On the Quantization of Reducible Gauge Systems

Reducible gauge theories with constraints linear in the momenta are quantized. The equivalence of the reduced phase space quantization, Dirac quantization and BRST quantization is established. The ghosts of ghosts are found to play a crucial role in the equivalence proof.


INTRODUCTION
Gauge theories can be quantized according to at least three different methods: (i) The reduced phase space method quantizes only the gauge invariant functions and is for that reason physically quite appealing. However, it is often not tractable because it requires the explicit finding of a complet set of gauge invariant functions. It is easy to check that the three different approaches to the quantization of gauge systems are equivalent in the case of simple constraints (see for instance Ref. [1]). The question of their equivalence for arbitrary systems is more subtle and has attracted recently a considerable amount of interest 1−14 . Because the problem of "quantization" is inherently ambiguous (many different quantum systems possess the sameh → 0 limit), the question of equivalence is actually ill-defined in the absence of a definite choice of quantization prescriptions. For this reason, a conclusive analysis should either exhibit correspondence rules that insure equivalence of the three quantization methods, or prove the inexistence of such rules.
The previous works on the equivalence question are all devoted to independent ("irre-ducible") first class constraints. The purpose of this paper is to investigate equivalence in the case of first class reducible constraints, for which some constraints are consequences of the others. The reducible case raises new problems with respect to the irreducible one. For instance, in order to get a consistent Dirac quantization, it is necessary not only that the constraints remain first-class quantum mechanically, but also that they remain dependent.
Otherwise, the number of degrees of freedom in the classical and quantum theories would be different. Furthermore, in the BRST formalism, ghosts of ghost are necessary besides the standard ghosts, and it is of interest to understand their role in definite quantum models.
As the general question of equivalence is quite intricate, we restrict in this paper the analysis to reducible first class systems with constraints that are linear, homogeneous in the momenta. This case is already of interest since it covers p-form gauge fields and illustrate very well the crucial role played by the ghosts of ghosts. The corresponding irreducible models have been investigated in Ref. [2,7,9].
In the framework of the quantization rules where the physical wave functions are taken to be densities of weight one -half in the configuration space, we show that the three methods of quantization yield the same physical spectrum, provided one transforms appropriately the Dirac wave functions under a redefinition of the constraints and of the reducibility coefficients. In order to get a consistent Dirac quantization, we also find it necessary to correct the naive Dirac operator condition by an extra term. This extra term, as well as the transformation properties of the Dirac wave functions, are quite natural from the BRST point of view. Our results generalize to the reducible case those derived by Tuynman in Ref. [9] for irreducible constraints.
Our paper is organized as follows. In the next section, we describe explicitly the models considered in this paper. We then derive the classical BRST charge that captures all the identities fulfilled by those models (Section 3). We turn next to the quantization of the models, first along the lines of the reduced phase space method (Section 4), and then along those of the Dirac approach (Section 5). We find it crucial to improve the naive Dirac quantum constraints by an appropiate term that makes them anomaly free, and we derive this term by geometric arguments. Section 6 establishes the equivalence of the reduced phase space and Dirac methods by developing further the geometric interpretation. The BRST quantization and its equivalence with the other methods of quantization are given in Section 7. The key role played by the ghosts and ghosts of ghosts is particularly stressed since they precisely yield the anomaly cancelling term of the Dirac quantization method.
Finally, Section 8 is devoted to concluding comments.

