BRST cohomology and Faddeev-Popov-ghost quantisation of gauge theories

Intuition Beginner

A gauge theory has a built-in redundancy: many different-looking field configurations describe the very same physical situation. They are linked by gauge transformations, the way two maps of the same city can use different grid lines yet show the same streets. When you try to add up over all configurations to compute a quantum answer, you keep counting the same physical situation again and again, once for every way of redrawing the grid.

That overcounting makes the sum infinite for the wrong reason. The fix is to count each physical situation exactly once. You pick one representative from each family of equivalent configurations, much as you might agree to always measure angles from due north. Choosing a representative is called gauge fixing.

The surprise is that the bookkeeping for this choice introduces new helper quantities, the ghosts, and a hidden symmetry that keeps everything consistent.

Visual Beginner

Picture a sheet of paper covered with curves that never cross. Each curve is one family of gauge-equivalent configurations, called a gauge orbit. Every point on a single curve is the same physics in a different costume.

A gauge-fixing choice is a single line drawn across the curves so that it meets each curve once. The crossing points are the chosen representatives. The ghosts are the accountants that keep the area measure honest when you slice this way.

Worked example Beginner

Suppose you want the average value of a quantity over a city, but your list of addresses repeats each house ten times. If you add up the quantity over the whole list and divide by the list length, you get the right average, because the tenfold repeat cancels top and bottom.

Now imagine the quantity itself depends on the house, and you forget about the repeats. You would weight every house ten times too heavily. To fix it, you divide the total by ten, the size of the repeat. In a gauge theory the "size of the repeat" is the volume of the gauge group, written $\mathrm{vol}(G)$, and you must divide by it.

What this tells us: dividing by the repeat count is the whole idea. The Faddeev-Popov trick is the careful version of "divide by ten" when the repeat factor changes from configuration to configuration, and the ghosts measure exactly how it changes.

Check your understanding Beginner

Formal definition Intermediate+

Let $A$ denote a connection on a principal $G$-bundle over a manifold $M$, with $G$ a compact Lie group and Lie algebra $\mathfrak{g}$ 03.07.05, 03.04.01. The gauge group $\mathcal{G}$ acts on the affine space $\mathcal{A}$ of connections, and the path integral of an observable $\mathcal{O}(A)$ formally reads $\int \mathcal{D}A\mathcal{O}(A)e^{i{S}_{\mathrm{YM}}(A)}$. Because $S_{\mathrm{YM}}$ and $\mathcal{O}$ are gauge-invariant, the integrand is constant along each gauge orbit, so the integral contains a factor $\mathrm{vol}(\mathcal{G})$ that must be divided out ^{[tong §2]}.

The Faddeev-Popov method introduces a gauge-fixing function $G(A)$ (for example $G(A)={\partial }^{\mu }{A}_{\mu }-\omega$) and the identity

$$1={\Delta }_{\mathrm{FP}}(A)\int \mathcal{D}\alpha \delta (G(A^{\alpha })),{\Delta }_{\mathrm{FP}}(A)=\det \left(\frac{\delta G(A^{\alpha })}{\delta \alpha }\right){|}_{G(A^{\alpha })=0},$$

where $A^{\alpha }$ is the gauge transform of $A$ by the parameter $\alpha$. Inserting this identity and using gauge invariance to absorb the $\alpha$-integral into $\mathrm{vol}(\mathcal{G})$ replaces the divergent volume by the determinant ${\Delta }_{\mathrm{FP}}$ and the delta function.

The determinant is exponentiated as a Berezin (Grassmann) integral over anticommuting ghost and antighost fields $c,\bar{c}$ valued in $\mathfrak{g}$:

$$\det \left(\frac{\delta G}{\delta \alpha }\right)=\int \mathcal{D}c\mathcal{D}\bar{c}\exp \left(i\int \bar{c}\frac{\delta G}{\delta \alpha }c\right).$$

The resulting gauge-fixed action is

$$S=S_{\mathrm{YM}}+S_{\mathrm{gf}}+S_{\mathrm{ghost}},S_{\mathrm{gf}}=-\frac{1}{2\xi }\int (G(A){)}^2,S_{\mathrm{ghost}}=\int \bar{c}\frac{\delta G}{\delta \alpha }c,$$

or, after introducing the auxiliary Nakanishi-Lautrup field $B$, the gauge-fixing term becomes $S_{\mathrm{gf}}=\int (BG+\frac{\xi }{2}{B}^2)$, whose $B$-integral reproduces the squared form. A non-example worth flagging: the determinant is not a number to be ignored. For non-abelian $G$ the operator $\delta G/\delta \alpha$ depends on $A$, so the ghosts couple to the gauge field and contribute to loop diagrams. They decouple only in the abelian case treated below.

