BV analysis for covariant and non-covariant actions

The equivalence between the covariant and the non-covariant version of a constrained system is shown to hold after quantization in the framework of the field-antifield formalism. Our study covers the cases of Electromagnetism and Yang-Mills fields and sheds light on some aspects of the Faddeev-Popov method, for both the coratiant and non-covariant approaches, which had not been fully clarified in the literature.


Introduction
In a recent paper [1] the BRST quantization of a class of constrained dynamical systems was performed in the framework of the Batalin-Vilkovisky (BV) formalism [2]. These systems were specified by a Lagrangian which is quadratic in the velocities and such that only primary first class constraints, linear in the momenta, appear in its Hamiltonian analysis. After solving the Classical Master Equation -which is straightforward due to its closed algebra structure-the problem of the ambiguity inherent to the resolution of the full Quantum Master Equation was addressed. It is well known [2] that this ambiguity, which can be drawn to the problem of defining the measure for the path integral, has no solution within the BV formalism by itself and one has to rely on other formulations -operator formalism, for instance-to get the correct answer. In this sense our result was promising: the physical requirement of making contact with the reduced path integral quantization procedure -which is very close to ensuring unitarityis equivalent to the internal requirement (i.e. without departing from the BV formalism) of choosing the solution of the Quantum Master Equation that makes the path integral reparametrization invariant.
But a wide class of constrained systems do not fit within the theories just considered. Many relevant physical examples, like Electromagnetism (EM) and Yang-Mills fields (YM), exhibit secondary as well as primary first-class Hamiltonian constraints. There is an important physical reason for the appearance of secondary constraints in these theories, and it is related to the way the Hamiltonian formalism, which is manifestly non-covariant, is able to provide us with gauge transformations which are Lorentz covariant. Consider, for instance, the case of EM. The infinitesimal gauge transformation δA µ = ∂Λ/∂x µ shows how to get a vector, ∂Λ/∂x µ , from a scalar, Λ: just by taking the gradient. The appearance of a time derivative of the arbitrary function Λ forces the Hamiltonian generator of the gauge transformation to have two pieces, one with the first time derivative of Λ and the other one without a time derivative. Let us be more specific; we know on theoretical grounds [3], that a generator of gauge transformations -acting through Poisson brackets-must have the following form straint. In the case of EM, G is the sum of two pieces, coming from one primary and one secondary constraints (two generations). In fact, from the Lagrangian L = − 1 4 F µν F µν we get a canonical Hamiltonian H c = dx 1 2 (π 2 + B 2 ) + π · ∇A 0 and a primary constraint (coming just from the definition of the momenta) π 0 ≃ 0. Stability of this constraint under the Hamiltonian dynamics leads to the secondary constraintπ 0 = {π 0 , H c } = ∇ · π ≃ 0. No more constraints arise. Both the primary and the secondary constraints are first-class and allow to write the gauge generator (1.1) in this case as where Λ is an infinitesimal arbitrary function. The gauge transformation of the gauge field is then δA µ = {A µ , G} = ∂Λ/∂x µ . We see therefore that a primary and a secondary constraints are necessary to get the gauge field A µ transformed covariantly. So, in principle, we are faced with the problem that the class of theories studied in [1], which exhibit only primary constraints, seems to exclude the important physical case of covariant theories. In fact this is not true, as we will see that our primary constraint theories can be conveniently covariantized simply by promoting the Lagrangian multipliers associated with the primary constraints to the status of dynamical variables, covering this way the cases of EM and YM fields [4]. With this covariant theory at hand we can proceed to study the new features that arise in this case and that were absent in the class of systems studied in [1], for instance the possibility of having an open gauge algebra. This process of covariantization and the study of the Hamiltonian and Lagrangian gauge algebra will be the topics of the next section. In section 3 we perform the BV quantization of the covariant theory by solving explicitly the Quantum Master Equation and, after that, implementing several gauge fixings. In this fashion we show the equivalence between this covariant quantization and the non-covariant approach used in [1]. Section 4 is devoted to conclusions. Finally, an Appendix is introduced to show how the covariantization procedure described in section 2 works in the YM case.
