Existence and Uniqueness of a Nilpotent BRST Charge
This appendix proves Theorem 30.51 of Section 30.7.3 in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism: for every regular, irreducible set of first-class constraints the series Equation (30.97) can be continued to a Grassmann-odd function \(\Omega\) of ghost number \(+1\) satisfying Equation (30.98) exactly; two such functions with the same leading term differ by a canonical transformation of the extended phase space; and when the structure functions of Equation (30.16) are constants the series stops at the two terms Equation (30.97) displays.
This is the statement that makes BRST a method rather than a device. Without it the definition Definition 30.50 would be a recipe that happens to work for Yang–Mills and is silent about general relativity, whose structure functions are genuinely functions (Proposition 30.43, proved in The Hypersurface-Deformation Algebra); with it, the cohomological characterization Equation (30.99) of the physical state space is available for every first-class system.
Scope. BRST is in this treatise because it is the machinery behind the perturbative calculations of Quantum Chromodynamics and the path-integral development of Path-Integral Quantization, whose predictions are measured; it is not here as a formal programme. Accordingly this section stops where the chapter stops — at the Hamiltonian BRST charge of a finite-dimensional constrained system — and does not go on to the antifield or Batalin–Vilkovisky formulation, which is a different construction with different inputs and is used nowhere in this book.
Attribution. The existence of a nilpotent charge for a general first-class system is due to Fradkin and Vilkovisky and, in the form with structure functions, to Batalin, Fradkin and Vilkovisky; the homological proof organized around the Koszul–Tate differential, which is the one given below, is due to Henneaux and Teitelboim. None of these works has an entry in this treatise's bibliography, so — exactly as Remark 30.2 records for the rest of the chapter — the attribution is made in words and nothing below rests on it. The original BRST papers [Becchi:1976] [Tyutin:1975] are cited in the chapter.
The extended phase space and its two gradings
Let \(z^{c}\), \(c=1,\ldots,2n\), be the original phase-space variables and \(\gamma_{A}\), \(A=1,\ldots,F\), the first-class constraints, with Equation (30.16), \(\pb{\gamma_{A}}{\gamma_{B}}=f_{AB}{}^{C}\gamma_{C}\). Adjoin the ghost pairs \(\left(\eta^{A},\mathcal{P}_{A}\right)\) of Definition 30.50. Functions of \(\left(z,\eta,\mathcal{P}\right)\) carry a Grassmann parity \(\varepsilon\in\set{0,1}\) and the graded bracket obeys
together with the graded Jacobi identity. It splits as
where \(\pb{\cdot}{\cdot}_{M}\) differentiates only the \(z^{c}\) — so that the ghosts pass through it as constants — and \(\pb{\cdot}{\cdot}_{\text{gh}}\) only the ghost pair, with \(\pb{\eta^{A}}{\mathcal{P}_{B}}=-\delta^{A}_{B}\).
Three gradings are used. The pure ghost number counts the \(\eta\)'s; the antighost number counts the \(\mathcal{P}\)'s,
and the ghost number of Definition 30.50 is their difference, \(\operatorname{gh}=\operatorname{pgh}-\operatorname{antigh}\), so that \(\operatorname{gh}\eta^{A}=+1\) and \(\operatorname{gh}\mathcal{P}_{A}=-1\). Parity is pgh \(+\) antigh modulo two. Rests on Definition 30.50, Equation (30.16) and Equation (30.97).
Two grading facts are used constantly and are recorded once. \(\pb{\cdot}{\cdot}_{M}\) preserves the antighost number, while \(\pb{\cdot}{\cdot}_{\text{gh}}\) removes one \(\eta\) and one \(\mathcal{P}\) and so lowers it by one; both add ghost numbers. And an \(\Omega\) that is odd with \(\operatorname{gh}\Omega=+1\) decomposes as
in which \(\Omega_{(k)}\) has pure ghost number \(k+1\): it is a polynomial carrying exactly \(k+1\) ghosts and \(k\) ghost momenta, hence \(2k+1\) Grassmann-odd factors, hence odd, as it must be. The first two terms are the ones Equation (30.97) displays,
The Koszul–Tate differential
\(\delta\) is the odd derivation of the algebra of functions on the extended phase space fixed by
It lowers the antighost number by one and raises the ghost number by one. Rests on Notation A54.1 and Equation (30.16).
