Finite anticanonical transformations in field-antifield formalism

We study the role of arbitrary (finite) anticanonical transformations in the field-antifield formalism, and the gauge-fixing procedure based on the use of these transformations. The properties of the generating functionals of the Green functions subjected to finite anticanonical transformations are considered.


Introduction
Field-antifield formalism [1,2], summarizing numerous attempts to find correct quantization rules for various types of gauge models [3,4,5,6,7], is a powerful covariant quantization method which can be applied to arbitrary gauge invariant systems. This method is based on fundamental principle of BRST invariance [8,9] and has a rich new geometry [10]. One of the most important objects of the field-antifield formalism is an odd symplectic structure called antibracket and known to mathematicians as Buttin bracket [11]. In terms of antibracket the master equation and the Ward identity for generating functional of vertex functions (effective action) are formulated. It is an important property that the antibracket is preserved under the anticanonical transformations which are dual to canonical transformations for a Poisson bracket. An important role and rich geometric possibilities of general anticanonical transformations in the field-antifield formalism have been realized in the procedure of gauge fixing [12] (see, also [13]). Original procedure of gauge fixing [1,2] corresponds in fact to a special type of anticanonical transformation in an action being a proper solution to the quantum master equation. That type of transformations is capable to yield admissible gauge-fixing conditions in the form of equations of arbitrary Lagrangian surfaces (constraints in the antibracket involution) in the field-antifield phase space. Thereby, the necessary class of admissible gauges was involved actually. The latter made it possible to describe in [12] the structure and renormalization of general gauge theories in terms of anticanonical transformations. As the authors [12] assumed the use of regularizations in which δ(0) = 0 in local field theories, they based on the use of general anticanonical transformations in an action being a proper solution to the classical master equation. In turn, the gauge dependence and the structure of renormalization of the effective action have been analyzed by using infinitesimal anticanonical transformations only.
In the present article, we extend the use of anticanonical transformations in the field-antifield formalism from the infinitesimal level to the finite one, and explore a gauge fixing procedure for general gauge theories, based on arbitrary anticanonical transformations in an action being a proper solution to the quantum master equation with fixed boundary condition. Now, it is worthy to notice the difference between the properties of the classical and quantum master equations under anticanonical transformations. The classical master equation is covariant under anticanonical transformations, as its left-hand side is the antibracket of the action with itself. In contrast to that, the form of the quantum master equation is not maintained under anticanonical transformations. One should accompany the anticanonical transformation by multiplying the exponential of i/ times the transformed action with the square root of the superjacobian of that anticanonical transformation. We will call such an operation for an anticanonical master transformation and the corresponding action for a master transformed action. Thus, one can say that the form of the quantum master equation is maintained under the anticanonical master transformation. We consider in all details the relationship between the two descriptions (in terms of the generating functions and the generators) for arbitrary finite anticanonical transformations.
Finally, let us notice the study [14], among the other recent developments, where it was found a procedure to connect generating functionals of Green functions for a gauge system formulated in any two admissible gauges with the help of finite field-dependent BRST transformations.

Field-Antifield Formalism
The starting point of the field-antifield formalism [1] is a theory of fields {A} for which the initial classical action S 0 (A) is assumed to be invariant under the gauge transformations δA = R(A)ξ. Here ξ are arbitrary functions of space-time coordinates , and {R(A)} are generators of gauge transformations. The set of generators is complete but, in general, maybe reducible and forms an open gauge algebra so that one works with general gauge theories. Here we do not discuss these points, referring to original papers [1,2]. The structure of the gauge algebra determines necessary content of total configuration space of fields {ϕ i (ε(ϕ i ) = ε i )} involving fields {A} of initial classical system, ghost and antighost fields, auxiliary fields and , in case of reducible generators, pyramids of extra ghost and antighost fields as well as pyramids of extra auxiliary fields. To each field ϕ i one introduces an antifield ϕ * i , whose statistics is opposite to that of the corresponding fields ϕ i , ε(ϕ * i ) = ε i + 1. On the space of the fields ϕ i and antifields ϕ * i one defines an odd symplectic structure ( , ) called the antibracket and the nilpotent Fermionic operator ∆, Here the notation is introduced. In terms of the antibracket and ∆-operator the quantum master equation is formulated as for a Bosonic functional S = S(ϕ, ϕ * ) satisfying the boundary condition S| ϕ * = =0 = S 0 (A) (2.5) and being the basic object of the field-antifield quantization scheme [1,2]. Among the properties of the antibracket and ∆-operator we mention the Leibniz rule, the Jacobi identity, and being the ∆-operator a derivative to the antibracket, There exists a generating functional Y = Y (ϕ, Φ * ), ε(Y ) = 1 of the anticanonical transformation, The invariance property of the odd symplectic structure (2.1) on the phase space of (ϕ, ϕ * ) is dual to the invariance property of an even symplectic structure (a Poisson bracket) under a canonical transformation of canonical variables (p, q) (for further discussions of relations between Poisson bracket and antibracket, see [15,16]).
The generating functional of Green functions Z(J) is defined in terms of functional integral as [1,2] where S e (ϕ) = S ϕ, ϕ * = ∂ ϕ ψ(ϕ) , (2.11) ψ(ϕ) is a Fermionic gauge functional, J i (ε(J i ) = ε i ) are usual external sources to the fields ϕ i and W (J) is the generating functional of the connected Green functions.
To discuss the quantum properties of general gauge theories, it is useful to consider, instead of the generating functional (2.10), the extended generating functionals Z(J, ϕ * ) and W (J, ϕ * ) defined by the relations Obviously, we have (2.14) The action S = S(ϕ, ϕ * ) (2.13) satisfies the quantum master equation It follows from (2.15) that the Ward identities hold for the extended generating functionals Z(J, ϕ * ) and W (J, ϕ * ) The generating functional of vertex function (effective action) is defined via the Legendre transformation with the properties It follows from (2.17) and (2.19) that the Ward identity for the effective action holds, which has the form of classical master equation in the field-antifield formalism.
As it was pointed out for the first time in [12], the gauge fixing procedure in the field-antifield formalism (2.13) can be described in terms of a special type of anticanonical transformation (2.9). Indeed, let us consider anticanonical transformations of the variables (ϕ, ϕ * ) with specific generating function We have so that the transformed actionS =S(ϕ, ϕ * ) S(ϕ, ϕ * ) = S(Φ, Φ * ) = S(ϕ, ϕ * + ∂ ϕ ψ(ϕ)) (2.23) coincides with (2.13). In particular, this fact made it possible to study effectively the gauge dependence and structure of renormalization of general gauge theories [12]. In what follows we explore a gauge fixing procedure in the field-antifield formalism as an anticanonical transformation of general type with the only requirement for anticanonically generalized action: supermatrix of second field derivatives of this action must be non-degenerate. An essential difference in this point with the approach used in [12] is that we work with general setting for an action (2.13) which satisfies the quantum master equation (not the classical master equation as in [12]).