2 Gauge algebra and covariantization of the action 2.1 General setting The non-covariant Lagrangians L N C (q,q) 1 of interest to us are those for which the tangent space is free from any constraints, yet L N C is still a singular Lagrangian. As is proven in [5] this is equivalent to having in phase space only primary first-class constraints. In this case, the canonical Hamiltonian H 0 (q, p) and the constraints T α (q, p), α = 1, ..., r, satisfy Let us now have a quick look at the issue of "covariantization". There is a simple way to get, from a canonical theory with only primary first-class constraints, a classically equivalent theory with primary and secondary first-class constraints. This can be done by promoting the Lagrangian multipliers λ α , associated with the original constraints, to the status of dynamical variables and to assume as a the new primary constraints its canonical conjugate momenta π α . Under such conditions the extended Hamiltonian will read H(q, p; λ) = H 0 (q, p) + λ α T α (q, p), and the stability of the new primary constraints π α ≃ 0 will lead to the -nowsecondary constraintsπ α = {π α , H} = −T α (q, p) ≃ 0, whose stability gives no new informatioṅ Using the well known algorithms briefly described in the introduction [3] it is straightforward to construct the gauge generator for this case and, consequently, the Hamiltonian gauge transformations for the coordinates q A , λ α where ε α is an infinitesimal arbitrary function of time (or space-time, in the case of field theory).
As is well known, to perform the covariant quantization of a gauge theory within the framework of the field-antifield formalism, knowledge of the gauge structure of the classical Lagrangian theory is of fundamental importance. Since for the systems under consideration much of this information is already contained in the Hamiltonian gauge structure, in what follows we briefly describe the derivation of quantities and relations defining this structure at the Hamiltonian level.
To begin with, let us derive some relations involving the quantities V α β , C α βγ , the constraints T α and the canonical hamiltonian H 0 , which appear as consequences of some Jacobi identities. Consider, for instance, the following Jacobi identity involving the constraints T α {T α , {T β , T γ }} + cyclic perm. of (α, β, γ) = 0.
The general solution of this equation leads to the existence of a new function B µρ αβγ , antisymmetric in its upper indices. In much the same way, from the Jacobi identity among the constraints and the canonical hamiltonian yielding the appearence of a new structure function, D σµ αβ , antisymmetric in its upper and lower indices.
Continuing this procedure, that is, taking an increasing number of Poisson brackets among the constraints and the canonical Hamiltonian and antisymmetrizing them in a convenient fashion, new quantities and new relations among the functions previously obtained are found. All these objects are the so-called structure functions and the relations among them determine the structure of the Hamiltonian gauge algebra. For a more exhaustive study of this Hamiltonian gauge structure we refer the reader to ref. [6].

The model: Quadratic Lagrangians
So far we have established the most general setting for theories we are interested in. Now we will apply this framework to the case of non-covariant quadratic Lagrangians of the type [1] where G AB (q) is a singular metric such that its null vectors U A α (q), G AB U B α = 0, α = 1, . . . , k, are Killing vectors for it, i.e.
and keep the potential V unchanged In (2.6), L α stands for the Lie derivative in theÛ α direction. These last two conditions enforce the non-existence of Lagrangian constraints [1]. The primary Hamiltonian constraints for this system are easily found and its first-class character, which is guaranteed by requirement (2.6) yields the commutation relations defining the structure functions C γ αβ which in the present case depend only on the coordinates q A .
On the other hand, the canonical Hamiltonian H 0 associated to (2.5) is where the metric M AB is a symmetric non-singular matrix satisfying Let us notice that the metric M AB defined through (2.10) displays a certain degree of arbitrariness. This corresponds to the fact that the canonical Hamiltonian is only unambiguously defined on the primary constraint surface 2 .
Taking into account (2.10) and the fact that the vector fieldsÛ α = U A α ∂/∂q A are Killing vectors for the metric G, (2.6), we immediately obtain This result leads to the following form for (L α M) Notice that the choice of M βA α (q) is again ambiguous. In fact there is a family of such possible objects, related to each other by

12)
G βγ α being an arbitrary array of coefficients antisymmetric in its upper indices. In the next subsection we will take advantage of this arbitrariness.