For every \(X\), \(\delta X=\pb{X}{\Omega_{(0)}}_{\text{gh}}\), and \(\delta^{2}=0\). Rests on Definition A54.2, Equation (A54.1) and Equation (A54.5).
Derives Lemma A54.3. \(\Omega_{(0)}\) contains no \(\mathcal{P}\), so \(\pb{\cdot}{\Omega_{(0)}}_{\text{gh}}\) annihilates \(z^{c}\) and \(\eta^{A}\), as \(\delta\) does. On \(\mathcal{P}_{A}\), the graded Leibniz rule Equation (A54.1) gives \(\pb{\mathcal{P}_{A}}{\eta^{B}\gamma_{B}}_{\text{gh}} =\pb{\mathcal{P}_{A}}{\eta^{B}}\gamma_{B}\), and graded antisymmetry between two odd objects gives \(\pb{\mathcal{P}_{A}}{\eta^{B}}=\pb{\eta^{B}}{\mathcal{P}_{A}} =-\delta^{B}_{A}\), so the value is \(-\gamma_{A}=\delta\mathcal{P}_{A}\). Both maps are odd derivations agreeing on generators, hence equal.
For nilpotency, \(\delta^{2}\) is the square of an odd derivation and is therefore an even derivation; it vanishes on \(z\) and \(\eta\) trivially and on \(\mathcal{P}_{A}\) because \(\delta^{2}\mathcal{P}_{A}=-\delta\gamma_{A}=0\), the constraints being functions of \(z\) alone. A derivation vanishing on generators is zero.
∎Remark 30.7 assumes two things about the constraint set, and Theorem 30.51 adds a third word, irreducible. Spelled out, what is assumed is: (i) the differentials \(\dd\gamma_{A}\) are linearly independent at every point of the surface \(\Sigma\) they define; (ii) every smooth function vanishing on \(\Sigma\) is \(c^{A}\gamma_{A}\) with smooth \(c^{A}\); and (iii) every \(\lambda^{A}\) with \(\lambda^{A}\gamma_{A}=0\) identically is of the form \(\lambda^{A}=\mu^{AB}\gamma_{B}\) with \(\mu^{AB}=-\mu^{BA}\) — there are no relations among the constraints beyond the trivial ones. Item (iii) is what irreducible means; a reducible set, such as the one obtained by writing a two-form constraint in components with a Bianchi identity among them, fails it, and the construction below then needs ghosts for ghosts and is not carried out in this treatise.
Lemma A54.5 shows that (i) already implies (ii) and (iii) locally, so the three are not independent assumptions; the content is entirely in (i), which is the same regularity that Proposition 30.5 and Theorem 30.17 rest on.
About each point of \(\Sigma\) there are local coordinates \(\left(y^{A},w^{\alpha}\right)\) on phase space in which \(\gamma_{A}=y^{A}\). In such coordinates, (ii) and (iii) of Remark A54.4 hold. Rests on Remark 30.4, Theorem A28.1 and Remark 30.7.
Derives Lemma A54.5. The map \(z\longmapsto\left(\gamma_{1}(z),\ldots,\gamma_{F}(z)\right)\) has surjective differential at each point of \(\Sigma\) by (i), hence constant maximal rank on a neighbourhood; the constant-rank theorem stated in Remark 30.4, itself a consequence of Theorem A28.1, supplies coordinates in which it is the projection onto the first \(F\) of them. Those coordinates are the \(\left(y^{A},w^{\alpha}\right)\).
For (ii), let \(F(y,w)\) vanish at \(y=0\). By the fundamental theorem of calculus applied along the segment \(t\longmapsto\left(ty,w\right)\),
and \(\lambda_{A}\) is smooth by differentiation under the integral sign. That is (ii). Item (iii) is the case \(k=1\) of Lemma A54.6 below, whose proof uses only Equation (A54.7) and the coordinates just constructed, so nothing is circular.
∎On the coordinate patch of Lemma A54.5: \(H_{k}(\delta)=0\) for every \(k\ge1\) — every \(\delta\)-closed function of antighost number \(k\ge1\) is \(\delta\) of a function of antighost number \(k+1\) — and \(H_{0}(\delta)\) is the algebra of smooth functions on \(\Sigma\) (tensored with the ghosts). Rests on Lemma A54.5, Definition A54.2 and Remark A54.4.
Derives Lemma A54.6. Work in the coordinates of Lemma A54.5, so that \(\delta\mathcal{P}_{A}=-y^{A}\). The \(\eta\)'s play no part: \(\delta\) annihilates them and they simply multiply everything, so the complex is a free module over the Grassmann algebra they generate and it suffices to treat \(\eta\)-independent coefficients.
