BRST Cohomology and Physical Observables

BRST cohomology is a kernel modulo an image, not automatically the physical observable algebra or state space. On a declared ghost-graded domain with $s^2=0$, it identifies closed representatives that differ by an exact term. Ghost-number-zero classes are natural observable candidates, but the word “physical” requires more: the differential must encode exactly the declared redundancies, exact terms must decouple, boundaries and zero modes must be controlled, and the quantum symmetry must be anomaly free. A state-space interpretation also needs a positive, nondegenerate, completed quotient.

This page makes those hypotheses explicit. Its worked example is the based Maxwell complex on the bounded spatial region established on the preceding page, followed by the corresponding local Yang–Mills check. The result is a controlled classical cohomology calculation, not a nonperturbative construction of the interacting physical Hilbert space.

Required background. The BRST Differential and Gauge-Fixed Complex supplies the declared graded field algebra, ghost numbers, boundary domain, and nilpotent differential used below. Chains, Homology, Cohomology, and Exact Sequences supplies the kernel–image quotient and exactness language.

Cohomology begins with a declared complex

Let $\mathscr F=\bigoplus_{g\in\mathbb Z}\mathscr F^g$ be a space preserved by the left BRST differential, where $g$ is ghost number and $s:\mathscr F^g\to\mathscr F^{g+1}$. Define

$$\begin{aligned} Z^g(s;\mathscr F) &=\ker\!\left(s:\mathscr F^g\to\mathscr F^{g+1}\right), \\ B^g(s;\mathscr F) &=\operatorname{im}\!\left(s:\mathscr F^{g-1}\to\mathscr F^g\right), \\ H^g(s;\mathscr F) &=\frac{Z^g(s;\mathscr F)}{B^g(s;\mathscr F)}. \end{aligned}$$

Nilpotency is precisely what puts $B^g$ inside $Z^g$. An element $\mathcal O$ is closed if $s\mathcal O=0$ and exact if $\mathcal O=sX$. Cohomology imposes the equivalence relation

$$\mathcal O\sim\mathcal O+sX, \qquad \operatorname{gh}(X)=\operatorname{gh}(\mathcal O)-1.$$

The grading is part of the answer. Ordinary gauge-theory observable candidates are normally sought at ghost number zero. A closed element at another degree can carry important consistency information without being an ordinary observable.

element in the four-field complex	immediate conclusion
$b=s\bar c$	closed and exact at ghost number zero
$A_\mu$	generally not closed because $sA_\mu=D_\mu c$
Maxwell $F_{\mu\nu}$	closed at ghost number zero
Maxwell $c$	closed at ghost number $+1$; whether it is exact depends on the chosen functional space

For a ghost-independent functional $\mathcal O[A]$, the equation $s\mathcal O=0$ says that its infinitesimal variation vanishes for every parameter represented by the ghost domain. If those infinitesimal actions integrate, this gives invariance under the identity component of the declared based redundancy group. It does not by itself test disconnected or large transformations, other bundle sectors, or transformations excluded from the ghost domain.

The familiar statement that ghost-number-zero BRST cohomology gives gauge-invariant functions also has an on-shell version in the BV resolution. There the Koszul–Tate part first restricts to the stationary surface and the longitudinal part then quotients gauge orbits Fuster, Henneaux, and Maas 2005, § 5, arXiv v2, pp. 10–12, eqs. (5.1)–(5.14), Open PDF. That result should not be silently attributed to the four-field off-shell complex: adding equations of motion changes the complex.

Functionals, densities, operators, and states are different

Several constructions are called “BRST cohomology,” but their spaces and equivalence relations are not interchangeable.

question	complex and quotient	additional issue
Global or nonlocal functionals	$H^g(s;\mathscr F)$	Which inverses, boundary data, regularity, and topology are allowed in $\mathscr F$?
Local $p$-form densities	$H^{g,p}(s\mid d)$	Total derivatives and descent equations depend on locality and the boundary.
Operators	Cohomology of $\delta_Q\mathcal O=i[Q,\mathcal O]_{\mathrm{gr}}$ in a declared convention	Domains, renormalized composite operators, and contact terms matter.
States	$H^g(Q;\mathcal D)$ on a common invariant domain $\mathcal D$	Conservation, the indefinite metric, positivity, closed range, and completion are extra requirements.