From the above results the form of the structure functions V α β is easily worked out. Indeed, taking into account its definition, the form of the constraints T α (2.7) and the Hamiltonian H 0 (2.9) we have where use of the relation (2.11) allows to factorize the constraints and write the structure functions V β α as Finally, let us write down the consequences of the Jacobi identities for the constraints and the canonical Hamiltonian in our particular model. From relation (2.3) we obtain the structure functions B µρ αβγ vanishing in this case, due to the fact that C γ αβ and U A α depend only on q A . On the other hand (2.4) turns out to be where now the structure functions D σµ αβ can be chosen to depend only on q A , as it is seen if the linear dependence of the constraints T α and the structure functions V α β on p A is taken into account. This analysis could be carried on to determine the higher order structure functions. Nevertheless, since these quantities will not appear in the situation we will consider, we do not pursue this direction any further. Rather, in what follows, we are going to undertake the study of the Lagrangian gauge structure using as background the above results.

Covariantization and Lagrangian gauge structure
Having studied the Hamiltonian gauge algebra we are ready for "covariantization". Using the Lagrangian multipliers as new variables, the extended Hamiltonian reads in our particular case To obtain the associated Lagrangian we should eliminate the momenta p A in terms of the velocitiesq A . Use of the equations of motion yieldṡ and since M AB is invertible, we have The corresponding "covariant" Lagrangian L C is In the Appendix we show that in the case of Yang-Mills theories (2.15) is the standard covariant Lagrangian for these systems. This Lagrangian L C (q,q; λ) will be the starting point of our analysis. First we can check that the pull-back of the transformations (2.2), given by are indeed, as was expected, gauge transformations for L C δL C = 0.
As we have said, the structure of the algebra of the Lagrangian gauge transformations plays a crucial role in solving the Master Equation in the field-antifield approach -which is the subject of the next section. In our case, for , and, after a lengthy calculation where D ασ βµ are the (pull-back of the) Hamiltonian structure functions defined through relation (2.14) and ∂L C /∂λ α the equations of motion for the fields λ α derived from the covariant Lagrangian L C (2.15), given by In the study of this gauge algebra we meet for the first time the new features introduced in the theory by the process of covariantization. Indeed, observe that the structure of (2.17) is, in general, that of an open algebra. This fact makes the computation of the proper solution of the Master Equation rather cumbersome and we will try to circumvent this problem. To this end we will use a result from ref. [8], to wit: any open algebra of gauge transformations may be closed by the addition of the appropiate antisymmetric combinations 3 of the equations of motion. In our specific case, since the openness of the algebra only concerns the λ sector and, moreover, its open algebra part -see (2.17)-only exhibits the equations of motion for the λ's, [L C ] λ α = ∂L C /∂λ α , (2.18), it seems to be very plausible that we can get the closed algebra structure just by leaving δq A unchanged and modifying δλ α (2.16) as follows with an appropriate F αβ γ antisymmetric in its upper indices. Using (2.16) and the explicit expression for the equations of motion of λ, (2.18), this can also be written as This last expression simply displays the freedom in the choice of M βA α we discovered in (2.12). We can therefore conclude that it is plausible that the freedom described by (2.12) allows for a choice of δ ′ λ α which satisfies the closed algebra structure. Strictly speaking we have not proven this, although it is very plausible, as we have argued.
Actually we may have considered from the beginning a more restrictive case: the assumption [9], for instance, that the regular metric tensor M AB be such that the action of the gauge group leads to isometries, i.e. (L α M) AB = 0, α = 1, . . . , k. In fact, this Killing condition implies that the vector fieldsÛ α form a Lie algebra (C γ αβ = constant). Indeed, eqs.(2.8) with C γ αβ = constant, are the integrability conditions for (L α M) AB = 0, α = 1, . . . , k. In that case, the treatment of the system greatly simplifies: from (2.11) we see that the quantities M βB α and, as a consequence, the structure functions V β α of (2.13) can be chosen to be zero. Then D σµ αβ defined in (2.14) can be put to zero as well. All these results together lead to the closure of the gauge algebra of (2.17) in this particular case. Note that the important cases of EM and YM, for which the structure functions C γ αβ are constants, fall into this last category and have a closed gauge algebra.