A homotopy. Let \(\sigma\) be the odd derivation with
so that on functions \(\sigma f=\mathcal{P}_{A}\,\pp f/\pp y^{A}\). The anticommutator of two odd derivations is an even derivation, so
is a derivation, and it is determined by its values on generators: \(\mathcal{N}y^{A}=-\delta\mathcal{P}_{A}=y^{A}\), \(\mathcal{N}\mathcal{P}_{A}=-\sigma\left(-y^{A}\right) =\mathcal{P}_{A}\), and \(\mathcal{N}w^{\alpha} =\mathcal{N}\eta^{A}=0\). So \(\mathcal{N}\) is the Euler operator counting the joint degree in \(y\) and \(\mathcal{P}\). Because \(\delta^{2}=0\), \(\mathcal{N}\) commutes with \(\delta\).
Inverting \(\mathcal{N}\) in positive antighost degree. A general element of antighost number \(k\) is \(a=\frac{1}{k!}a^{A_{1}\ldots A_{k}}(y,w)\, \mathcal{P}_{A_{1}}\cdots\mathcal{P}_{A_{k}}\), on which \(\mathcal{N}=k+\mathcal{N}_{y}\) with \(\mathcal{N}_{y}=y^{A}\pp/\pp y^{A}\). For \(k\ge1\) define
acting on the coefficient functions and leaving the \(\mathcal{P}\)'s alone; it is smooth by differentiation under the integral sign. It is the inverse: from \(\dv{}{t}\left[t^{k}a(ty,w)\right] =t^{k-1}\left[k\,a+\mathcal{N}_{y}a\right](ty,w)\), integrating from \(0\) to \(1\) — where the boundary term at \(t=0\) vanishes because \(k\ge1\) — gives \(a=\mathcal{N}^{-1}\left(k+\mathcal{N}_{y}\right)a =\mathcal{N}^{-1}\mathcal{N}a\). Since \(\mathcal{N}\) commutes with \(\delta\) and is invertible in every antighost degree \(\ge1\), so does \(\mathcal{N}^{-1}\).
Acyclicity. Let \(\delta a=0\) with \(\operatorname{antigh}a=k\ge1\). Then by Equation (A54.9), \(\mathcal{N}a=-\delta\sigma a-\sigma\delta a=-\delta\sigma a\), and applying \(\mathcal{N}^{-1}\), which commutes with \(\delta\),
so \(a\) is \(\delta\)-exact, with a primitive of antighost number \(k+1\).
Degree zero. At antighost number \(0\) every element is closed, and the image of \(\delta\) from degree one is \(\set{\lambda^{A}\gamma_{A}}\), which by Equation (A54.7) is exactly the ideal of functions vanishing on \(\Sigma\). Hence \(H_{0}(\delta)=C^{\infty}(\Sigma)\).
This is the mathematical heart of the section, and it is worth naming what was consumed: the coordinates supplied by regularity, and nothing else. Irreducibility — item (iii) of Remark A54.4 — is the case \(k=1\) of what has just been proved, since \(\lambda^{A}\mathcal{P}_{A}\) is \(\delta\)-closed precisely when \(\lambda^{A}\gamma_{A}=0\), and a primitive of antighost number \(2\) is an antisymmetric \(\mu^{AB}\) with \(\lambda^{A}=\mu^{AB}\gamma_{B}\).
∎The recursion
With \(\Omega\) expanded as in Equation (A54.4), the antighost-\(k\) component of \(\pb{\Omega}{\Omega}\) is
where
depends on \(\Omega_{(0)},\ldots,\Omega_{(k)}\) only. So Equation (30.98) is the sequence of equations \(\delta\Omega_{(k+1)}=D_{(k)}\), \(k\ge0\). Explicitly
and \(\Omega_{(1)}\) of Equation (A54.5) solves \(\delta\Omega_{(1)}=D_{(0)}\). Rests on Lemma A54.3, Notation A54.1 and Equation (30.98).
