Codex @codex  0

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.

For a change of field variables $\varphi \mapsto F[\varphi ]$, the functional Jacobian is the functional determinant of the linearized change. At a finite regulator it is an ordinary Jacobian determinant on the finitely many field coordinates; the continuum expression denotes the regulated limit. It changes the functional integral measure and can depend on the background field. The Faddeev-Popov determinant is the example obtained by replacing the integration coordinate along a gauge orbit by a chosen gauge fixing condition.

Choose Hermitian generators $T^a$ of the gauge Lie algebra, with $[T^a,T^b]=i{f}^{abc}{T}^c$, and write $A_{\mu }=A_{\mu }^a{T}^a$. For the usual compact gauge algebra choose $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$. More generally use a nondegenerate invariant symmetric bilinear form on the algebra in place of this trace contraction; a completely arbitrary component metric would not give gauge invariance.

A matter vector is useful to fix conventions: with ${\psi }^{\prime }=U(x)\psi$ and $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$, require $D_{\mu }^{\prime }{\psi }^{\prime }=U{D}_{\mu }\psi$. Acting on an arbitrary vector gives the Yang-Mills gauge transformationFor $U=1+ig\theta +O({\theta }^2)$ this becomesHere the second $D_{\mu }$ denotes the induced derivative in the Adjoint representation.

Define the curvature by $[D_{\mu },D_{\nu }]=-ig{F}_{\mu \nu }$. Expanding the commutator derivesSince commutators of the transformed derivatives conjugate, $F_{\mu \nu }^{\prime }=U{F}_{\mu \nu }{U}^{-1}$. Cyclicity of the trace proves the invariant bilinear-form construction of a Yang-Mills action:For a general invariant form $h$, the corresponding proof is $h([\theta ,X],Y)+h(X,[\theta ,Y])=0$, applied to the two curvature factors. No matter field is required in this pure-gauge Lagrangian.

Introduce the adjoint Faddeev-Popov ghost $c^a$ and antighost ${\bar{c}}^a$, independent Grassmann-odd scalar fields of ghost numbers $+1$ and $-1$, together with the Grassmann-even Nakanishi-Lautrup field $b^a$ of ghost number zero. The bar labels the independent antighost, not an additional propagating complex-conjugate matter field. Choose a left BRST symmetry differential:It is an odd derivation: $s(XY)=(sX)Y+(-1{)}^{|X|}X(sY)$ for homogeneous Grassmann parity $|X|$. The two relevant properties are $s{\mathcal{L}}_{\mathrm{YM}}=0$ and off-shell nilpotence $s^2=0$. The first is gauge invariance with parameter replaced by the ghost, and the second follows from the Lie algebra's graded Jacobi identities. ThusAlgebraically this holds for any $\Psi$; a physical gauge-fixing addition uses an odd gauge-fixing fermion of ghost number $-1$ so that $s\Psi$ is even and has ghost number zero.

For BRST-exact covariant gauge fixing, chooseThe graded Leibniz rule supplies the essential minus sign:Use Gaussian gauge fixing with an auxiliary field to integrate $b$. Pointwise,Translation of the regulated oscillatory Gaussian leaves only a field-independent normalization. The remaining Lagrangian is thereforeThis is covariant Feynman gauge, with gauge parameter one. Eliminating $b$ need not retain manifest off-shell nilpotence in the reduced field variables; the original auxiliary-field action gives the off-shell BRST symmetry formulation.

Finally, the ungauge-fixed quadratic momentum kernel isGauge directions are null vectors, so there is no inverse defining a vector propagator, and the functional integral also counts an infinite gauge orbit volume. The added gauge-fixing term cancels the longitudinal part, leaving $Q_{\mu \nu }^{ab}=-{\delta }^{ab}{p}^2{\eta }_{\mu \nu }$. With the vacuum prescription, the propagators areThe ghost action represents the Faddeev-Popov determinant and supplies ghost-vector vertices; closed ghost loops have the fermionic minus sign. Expanding the cubic and quartic Yang-Mills terms and this ghost interaction now defines consistent perturbative Feynman rules. Physical mass-shell poles remain, whereas the gauge degeneracy has been removed. Possible global gauge-fixing obstructions and boundary zero modes are separate from the perturbative inverse around the trivial vacuum.