THE MODELS
The systems considered in this paper are described by n pairs of canonically conjugate variables (q i , p i ). They are subject to m 0 bosonic constraints, which we take to be linear in the momenta The constraints are first class, i.e.
where {, } stands for the Poisson bracket in the phase space spanned by the variables (q i , p i ). Since the constraints are linear in the momenta, the structure functions C c 0 a 0 b 0 can be taken to depend only on the coordinates q i . Furthermore, the gauge transformation of a function f (q) defined on the configuration space Q ,is and depends also only on q. The vector fields ξ a 0 define the gauge transformations in the configuration space and are tangent to the gauge orbits. By inserting eq.(2.2) in eq.(2.3) we obtain where ,j denotes differentiation with respect to q j . The equation (2.5.a) can be rewritten as where [ , ] is the Lie bracket and L ξ a 0 is the Lie derivative operator along ξ a 0 .
The gauge transformations generated by the constraints are said to be reducible when Because the constraints are linear and homogeneous in the momenta, we may assume the reducibility functions Z a 0 a 1 to depend only on q i ; eq.(2.6) is then equivalent to Z a 0 a 1 (q) ξ a 0 (q) = 0, a 1 = 1, ..., m 1 . (2.7) The functions Z a 0 a 1 are required to exhaust all the relations among the fields ξ a 0 .
It might happen that the set {Z a 0 a 1 } is overcomplete, i.e., there exists a set of functions Eq. (2.8) means that Z a 1 a 2 Z a 0 a 1 can be written as a combination of the constraints. Again, the functions Z a 1 a 2 may be taken to depend only on q. Since the constraints depend on the momenta but the Z's do not, the only possibility is that eq.(2.8) is valid strongly. In general one finds a tower of reducibility equations: The tower stops with functions Z a L−1 a L (q) that are linearly independent, The theory has then order of reducibility equal to L. The number of independent gauge generators is: It will be convenient to choose the Z's such that Non linearly independent gauge generators appear in a physical theory when one cannot isolate a subset of independent constraints without violating explicit covariance, locality, or global conditions. A well known example is the case of a p-form gauge field For such a field the constraints are: (the generalization of the Gauss law) which are not independent because of the antisymmetrization in the indices i 1 , ..., i p (π i 1 i 2 ...i p ,i 1 i 2 ≡ 0).

IDENTITIES
The functions ξ i a 0 (q) and Z a k−1 a k (q) fulfill a series of identities that can be derived from eqs.(2.5) and (2.6) by differentiation and use of the symmetry of the second partial derivatives. For instance, the Jacobi identity, leads to the equation: For a reducible theory, this means that there exist functions M a 1 a 0 b 0 c 0 (q) such that the last term being not present when the theory is irreducible.
Similarly, by differentiating the identity (2.7) along the orbits, one gets or, in terms of the vector fields ξ a 0 , Because of eq.(2.5.a), this is equivalent to The completeness of the functions Z a 0 a 1 implies then the existence of functions D b 1 b 0 a 1 (q) such that: If one contracts this identity with Z b 0 c 1 , sums over b 0 and uses eq.(2.7), one gets The symmetric part of this equation in a 1 , c 1 , reads The completeness of Z a 0 b 1 implies then the existence of functions the right-hand side of this equation being absent for reducible theories of order L = 1.

