts on creator polynomials as $-{\partial }_{a^{0†}}$ and $a^3$ as ${\partial }_{a^{3†}}$, so the constraint kernel consists of transverse creator polynomials and polynomials in $C^{†}=a^{0†}-a^{3†}$. Since $[C,C^{†}]=0$, these latter excitations remain constrained. They are orthogonal to every constrained state because $⟨\Phi |C^{†}=⟨C\Phi |=0$. Quotienting this radical of a Hermitian form leaves only the transverse bosonic Fock space with a positive inner product. The constraint alone gives a positive semidefinite Hermitian form; quotienting removes its null directions. If starting from finite-particle creator polynomials, take the Hilbert space completion of this positive quotient to obtain the physical Hilbert space.

Choose a temporal polarization vector, two transverse polarization vectors and a spatial longitudinal polarization vector. The Gupta-Bleuler quantization condition is $(a^0-a^3)|\chi ⟩=0$. Applied to ${\sum }_{\lambda }{c}_{\lambda }{a}^{\lambda †}|0⟩$, it gives $c_3=-c_0$. The remaining temporal-longitudinal excitation $a^{0†}-a^{3†}$ has zero norm and is orthogonal to every constrained state. Its field wavefunction is proportional to the null four-momentum, hence is pure gauge. Quotienting this direction leaves the two positive transverse photon states. Use wave packets or a box normalization to avoid treating continuum momentum eigenstates as normalizable vectors.

The covariant free-photon oscillator construction uses four polarization vectors and $[a^{\lambda },a^{{\lambda }^{\prime }†}]=-{\eta }^{\lambda {\lambda }^{\prime }}$. Starting from a positive Fock vacuum, it induces an indefinite Hermitian form on the multiparticle state space: the temporal oscillator has negative norm while the three spatial oscillators have positive norm. It is therefore not the physical positive Hilbert space. The Gupta-Bleuler null-state quotient selects a positive physical space with two transverse photon polarizations. Continuum momentum oscillators and their states are understood after smearing or finite-volume regularization.

With real complete polarization vectors, the covariant photon Fock space oscillator commutators and ${\sum }_{\lambda ,{\lambda }^{\prime }}{ϵ}_{\mu }^{\lambda }{ϵ}^{\nu {\lambda }^{\prime }}{\eta }_{\lambda {\lambda }^{\prime }}={\delta }_{\mu }{}^{\nu }$ imply the equal-time canonical commutation relations $[A_{\mu }(\mathbf{x}),-{\dot{A}}^{\nu }(\mathbf{y})]=i{\delta }_{\mu }{}^{\nu }{\delta }^3(\mathbf{x}-\mathbf{y})$. The two mixed creation-annihilation terms add; the two coordinate-coordinate or momentum-momentum terms cancel after their polarization sums. The sign of the time-component oscillator is essential to the identity. Boundary-induced canonical transformation in Feynman gauge explains the momenta for the unsubtracted density.

For a temporal photon wave packet $|f,0⟩=\int d^{3}pf(\mathbf{p})a_{\mathbf{p}}^{0†}|0⟩/(2\pi {)}^3$, the oscillator commutator gives $⟨f,0|f,0⟩=-\int d^{3}p|f(\mathbf{p}){|}^2/(2\pi {)}^3<0$ when $f\ne 0$. This is the negative direction of the indefinite Hermitian form, not the divergent normalization of an unsmeared momentum eigenstate. It is excluded by the physical condition in Gupta-Bleuler quantization.

For the Euclidean covariant gauge term $({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$, the free gauge-boson propagator is

With a mostly-plus metric signature, $e^{iS}$, and gauge-fixing term $-(\partial \cdot A{)}^2/2$, the free adjoint gauge-field kernel is $-p^2{\eta }_{\mu \nu }{\delta }_{ab}$. Inverting it gives the displayed Feynman propagator. This convention uses the same denominator and sign as a massless real scalar propagator.

At $\xi =1$, the Euclidean photon propagator is $D_{\mu \nu }(k)={\delta }_{\mu \nu }/k^2$. Its Lorentz-index contraction simplifies loop numerators in quantum electrodynamics.

At a nonzero Euclidean momentum $k$, the transverse projector of a vector field is $P_T^{ab}={\delta }^{ab}-k^a{k}^b/k^2$. In a Lorentzian Minkowski metric, its mixed-index form is $(P_T{)}^a{}_b={\delta }_b^a-p^a{p}_b/p^2$, defined for $p^2\ne 0$. It obeys $P_{T}p=0$ and $P_T^2=P_T$. A nonzero null four-momentum is insufficient for this particular decomposition because the denominator vanishes.