Key theorem with proof Intermediate+

Theorem (nilpotency of the BRST operator). Define the BRST transformation $s$, an odd derivation on the algebra of fields $\{A,c,\bar{c},B\}$ with $\mathfrak{g}$-bracket $[\cdot ,\cdot ]$, by

$$sA=D_{A}c=dc+[A,c],sc=-\frac{1}{2}[c,c],s\bar{c}=B,sB=0.$$

Then $s$ is nilpotent: $s^2=0$ on every field.

Proof. Compute $s^2$ on each generator, using that $s$ is an odd (degree $+1$) derivation and that $c$ is Grassmann-odd, so $[c,c]=2cc$ does not vanish.

On $\bar{c}$ and $B$: $s^2\bar{c}=sB=0$ and $s^{2}B=0$ by definition.

On $c$: since $sc=-\frac{1}{2}[c,c]$,

$$s^{2}c=-\frac{1}{2}s[c,c]=-\frac{1}{2}([sc,c]-[c,sc])=-[sc,c]=\frac{1}{2}[[c,c],c],$$

where the sign in the middle step is the odd Leibniz rule for the bracket of two odd objects. The expression $[[c,c],c]$ vanishes by the graded Jacobi identity of $\mathfrak{g}$: the cyclic sum $[[c,c],c]+[[c,c],c]+[[c,c],c]$ with all entries equal collapses, and the Jacobi identity for three odd copies of $c$ states exactly $[[c,c],c]=0$. Hence $s^{2}c=0$.

On $A$: using $sA=dc+[A,c]$ and the derivation property,

$$s^{2}A=s(dc)+s[A,c]=d(sc)+[sA,c]-[A,sc].$$

Substitute $sc=-\frac{1}{2}[c,c]$ and $sA=dc+[A,c]$:

$$s^{2}A=-\frac{1}{2}d[c,c]+[dc+[A,c],c]+\frac{1}{2}[A,[c,c]].$$

The term $-\frac{1}{2}d[c,c]=-[dc,c]$ cancels $[dc,c]$. The remaining $[[A,c],c]+\frac{1}{2}[A,[c,c]]$ vanishes by the graded Jacobi identity, since $[[A,c],c]=-\frac{1}{2}[A,[c,c]]$. Therefore $s^{2}A=0$. Nilpotency on all generators, together with the derivation property, gives $s^2=0$ on the whole algebra. $\square$

Bridge. The nilpotent operator $s$ builds toward the identification of physical observables with its degree-zero cohomology $H^0(s)$, where gauge-invariant quantities are precisely the BRST-closed classes modulo BRST-exact ones; it appears again in the Chevalley-Eilenberg reading of $sc=-\frac{1}{2}[c,c]$ as the Lie-algebra cohomology differential of $\mathfrak{g}$ 03.04.01, with $c$ the dual generator; it connects to the Yang-Mills action 03.07.05, whose gauge-fixed completion $S_{\mathrm{gf}}+S_{\mathrm{ghost}}$ is $s$-exact and therefore leaves the physical content untouched; and it underlies the cohomological characterisation of anomalies, where a candidate breaking of gauge symmetry is a ghost-number-one $s$-cocycle subject to the Wess-Zumino consistency condition.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — the graded-superalgebra and gauge-orbit infrastructure required to state BRST cohomology intrinsically is not yet present in Mathlib.

-- Pseudocode only: the BRST complex of a gauge theory is not yet available.
axiom Fields : Type*                         -- graded algebra of A, c, cbar, B
axiom ghostNumber : Fields → ℤ
axiom brst : Fields → Fields                 -- the odd derivation s

axiom brst_nilpotent (x : Fields) : brst (brst x) = 0     -- s² = 0
axiom brst_raises_ghost (x : Fields) :
    ghostNumber (brst x) = ghostNumber x + 1

-- physical observables = degree-0 BRST cohomology
axiom PhysicalObservables : Type*            -- ker s / im s at ghost number 0

The missing formalization work includes a ghost-number-graded superalgebra of fields, the odd nilpotent derivation $s$, the Chevalley-Eilenberg differential of the gauge Lie algebra, the Berezin integral producing the Faddeev-Popov determinant, and the cohomology $H^0(s)$ as the physical-state space.