THE BRST GENERATOR
The identities ( where Ω = Ω(q i , p j , η a k , P b k ), the BRST generator, is a fermionic function in an extended phase space E including canonically conjugate pairs of ghosts (η a k , P a k ), k = 0, ..., L, besides the original canonical variables,
η a k+1 Z a k a k+1 P a k + "more", (3.6) where "more" does not contains terms of the already indicated form.
In our case the G's and the Z's are bosonic; in order that Ω be fermionic, the ghosts belonging to an even generation (η a 0 , P a 0 , η a 2 , P a 2 , ...) must be fermionic, while those belonging to an odd generation must be bosonic. Due to the choice (2.11), Ω turn out to be real if The generator Ω can be built by means of a recursive method (see Ref. [1]): where Ω , Ω , Ω { , } η a s ,P a s is the bracket with respect to the pair (η a s , P a s ). In eq.(3.9) δ is the Koszul-Tate operator. In our case δ reads explicitly 11.c) and is clearly nilpotent (δ 2 = 0), because the reducibility eq.(2.9.a) holds strongly when the constraints are linear in the momenta (in the general case, additional terms are needed in (3.11) to achieve nilpotency).
The existence of Ω is established in Ref. [17]. Its explicit form for arbitrary L is cumbersome and will not be needed here. We shall only need: (i) the crucial fact that Ω is linear in the momenta (p i , P a k ) (Proposition 1); and (ii) the identities in Propositions 2, 3 and 4 below.
In the case of constraints linear in the momenta p j , the BRST generator Ω can be taken to be linear in the momenta (p j , P a k ).
Proof. One has , So let us suppose that all the Ω (k) 's for k ≤ p are linear in the momenta (this is true for is linear in the momenta from (3.10). Because of the definition (3.11) for δ, we find then that Ω (p+1) in eq.(3.9) can be taken to be linear in the momenta.♦ By expanding out {Ω, Ω} = 0, one finds at the lowest orders in the ghosts and their momenta the identities (2.5), (2.7) and (2.9), since C c 0 in Ω, as we just got. Similarly, by calling M a 1 a 0 b 0 c 0 /3, the coefficient of η a 0 η b 0 η c 0 P a 1 , one finds the identity (3.1). In addition one has: The proof of these propositions goes as follows.
in Ω. Then, the vanishing of the coefficients of η a k η b 0 P a k−1 , η d 1 η c k P a k , and η a k η b 0 η a 0 P c k in {Ω, Ω} yield respectively (3.12), (3.13), and (3.14). Note that since the ghosts η a 1 are bosonic, the coefficients B a 2 c 1 d 1 are symmetric in (c 1 , d 1 ), Note also that the identity (3.12) reduces to the identity (3.2) for k = 1, and (3.13) (with It is of interest to write explicitly the BRST charge for a reducible theory of order (3.16)

REDUCED PHASE SPACE QUANTIZATION
Because the gauge transformations are defined within the space Q of the q's, one can introduce the reduced configuration space Y ≡ Q/G as the quotient of the configuration space Q by the gauge orbits in Q. Let y α , α = 1, ..., N , be coordinates in the reduced configuration space. N is equal to n minus the number m of independent constraints. One Let π α (q, p) be the gauge invariant momenta conjugate to y α , The variables y α and π α define a standard unconstrained system, the "reduced system" associated with the original gauge system. The reduced phase space quantization consists in quantizing this reduced system without worring about its origins.
So, let us consider a non-constrained system described classically by coordinates and momenta (y α , π α ), α = 1, ..., N . At a given time, the quantum state of the system is given by a wave function ϕ(y α ) belonging to a Hilbert space. It is convenient to define the inner product in this space as: In order that the inner product (4.3) be invariant under coordinate changes, the wave functions must behave as scalar densities of weight 1/2: The product ϕ * ψ is then a density of weight 1, i.e., defines a N -form in Y. Since the integral of a N -form over a N -dimensional manifold is intrinsically defined (does not require an extra integration measure), the convention of taking the wave functions to be densities of weight 1/2 is convenient in the absence of a natural integration measure † .
The observables that are linear in the momenta π α conjugate to y α , a = a α (y) π α (classically), (4.5) possess a natural geometric interpretation since they define vector fields on the manifold of the y's. Their quantum version reads a = 1 2 (a α (y) π α + π α a α (y)) , (4.6) † In practice, however, the manifold comes equipped with an integration measure ν. For instance, it is a Riemannian manifold and ν = g 1/2 . In that case, one can replace the wave functions by scalars, by redefining them as ϕ → ν −1/2 ϕ. Of course, this procedure also requires a redefinition of the operators in order to keep the matrix elements unchanged. and is formally Hermitian for the scalar product (4.3) whenever a α is real. The action of a on a wave function yields −i times its Lie derivative (as a density of weight 1/2),

DIRAC QUANTIZATION
We now turn to the Dirac quantization, where the wave funtions are taken to depend on all the coordinates q i and not just on the gauge invariant ones. In order to remove the unphysical degrees of freedom, one imposes on the physical states the condition whereĜ a 0 is the realization of each constraint as an operator in the space of the wave functions. Since the quantum realization of any classical function is ambiguous (factor ordering problem), we should carefully define the operatorĜ a 0 . Because the constraints are linear in the momenta, it is natural to takeĜ a 0 ψ to be the Lie derivative of ψ along ξ a 0 .
However, the Lie derivative of ψ is ill-defined as long as one does not give the transformation rules for ψ. So the question is: which object is ψ? In order to gain insight into this question, let us first consider the simple abelian case.