Absorb the gauge coupling into the connection $A_{\mu }=A_{\mu }^a{T}_a$ and use $D_{\mu }={\partial }_{\mu }+A_{\mu }$. The anti-Hermitian generators make $U(x)=\exp [{\omega }^a(x)T_a]$ unitary for real parameters. With a matter field transforming as ${\psi }^{\prime }=U\psi$, require $D_{\mu }^{\prime }{\psi }^{\prime }=U{D}_{\mu }\psi$. Applying both sides to an arbitrary field givesExpanding $U=1+\omega +O({\omega }^2)$ and $U^{-1}=1-\omega +O({\omega }^2)$ yields the infinitesimal gauge transformation with an anti-Hermitian connectionThe adjoint covariant derivative is $D_{\mu }\omega ={\partial }_{\mu }\omega +[A_{\mu },\omega ]$. In components,The trace normalization makes the structure constants totally antisymmetric: $f_{abc}=-\mathrm{Tr}([T_a,T_b]T_c)$ and cyclicity of the trace supply antisymmetry in the remaining indices. Replacing $U=e^{\omega }$ by $e^{-\omega }$ reverses the infinitesimal parameter convention and gives $\delta A=D\omega$; mixing the two conventions would change the ghost operator incorrectly. The trace normalization also gives the invariant quadratic form $-\mathrm{Tr}(A_{\mu }{A}_{\nu })=A_{\mu }^a{A}_{\nu }^a$.

A gauge-invariant functional integral counts all representatives along each gauge orbit. The gauge condition $\mathcal{F}[A]=0$ selects a representative, and its delta functional enforces that selection. Assume a local patch where each orbit intersects the gauge slice once and the gauge-condition derivative has no residual zero modes. DefineChanging the orbit integration variable from $\omega$ to $\mathcal{F}[A^{\omega }]$ givesInsert this Faddeev-Popov gauge-orbit identity into the unfixed path integral. The action and measure are invariant under a change $A\mapsto A^{\omega }$, so integration over $\omega$ factors out as the gauge-group volume. Dividing by that redundant volume leaves the fixed integral and its Faddeev-Popov determinant. In a perturbative patch with fixed determinant sign or phase, the absolute value is replaced by the oriented determinant, up to a field-independent normalization:Here $D_{\mu }^{cb}={\delta }^{cb}{\partial }_{\mu }+f_{dbc}{A}_{\mu }^d$. This is a functional determinant of the gauge-condition Jacobian, not an arbitrary extra weight. The local invertibility assumption matters: residual gauge transformations or multiple orbit intersections, described by the Gribov ambiguity, prevent treating this derivation as a global unique gauge slice without further qualifications.

Both extra factors can be incorporated into an enlarged action. An auxiliary real field $b^a$ Fourier-represents the delta functional, while a Grassmann Gaussian integral over independent Faddeev-Popov ghost fields $c,\bar{c}$ represents the determinant:Thus the original fixed integral can be written with action $S+\int b\mathcal{F}+\int \bar{c}Mc$ and ordinary integration over the extra fields, with no explicit delta functional or determinant. The anticommuting fields give a determinant rather than the inverse determinant obtained from commuting Gaussian fields.

For the usual Gaussian family of gauges, add $\xi b^a{b}^a/2$ and integrate out $b$. Completing the square producesThis is Gaussian gauge fixing with an auxiliary field; equivalently one averages the condition $\mathcal{F}=f$ with its Gaussian weight. Gauge-invariant observables are unchanged apart from normalization. For the Lorenz gauge ${\mathcal{F}}_a={\partial }^{\mu }{A}_{\mu }^a$, the chosen transformation convention gives $M=-{\partial }^{\mu }{D}_{\mu }$. The ghost action is $-\int \bar{c}{\partial }^{\mu }{D}_{\mu }c$, or $\int ({\partial }^{\mu }\bar{c})D_{\mu }c$ after integration by parts. The dependence of $D$ on $A$ accounts for the ghost interactions.