At a nonzero Euclidean momentum $k$, the longitudinal projector of a vector field is $P_L^{ab}=k^a{k}^b/k^2$. Its Lorentzian mixed-index form is $(P_L{)}^a{}_b=p^a{p}_b/p^2$ for non-null four-momentum. Together with the transverse projector of a vector field, it obeys $P_T+P_L=I$, $P_T{P}_L=0$, $P_T^2=P_T$, and $P_L^2=P_L$. This formula is not a pointwise decomposition at null four-momentum.

The Faddeev-Popov determinant is the functional Jacobianthat compensates for the change from integration along a gauge orbit to a gauge-fixing condition $G(A)=0$. It can be represented by a path integral over a Faddeev-Popov ghost field pair.

Suppose a gauge fixing condition intersects each gauge orbit once in the local region under consideration, and its linearized Faddeev-Popov operator has no residual zero modes after boundary conditions are imposed. The functional Jacobian for changing from the gauge transformation parameter to the gauge condition gives the displayed identity. Inserting it in a gauge-invariant functional integral allows the integration along the gauge orbit to factor out and cancel the formal gauge-group volume. In a perturbative neighborhood the Jacobian's sign or phase is fixed; a field-independent determinant factor is absorbed into normalization. The Gribov ambiguity prevents treating this local identity as a globally unique gauge choice without further qualifications.

Assume an invariant volume measure, an invariant integrand and one regular gauge slice representative per orbit. At that representative, let $M$ be the Jacobian matrix of $F$ along the group coordinates, whose Haar measure is normalized to ordinary coordinate measure at the identity. Changing coordinates from the group parameter to $F$ gives ${\int }_{G}d\mu (g)\delta (F(x_g))=1/|\det M(x_0)|$. Multiply by its reciprocal and insert this identity into the original integral. Invariance permits a change of variables $x\mapsto x_g$ for each $g$, factoring out the group volume. The absolute determinant is required for ordinary unoriented real integration; a consistently positive orientation permits writing $\det M$. Infinite group volume is divided out formally or treated with a regulator.

For a gauge condition $\chi [A]=0$, the linearization of $\chi [A+D\omega ]$ in the gauge parameter is the Faddeev-Popov operator. Its determinant is represented by Faddeev-Popov ghost fields.

A Faddeev-Popov ghost field is a Grassmann-valued scalar field whose Gaussian functional integral represents the Faddeev-Popov determinant. Ghosts occur only on internal lines and cancel unphysical gauge-field contributions.

A ghost propagator is the two-point function $⟨Tc(x)\bar{c}(y)⟩$ of a Faddeev-Popov ghost field and its independent Faddeev-Popov antighost field. For quadratic action $\int \bar{c}Mc$ and source ordering consistent with that field ordering, the Grassmann integral gives $G_c=i{M}_F^{-1}$. The subscript specifies the causal pole prescription. For the negative derivative kinetic term $-\int \partial \bar{c}\cdot \partial c$, the Fourier kernel is $M=-p^2$. With signature $(-,+,+,+)$ this gives $G_c=-i/(p^2-i0)$. Field normalization multiplies the inverse kernel accordingly, while an antighost sign redefinition changes propagator and vertex signs together. Anticommuting statistics and loop signs do not turn these auxiliary fields into physical asymptotic particles.

With signature $(+---)$ and action $-\int {\partial }^{\mu }\bar{c}{\partial }_{\mu }c$, integration by parts gives the quadratic kernel $K_c=\square$, or $-p^2$ in Fourier space. The Grassmann Gaussian integral with odd sources gives $Z=\exp [-i\bar{\eta }{K}_{c,F}^{-1}\eta ]$. Left and right source derivatives, including their insertion factors, yield $⟨Tc\bar{c}⟩=i{K}_{c,F}^{-1}$ and hence the displayed sign. Replacing $\bar{c}$ by $-\bar{c}$ changes this propagator and the ghost-gluon vertex together; it is a convention change, not a physical discrepancy.

Expanding the ghost covariant derivative in Yang-Mills theory produces an interaction between the Faddeev-Popov ghost field, Faddeev-Popov antighost field, and gauge boson. For the displayed kinetic convention, Fourier factors $e^{-ipx}$ and all momenta incoming give vertex $-g{f}^{abc}{k}_{\mu }$, where $k$ is the antighost momentum. The structure constants encode the Adjoint representation. Propagator, vertex, Fourier, and ghost-order conventions must be chosen consistently.