For local forms, closure and equivalence take the relative form

$$s a^{g,p}+d b^{g+1,p-1}=0,$$ $$a^{g,p} \sim a^{g,p}+s m^{g-1,p}+d n^{g,p-1}.$$

On a manifold without boundary, or for support and boundary conditions that kill the surface term, integrating $dn$ gives no contribution. On a bounded region, $\int_\Sigma dn=\int_{\partial\Sigma}n$ is not automatically zero. A boundary observable can therefore be lost by an unjustified use of “modulo $d$.” The local-form and descent complexes are developed in Barnich, Brandt, and Henneaux 2000, §§ 4.1–4.4 and 9.1–9.2, arXiv v3, pp. 22–24 and 70–72, Open PDF.

State cohomology begins instead with a nilpotent BRST charge $Q$ on a common graded domain $\mathcal D$:

$$H^g(Q;\mathcal D) = \frac{ \ker\!\left(Q:\mathcal D^g\to\mathcal D^{g+1}\right) }{ \operatorname{im}\!\left(Q:\mathcal D^{g-1}\to\mathcal D^g\right) }.$$

Usually $[N_{\mathrm{gh}},Q]=Q$ and the physical candidate sector is selected at relative ghost number zero after fixing a ghost-vacuum convention. That is a convention-dependent selection rule, not a reason to erase the other degrees. In particular, local ghost-number-one classes supply candidate quantum consistency obstructions; only the relative top-form cohomology, regulator, and allowed counterterms decide whether a class is a realized anomaly.

The charge construction and its free gauge-field test are given in Weinberg 1996, vol. II, § 15.7, pp. 32–36, eqs. (15.7.27)–(15.7.40) and Srednicki 2007, § 74, pp. 452–455, eqs. (74.25)–(74.44). Both are perturbative teaching constructions; neither supplies the boundary and nonperturbative analytic hypotheses by itself.

Contractible pairs remove gauge-fixing variables

The nonminimal pair $(\bar c,b)$ illustrates why adding gauge-fixing variables need not change cohomology. More generally, suppose

$$su=v, \qquad sv=0,$$

and suppose the transformations of all other variables are independent of $u$ and $v$. On polynomials in the pair, introduce the doublet-number operator and an odd contracting homotopy,

$$\begin{aligned} N&=u\frac{\partial}{\partial u} +v\frac{\partial}{\partial v}, \\ \kappa&=u\frac{\partial}{\partial v}, \qquad \{s,\kappa\}=N. \end{aligned}$$

If $sX_r=0$ and $NX_r=rX_r$ with $r>0$, then

$$X_r = \frac1r\{s,\kappa\}X_r = s\!\left(\frac1r\kappa X_r\right).$$

Every positive-doublet-degree closed term is therefore exact, and each class has a representative independent of the pair. For local jets, $N$ and $\kappa$ are summed over the derivatives of $u$ and $v$ as well. Applying the argument to $u=\bar c$ and $v=b$ shows why the retained nonminimal pair does not add classes Barnich, Brandt, and Henneaux 2000, § 2.7, arXiv v3, pp. 18–19, eqs. (2.48)–(2.51), Open PDF.

This proof has hypotheses. The algebra must admit the nonnegative $N$-decomposition, the homotopy must preserve its locality, regularity, and boundary domain, and the relevant expansion or filtration must converge or be used formally. Singular functions of the doublet, an incompatible completion, or unpaired boundary and zero modes require a new proof. Eliminating $b$ also removes the off-shell doublet relation $s\bar c=b$, so the argument here keeps $b$.