For a bosonic quadratic action $S_2=\frac{1}{2}\int \varphi K\varphi$, completing the square in the generating functional gives $Z_0[J]/Z_0[0]=\exp (-iJ{K}^{-1}J/2)$. Two functional derivatives therefore give the Feynman propagator $\Delta =i{K}^{-1}$, with boundary conditions and a pole prescription included in the inverse. Zero modes must be removed or fixed before an inverse exists.

Take Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$, Fourier convention $e^{-ikx}$, and the Yang-Mills field strength $F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$. The interaction terms do not contribute to the quadratic action. Integration by parts in $-F_{\mu \nu }^a{F}^{a\mu \nu }/4-(\partial \cdot A^a{)}^2/(2\alpha )$ yieldsThe color kernel is diagonal. For $k^2\ne 0$, use the projectors $P_L^{\mu \nu }=k^{\mu }{k}^{\nu }/k^2$ and $P_T^{\mu \nu }={\eta }^{\mu \nu }-P_L^{\mu \nu }$. They obey $P_L^2=P_L$, $P_T^2=P_T$, $P_L{P}_T=0$, soMultiplication explicitly gives the identity on both transverse and longitudinal components. Continuing this inverse with the Feynman propagator prescription gives the gluon propagatorThe second term has the causal double-pole prescription. Away from the pole this is the usual $-i{\delta }^{ab}[{\eta }_{\mu \nu }-(1-\alpha )k_{\mu }{k}_{\nu }/k^2]/(k^2+i0)$. For $\alpha =1$ it becomes the Feynman-gauge adjoint propagator; $\alpha \to 0$ gives a transverse covariant Landau gauge propagator.

The limit $\alpha \to \infty$ removes gauge fixing and has no finite inverse on the full field space. The longitudinal eigenvalue $-k^2/\alpha$ tends to zero and its inverse diverges. A contraction with conserved external currents kills the longitudinal term, but that does not define a gauge-unfixed Gaussian integral. This massless limit is not the massive-vector unitary-gauge limit: here it exposes the unfixed gauge orbit degeneracy.

Let a gauge transformation act infinitesimally as $\delta A_{\mu }^a=(D_{\mu }\omega {)}^a$, and define the Faddeev-Popov operator by variation of the chosen gauge functional:Work in a perturbative regular gauge slice, with boundary conditions that remove residual zero modes and with one local representative of each gauge orbit. The Faddeev-Popov gauge-orbit identity is $1={\Delta }_{\mathrm{FP}}[A]\int \mathcal{D}\omega \delta [\mathcal{F}[A^{\omega }]]$. Inserting it and factoring out the formal gauge-group volume givesThe functional delta imposes the gauge condition at every point and for every color. The determinant compensates the gauge-orbit Jacobian. For ordinary unoriented real gauge-orbit integration the local Jacobian is $|\det \mathcal{M}|$; perturbatively its sign is fixed and absorbed into $\mathcal{N}$, leaving the displayed determinant. Multiple global intersections, or a Gribov ambiguity, require more care; the local perturbative formula is not a global uniqueness assertion.

The complex Grassmann Gaussian integral now generalizes to spacetime and color indices. Introduce independent anticommuting Faddeev-Popov ghost fields $c^a(x),{\bar{c}}^a(x)$ and writeThe factors $i$ in the functional determinant differ from a real exponential only by a regulated, field-independent constant. These Grassmann fields are Lorentz scalars in the adjoint color representation; the antighost is independent of the ghost rather than an ordinary complex-conjugate commuting variable. They are unphysical fields, not external observable particles. Representing the functional determinant by a local ghost action when $\mathcal{F}$ is local makes ordinary Feynman rules available. Anticommutation gives the minus sign of a closed ghost loop, enabling cancellation of unphysical gauge contributions and maintaining the gauge identities in perturbation theory.