From now on, whatever be the case we are dealing with, we will assume that we have met the conditions to get the gauge algebra in the closed form This assumption greatly simplifies the determination of the solutions of the Classical and Quantum Master Equations, which we are going to undertake in the next section.

BV quantization of the covariant action
In the case of an irreducible closed algebra of gauge transformations, the field-antifield formalism starts by enlarging the original configuration space with the introduction of a ghost c α for each gauge parameter ε α , of opposite parity. These ghosts, together with the classical fields {φ a } = {q A , λ α }, form the minimal sector of fields {φ i }. A new set of "antifields", {q * A , λ * α , c * α } = {φ * i }, with parities opposite to those of its associated fields, is introduced as well. Then, in the space of functionals of the fields and their antifields, some new structures, the antibracket For an irreducible, closed gauge algebra the Classical Master Equation (3.2) has the well known (minimal) proper solution where S 0 , R a α and T α βγ are the classical action, the generators of the gauge transformations and the structure functions of the gauge algebra, respectively. In our case, taking into account the results obtained in the preceding section and assuming to have met the conditions to get the algebra in closed form, we obtain the following expression for the proper solution where now S C is the classical action associated with the covariant Lagrangian (2.15), U A α (q) the gauge generators for the fields q A and V α β , C α βγ the pull back of the corresponding Hamiltonian structure functions (2.8) and (2.13).
Let us consider now the equations for the quantum corrections (3.3). Since for the type of theories we are analyzing the proper solution is linear in the antifields, the quantity ∆S does not depend on them. As a consequence, the term W 1 can be chosen to be a function of the classical fields only and eqs.(3.3) are immediately solved by taking W p = 0 for p ≥ 2. Therefore, no higher order terms inh appear apart from the W 1 term.
To compute the first quantum correction W 1 we need the explicit expression of ∆S. In our case it is easily seen that Use of the Lie derivative of M AB in theÛ α direction (2.11) together with the symmetry property of this metric allows to write the last term of the above expression as where in the last equation use has been made of the definition of (L α M) AB . Therefore, after these manipulations, ∆S can be written in the more useful form Notice that the trace over continuous indices imply, in our case of a local gauge theory, that ∆S is proportional to δ(0). Therefore, in order to make sense out of this construction, some scheme to regularize the above expression must be considered. Expression (3.4) is already spelling out the formal solution for W 1 . Indeed, we can simply take (3.5) where by "BRST-invariant terms" we mean terms with vanishing antibracket with S. As we have said, the above ambiguity in W 1 is inherent to the fieldantifield formalism. In the present case we solve this ambiguity just by dropping the second term in the lhs of (3.5). As will be shown below, this is the correct choice that makes contact with the reduced path integral formalism. Now, to proceed to fix the gauge within the field-antifield approach, some auxiliary fields, {c α , B α }, and its corresponding antifields, {c * α , B * α }, are introduced. After that, the minimal proper solution S is modified by the addition of an extra term in these new fields as S n.m. = S +c * α B α . Then, if we call Φ i the whole set of fields, a "gauge fermion" Ψ will eliminate the antifields through the requirement

The Batalin-Vilkovisky path integral is then defined as
where W Σ stands for W (Φ, Φ * = ∂Ψ/∂Φ). It should be noted that, in our case, as W 1 does not have any dependence on the antifields Φ * , its expression will not depend on the choice of the gauge fixing. Therefore, we will have Now, a customary lattice regularization for δ(0), δ(0) → 1 ε , ε → 0, allows us to write part of the exponential in (3.6) as Therefore, from the above expression for Z Ψ it is evident that while the gauge fixed proper solution of the classical master equation ( (3.6) under deformations of the gauge fixing fermion, is a well known result in the context of the field-antifield formalism, and has been proven by Batalin and Vilkovisky in [2]. In fact, what they do in this reference is to prove the theorem for gauges that differ infinitesimally. More suited to our purposes, a direct proof of this invariance under arbitrary deformations of Ψ for the case of theories with closed, irreducible gauge algebras, can be found in ref. [10] and will not be repeated here. From now on, we will take for granted this invariance of the path integral (3.6) under changes of the gauge fixing fermion 4 .