Advanced results Master

The physical content of the quantised gauge theory is the BRST cohomology at ghost number zero. An observable $\mathcal{O}$ is admissible when it is BRST-closed, $s\mathcal{O}=0$, and two observables differing by a BRST-exact term $s\Lambda$ are physically identified. The space of physical observables is therefore

$$H^0(s)=\frac{\ker s{|}_{\text{gh}=0}}{\mathrm{im}s{|}_{\text{gh}=0}}.$$

At ghost number zero, $s\mathcal{O}=0$ for $\mathcal{O}=\mathcal{O}(A)$ reads $D_{A}c\cdot \delta \mathcal{O}/\delta A=0$ for all $c$, which is the statement that $\mathcal{O}$ is gauge-invariant. The exact part removes the gauge-fixing ambiguity. The construction realises the gauge-orbit space $\mathcal{A}/\mathcal{G}$ cohomologically: the ghost $c$ is the Maurer-Cartan form of the gauge group, and $s$ restricted to the ghost sector is the Chevalley-Eilenberg differential, so the BRST complex computes the equivariant cohomology of $\mathcal{A}$ under $\mathcal{G}$, equivalently the de Rham theory of the quotient where the action is free ^{[Sternberg Ch. 12]}.

The Kugo-Ojima quartet mechanism explains why the unphysical modes cancel. The fields organise into BRST quartets: a closed-but-not-exact pair and a BRST-doublet $(\bar{c},B)$ together with a longitudinal gauge mode and its ghost partner. Quartets contribute to the indefinite-metric Fock space but pair off in the cohomology, leaving only the transverse physical states. The condition that physical states $|\psi ⟩$ satisfy $Q_{\mathrm{BRST}}|\psi ⟩=0$ with the identification $|\psi ⟩\sim |\psi ⟩+Q_{\mathrm{BRST}}|\chi ⟩$ selects a positive-norm physical subspace, the operator analogue of $H^0(s)$ ^{[Kugo-Ojima 1979]}.

Anomalies appear as obstructions in BRST cohomology one degree up. A candidate anomaly is the BRST variation of the effective action, $s\Gamma =\mathcal{A}[c,A]$, a local functional of ghost number one. Consistency of the second variation forces the Wess-Zumino condition $s\mathcal{A}=0$, so $\mathcal{A}$ is a ghost-number-one $s$-cocycle. A genuine anomaly is a nontrivial class in $H^1(s)$: when $\mathcal{A}=s(\text{local counterterm})$ it can be removed, and when the class is nonzero the gauge symmetry is broken at the quantum level. The consistent anomalies are classified by the Lie-algebra cohomology $H^1(\mathfrak{g},\cdot )$ through the descent equations, tying the obstruction directly to the Chevalley-Eilenberg structure of $\mathfrak{g}$ ^{[Becchi-Rouet-Stora 1976]}.

The antifield (BV) formalism extends the construction when the gauge algebra closes only on-shell or has reducible generators. Each field $\varphi$ is paired with an antifield ${\varphi }^{\ast }$ of opposite statistics and complementary ghost number, and the BRST operator is generated by an antibracket with a master action $S$ solving the master equation $(S,S)=0$. The classical BRST differential is recovered as $s\cdot =(S,\cdot )$, and the master equation is the antibracket form of nilpotency ^{[Henneaux-Teitelboim 1992]}.

Synthesis. The Faddeev-Popov procedure divides the path integral by the gauge-orbit volume by inserting $1={\Delta }_{\mathrm{FP}}\int \mathcal{D}\alpha \delta (G(A^{\alpha }))$ and exponentiating ${\Delta }_{\mathrm{FP}}=\det (\delta G/\delta \alpha )$ with anticommuting ghosts; the residual nilpotent BRST symmetry $s$ with $sA=D_{A}c$, $sc=-\frac{1}{2}[c,c]$, $s\bar{c}=B$, $sB=0$ encodes the gauge structure and makes $S_{\mathrm{gf}}+S_{\mathrm{ghost}}$ manifestly $s$-exact. This construction builds toward the cohomological definition of physical observables as $H^0(s)$ at ghost number zero, where BRST-closedness modulo BRST-exactness is gauge invariance; it appears again in the Chevalley-Eilenberg identification of $sc=-\frac{1}{2}[c,c]$ with the Lie-algebra cohomology differential of $\mathfrak{g}$ 03.04.01, so the ghost sector is the cochain complex ${\Lambda }^{\bullet }{\mathfrak{g}}^{\ast }$; it connects to the Yang-Mills action 03.07.05, whose gauge invariance is exactly the $s$-closedness of $S_{\mathrm{YM}}$; and it underlies the cohomological theory of anomalies, where a consistent anomaly is a nontrivial ghost-number-one class in $H^1(s)$ obeying the Wess-Zumino condition. The single structural fact organising the subject is that gauge fixing trades a quotient by a group action for an odd nilpotent differential, so the geometry of the gauge-orbit space $\mathcal{A}/\mathcal{G}$ becomes the algebra of a cohomology.