For canonically normalized fields with action $-\int [\frac{1}{4}{F}^2+\frac{1}{2}(\partial \cdot A{)}^2+\partial \bar{c}\cdot Dc]$, use signature $(-,+,+,+)$ and incoming Fourier modes $e^{ipx}$. The ghost-gluon vertex is $g{p}_{\mu }{f}_{abc}$ and the ghost propagator is $-i/(p^2-i0)$. The open ghost-line one-particle-irreducible two-point vertex at one loop is $i{g}^{2}C{I}_d(p)$, whereA counterterm ${\mathcal{L}}_{\mathrm{ct}}=-\delta Z_c\partial \bar{c}\cdot \partial c$ inserts $-i\delta Z_c{p}^2$ and cancels this pole. Equivalently, writing $d=4-2ϵ$ gives $\delta Z_c=g^{2}C/(32{\pi }^2ϵ)$. There is no closed ghost-loop sign for this open-line graph. The Feynman i-epsilon prescription, Fourier convention and antighost sign must be kept consistent throughout.

The Grassmann-odd partner of the Faddeev-Popov ghost field in the gauge-fixing determinant representation. In an auxiliary-field BRST symmetry description it obeys $sb=h$, where $h$ is the Nakanishi-Lautrup field. Its ordinary spacetime derivative remains odd, which matters when moving an anticommuting BRST parameter past it.

The additive grading assigns ghost number $+1$ to $c$ and $-1$ to $\bar{c}$. Physical observables have ghost number zero. This grading is distinct from spin or the Adjoint representation.

For ghost term $-({\partial }^{\mu }\bar{c})\cdot D_{\mu }c$ and the current convention $\delta S=-\int ({\partial }_{\mu }\theta )j_G^{\mu }$, localize $\delta c=\theta c$, $\delta \bar{c}=-\theta \bar{c}$. The two differentiated parameters give this current. Its Noether charge grades ghosts by $+1$ and antighosts by $-1$, up to the corresponding overall generator normalization.

A ghost loop is a closed cycle of ghost propagators in a Feynman diagram. Its Grassmann statistics contribute an additional minus sign.

An axial gauge imposes $n^{\mu }{A}_{\mu }=0$ for a fixed vector $n^{\mu }$. Its Faddeev-Popov operator is $n^{\mu }{D}_{\mu }$; on the strict gauge slice its gauge-field-dependent part vanishes, so its ghosts decouple.

The inverse of $n\cdot \partial$ and the axial-gauge propagator are singular when $n\cdot k=0$. A pole prescription and residual-gauge boundary conditions specify these inverses. They must be compatible throughout a calculation; formal ghost decoupling alone does not specify them.

In strict axial gauge, the gauge-boson propagator obeys $n^{\mu }{D}_{\mu \nu }(k)=0$ and contains poles at $n\cdot k=0$. Its contraction with conserved external currents removes the gauge-choice terms. Consistent calculations require an axial-gauge pole prescription.

Temporal gauge sets the time component of a gauge potential to zero, $A_0=0$. The mixed curvature components then reduce to time derivatives of the spatial connection, $F_{0i}={\partial }_0{A}_i$.

BRST symmetry is a nilpotent fermionic symmetry of a gauge-fixed action. Its differential $s$ replaces an infinitesimal gauge parameter by the ghost field and satisfies $s^2=0$.

For adjoint fields whose coefficients have Grassmann parity, $X\times Y=-(-1{)}^{|X||Y|}Y\times X$. Thus two odd ghost fields have a symmetric adjoint bracket, and $c\times c$ need not vanish. The graded Jacobi identity and this symmetry are the algebraic mechanisms behind BRST nilpotence.

A gauge quantization prescription using a nilpotent BRST charge on the gauge-fixed state space. Physical states are represented by its BRST cohomology, usually at ghost number zero. Exact states decouple under suitable invariant-measure, boundary and positivity assumptions. Nilpotence alone does not prove positivity of the quotient or select its physical ghost-number sector.

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 $\{b,c\}=-i$.

A BRST charge is nilpotent when $Q^2=0$, making BRST cohomology well-defined. For the critical bosonic string theory, the matter and ghost central charges cancel, $D-26=0$, and the intercept is one. In a ghost convention with a zero-mode linear shift, the shifted total constraint generators, rather than the unshifted ones, obey the centerless Witt algebra.

Using the graded Leibniz rule, $s^{2}A=D(sc)+g(Dc)\times c=0$ because $D(c\times c)=2(Dc)\times c$. The graded Jacobi identity gives $s^{2}c=g^2(c\times c)\times c/2=0$. BRST nilpotence on the antighost and Nakanishi-Lautrup field is immediate. No field equations are required.

Writing a transformation as $\delta \Phi =ϵs\Phi$ with the odd parameter on the left defines this graded derivation convention. For Yang-Mills fields, $sA=Dc$, $sc=-[c,c]/2$, $sb=h$, and $sh=0$. The Jacobi identity supplies nilpotence; the placement of the odd parameter controls signs in a Noether current.