ABELIAN CASE
Let us thus assume that the coordinates q i can be split as q i ≡ (y α , Q A ), in such a way that the constraints G a 0 ≈ 0 are equivalent to P A being the momenta conjugate to Q A . Locally, this can always be achieved 1 . The reduced phase space for the system is then the space of the y α and the π α . The discussion of Section 4 shows that the physical wave functions depend only on y α , i.e. are annihilated byP A . Furthermore, the scalar product (4.3) can be rewritten as This expression is invariant if we transform the wave functions not only as densities of weight 1/2 under changes of the physical coordinates y α (as we already pointed out), but also as scalars under changes of the pure gauge coordinates Q A .
This asymmetric behaviour of the wave functions under change of coordinates is undesirable since in practice, one cannot split the q i into the y α and the Q A . It is thus necessary to reformulate the transformation properties of the wave functions in a manner that treats the coordinates more uniformly. To that end, we rewrite (5.3) as where χ A = 0 define good gauge conditions. The inclusion of the "Faddeev-Popov" determinant det{χ A , G B } makes (5.4) independent of χ A , and equal to (5.3) (to see this, take . We then observe that under a redefinition of the Q's, the momenta P A conjugate to Q A transform as Hence, the original constraints (5.2) are not identical with P ′ A but differ from the con- if and only if the wave functions transforms as densities of weight −1/2 for (5.7). Accordingly, we shall postulate that the wave functions transforms as densities of weight 1/2 for q i −→ q ′i = q ′i (q j ) (5.8) and as densities of weight −1/2 for (5.7), This automatically guarantees that ψ is a scalar under changes of coordinates along the gauges orbits, since the redefinition of the constraints (5.6) (P A → P ′ compensates the Jacobian coming from the density weight of ψ, The conclusion is that in order to treat uniformly the coordinates, one must require the Dirac wave functions to transform non trivially as in (5.9) under a redefinition of the constraints.

GENERAL CASE -DEFINITION OF LIE DERIVATIVE OF ψ
In the general case (2.1)-(2.9) of reducible constraints, not only can one "rotate" the constraints, but one can also transform the reducibility functions Z We shall generalize the previous transformation laws by requiring that the Dirac wave We shall show in Section 6 that this is the correct choice in that it yields a Dirac quantization equivalent to the reduced phase space one. In this section, we shall verify † Given the new and the old constraints and reducibility functions, one cannot read off it is easy to convince oneself that this ambiguity in the A's does not affect ψ ′ in (5.12).
that the Dirac quantization based on (5.12) is consistent. Namely, that it leads to quantum constraints (5.1) that are still first class, Therefore the Lie derivative of ψ involves not only the term (1/2)ξ i a 0 ,i ψ, reflecting the weight of ψ under coordinate changes, but also terms arising from its variance under (5.10) and (5.11). Let R be the point of coordinates q i and S the point of coordinates (for fixed a 0 ). The Jacobian matrix of (5.16) is while the matrices A a k b k induced by (5.16) are Thus, one gets where ψ S is the wave function at S, ψ R the wave function at R and ψ R→S the transformed (at S) of the wave function at R under the diffeomorphism (5.16) mapping R in S. Thus yields finally However, the alternating trace in (5.17) is unambiguous, so that (5.17) is well defined.