Full proof set Master

Proposition (BRST-invariance of the gauge-fixed action). Let $S=S_{\mathrm{YM}}+s\Psi$ with gauge-fixing fermion $\Psi =\int \bar{c}(G(A)+\frac{\xi }{2}B)$ of ghost number $-1$. Then $sS=0$.

Proof. The BRST variation of $A$ is $sA=D_{A}c$, which is an infinitesimal gauge transformation of $A$ with the ghost $c$ in the role of the gauge parameter. Since $S_{\mathrm{YM}}$ is gauge-invariant 03.07.05, its variation under any gauge transformation is zero, and replacing the commuting parameter by the anticommuting $c$ preserves this because $S_{\mathrm{YM}}$ depends on $A$ only and the variation is linear in the parameter. Hence $s{S}_{\mathrm{YM}}=0$. For the second term, $s(s\Psi )=s^2\Psi =0$ by nilpotency, proved on each generator above and extended to $\Psi$ by the derivation property of $s$. Therefore $sS=s{S}_{\mathrm{YM}}+s^2\Psi =0$. $\square$

Proposition (gauge independence of BRST-cohomology expectation values). Let $\mathcal{O}$ be BRST-closed, $s\mathcal{O}=0$, and let the gauge-fixing fermion change by $\delta \Psi$. Then $⟨\mathcal{O}⟩$ is unchanged, provided the BRST Ward identity $⟨sX⟩=0$ holds for every $X$.

Proof. The action shifts by $\delta S=s\delta \Psi$ because only the $s$-exact part of $S$ depends on $\Psi$. To first order,

$$\delta ⟨\mathcal{O}⟩=i⟨\mathcal{O}s\delta \Psi ⟩.$$

Since $\mathcal{O}$ is BRST-closed, $s(\mathcal{O}\delta \Psi )=(s\mathcal{O})\delta \Psi \pm \mathcal{O}(s\delta \Psi )=\pm \mathcal{O}s\delta \Psi$, the sign being the Koszul sign from the ghost number of $\mathcal{O}$. Therefore $\mathcal{O}s\delta \Psi =\pm s(\mathcal{O}\delta \Psi )$, and

$$\delta ⟨\mathcal{O}⟩=\pm i⟨s(\mathcal{O}\delta \Psi )⟩=0$$

by the Ward identity. Likewise, replacing $\mathcal{O}$ by $\mathcal{O}+s\Lambda$ shifts $⟨\mathcal{O}⟩$ by $⟨s\Lambda ⟩=0$. The expectation value depends only on the BRST-cohomology class of $\mathcal{O}$, and not on the gauge choice encoded in $\Psi$. $\square$

Proposition (ghost-number-zero cohomology is gauge-invariant observables). Let $\mathcal{O}=\mathcal{O}(A)$ have ghost number zero. Then $s\mathcal{O}=0$ holds if and only if $\mathcal{O}$ is invariant under infinitesimal gauge transformations.