Derives Lemma A54.7. Expand \(\pb{\Omega}{\Omega}\) bilinearly and sort by antighost number, using that \(\pb{\cdot}{\cdot}_{M}\) preserves it and \(\pb{\cdot}{\cdot}_{\text{gh}}\) lowers it by one. The terms of the ghost sum with \(i=0\) or \(j=0\) are \(\pb{\Omega_{(0)}}{\Omega_{(k+1)}}_{\text{gh}} +\pb{\Omega_{(k+1)}}{\Omega_{(0)}}_{\text{gh}}\), and the bracket of two odd functions is graded-symmetric by Equation (A54.1), so these are equal and their sum is \(2\pb{\Omega_{(k+1)}}{\Omega_{(0)}}_{\text{gh}} =2\delta\Omega_{(k+1)}\) by Lemma A54.3. Everything else is Equation (A54.13). That \(D_{(k)}\) involves no \(\Omega_{(j)}\) with \(j>k\) is immediate from the ranges of the two sums.
For Equation (A54.14): since \(\pb{\cdot}{\cdot}_{M}\) differentiates only the \(z^{c}\), the ghosts pass through it, and moving one odd \(\eta\) past the even object \(\pp_{c}\gamma_{A}\) costs no sign, so
by Equation (30.16). Halving and negating gives Equation (A54.14). Equation (A54.15) is the case \(k=1\) of Equation (A54.13), the ghost sum having only the term \(i=j=1\) and the \(M\) sum only the two equal terms \(\pb{\Omega_{(0)}}{\Omega_{(1)}}_{M}\).
Finally, \(\delta\) passes through the even coefficient \(-\tfrac{1}{2}\eta^{B}\eta^{A}f_{AB}{}^{C}\) without a sign and \(\delta\mathcal{P}_{C}=-\gamma_{C}\), so
the middle step by anticommutativity of the ghosts. So the term Equation (30.97) displays is not a guess: it is the unique solution of the first equation of the recursion, up to the \(\delta\)-closed ambiguity that Uniqueness disposes of.
∎Notice what the first step already required. \(D_{(0)}\) has antighost number \(0\), where \(\delta\) is not acyclic, so \(\delta\Omega_{(1)}=D_{(0)}\) is solvable only if \(D_{(0)}\) lies in the image of \(\delta\) — that is, by Lemma A54.6, only if it vanishes on \(\Sigma\). Equation (A54.14) shows it is proportional to the constraints, and that is precisely the statement that the constraints are first class. The whole construction is powered by Equation (30.16), and it fails at the first step for a set that is not first class.
Existence
Let \(\gamma_{A}\) be regular and irreducible in the sense of Remark A54.4. Then on a neighbourhood of \(\Sigma\) there is an odd \(\Omega\) of ghost number \(+1\), of the form Equation (A54.4) with the first two terms Equation (A54.5), satisfying \(\pb{\Omega}{\Omega}=0\). Rests on Lemma A54.6, Lemma A54.7 and Equation (30.16).
Derives Theorem A54.8. Induction on the antighost number. By Equation (A54.17), \(\Omega_{(0)}\) and \(\Omega_{(1)}\) are constructed and \(\pb{\Omega}{\Omega}\) has no antighost-\(0\) component. Suppose \(\Omega_{(0)},\ldots,\Omega_{(m)}\) have been found, \(m\ge1\), such that \(R:=\pb{\Omega^{[m]}}{\Omega^{[m]}}\), with \(\Omega^{[m]}:=\sum_{j\le m}\Omega_{(j)}\), has vanishing components of antighost number \(\le m-1\).
The obstruction is \(\delta\)-closed. For an odd \(\Omega\) the graded Jacobi identity applied to the triple \(\left(\Omega^{[m]},\Omega^{[m]},\Omega^{[m]}\right)\) collapses to the identity
which holds whatever \(\Omega^{[m]}\) is, nilpotent or not: the three cyclic terms of the graded Jacobi identity are equal, and their common sign is such that three times one of them must vanish. Take the antighost-\(\left(m-1\right)\) component of Equation (A54.18). The \(M\)-part contributes \(\sum_{i+j=m-1}\pb{\Omega_{(i)}}{R_{(j)}}_{M}\), in which every \(j\) is at most \(m-1\) and every \(R_{(j)}\) therefore vanishes by hypothesis. The ghost part contributes \(\sum_{i+j=m}\pb{\Omega_{(i)}}{R_{(j)}} _{\text{gh}}\), in which the only surviving term is \(i=0\), \(j=m\). Hence \(\pb{\Omega_{(0)}}{R_{(m)}}_{\text{gh}}=0\), and since \(R\) is even and \(\Omega_{(0)}\) odd this is \(-\delta R_{(m)}\). So \(\delta R_{(m)}=0\).