GENERAL CASE -CONSISTENCY OF DIRAC QUANTIZATION
In view of the above discussion, we take the operatorĜ a 0 in (5.1) to be −iL ξ a 0 , i.e., Because the second term in the right-hand side of (5.19) is multiplied byh (set equal to one here), one can view it as arising from an ordering ambiguity † . In the limith → 0, G a 0 goes over into ξ i a 0 p i and so, possess the correct classical limit.
To verify the consistency of the Dirac quantization, one must check (5.13) and (5.14).
This is direct because the Lie derivative (5.17) has the following crucial properties: (i) L ξ a 0 ψ transforms as ψ. (5.20) (ii) L µ ξ a 0 ψ = µ L ξ a 0 ψ for any µ(q). can be established by using Proposition 4. We leave the details of the calculations to the reader.
From (5.21), one gets and from (5.22) and (5.21), This proves (5.13) and (5.14) † . In addition, the property (5.20) guarantees the covariance of the Dirac conditions under changes of coordinates and redefinitions of the constraints and of the reducibility functions.

OBSERVABLES LINEAR IN THE MOMENTA
The quantum definition of an observable A(q, p) linear in the momenta
can be done along the same geometrical lines. Indeed, it follows from (5.23) and (5.24 and thus also where α as the vector field on Q defined by The functions X b k a k (q) are subject to identities similar to that fulfilled by D b k a 0 a k . By repeating the steps leading to (5.17), one gets, for any vector field α The group property of diffeomorphisms implies and (from (5.25) and (5.21)). Note that L µ α ψ = µL α ψ in general (unless α = α a 0 ξ a 0 ).
One can take for the quantum operatorÂ associated with A minus i times the Lie derivative along α,Â ψ = −iL α ψ. Because of (5.30),Â maps Dirac states on Dirac states so that (5.31) is consistent.

CONCLUSIONS
In this section, we have shown that the transformation law (5.12) for the Dirac wave functions leads to a quantization procedure that is consistent. The absence of anomaly in the algebra of the quantum constraints would not have been achieved had we taken ψ(q) to be merely a density of weight one-half in Q, without weight for the redefinitions of the constraints. Indeed for densities of weight 1/2, one does not have L µ ξ a 0 ψ = µL ξ a 0 ψ, (unless ξ i a 0 ∂ i µ = 0) and thus, (5.13) and (5.14) would fail. The extra terms in (5.17) containing the structure functions are therefore crucial.
The quantization procedure allows also for a geometrical consistent definition of the observables that are linear in the momenta. The resulting expression forÂ is not formally Hermitian in the scalar product dqχ * (q)ψ(q). However this is no harm because dqχ * (q)ψ(q) is not the physical scalar product. We shall derive the correct physical scalar product and prove its equivalence with the reduced phase space quantization scalar product in the next section. This requires a better understanding of the transformation law (5.12).