For example, the linear covariant choice ${\mathcal{F}}^a={\partial }^{\mu }{A}_{\mu }^a$ gives ${\mathcal{M}}^{ab}={\partial }^{\mu }{D}_{\mu }^{ab}$, so non-abelian ghost-gauge interactions remain. In an axial gauge ${\mathcal{F}}^a=n^{\mu }{A}_{\mu }^a$ with constant $n$, however,on the exact gauge slice. Its determinant is field-independent, so axial-gauge ghosts decouple and contribute only a normalization factor. This statement uses the strict delta-functional constraint; a finite-width Gaussian gauge weight is not the same exact slice.

The disadvantage is that choosing $n$ obscures manifest Lorentz covariance and the axial-gauge propagator contains spurious $1/(n\cdot k)$ poles. An axial-gauge pole prescription and control of residual transformations satisfying $n\cdot \partial \omega =0$ are necessary; formal decoupling alone does not fix those problems.

Abelian ghosts do not always decouple. In a field-independent linear Lorenz gauge, $\delta A_{\mu }={\partial }_{\mu }\omega$ gives $\mathcal{M}=\square$ and they are free. But the quadratic Abelian gauge fixing $\mathcal{F}[A]=\partial \cdot A+\kappa A_{\mu }{A}^{\mu }$ givesIts Faddeev-Popov operator depends on $A$, and the ghost action contains $2\kappa \bar{c}{A}^{\mu }{\partial }_{\mu }c$. This is an explicit abelian ghost-gauge interaction. Abelianity removes the commutator term in the gauge transformation, not the field dependence of an arbitrary nonlinear gauge condition.

Choose a definite matrix convention: $A_{\mu }=A_{\mu }^a{T}_a$, $\omega ={\omega }^a{T}_a$, and $D_{\mu }={\partial }_{\mu }+e{A}_{\mu }$. Matter transforms as ${\psi }^{\prime }=U\psi$, so gauge covariance requires $D_{\mu }^{\prime }=U{D}_{\mu }{U}^{-1}$. Applying both sides to a test field yields the Yang-Mills gauge transformationDefine the Yang-Mills field strength by $[D_{\mu },D_{\nu }]=e{F}_{\mu \nu }$:Conjugating the gauge covariant derivative commutator proves $F_{\mu \nu }^{\prime }=U{F}_{\mu \nu }{U}^{-1}$: unlike the connection, the gauge field strength has no inhomogeneous derivative term. The trace normalization supplies an invariant color metric; its Lie algebra structure constants with all indices lowered are antisymmetric.

For $U=1+e\omega +O({\omega }^2)$ and $U^{-1}=1-e\omega +O({\omega }^2)$, expand the finite transformations:In components this isA convention using the opposite matter transformation or gauge covariant derivative sign changes these signs consistently. The formulas above all follow from the explicitly chosen $D_{\mu }$.

Gauge reduction and the determinant. A direct integral $\int \mathcal{D}A{e}^{iS[A]}$ integrates repeatedly over gauge-equivalent configurations. The gauge group volume is infinite, and the ungauge-fixed quadratic operator has longitudinal zero directions, so it has no ordinary inverse quantum field theory propagator. Gauge-invariant observables require dividing out this redundancy rather than treating it as an ordinary physical fluctuation.

The finite-dimensional change-of-variables identity underlying gauge fixing is precise. Suppose a condition $\chi (x^{\alpha })=f$ has one solution in the group-coordinate patch and its derivative matrix at the solution is nonsingular. ThenFor several isolated solutions it instead counts those solutions. On the slice, the Jacobian determinant can be factored into the reciprocal of the orbit integral of the delta function. An invariant measure and invariant integrand then allow the group volume to factor out. In the functional version, group coordinates carry Haar measure and the corresponding operator determinant replaces the matrix determinant.