Solve. \(R_{(m)}\) has antighost number \(m\ge1\), so by Lemma A54.6 there is a \(c\) with \(\delta c=R_{(m)}\) and \(\operatorname{antigh}c=m+1\). Counting gradings: \(R\) is even with ghost number \(2\), so \(R_{(m)}\) has pure ghost number \(m+2\); \(c\) therefore has pure ghost number \(m+2\) and antighost number \(m+1\), hence ghost number \(+1\) and odd parity — exactly the type of an \(\Omega_{(m+1)}\). Put \(\Omega_{(m+1)}:=-\tfrac{1}{2}c\). Adding it to \(\Omega^{[m]}\) changes nothing of antighost number \(<m\), because the new term enters \(\pb{\Omega}{\Omega}\) only at antighost number \(\ge m\), and it changes the antighost-\(m\) component by \(2\delta\Omega_{(m+1)}=-\delta c=-R_{(m)}\), cancelling it. This is the statement \(\delta\Omega_{(m+1)}=D_{(m)}\) of Lemma A54.7.
The induction produces \(\Omega_{(k)}\) for every \(k\). It terminates for a reason of degree: \(\Omega_{(k)}\) carries \(k\) factors \(\mathcal{P}_{A}\), which anticommute, so \(\Omega_{(k)}=0\) identically for \(k>F\). The series Equation (A54.4) is therefore a finite sum, and \(\Omega\) is a genuine function on the extended phase space, not a formal series.
∎Termination at the displayed terms
If the \(f_{AB}{}^{C}\) of Equation (30.16) are constants, then \(D_{(1)}=0\) and \(\Omega=\Omega_{(0)}+\Omega_{(1)}\) is exactly nilpotent: the series Equation (30.97) stops at the two terms displayed. Rests on Lemma A54.7, Equation (30.16) and Equation (26.46).
Derives Proposition A54.9. Both terms of Equation (A54.15) vanish.
The second does so for a trivial reason: with constant \(f\), the function \(\Omega_{(1)}\) contains no \(z^{c}\) at all, and \(\pb{\cdot}{\cdot}_{M}\) differentiates only the \(z^{c}\), so \(\pb{\Omega_{(0)}}{\Omega_{(1)}}_{M}=0\). This is exactly the place where structure functions would obstruct: for non-constant \(f\) this bracket is a nonzero function of antighost number \(1\), and it is what forces \(\Omega_{(2)}\) to exist.
The first is the Jacobi identity. The ghost bracket pairs the \(\pp/\pp\eta\) of one \(\Omega_{(1)}\) with the \(\pp/\pp\mathcal{P}\) of the other, producing a term with three \(\eta\)'s, one \(\mathcal{P}\) and two factors \(f\); because the three ghosts anticommute, the coefficient is the total antisymmetrization \(f_{[AB}{}^{D}f_{C]D}{}^{E}\). That vanishes: applying the Poisson Jacobi identity Equation (26.46) to the triple \(\left(\gamma_{A},\gamma_{B},\gamma_{C}\right)\) and using Equation (30.16) twice with constant coefficients gives \(f_{[AB}{}^{D}f_{C]D}{}^{E}\gamma_{E}=0\), and the \(\gamma_{E}\) are independent by irreducibility, so the coefficient itself vanishes.
With \(D_{(1)}=0\) one may take \(\Omega_{(2)}=0\), and then every later \(D_{(k)}\) is built from \(\Omega_{(0)}\), \(\Omega_{(1)}\) and vanishing terms and is itself zero by the same two arguments, so \(\Omega_{(k)}=0\) for all \(k\ge2\) solves the recursion. That is the case of Section 30.6.4; general relativity, whose structure functions are genuinely functions by Proposition 30.43, is not it, and the omitted terms of Equation (30.97) are exactly what Theorem A54.8 supplies there.
∎Uniqueness
Let \(\Omega\) and \(\Omega'\) both satisfy the hypotheses and conclusion of Theorem A54.8 with the same \(\Omega_{(0)}=\Omega'_{(0)}=\eta^{A}\gamma_{A}\). Then there is a canonical transformation of the extended phase space, generated by an even function of ghost number \(0\), carrying \(\Omega\) to \(\Omega'\). Rests on Theorem A54.8, Lemma A54.6 and Lemma A54.7.