One of the main purposes of this paper is to make contact with the noncovariant path integral quantization presented in [1]. To this end, we will use a gauge fixing fermion implementing unitary or, more generally, non-covariant gauge fixing conditions. However, other types of gauge fixing conditions, for instance "covariant" gauges, can be chosen as well. In what follows, we are going to work out the form of the Batalin-Vilkovisky path integral in both classes of gauges.
Unitary or non-covariant gauge conditions are necessary in order to make contact with the reduced path integral quantization. In this formulation, unitarity is obvious once the usual assumptions about the positivity of the spectrum of the reduced theory are made. Gauge fixing fermions which implement these gauges are taken to be of the form Ψ 1 =c α F α (q), where the gauge fixing conditions F α (q) do not involve the Lagrange multipliers λ α . For such gauge fixing fermions S Σ 1 becomes The non-covariant formulation is recovered by integrating out the Lagrange multipliers λ, which appear quadratically in S C . Once this is done we get where θ αβ is defined by θ αβ = U A α M AB U B β , and the new classical action S ′ 0 is It can be shown that the metric defining the kinetic term of S ′ 0 is nothing but the one which appears in the original quadratic non-covariant Lagrangian (2.5), i.e.
Therefore, we conclude that S ′ 0 is equal to S N C -the action for the non-covariant Lagrangian (2.5)-and that expression (3.9) for Z Ψ 1 in non-covariant gauges is in complete agreement with the one obtained in [1]. This result proves that the equivalence between the non-covariant (with variables q only) and the covariant (with variables q and λ) formulations, which is easily seen at the classical level, still holds after quantization of the theory within the framework of the fieldantifield formalism when a non-covariant gauge fixing is imposed.
At this point it is worth comparing the two versions, (3.8) and (3.9), of the path integral Z Ψ 1 . In fact, they are two different, although equivalent, Faddeev-Popov (FP) formulas. On the one hand, expression (3.8) for the covariant theory corresponds to the standard FP formula as used in the literature (with two extensions: the presence of a non-trivial determinant in the measure and also the fact that we are dealing with the so-called quasigroup structure [10] rather than a Lie group). On the other hand, the equivalent expression (3.9) uses a non-covariant action and corresponds to the correct FP formula for systems with first-class primary constraints only (strictly speaking, systems with quadratic kinetic term and constraints linear in the momenta), as it was proven in [7]. In this second case, it should be noted the presence of a new determinant, (det θ αβ ) 1/2 , which makes the path integral invariant under rescaling of the constraints. In summary, the above discussion points out that the structure of the constraints of the theory (primary constraints in (3.9); primary and secondary in (3.8)) makes a difference with regard to the final form of the FP formula.
In connection with the measure of our path integral (3.7) another comment is in order. As we have said in the preceding section, there is a certain amount of arbitrariness in the selection of the metric M AB fulfilling (2.10). One may then wonder how this arbitrariness affects the path integral (3.7). In fact, using expression (3.9) of Z Ψ in a non-covariant gauge, one can see that it does not affect it at all. This expression was obtained in [7] starting from the reduced path integral quantization, in which this kind of ambiguity was not present 5 . Therefore, in spite of this apparent dependence of (3.7) on the particular choice of M AB , the measure and the action depend on it in such a way that (3.7), in the end, does not suffer from this arbitrariness. In the measure, this feature is neatly displayed as a cancellation of the dependence on the gauge part of (det M AB ) 1/2 , namely, (det M αβ ) 1/2 , and the similar dependence in (det θ αβ ) 1/2 ( in [7] it was shown that (det M AB ) 1/2 factorizes into a physical piece -dependent only on gauge invariant degrees of freedom -times (det M αβ ) 1/2 . This was achieved in the so-called adapted coordinates; the gauge dependent piece of (det θ αβ ) 1/2 is also explicitely displayed this way).
To conclude, for the sake of completeness, let us study the form of Z Ψ (3.7) in covariant gauges. As is well known, covariant gauge fixings are more convenient in obtaining Feynman rules which describe the perturbative sector of the quantum theory. This class of gauges is constructed so that all the fields become propagating. Gauge fixing fermions enforcing covariant gauges are usually of the where the maximum rank metric ω αβ is usually taken to be independent of the fields. The gauge fixed action S Σ 2 reads in this case The auxiliary fields B α can be integrated out of the path integral or, equivalently, eliminated algebraically in terms of their equations of motion where ω αβ is the inverse of the metric ω αβ , yielding in this way a gauge fixed action of the form in which the kinetic terms of all the fields are invertible, so that they become propagating. This is an important feature which distinguish the covariant formulation (i.e. with variables λ) from the non-covariant one. Finally, the partition function in covariant gauges is written as this expression being the starting point in the construction of covariant Feynman rules.