For example, choose the Lorenz gauge ${\chi }^a(A)={\partial }^{\mu }{A}_{\mu }^a=f^a$. Its variation is $\delta {\chi }^a=-{\partial }^{\mu }(D_{\mu }^{\mathrm{ad}}\omega {)}^a$. Hence on the gauge slice the Faddeev-Popov operator and Faddeev-Popov determinant areIts extension along each local gauge orbit yields the Faddeev-Popov gauge-orbit identityFor an unoriented finite-dimensional real integral the absolute determinant is required. In a perturbative gauge patch, its sign or phase is fixed and the constant convention is absorbed in normalization. Residual zero modes must be removed by boundary conditions or separate treatment. Multiple intersections, the Gribov ambiguity, prevent interpreting this as a globally unique gauge slice without additional work.

Insert the identity, change variables along the orbit using invariance of the action and measure, and divide out the common gauge group volume. Averaging $f$ with Gaussian weight $\exp [-i\int f^a{f}^a/(2\xi )]$ produces the covariant gauge-fixed functionalThe added term makes the quadratic gauge-field operator invertible after the stated residual-mode treatment.

Ghost representation. For a finite matrix $M$, the Grassmann Gaussian integral over independent anticommuting variables is proportional to its determinant:The constant is independent of $M$ and depends on ordering and factors of $i$. The functional counterpart introduces Faddeev-Popov ghost fields $c^a$ and antighost fields ${\overline{c}}^a$, withIntegration by parts writes its density as $({\partial }^{\mu }{\overline{c}}^a)(D_{\mu }c{)}^a$, displaying a kinetic term and a gauge-field/ghost interaction. These are Grassmann-valued fields with scalar Lorentz transformation law. They represent the orbit Jacobian determinant; their closed loops carry the fermionic minus sign, and they are not physical asymptotic particles. In the abelian case $D^{\mathrm{ad}}=\partial$, the determinant is gauge-field independent and the ghosts decouple. Nonabelian ghosts interact and are necessary for the consistent cancellation of unphysical gauge contributions.

Physical-state reduction. Let $Q$ be the Hermitian BRST charge. Its key algebraic property is BRST nilpotence, $Q^2=0$, and the gauge-fixed dynamics preserves it. Physical states are its BRST cohomology at ghost number zero:Here $|\chi ⟩$ has ghost number $-1$. BRST nilpotence ensures that exact vectors lie in the kernel, so the quotient is defined. Because $Q$ is Hermitian, an exact vector is orthogonal to every closed vector, $⟨\psi |Q\chi ⟩=⟨Q\psi |\chi ⟩=0$, and exact vectors are null, $⟨Q\chi |Q\chi ⟩=⟨\chi |Q^2\chi ⟩=0$. They therefore represent no independent physical state. Commutation of $Q$ with the Hamiltonian preserves these classes under time evolution.

The covariant gauge/ghost space before reduction has an indefinite inner product. This is essential: a Hermitian nilpotent BRST charge requires an indefinite auxiliary space if it is nonzero. On an ordinary positive-definite Hilbert space the same norm identity would force $Q=0$. Positivity of the resulting physical space uses the gauge-theory structure in addition to the abstract BRST nilpotence identities.

For a nonzero on-shell photon momentum, the one-particle conclusion isThe second relation preserves the first because $k^2=0$. In four dimensions the transverse kernel has dimension three, and quotienting by its null longitudinal direction leaves two physical photon polarizations. This is the positive one-particle BRST cohomology of the photon sector.

For a constant unitary matrix $h$, ${\varphi }^{\prime }=h\varphi$ gives ${\varphi }^{\prime †}{\varphi }^{\prime }={\varphi }^{†}{h}^{†}h\varphi ={\varphi }^{†}\varphi$. Also ${\partial }_{\mu }{\varphi }^{\prime }=h{\partial }_{\mu }\varphi$, so the kinetic contraction is unchanged. The mass term and the squared norm in the potential are unchanged for the same reason. The original scalar action has the stated global symmetry. For a spacetime-dependent $h$, however, ${\partial }_{\mu }(h\varphi )=h{\partial }_{\mu }\varphi +({\partial }_{\mu }h)\varphi$, and the extra term prevents local invariance of the ordinary-derivative kinetic term.

Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$, with $A_{\mu }=A_{\mu }^{\alpha }{T}_{\alpha }$ Hermitian. Demand $D_{\mu }^{\prime }(h\varphi )=h{D}_{\mu }\varphi$. Expanding both sides yieldsso the necessary Yang-Mills gauge transformation isUnitarity makes the transformed connection Hermitian. This formula includes the inhomogeneous derivative term; mere conjugation would be insufficient for a local transformation.

Define the Yang-Mills field strength by $[D_{\mu },D_{\nu }]=ig{F}_{\mu \nu }$. Direct expansion givesChoose an orthonormal Hermitian generator basis with $\mathrm{tr}(T_{\alpha }{T}_{\beta })=\kappa {\delta }_{\alpha \beta }$. The compact simple Lie algebra has an invariant positive trace metric, so this basis can be chosen; its structure constants are totally antisymmetric. With these conventions a locally invariant dynamical action isEach matter contraction is invariant because $D\varphi$ transforms like $\varphi$, and the gauge kinetic term is invariant by cyclicity of the trace. It is $-\frac{1}{4}{F}_{\mu \nu }^{\alpha }{F}_{\alpha }^{\mu \nu }$ in components. Choosing $\kappa =1/2$ gives the familiar $-\frac{1}{2}\mathrm{tr}{F}^2$ convention.

Formally introduce a gauge-field source $J_{\alpha }^{\mu }$ and a normalized generating functionalA gauge-field Green function is obtained by source differentiation:Here $\mathcal{N}$ is the unnormalized integral at zero source, so the normalized functional obeys $Z[0]=1$. This expression needs gauge fixing: the integral repeats every physical configuration over its whole gauge orbit, producing an irrelevant but divergent gauge-group volume, and the free gauge kinetic operator has longitudinal zero modes, so it has no inverse propagator on that unrestricted space. Ordinary gauge-field Green functions are gauge-dependent; physical gauge-invariant quantities must be obtained consistently from the gauge-fixed theory.

Use the Lorenz condition ${\chi }^{\alpha }[A]={\partial }^{\mu }{A}_{\mu }^{\alpha }$. To keep signs explicit, parametrize a small transformation by $h=1-ig{\omega }^{\alpha }{T}_{\alpha }+O({\omega }^2)$, equivalently the original parameter is ${\theta }^{\alpha }=g{\omega }^{\alpha }$. The finite transformation above gives the adjoint gauge variation for a positive-sign covariant derivative:Therefore $\delta {\chi }^{\alpha }={\partial }^{\mu }{\mathcal{D}}_{\mu }^{\alpha \beta }{\omega }^{\beta }$. Define the Faddeev-Popov operator as $\mathcal{M}=-{\partial }^{\mu }{\mathcal{D}}_{\mu }$, absorbing the field-independent minus sign of its determinant into normalization.

In a perturbative gauge patch, with residual zero modes removed by boundary conditions, the Faddeev-Popov gauge-orbit identity isIt follows by changing coordinates from the gauge parameter to the gauge condition: the functional Jacobian is precisely the determinant of its linearization. Insert the identity into the integral and change variables along the orbit. The action and measure are invariant, so the gauge-group volume factors out and is divided away. Averaging the gauge-slice value $f$ with Gaussian weight $\exp [-i\int f^2/(2\xi )]$ changes the delta-functional restriction to the covariant gauge-fixing termThe remaining determinant depends on $A$ through ${\mathcal{D}}_{\mu }$. Omitting it while adding only ${\mathcal{L}}_{\mathrm{gf}}$ would weight different orbits incorrectly.

