BRST symmetry

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.

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.

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.

// Target: relativistic-quantum-field.bigb

For the left graded Leibniz rule and $s\bar{c}=-b$, $sb=0$, the displayed gauge-fixing fermion gives $b\partial \cdot A+\xi b^2/2+\bar{c}\partial \cdot Dc$. The ghost signs follow from odd parity. BRST nilpotence implies invariance of this density without using field equations. Eliminating the Nakanishi-Lautrup field gives $-(\partial \cdot A{)}^2/(2\xi )$ for nonzero $\xi$.

For the convention $s\bar{c}=-b$, the action $S_{\mathrm{YM}}-s\Psi$ contains $-b\partial \cdot A+\xi b^2/2-\bar{c}\partial \cdot Dc$ after integration by parts. Integrating the Nakanishi-Lautrup field gives $-(\partial \cdot A{)}^2/(2\xi )$. The Grassmann Gaussian integral represents the Faddeev-Popov determinant.

BRST cohomology identifies physical states and observables with BRST-closed objects modulo BRST-exact ones. A change of gauge-fixing fermion changes the action by a BRST-exact term and therefore leaves BRST-cohomology classes unchanged when the measure has no BRST anomaly.

For a nonzero null momentum and a nontrivial BRST charge on the gauge/ghost one-particle sector, closed vector states obey $k\cdot \epsilon =0$ and exact vector states have polarization proportional to $k$. The Faddeev-Popov ghost field state is exact and the antighost field state is not closed. With mostly-plus Minkowski metric, choose $k=(\omega ,0,0,\omega )$; closure sets ${\epsilon }^3={\epsilon }^0$, so the norm is $|{\epsilon }^1{|}^2+|{\epsilon }^2{|}^2$. Quotienting by the null longitudinal polarization leaves two positive-norm transverse polarizations per group index. Self-adjointness and nilpotence alone do not establish positivity on an arbitrary state space: the nonzero momentum, nontrivial charge and stated indefinite inner products are essential here.

A BRST-closed operator $O$ satisfies $QO=0$. It represents a physical observable when it is not also BRST exact.

A BRST-exact operator has the form $O=QX$. It is BRST closed by nilpotence and represents the zero class in BRST cohomology.

When separated operators $G_i$ are BRST-closed, the graded BRST Ward identity makes an exact insertion vanish. This is the correlator form of identifying closed representatives that differ by exact terms in BRST cohomology. The statement presupposes an invariant measure and suitable boundary conditions.