Derives Proposition A54.10. Induction on the lowest order at which the two differ. Suppose \(\Omega_{(j)}=\Omega'_{(j)}\) for \(j<k\) and put \(\Delta_{(k)}:=\Omega'_{(k)}-\Omega_{(k)}\), with \(k\ge1\). By Lemma A54.7 both satisfy \(\delta\Omega_{(k)}=D_{(k-1)}\) with the same right-hand side, since \(D_{(k-1)}\) is built from the orders below \(k\), which agree. Hence \(\delta\Delta_{(k)}=0\).
By Lemma A54.6 there is a \(K\) with \(\delta K=\Delta_{(k)}\) and \(\operatorname{antigh}K=k+1\). Its type is forced: \(\Delta_{(k)}\) is odd with ghost number \(+1\) and pure ghost number \(k+1\), so \(K\) has pure ghost number \(k+1\) and antighost number \(k+1\), hence ghost number \(0\) and even parity — a legitimate generator of a canonical transformation preserving both gradings.
Let \(T_{K}\) be the canonical transformation generated by \(K\),
which is a finite sum here because \(K\) has positive antighost number and each bracket lowers it by at most one while raising the number of \(\eta\)'s, so the series terminates. Since \(K\) has antighost number \(k+1\) and the ghost bracket lowers antighost number by one, \(\pb{\Omega}{K}\) has antighost number \(\ge k\), and its antighost-\(k\) part is \(\pb{\Omega_{(0)}}{K}_{\text{gh}}=-\delta K=-\Delta_{(k)}\). So \(T_{K}\Omega'\) agrees with \(\Omega\) through antighost number \(k\) and, being the image of a nilpotent charge under a canonical transformation, is itself nilpotent — a canonical transformation preserves the graded bracket, so it carries \(\pb{\Omega'}{\Omega'}=0\) to \(\pb{T_{K}\Omega'}{T_{K}\Omega'}=0\).
Repeat at the next order at which the two still differ. The generators so produced have strictly increasing antighost number, and all vanish beyond \(F\) by the degree argument of Theorem A54.8, so their composition is a single canonical transformation carrying \(\Omega'\) to \(\Omega\).
∎Two things, and one limitation.
Quoted. The constant-rank theorem, in the form Remark 30.4 states it, is used once, in Lemma A54.5, to flatten the constraints. That theorem is owed by Real Analysis and Differentiable Manifolds, Tensors, and Curvature and is a standing debt of Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism, not a new one; the analytic input from which it is proved, Theorem A28.1, is in this appendix.
Not quoted, though it usually is. The acyclicity of the Koszul complex in positive degree (Lemma A54.6) is normally imported from homological algebra, where it is the statement that the Koszul complex of a regular sequence is a resolution. This treatise carries no homological algebra, and opening a chapter of it to prove one lemma would be out of proportion; the case at hand is therefore proved here outright, by an explicit contracting homotopy Equations (A54.8) and (A54.10), and the general theorem is recorded as a debt of Part II rather than used.
The limitation. Lemma A54.5 is local: the coordinates in which \(\gamma_{A}=y^{A}\) exist on a neighbourhood of a point of \(\Sigma\), so Lemma A54.6 and with it Theorem A54.8 are local statements. Passing to a global \(\Omega\) requires patching the local solutions with a partition of unity, which works whenever the \(\gamma_{A}\) are globally defined and the surface admits a tubular neighbourhood — the case in every application in this book, where the constraints are given by global formulas — but it is a further step and it is not carried out here. This is the same locality that Remark 30.27 records for gauge fixing, and for the same reason: what is easy near a point of the constraint surface can be obstructed in the large.
Existence and Uniqueness of a Nilpotent BRST Charge discharges the derivation owed at Theorem 30.51 of Section 30.7.3 in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism. Three consequences are worth carrying back. The construction shows that the two terms Equation (30.97) displays are not an ansatz but the first two solutions of a recursion, the second of them forced; Proposition A54.9 shows that the series stops there exactly when Equation (30.16) closes on constants, which is the Yang–Mills case of Section 30.6.4 and is why the chapter can display two terms and say “\(+\cdots\)”; and Proposition A54.10 is what makes Equation (30.99) well posed, since a cohomology defined by a charge that was only unique up to nothing in particular would not be a property of the theory. The first-class algebra that the whole construction consumes is Proposition 30.13; for general relativity it is the hypersurface-deformation algebra of The Hypersurface-Deformation Algebra, whose structure functions are the reason a general existence theorem was needed at all.