Quartets need an indefinite state space and a positivity theorem

An algebraic doublet is not the same object as the state-space quartet mechanism. In covariant quantization, let $Q$ be conserved on an invariant domain,

$$Q^2=0, \qquad [H,Q]=0,$$

so that its cohomology is stable under time evolution. The auxiliary covariant state space is normally indefinite—a Krein space—not already the physical Hilbert space. This is unavoidable for a nonzero charge that is both nilpotent and “Hermitian”: on a positive Hilbert space, an ordinary self-adjoint $Q$ would obey

$$\lVert Q\psi\rVert^2 = \langle\psi,Q^2\psi\rangle =0,$$

and hence would vanish. Hermiticity of the covariant BRST charge must instead be interpreted with the indefinite adjoint.

Write the Krein form as $[\mathord\cdot,\mathord\cdot]_K$ and assume $Q^\times=Q$. If $Q\phi=0$, then

$$\begin{aligned} [\phi,Q\chi]_K&=[Q\phi,\chi]_K=0, \\ [Q\chi,Q\chi]_K&=[\chi,Q^2\chi]_K=0. \end{aligned}$$

Thus exact states are null and orthogonal to closed states, so the form can descend to cohomology. Positivity still has not been proved.

The quartet orientation can be expressed by an odd state-space homotopy $R$. If, on a common domain,

$$N_{\mathrm{unphys}}=\{Q,R\}, \qquad [N_{\mathrm{unphys}},Q]=0,$$

and $N_{\mathrm{unphys}}$ is diagonalizable with nonnegative spectrum, then a closed eigenstate of eigenvalue $r>0$ is exact:

$$Q|\psi_r\rangle=0 \quad\Longrightarrow\quad |\psi_r\rangle = Q\!\left(\frac1rR|\psi_r\rangle\right).$$

When two $Q$-doublets occur with their metric-conjugate partners, this is the quartet mechanism: nonzero unphysical-number sectors disappear from cohomology. The doublet/quartet representation and projector proof are given in Kugo and Ojima 1979, ch. III, §§ 3.1–3.2, pp. 24–33, especially eqs. (3·15)–(3·16) and (3·25)–(3·32), with the § 3.1 graded-bracket and normalization corrections in Kugo and Ojima 1984, p. 1121, Erratum.

To obtain a physical Hilbert space one must still prove that the remaining singlet sector is positive and that $\operatorname{rad}(\ker Q)=\operatorname{im}Q$ when the image is closed. If it is not closed, a topology and a quotient-by-closure prescription must be declared before completing the quotient. Unpaired null, zero, or boundary modes can defeat that conclusion. The classic operator framework is the Kugo–Ojima construction; here it is used only as a free or asymptotic orientation, not as a theorem about the nonperturbative Yang–Mills spectrum.

On a bounded region there is a further condition. If $Q=\int_\Sigma j_B^0$, current conservation gives

$$\frac{dQ}{dt} = -\int_{\partial\Sigma}dS_i\,j_B^i.$$

The BRST-stable field domain of the preceding page does not by itself prove that this flux vanishes. A bounded state-space theorem would also have to specify canonical domains, the boundary data for the electric and temporal sectors, and any additional boundary degrees of freedom. The worked example below therefore computes functional cohomology rather than claiming a bounded-space quartet theorem.

Based Maxwell theory separates local from nonlocal cohomology

Return to the preceding page’s smooth, bounded, connected spatial region $\Sigma$, trivial $U(1)$ bundle, zero tangential pullback of $A$, and the identity component of the based redundancy group. Equivalently, work in the affine sector generated by real Dirichlet parameters. The ghost has Dirichlet trace, Coulomb gauge uses $M_0=\Delta_D$, and the Dirichlet scalar Laplacian has no zero mode. These are exactly the hypotheses that make the following inverse meaningful.