For a finite matrix $M$, expansion of a Grassmann integral selects the product containing every $c$ and $\bar{c}$. Anticommuting the variables into a fixed order attaches the sign of each permutation, giving $\int d\bar{c}dc{e}^{i\bar{c}Mc}$ proportional to $\det M$, with only a matrix-independent phase. The regulated functional version therefore represents the Faddeev-Popov determinant asThe independent Faddeev-Popov ghost fields and antighost fields obey $c_{\alpha }{c}_{\beta }=-c_{\beta }{c}_{\alpha }$, with the corresponding mixed and antighost anticommutation. A commuting complex Gaussian integral instead gives an inverse determinant, so it cannot supply this Jacobian. The required local ghost term isAfter integration by parts,This exhibits the free ghost kinetic term and the ghost-gluon vertex. Ghosts are Grassmann-valued scalar auxiliary fields in the Adjoint representation, not physical fermionic particles or external asymptotic states. Their loop signs arise from anticommutation. In an Abelian theory the adjoint derivative loses the field-dependent part, so the determinant is field-independent and the ghosts decouple; in the non-Abelian theory they must be retained.

The correctly gauge-fixed source functional thus uses $S_{\mathrm{inv}}+S_{\mathrm{gf}}+S_{\mathrm{gh}}$ and integrates over $A,\varphi ,{\varphi }^{†},c,\bar{c}$. Expanding its interaction terms gives the gauge, matter and ghost vertices, and differentiating its source functional computes the gauge-field Green functions. For example, at non-null momentum the quadratic gauge kernel iswhere $P_T$ and $P_L$ are the transverse projector of a vector field and longitudinal projector of a vector field. With the Feynman prescription this gives the gauge-boson propagatorFor the chosen ghost normalization the quadratic kernel is $-{\partial }^2$, so the ghost propagator is $i{\delta }^{\alpha \beta }/(k^2+i0)$. Redefining the antighost sign changes ghost propagator and vertex signs together, without changing the determinant or physical conclusions. Gauge fixing requires the accompanying determinant, and its local representation requires anticommuting ghost and antighost fields. The gauge-orbit argument is local/perturbative: global gauge copies, the Gribov ambiguity, prevent interpreting it as a globally unique gauge slice without further treatment.

The gauge-invariant Yang-Mills action integrates over physically equivalent configurations along each gauge orbit. At the quadratic level its gauge-field kernel has longitudinal zero directions: in momentum space $K_{\mu \nu }(k)=-k^2{\eta }_{\mu \nu }+k_{\mu }{k}_{\nu }$ satisfies $K_{\mu \nu }{k}^{\nu }=0$. It therefore has no inverse from which to construct a gauge-boson propagator. Gauge fixing both removes the redundant orbit integration and supplies an invertible quadratic operator away from the ordinary physical propagator poles.

For the covariant condition ${\partial }^{\mu }{A}_{\mu }=0$, the Faddeev-Popov determinant is that of ${\partial }^{\mu }{D}_{\mu }$. Its representation by odd Faddeev-Popov ghost fields and antighost fields produces the displayed ghost term. In a nonabelian theory $D_{\mu }$ depends on the gauge field, so the determinant is not an irrelevant constant: ghost interactions must be retained. The even Nakanishi-Lautrup field $b$ writes gauge fixing in a form with off-shell BRS symmetry. For $\xi \ne 0$, its algebraic equation is $b=-\partial \cdot A/\xi$, giving $-(\partial \cdot A{)}^2/(2\xi )$ after elimination. At $\xi =0$ it imposes the gauge condition as a Lagrange multiplier.

For the nilpotence proof, write the adjoint bracket as $[X,Y{]}^a=f^{abc}{X}^b{Y}^c$, with coefficients in a Grassmann algebra. If $|X|,|Y|$ are their parities, this is a graded Lie bracket:Thus two odd fields have a symmetric bracket, in particular $[c,c]$ need not vanish. The transformations become $s{A}_{\mu }=D_{\mu }c$, $sc=-[c,c]/2$, $s\bar{c}=-b$, $sb=0$, and $s$ obeys the left graded Leibniz rule.