Conclusions
In this paper we have extended to Yang-Mills type systems some previous work on the quantization of constrained systems which exhibited, in the canonical formalism, only primary first-class constraints linear in the momenta. This extension can be understood as the covariantization of the original system by introducing new degrees of freedom to it. At this point it is worth noticing that the marriage of covariance (for a constrained system like YM, for instance) with the Hamiltonian formalism immediately implies the appearance of secondary constraints. Due to this fact, there are some differences in the application of the Batalin-Vilkovisky formalism to both the non-covariant and the covariant case which deserve some specific comments. The main difference is perhaps that the algebra of gauge transformations will be generally open in the covariant case, even though it was closed in the non-covariant one. In this paper we have dealt with this eventuality by arguing that it is possible to set up the covariant formalism in such a way that the algebra is still closed, and this is in fact the only case we have studied and where the equivalence with the non-covariant formulation has been shown.
Another difference, which can be traced to the different structure of the constraints in both cases, is the following: in the non-covariant case (which in our terminology corresponds to a system without Lagrangian constraints or, in other words [5], with only primary first-class Hamiltonian constraints), there appears [7] in the Faddeev-Popov formula a new determinant, unrelated to the gauge fixing procedure, that keeps the path integral invariant under rescaling of the constraints (which we emphasize are linear in the momenta). Instead, in the covariant case (which is achieved by promoting the old Lagrangian multipliers to the status of dynamical variables, thus creating two generations, primary and secondary, of constraints), the new determinant is absorbed in the definition of the covariant Lagrangian, and the usual Faddeev-Popov formula is obtained. Our result, however, is an extension of the Faddeev-Popov formula because now the generators of the gauge group do not span a Lie algebra. The structure defined by these generators -whose commutation relations give rise to structure functions, unlike the structure constants that appear in a Lie algebra-has been called a quasigroup [10].
In conclusion, our results establish the equivalence, at the quantum level, of the non-covariant and the covariant version of a constrained dynamical system of Yang-Mills type. This equivalence is a fundamental issue because, in terms of path integrals, unitarity is best checked in the reduced quantization (classical elimination of the gauge degrees of freedom). This reduced quantization corresponds, as it is proven in [7], to the quantization of the non-covariant version of the system.
During the preparation stages of this manuscript we received a preprint by Epp et al. [12], which deals with some of the topics raised here, as well as with other work by some of us. We completely agree with their results. The crucial point first raised in [7], and clarified to some extent in [12] using scalar QED as an example, is the need to distinguish between different forms of the Faddeev-Popov ansatz when both primary and secondary constraints are present classically (i.e. before the Lagrange multipliers are integrated out) and when only primary constraints are present (i.e. after the Lagrange multipliers are integrated out). Thus for example the usual, covariant form of the Faddeev-Popov ansatz, ( and consequently formula (3.31) of [13] is correct only when secondary constraints are present. Ref. [7] dealt specifically with primary constraints only, while the present work extends the results to the case in which secondary constraints are also present.

Appendix
Here we derive the standard covariant Yang-Mills Lagrangian from its noncovariant version as an example of the procedure of "covariantizing" (2.5) to get (2.15). Actually we can directly start from the Hamiltonian H 0 of (2.9) which for Yang-Mills takes the form: H 0 = 1 2 π k a π k a + 1 4 F ij a F ij a . We hence identify, in the notation of sect.2 M AB = δ(x − y)δ ab , V (A i ) = 1 4 F ij a F ij a , and, from equations (2.16) where we have used the notation λ a = A a 0 for the Lagrange multipliers, and taken into account that now V β α (p, q), (2.13), can be chosen to be zero. Then, we have, for (2.15) and since (Ȧ a i − D ab i A b 0 ) = F a 0i , we finally get that is, the covariant action for Yang-Mills theories.