A right-acting BRST differential obeys the right graded Leibniz rule. This convention permits $Q\varphi =[c,\varphi ]$ together with $Qc=-[c,c]/2$. Using a left graded Leibniz rule with those same two signs does not give a nilpotent operator; one of the signs must change.

The Nakanishi-Lautrup field is an auxiliary bosonic field used to write gauge fixing as a BRST-exact term. Its algebraic field equation produces the usual quadratic gauge-fixing term.

Complete the square as $bF+\xi b^2/2=\xi (b+F/\xi {)}^2/2-F^2/(2\xi )$. Translation of the Gaussian functional integral gives the displayed identity for nonzero $\xi$, with the appropriate oscillatory contour prescription. At $\xi =0$, the Nakanishi-Lautrup field is a Lagrange multiplier imposing the gauge-condition delta functional. This field has no kinetic term and no independent propagating degree of freedom.

The BRST charge generates BRST transformations. It has odd Grassmann parity and is nilpotent, $Q^2=0$.

On a positive-definite Hilbert space, these two identities imply $\Vert Q\psi {\Vert }^2=⟨\psi ,Q^2\psi ⟩=0$, hence $Q=0$ on its invariant domain. A nontrivial covariant BRST charge therefore acts before reduction on an indefinite-inner-product gauge/ghost space. Exact vectors are null and orthogonal to closed vectors, and BRST cohomology quotients them out. Positivity of the reduced physical space requires the additional gauge-theory structure; it does not follow from abstract nilpotence alone.

The FP ghost square vanishes, so this BRST charge is nilpotent. On a ghost-number-zero wavefunction, BRST cohomology implements the Klein-Gordon equation. Closure of an unrestricted ghost-extended wavefunction does not independently constrain every component: a pure FP ghost component can lie in the kernel without satisfying the same scalar equation. The FP ghost sector is part of the physical prescription.

Use gauge-fixing term $({\partial }^{\mu }b)\cdot A_{\mu }$, ghost term $-({\partial }^{\mu }\bar{c})\cdot D_{\mu }c$, and the left-acting BRST differential with $s\bar{c}=b$. Localizing the odd parameter on the left yields $\delta S=-\int ({\partial }_{\mu }ϵ)j_B^{\mu }$. Moving it through the antighost derivative determines the last sign. Changing the overall Noether convention reverses this current; rewriting gauge fixing by integration by parts changes its improvement term. Every summand has ghost number one.

For the left-parameter convention and ghost kinetic term $(\partial b)Dc$, localization of the BRST symmetry yields this conserved Noether current. Subtract the total-derivative contribution $K^{\mu }=h^a(D^{\mu }c{)}^a$ in the Lagrangian variation. The BRST charge is the spatial integral of $j^0$. The antighost term has the displayed sign because the parameter and antighost field derivative anticommute.

For a BRST-invariant action and functional measure, the expectation of a BRST-exact insertion vanishes: $⟨QX⟩=0$, subject to anomaly and boundary qualifications.

For auxiliary-field covariant gauge fixing, varying the parameter inserts a BRST-exact operator. Its correlations with separated physical BRST-closed operators vanish by the BRST Ward identity. Normalized physical correlators are therefore independent of the parameter under the invariant-measure and boundary assumptions.

For an admissible family of gauge-fixing fermions, variation inserts a BRST-exact operator into normalized correlations. If the measured operators are BRST-closed, the graded BRST Ward identity makes that insertion vanish. This extends gauge-parameter independence to changes of the gauge functional itself, provided the action and measure preserve BRST symmetry and the change produces no boundary contribution. It is the perturbative statement; global gauge-orbit problems require additional analysis.

An invariant path-integral measure and BRST symmetry action imply this identity when every separated insertion $G_i$ is BRST-closed. The bracket is the graded commutator, so it becomes an anticommutator for an odd $\mathcal{O}$. The arbitrary operator need not itself be closed. Exact insertions therefore decouple from physical correlators.

A gauge-fixing fermion is a Grassmann-odd functional whose BRST transformation $s\Psi$ supplies the gauge-fixing and ghost terms. Nilpotence gives $s(s\Psi )=0$ immediately.

For a left left-acting BRST differential with $sA=Dc$, $s\bar{c}=b$, $sb=0$, choose $\Psi ={\bar{c}}^a(\partial \cdot A^a+\xi b^a/2)$. The graded Leibniz rule gives $s\Psi =b\partial \cdot A+\xi b^2/2-\bar{c}\partial \cdot Dc$. Integrating the auxiliary Gaussian yields $-(\partial \cdot A{)}^2/(2\xi )-\bar{c}\partial \cdot Dc$. At $\xi =1$ this is Feynman gauge. Opposite antighost or exact-action sign conventions require a corresponding change of fermion.

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