EQUIVALENCE OF THE REDUCED PHASE SPACE
AND DIRAC QUANTIZATIONS

EQUIVALENCE OF PHYSICAL SPECTRUM
The reduced configuration space Q/G is the quotient of the configuration space Q by the gauge orbits. There is a natural map π : Q → Q/G that maps any point of Q on its equivalence class in Q/G. If the wave functions of the reduced phase space quantization were scalars in Q/G, they would induce by pull-back scalars in Q that are constants along the gauge orbits: However, the wave functions of the reduced phase space quantization are not scalars in Q/G. Rather, they are densities of weight 1/2. We show here that they induce objects on Q with the transformation law (5.12) which have, furthermore, zero Lie derivative (5.17) along the gauge generators ξ a 0 .
Consider the vectors ∂/∂y α tangent to the coordinate lines (y α ) in a local chart of Q/G. Let (∂/∂y α ) R ≡ Y α be vector fields in Q that project down to ∂/∂y α . At each point of Q, the vectors Y α and ξ a 0 provide an overcomplete set of vectors, which is a basis if and only if the constraints are independent. Among the ξ a 0 , one can choose locally m independent vectors ξ A , which, together with the Y α , form a basis of the tangent space to Q. Let us expand the vectors ∂/∂q i in terms of (Y α , ξ A ), and let us define , the determinant is unambiguous since the ambiguity in Y α simply modifies the rows µ α i by linear combinations of the rows µ A i .
Proposition 5. The determinant µ(q) transforms as under the transformations Proof. This simply follows from standard properties of determinants.
Corollary. Let ψ(q) be a density of weight one-half in the reduced configuration space, and π : q i → y α = y α (q) the projection from Q to Q/G. Then, the function ψ(q) = |µ(q)| 1/2 ψ (y α (q)) is a density of weight 1/2 in q-space that transform with Proof. Obvious.
If the vectors ξ a 0 are independent (i.e., m 0 = m and ξ a 0 ≡ ξ A ) the analysis is done.
But if they are dependent, this is not the whole story. In that case, the dimension m of the tangent space T G to the orbits is given by eq.
This is the case if we take V 1 to be the space generated by the vectors Ξ a 1 ≡ Z a 0 a 1 Ξ a 0 , a 1 = 1, ..., m 1 . Indeed, these vectors are mapped on zero, as they should. Since the order of reducibility is L = 1, then eq.(2.9.b) tell us that { Ξ a 1 } is a set of m 1 linearly independent vectors. So we can replace the basis where the Ξ A 's are the m vectors associated with the basis { ξ A } of T G via the eq.(6.7).
Therefore it is clear that T G can be regarded as V 0 /V 1 , where the equivalence relation is such that the vectors Ξ a 1 ∈ V 1 are identified with zero. One has Define µ 0 (q) ≡ det(µ A a 0 , µ a 1 a 0 ) . (6.10) Again µ 0 (q) does not depend on how one choose to lift ξ A .

Proof. Obvious.
Corollary. The function ψ(q) = µ 1/2 µ −1/2 0 ϕ (y α (q)) is a density of weight 1/2 in Q that transforms with det A a 1 b 1 1/2 det A a 0 b 0 −1/2 under redefinitions of the constraints and of the Ξ a 1 . In particular, it does not depend on the choice of the intermediate vectors This corollary is a direct consequence of Propositions 5 and 6, and of the fact that a redefinition of the constraints G a 0 yields a redefinition of the vectors ξ a 0 .
For L = 1 the analysis is done. If the Z a 0 a 1 are, however, not independent, one should keep going and regard the vector space V 1 itself as a quotient, etc..., until one reaches the last reducibility stage. For L > 1 the set { Ξ a 1 } is not linearly independent, so no longer we define V 1 to be the space that they generate. Rathe, we denote that subspace of V 0 by W 0 . Let be { Ξ A 0 } a basis for W 0 ; then eq.(6.9) now reads, The relations (2.9) among the Z's mean which suggests again to regard W 0 as a multiple quotient space. So, define a vector space for W 1 . The quotient space V 1 /W 1 can be identified with W 0 , and the image of ℵ a 1 in the mapping V 1 → W 0 (which we denote by ℵ a 1 | W 0 ) can be identified with Ξ a 1 since one has Then a basis for The ℵ a 1 's can be expanded in this basis: By using the same argument for W 1 and so on, we will obtain One can define where the µ's are the coefficients that appear in the equations generalizing (6.17).
These determinants do not depend on how one lifts the basis vectors of W k to V k+1 . One has, Proposition 7. Let ϕ(y α ) be a density of weight one-half on the reduced configuration space Q/G, and let ψ(q) be defined through Proof. It is enough to check (6.20) in a particular coordinate system and with a particular choice of the constraints and of the reducibility functions. We take the q i coordinates to split into gauge invariant coordinates y α and pure gauge invariant Q A , as in Section 4, q i ≡ (y α , Q A ). The constraints can then be taken to be Similarly, the reducibility functions can be taken to be zero or one 17,1 . With that choice, the Lie derivative of ψ reduces to Furthermore, the vectors ξ A , ξ a 0 , Ξ a 1 , ℵ a 2 , etc., can be taken in such a way that the determinants µ, µ 0 , µ 1 ,...,µ L−1 , are all equal to one. Hence ψ(q) = ψ(y α ) does not depend on Q A , establishing (6.20) (⇔ (6.21)). This proves Proposition 7.
Conversely, let ψ(q) be an object that transforms as in (5.12) and that fulfills (6.20).
We thus conclude that a density of weight 1/2 in Q/G induces naturally a Dirac state as defined in Section 5 and vice-versa. The Dirac and reduced phase-space quantizations give the same spectrum of physical states.