Define the based orbit coordinate and its transverse representative by

$$\begin{aligned} \varphi[A] &=\Delta_D^{-1}\partial_iA_i, \\ A_i^\perp &=A_i-\partial_i\varphi[A]. \end{aligned}$$

Then $\varphi|_{\partial\Sigma}=0$ and $\partial_iA_i^\perp=0$. Since $sA_i=\partial_i c$ and $sc=0$,

$$\begin{aligned} s\varphi &=\Delta_D^{-1}\Delta_Dc=c, \\ sA_i^\perp &=\partial_i c-\partial_i(s\varphi)=0. \end{aligned}$$

Thus $(\varphi,c)$ and $(\bar c,b)$ are two contractible pairs. In a modewise finite regulator chosen to preserve the linear differential and both homotopies, or in a smooth cylindrical functional algebra that explicitly admits $\Delta_D^{-1}$ and is preserved by these homotopies, the spatial gauge-field/nonminimal sector has

$$H^0(s;\mathscr F_{\mathrm{nonlocal}}) \cong \mathscr F(A^\perp).$$

Indeed, under the based transformation $A\mapsto A+d\epsilon$, $\varphi\mapsto\varphi+\epsilon$ while $A^\perp$ is unchanged; Coulomb gauge sets the orbit coordinate $\varphi$ to zero. This is a global statement only for the controlled affine Maxwell sector, or equivalently the based identity component, because the Dirichlet Poisson problem is unique there. It is not a global non-Abelian slice, a Gauss-law-reduced phase space, or a positive state-space construction. Harmonic one-form modes, when the topology permits them, survive this small based quotient and are not scalar Faddeev–Popov zero modes; large compact-$U(1)$ transformations may further identify them.

The answer changes when the functional space changes. In the interior local polynomial jet algebra, $\Delta_D^{-1}$ is not allowed, so $c=s\varphi$ is not an admissible contraction. Symmetrized derivatives of $A$ pair with derivatives of $c$, while the curvature and the undifferentiated Abelian ghost remain unpaired. Let $\mathscr I(F,\partial F,\ldots)$ denote the local polynomial algebra generated by curvature jets, with smooth interior coefficients and compact support when a representative is integrated. For one $U(1)$ field,

$$H_{\mathrm{loc}}(s) \cong \mathscr I\!\left(F_{ij},\partial F_{ij},\ldots\right) \otimes\Lambda(c),$$

and hence

$$\begin{aligned} H^0_{\mathrm{loc}}(s) &\cong\mathscr I(F,\partial F,\ldots), \\ H^1_{\mathrm{loc}}(s) &\cong c\,\mathscr I(F,\partial F,\ldots). \end{aligned}$$

This is an off-shell, interior-jet statement; imposing equations of motion, passing to $H(s\mid d)$, or admitting boundary-supported representatives changes the calculation. The adapted jet coordinates and elimination of ghost derivatives are described in Barnich, Brandt, and Henneaux 2000, §§ 8.1–8.2, arXiv v3, pp. 61–63, eqs. (8.1)–(8.8), Open PDF.

The basic ghost-number-zero representative is visible without the general classification:

$$sF_{ij} = \partial_i\partial_jc-\partial_j\partial_ic =0.$$

After the nonminimal doublet is removed, there is no local ghost-number $-1$ minimal generator whose variation is $F_{ij}$. For a nonzero, compactly supported antisymmetric test tensor $f^{ij}$, a functional such as $\int_\Sigma f^{ij}F_{ij}$ therefore supplies a concrete class. The ghost-number-one factor $c$ is not itself a physical observable; in relative top form it can participate in a candidate consistency or anomaly class. First-order consistent action deformations instead live in the integrated relative group $H^{0,d}(s\mid d)$ of the BV complex in the standard grading, while ghost-number-one top-form classes are candidate anomalies or higher consistency obstructions. Any different deformation-theory degree shift must be declared explicitly.

For compact Yang–Mills theory, the curvature transforms covariantly:

$$sF_{\mu\nu}=-ig[F_{\mu\nu},c], \qquad s\operatorname{tr}(F_{\mu\nu}F^{\mu\nu})=0.$$

The trace supplies a ghost-number-zero closed representative, but this short calculation neither classifies all cohomology nor proves a nonperturbative state-space result.