Proof. For $\mathcal{O}(A)$ of ghost number zero, the chain rule gives $s\mathcal{O}=\int \frac{\delta \mathcal{O}}{\delta A}sA=\int \frac{\delta \mathcal{O}}{\delta A}{D}_{A}c$. The expression $\int \frac{\delta \mathcal{O}}{\delta A}{D}_{A}c$ is the infinitesimal gauge variation of $\mathcal{O}$ with parameter $c$. It vanishes for every ghost field $c$ if and only if the variation ${\delta }_{ϵ}\mathcal{O}=\int \frac{\delta \mathcal{O}}{\delta A}{D}_Aϵ$ vanishes for every gauge parameter $ϵ$, because $c$ ranges over the same Lie-algebra-valued functions as $ϵ$. Thus $\mathcal{O}\in \ker s$ at ghost number zero exactly when $\mathcal{O}$ is gauge-invariant. There are no ghost-number-zero $s$-exact terms built from $A$ alone, since $s$ raises ghost number, so $H^0(s)$ in the matter-gauge sector is the algebra of gauge-invariant functionals. $\square$

Connections Master

• Yang-Mills action 03.07.05 — the gauge invariance of $S_{\mathrm{YM}}$ is precisely its BRST-closedness $s{S}_{\mathrm{YM}}=0$, and the gauge-fixed quantisation that makes the Yang-Mills path integral well-defined is the Faddeev-Popov construction whose residual symmetry is the BRST operator developed here.

• Lie algebra 03.04.01 — the ghost transformation $sc=-\frac{1}{2}[c,c]$ is the Chevalley-Eilenberg differential of the gauge Lie algebra $\mathfrak{g}$, with the ghost $c$ the degree-one generator of ${\Lambda }^{\bullet }{\mathfrak{g}}^{\ast }$, so nilpotency $s^2=0$ is the Jacobi identity and BRST cohomology contains the Lie-algebra cohomology of $\mathfrak{g}$.

• Electromagnetism as a U(1) Yang-Mills theory 03.07.29 — in the abelian case the bracket vanishes, the ghosts decouple as free fields, and BRST quantisation reduces to the gauge-fixed Maxwell theory, showing the ghost coupling is a strictly non-abelian phenomenon driven by the structure constants of $\mathfrak{g}$.

Historical & philosophical context Master

Feynman observed in the early 1960s that a naive Feynman-diagram expansion of non-abelian gauge theory violated unitarity at one loop, and Faddeev and Popov resolved this in 1967 by introducing the determinant and the auxiliary anticommuting fields that now carry their names, exponentiating the gauge-orbit measure into a local ghost action ^{[Faddeev-Popov 1967]}. The procedure was reformulated as a symmetry by Becchi, Rouet, and Stora in 1976, and independently by Tyutin in an unpublished 1975 Lebedev preprint, who recognised that the gauge-fixed action retains a global odd nilpotent symmetry whose square vanishes ^{[Becchi-Rouet-Stora 1976]}. Kugo and Ojima developed the operator formulation in 1979, defining physical states as BRST-cohomology classes in an indefinite-metric Fock space and using the quartet mechanism to prove that unphysical modes decouple ^{[Kugo-Ojima 1979]}. The geometric reading, in which the ghost is the Maurer-Cartan form of the gauge group and $s$ is the Chevalley-Eilenberg differential on the gauge-orbit space, was systematised by Sternberg among others, and the antifield extension handling open and reducible gauge algebras was given its definitive treatment by Henneaux and Teitelboim in 1992 ^{[Henneaux-Teitelboim 1992]}.

Bibliography Master

@article{FaddeevPopov1967,
  author  = {Faddeev, L. D. and Popov, V. N.},
  title   = {Feynman Diagrams for the Yang-Mills Field},
  journal = {Physics Letters B},
  volume  = {25},
  pages   = {29--30},
  year    = {1967}
}

@article{BecchiRouetStora1976,
  author  = {Becchi, C. and Rouet, A. and Stora, R.},
  title   = {Renormalization of Gauge Theories},
  journal = {Annals of Physics},
  volume  = {98},
  pages   = {287--321},
  year    = {1976}
}

@techreport{Tyutin1975,
  author      = {Tyutin, I. V.},
  title       = {Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism},
  institution = {Lebedev Physics Institute},
  number      = {39},
  year        = {1975},
  note        = {arXiv:0812.0580}
}

@article{KugoOjima1979,
  author  = {Kugo, T. and Ojima, I.},
  title   = {Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem},
  journal = {Progress of Theoretical Physics Supplement},
  volume  = {66},
  pages   = {1--130},
  year    = {1979}
}

@book{HenneauxTeitelboim1992,
  author    = {Henneaux, Marc and Teitelboim, Claudio},
  title     = {Quantization of Gauge Systems},
  publisher = {Princeton University Press},
  year      = {1992}
}

@book{Sternberg2012,
  author    = {Sternberg, Shlomo},
  title     = {Curvature in Mathematics and Physics},
  publisher = {Dover Publications},
  year      = {2012}
}