OBSERVABLES
Similarly, the action of the observables that are linear in the momenta are equivalent in both quantization methods. Namely, if ψ is a reduced phase space state and ψ D the corresponding Dirac state, and if α is a vector field in Q/G and α D a vector field that project down to α (fulfilling accordingly [ α D , ξ a 0 ] ∼ ξ b 0 ), then (L α ψ) D = L α D ψ D . This simply follows from the group property of mappings and the commutativity of the following in which γ t and γ t D are elements of the one-parameter group of diffeomorphisms generated respectively by α and α D . The equivalence of the linear observables was already pointed out by Tuynman 9 in the irreducible case.

SCALAR PRODUCT
Finally, we turn to the scalar product. One defines the scalar product of two Dirac states as

BRST CHARGE
In the standard BRST method for quantizing a constrained system, the classical BRST generator is realized as an Hermitian operator on the Hilbert space of the functions depending on the original variables q i and the ghosts η a k . We say that the theory is free from BRST anomalies, if a realizationΩ =Ω † can be found such that the classical property (3.4) becomes i.e.,Ω As it was already proved in Section 3, the BRST generator is linear in the original momenta and the ghost momenta, when the constraints are linear in the momenta. So it has the generic structure with since Ω is real. We will prove that the Hermitian orderinĝ leads to a theory free from BRST anomalies. In fact, so that The first term in the right-hand side of (7.6) is zero (it is the same calculation as in the classical case). Hence one gets Now the odd vector field Ω a k , defined on the configuration space of the q's and the η's, has vanishing Lie bracket with itself, This is just the expression of the classical nilpotency of Ω, {Ω, Ω} = 0.

Proposition 8.
Let Ω a k be an odd vector field that has vanishing Lie bracket with itself. Then Proof. By differentiating (7.8) with respect to η b j , But it is easy to prove that the first term in the right hand side of this equation is zero by itself: This proves Proposition 8, and as a corollary, the nilpotency of the quantumΩ (eq.(7.1)).
Let us now expand explicitly (7.5). One finds (recall that −D b k a 0 a k is the coefficient of with the same operatorĜ a 0 as in the Dirac quantization method (equation (5.19)). That is,Ĝ a 0 is the coefficient of η a 0 in the η − P ordering of the Hermitian BRST charge (7.4).
In this view, the term ξ i a 0 ,i comes from the reordering of the original degrees of freedom , the term C b 0 a 0 b 0 comes from the reordering of the ghosts degrees of freedom (η a 0 , P a 0 ), while the terms D b k a 0 a k comes from the reordering of the ghosts of ghosts (η a k .P a k ). Thus each generation of ghosts contributes to the Dirac constraint operatorsĜ a 0 .
The consistency of the Dirac quantization scheme -i.e., no anomaly in (5.13) and fulfillment of (5.14) -can also be viewed as a direct consequence of the absence of BRST anomaly. Indeed, the nilpotency of (7.1) implies straightforwardly (5.13) and (5.14) as one can see by examining the first terms ofΩ 2 . One can thus say that the absence of anomaly in the algebra of the Dirac constraints follows from the inclusion inĜ a 0 of the ghost contribution as well as of the contribution from the ghosts of ghosts. It should be noted in that respect that one could have achieved the classical nilpotency condition {Ω, Ω} = 0 without the ghosts of ghosts. But then, one would not have found [Ω,Ω] = 0 quantum mechanically, since the D b k a 0 b k -terms in G a 0 are essential.
We leave to the reader to check that similar considerations apply to the the observables that are linear in the momenta, which, in the BRST quantization scheme, fulfill [Â,Ω] = 0, see e.g. Ref. [1].