The bounded example has three complementary readings:

reading	what the cohomology calculation says
Orbit	$s$ tests constancy along the identity-component based orbit; $\varphi$ is its Maxwell coordinate and $A^\perp$ labels that controlled quotient. Large compact-$U(1)$ transformations require a separate quotient.
Charge	Boundary-nontrivial transformations are absent from $c$. BRST closure relative to the based group need not mean invariance under every boundary symmetry, which may carry a surface charge.
Gauge fixed	Coulomb gauge sets $\varphi=0$, and the nonminimal pair is contractible. Equality of quantum predictions in different gauges still needs the BRST functional identity.

BRST-compatible Maxwell boundary data with $b$ retained are exhibited in Moss and Silva 1997, § III, pp. 7–8, eqs. (30), (31), (33), and (37)–(38), Open PDF. The possible charge carried by a nonzero-boundary transformation depends on the boundary phase-space setup Assanioussi, Kowalski-Glikman, Mäkinen, and Varrin 2024, §§ 3.1–3.3, arXiv v2, pp. 13–16, especially eqs. (3.24)–(3.26), Open PDF.

Physical interpretation is a theorem with hypotheses

The slogan “physical quantities are BRST cohomology” is justified only after the following questions have affirmative, compatible answers.

1. Which complex? The functional or state space, ghost grading, topology, boundary conditions, regularity, and operator domains are declared.
2. Which redundancies? The nilpotent $s$ or $Q$ encodes exactly the transformations to be quotiented, not charged boundary symmetries or unexamined disconnected transformations.
3. Which zero modes? Stabilizers, residual Faddeev–Popov modes, harmonic modes, and ghost zero modes are separately removed, retained, or saturated.
4. Which quotient? The question really calls for $H(s)$, $H(s\mid d)$, operator cohomology, or $H(Q)$, and any use of equations of motion is made explicit.
5. Why do exact terms decouple? A valid Ward identity or charge argument shows that exact insertions or states have no physical effect.
6. Does the symmetry survive quantization? The regulator, measure, contour, renormalization prescription, time evolution, and boundary domain preserve BRST, with no nonremovable anomaly.
7. Is the state quotient physical? The induced form is nondegenerate and positive at the selected ghost number, the range and closure prescription are controlled, and the quotient is completed.
8. Is the claim local or global? A regular Faddeev–Popov patch is not mistaken for a global construction of orbit space.

The last two qualifications are logically independent of classical nilpotency. A local ghost-number-one consistency class is only a candidate anomaly until the regulator and counterterm problem are fixed Barnich, Brandt, and Henneaux 2000, § 2.6 and § 12.3, arXiv v3, pp. 16 and 119–121, eqs. (2.36)–(2.38), Open PDF. Conversely, a Faddeev–Popov zero mode can destroy the local gauge-fixed inverse without changing the algebraic calculation $s^2=0$ Vandersickel and Zwanziger 2012, § 2.1.5 and § 2.2.1, arXiv v2, pp. 18 and 24–25, Open PDF.

Slavnov–Taylor and Zinn-Justin Identities next supplies the functional identity needed to compare exact insertions. BRST Cohomology as Derived Invariants owns theorem-level regularity and derived-invariance questions; Local BRST Cohomology, Consistent Deformations, and Currents owns the full local classification. Equations of motion and reducibility enter through the Koszul–Tate Resolution and the BRST Bicomplex, while global slice failure remains with Gribov Copies and the Limits of Local Gauge Fixing.

Common pitfalls

Calling every closed expression physical. Closure must be interpreted at a fixed ghost number in a declared complex. Maxwell $c$ is closed in the local algebra, but it is not an ordinary ghost-number-zero observable.

Interchanging functional, local, and state cohomology. Their domains and equivalence relations differ. In particular, a total derivative need not be trivial at a boundary, and a functional contraction using $\Delta_D^{-1}$ is not a local-jet contraction.

Treating a doublet theorem as a positivity theorem. The homotopy $\{s,\kappa\}=N$ removes a contractible algebraic pair under its domain hypotheses. It neither constructs a state quartet nor proves that the remaining state cohomology has positive norm.

Assuming exact insertions vanish without a quantum identity. Exactness means zero in the algebraic quotient. Decoupling from correlators also needs an invariant measure, domain, contour, regulator, and renormalization prescription.

