BRST charge ($Q$)

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.

A Slavnov-Taylor identity is a BRST symmetry constraint relating correlation functions or interaction vertices in a gauge-fixed Non-Abelian gauge theory. Unlike a simple statement that every open gauge index is transverse, the general identities include Faddeev-Popov ghost fields. After putting external legs on shell and making all the other external states physical, they imply decoupling when one massless vector polarization vector is replaced by its four-momentum. They also control cancellations of auxiliary covariant components needed for physical unitarity.

For massless $gg\to q\bar{q}$, define $i\mathcal{M}$ by the Feynman-rule product with quark vertex $ig{T}^a{\gamma }^{\mu }$. The ordered quark exchanges contribute $-g^2\bar{u}/k_1[T^a,T^b]v$ after double contraction. Three-gluon exchange contributes $i{g}^2{f}^{abc}\bar{u}/k_1{T}^{c}v$, so the sum vanishes by the Lie algebra. This does not assert off-shell transversality of each open index; the full Slavnov-Taylor identity includes Faddeev-Popov ghost terms.

For a Dirac field annihilating with its antiparticle into two massless gauge bosons, the two ordered fermion-exchange diagrams leave a colour matrix commutator when one outgoing momentum contracts the scattering amplitude. The three-boson exchange diagram cancels this current term but leaves a term proportional to the other outgoing momentum. With $D=\partial -igA$, $J_c^{\lambda }=\bar{v}{\gamma }^{\lambda }{t}^{c}u$, $q=k_1+k_2$, and the source's positive-sign three-boson vertex, the tree result is $k_{2\nu }{\mathcal{M}}^{ab\mu \nu }=i{g}^2{f}^{abc}(J_c\cdot k_2)k_1^{\mu }/q^2$. It vanishes after contraction with a physical polarization vector on leg 1, although the open-index tensor is generally nonzero.

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.