BRST PHYSICAL STATES
In the BRST method, the physical states are annihilated by the BRST charge, The link with the Dirac method as developed in the previous sections, is obtained by demanding, in addition, that the physical states be annihilated by the ghost momenta P a k ψ = 0, k = 0, ..., L.
Since we have adopted an Hermitian ordering forΩ, adapted to the formal scalar product dqdη ψ * (q, η)χ(q, η), we shall require the BRST wave functions to transform as superdensities of weight 1/2 under changes of coordinates in the configuration space of the q i and the ghosts, q i , η a k → q ′i , η ′a k , (7.14.a) This leaves the integral dqdη ψ * χ invariant. For the states (7.11), the rule (7.14) exactly yields the transformation law (5.12) for the Dirac states. Indeed, the redefinition G ′ a 0 = A a 0 b 0 G b 0 induces the transformation η a 0 = A a 0 b 0 η ′b 0 of the ghosts, in order to leave G a 0 η a 0 invariant, G a 0 η a 0 = G ′ a 0 η ′a 0 . Similarly, the redefinitions of the reducibility functions are equivalent to a transformation of the higher order ghosts. Hence, the transformation law (5.12) has a very direct explanation in terms of the BRST quantization.
We again leave it to the reader to check that the BRST observables that are linear in the momenta reproduce correctly the Lie derivative when acting on the states (7.11) and (7.13).

BRST INNER PRODUCT
To complete the proof of equivalence of the BRST method with the Dirac method, it remains to discuss the scalar product. Now, if one computes the integral dqdη ψ * (q, η)χ(q, η) for the BRST physical states (7.11), (7.12), one obtains an ill-defined result. The way out is to introduce a so-called "non minimal sector" (i.e., further variables that do not change the physics) and to regularize dqdη ψ * (q, η)χ(q, η) by inserting the operator exp −[K,Ω] between ψ * and χ, and integrating also over the non minimal variables, with the natural measure dq dη d(non minimal variables). Here,K is a ghost minus one operator depending on all the variables and chosen so that the integral dqdηd(nonminimal) ψ * (q, η) exp [K,Ω]χ(q, η) is well defined. The operator exp −[K,Ω] is (formally) equivalent to the unit operator between physical states becauseΩ is (formally) Hermitian with that natural measure. When this regularization is appropriately carried through, one finds that the BRST scalar product coincides with the Dirac scalar product for the states obeying (7.11) and (7.12).
This result is derived in detail in Ref. [1] for the irreducible case (Chapter 14). It is easy to see, by using the invariance of the scalar product under changes of coordinates q i , η a k → q ′i , η ′a k , that the same result applies to the reducible case as well.

CONCLUSIONS
In this paper we have established the equivalence of the reduced phase space, Dirac and BRST quantization methods for reducible gauge systems described by constraints linear in the momenta. We have shwon that densities of weight one-half in the reduced configuration space define densities of weight one-half in the original configuration space, which have a non trivial weight under redefinitions of the constraints and of the reducibility functions.
Because of this extra variance, the Lie derivative of the Dirac wave functions contains extra terms besides those characteristic of ordinary density of weight one-half. These terms guarantee the absence of anomalies in the Dirac quantization scheme, as well as the reducibility of the quantum constraints. Finally, we have given a BRST interpretation of the Dirac analysis. In particular, we have shown that the extra anomaly-cancelling terms in the quantum constraints could be thought of as arising from the ghosts and the ghosts of ghosts, which play thus a fundamental role.