Using BRST to erase global gauge-fixing problems. Nilpotency does not select a unique representative, remove Gribov copies, or turn a charged boundary transformation into a redundancy.

Check your understanding

1. Classify $A_\mu$, $F_{\mu\nu}$, $b$, and the Maxwell ghost $c$ by ghost number, closure, and exactness.

   Check

   $A_\mu$ has ghost number zero but is not closed. $F_{\mu\nu}$ is closed at ghost number zero and is nontrivial in the local minimal algebra. The field $b=s\bar c$ is exact at ghost number zero. The ghost $c$ is closed at ghost number $+1$; it is unpaired in the local jet algebra but becomes exact as $c=s\varphi$ in the declared nonlocal bounded-Maxwell algebra.

2. Use $\{s,\kappa\}=N$ to remove a closed polynomial of positive $(\bar c,b)$ degree.

   Check

   Decompose the polynomial into $N$-eigenvectors. For a closed component $X_r$ with $r>0$, $rX_r=NX_r=(s\kappa+\kappa s)X_r=s(\kappa X_r)$. Hence $X_r=s(\kappa X_r/r)$ is exact. Only the degree-zero component can represent a class.

3. Explain why $sa+db=0$ does not automatically make $\int_\Sigma a$ BRST closed.

   Check

   Integrating gives $s\int_\Sigma a=-\int_\Sigma db=-\int_{\partial\Sigma}b$. The result vanishes only if support, boundary conditions, or added boundary degrees make that surface term zero or cancel it.

4. Compute the cohomology of one nonzero Maxwell cavity mode with $sA_{\mathrm L}=c$, $sc=0$, $s\bar c=b$, $sb=0$, and $sA_{\mathrm T}=0$.

   Check

   $(A_{\mathrm L},c)$ and $(\bar c,b)$ are contractible pairs, so they add no classes. Functions of the transverse amplitude $A_{\mathrm T}$ remain at ghost number zero. The normalization that puts the longitudinal pair in this form uses a nonzero Dirichlet eigenvalue and cannot be applied to a residual zero mode.

5. A nilpotent, Krein-self-adjoint charge has been constructed. List what is still missing before its ghost-number-zero cohomology is a physical Hilbert space.

   Check

   One still needs conservation and a common invariant domain, exact equality between the radical and the exact subspace, positivity of the induced form, control of the closure of $\operatorname{im}Q$, completion of the quotient, absence or treatment of unpaired zero and boundary modes, and a nonanomalous quantum implementation.

References

• Assanioussi, Mehdi, Jerzy Kowalski-Glikman, Ilkka Mäkinen, and Ludovic Varrin. “On the Covariant Formulation of Gauge Theories with Boundaries.” Classical and Quantum Gravity 41, no. 11 (2024): 115007. DOI. Open PDF, arXiv v2.
• Barnich, Glenn, Friedemann Brandt, and Marc Henneaux. “Local BRST Cohomology in Gauge Theories.” Physics Reports 338, no. 5 (2000): 439–569. DOI. Open PDF, arXiv v3.
• Fuster, Andrea, Marc Henneaux, and Axel Maas. “BRST-Antifield Quantization: A Short Review.” International Journal of Geometric Methods in Modern Physics 2, no. 5 (2005): 939–964. DOI. Open PDF, arXiv v2.
• Kugo, Taichiro, and Izumi Ojima. “Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem.” Progress of Theoretical Physics Supplement 66 (February 1979): 1–130. DOI.
• Kugo, Taichiro, and Izumi Ojima. “Errata.” Progress of Theoretical Physics 71, no. 5 (May 1984): 1121. DOI.
• Moss, Ian G., and Pedro J. Silva. “BRST-Invariant Boundary Conditions for Gauge Theories.” Physical Review D 55, no. 2 (1997): 1072–1078. DOI. Open PDF.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI. Author page and errata.
• Vandersickel, Nele, and Daniel Zwanziger. “The Gribov Problem and QCD Dynamics.” Physics Reports 520, no. 4 (2012): 175–251. DOI. Open PDF, arXiv v2.
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge: Cambridge University Press, 1996. DOI.