Because $D_{\mu }$ is even, $D_{\mu }[c,c]=[D_{\mu }c,c]+[c,D_{\mu }c]=2[D_{\mu }c,c]$. Varying both the field and the connection in $D_{\mu }c$ now givesFor the ghost,The last equality is the graded Jacobi identity applied to three copies of the odd field $c$. In components the remaining expression is $\frac{1}{2}{f}^{abc}{f}^{bde}{c}^d{c}^e{c}^c$; its coefficient is antisymmetrized over $c,d,e$, and the ordinary Jacobi identity for the structure constants makes that antisymmetrization zero. Finally $s^2\bar{c}=-sb=0$ and $s^{2}b=0$. HenceThis is off-shell nilpotence of the Yang-Mills BRST quartet.

To verify invariance of the complete Lagrangian, $s{F}_{\mu \nu }=[F_{\mu \nu },c]$, obtained from $[D_{\mu },D_{\nu }]c=[F_{\mu \nu },c]$. Thus $s(F_{\mu \nu }^a{F}^{\mu \nu a})=0$ because the product of the two curvature components is symmetric in their color labels while $f^{abc}$ is antisymmetric. The remaining terms vary asThe last term vanishes by the already proved $s^{2}A=0$, and the first and third terms cancel pointwise. ThereforeEquivalently, the covariant gauge-fixing density as a BRST variation is $s\Psi$ with $\Psi =-{\bar{c}}^a(\partial \cdot A^a+\xi b^a/2)$. The graded sign from moving $s$ past $\bar{c}$ produces the positive ghost term, and $s^2\Psi =0$ proves its invariance. No integration-by-parts qualification is needed for the density as written.

Let $Q=Q_{\mathrm{BRS}}$ be the BRS charge. The physical state space is its BRST cohomology, usually at ghost number zero:Nilpotence ensures $\mathrm{im}Q\subseteq \ker Q$. A state $Q|\chi ⟩$ is orthogonal to every closed state because $⟨\psi |Q|\chi ⟩=⟨Q\psi |\chi ⟩=0$, and its norm is $⟨\chi |Q^2|\chi ⟩=0$. Thus adding an exact state changes neither physical inner products nor amplitudes. Self-adjointness here is relative to the stated indefinite Hermitian form; a nonzero self-adjoint nilpotent operator could not exist on a positive-definite state space. Positivity must be proved on the quotient, not assumed on the gauge/ghost space.

Fix a color label and a nonzero momentum $k$, and take the usual nontrivial asymptotic charge, $\alpha \ne 0$. Nilpotence on a vector state gives $Q|c⟩=0$, because at least one component of $k$ is nonzero. Nilpotence on the antighost state then givesThe two coefficients are complex conjugates as required by self-adjointness: using the given ghost pairing,Thus the antighost-to-vector coefficient is ${\alpha }^{\ast }$, as the printed bar notation indicates. These conditions are the massless shell condition and the consistency of the charge with the inner products.

A general one-particle vector representative is $|\epsilon ⟩={\epsilon }^{\mu }|A_{\mu }⟩$. It is closed precisely when $k_{\mu }{\epsilon }^{\mu }=0$. The vector state $k^{\mu }|A_{\mu }⟩=Q|\bar{c}⟩/\bar{\alpha }$ is exact, so representatives obeyThe ghost state is also exact: choose $v^{\mu }$ with $\alpha v\cdot k=1$, then $Q(v^{\mu }|A_{\mu }⟩)=|c⟩$. The antighost state is not closed because its image is a nonzero longitudinal vector. More explicitly, closure of a general sum of vector, ghost and antighost states forces the antighost coefficient to vanish and the vector polarization to be orthogonal to $k$; its ghost component disappears in the quotient. Thus the full one-particle cohomology already reduces toWith the mostly-plus Minkowski metric, choose a Lorentz frame with $k^{\mu }=(\omega ,0,0,\omega )$, $\omega >0$. Closure gives ${\epsilon }^3={\epsilon }^0$, and the norm isSubtracting a multiple of $k$ sets ${\epsilon }^0={\epsilon }^3=0$. Consequently positive one-particle BRST cohomology consists of two transverse positive-norm polarizations per color, represented by $|A_1⟩,|A_2⟩$. A nonzero class has strictly positive norm; the zero-norm longitudinal direction was precisely the exact direction removed.