Infinitesimal anticanonical transformations
As the first step in our study of anticanonical transformations in the field-antifield formalism, we consider the properties of the main objects subjected to infinitesimal anticanonical transformations. In the latter case, the generating functional Y reads The functional X is considered as the infinitesimal one. Then, anticanonical transformations of the variables, can be written down to the first order in X as or, in terms of the antibracket (2.1), The anticanonically transformed actionS, does not satisfy the quantum master equation to the first approximation in X, Consider now the superdeterminant of the anticanonical transformation whereZ To the first order approximation in X, the J reads In contrast to the notation used in [17,18], now we refer to S ′ = S ′ (ϕ, ϕ * ) constructed from S = S(ϕ, ϕ * ) via the anticanonical master transformation, as the master-transformed action.
Note that by itself, the anticanonical master transformation can be defined without reference on solutions of the quantum master equation. Namely, let us define a transformation of the form 4 where G, F (ε(G) = 0, ε(F ) = 1) are arbitrary functions of ϕ, ϕ * , and [ , ] stands for the supercommutator. Then we can prove (see Appendices C and D) the relation (3.12) which repeats the relation (3.10). In (3.12) the notation f (x) = (exp x − 1)x −1 is used.
The action S ′ (3.10) to the first order in X does satisfy the quantum master equation (3.14) Note that, due to the results of [17,18], the action (3.10) by itself satisfies the quantum master equation in the case of arbitrary anticanonical transformation, as well (see, also [19,13]).

(3.22)
One can rewrite the equation (3.21) in terms of Γ = Γ(ϕ, ϕ * ) as This result is proved in Appendix A. The equation (3.23) means that any infinitesimal anticanonical master transformation of the action S (3.5) with a generating functional X induces an infinitesimal anticanonical transformation in the effective action Γ (3.23) with a generating functional X , provided the generating functional of Green functions is constructed via the master-transformed action. An important goal of our present study is a generalization of this fact (for the first time known among results of paper [12]) to the case of arbitrary (finite) anticanonical transformation.

Finite anticanonical transformation
Consider an arbitrary (finite) anticanonical transformation described by a generating func- where a is a parameter. Then solution of equations (4.1) up to second order in a can be written as where f ≡ f (ϕ, ϕ * ). Let us denote and Then we haveZ We call F in (4.6) for a generator of anticanonical transformation to the second order. It should be noticed that the generator of anticanonical transformation does not coincide with the generating functional of this transformation already to the second order. A natural question arises: Does a generator exist for a given anticanonical transformation, actually? To answer this question, we begin with the claim that an operator exp{ad(F )} generates anticanonical transformation. Indeed, let Z A be anticanonical variables so that the antibracket (2.1) can be presented in the form where E AB is a constant supermatrix with the properties Then the transformation is anticanonical, To prove this fact we introduce an one-parameter family of transformations and quantitiesZ AB (Z, a), (4.14) It follows from the definitions (4.13), (4.14), that the relations  17) and the transformation (4.11) is really anticanonical. An inverse to this statement is valid as well: an arbitrary set of anticanonical variablesZ A (Z) can be presented in the form with some generator functional F (Z), ε(F (Z)) = 1. In Appendix B, a proof of this fact is given.
Consider now a master-transformed action S ′ = S ′ (ϕ, ϕ * ) (3.10). It was pointed out in [14] that there are presentations of S ′ in the following forms where S = S(ϕ, ϕ * ), and F = F (ϕ, ϕ * ) is a generator functional of an anticanonical transfor- In accordance with (3.10), the first term in the right-hand side in (4.20) describes an anticanonical transformation of S with an odd generator functional F , while the second term is a half of a logarithm of the Jacobian (3.7) of that transformation, up to (−i ). In Appendix D, we give a proof of the latter statement.
Now we are in a position to study the properties of generating functionals of Green functions subjected to an arbitrary anticanonical transformation. We start with the generating functionals of Green and connected Green functions where S ′ is defined in (4.19). Constructed generating functionals (4.21) obey very important property of independence of F for physical quantities on-shell. 6 Indeed, for infinitesimal δF the variation of Z ′ (4.21), is proportional to the external sources J. Due to the equivalence theorem [21], it means that Green functions calculated with the help of the generating functionals Z(J, ϕ * ) and Z ′ (J, ϕ * ) give the same physical answers on-shell. In deriving (4.22), the results of calculation (3.17) is used and the notation is introduced.
In the case of finite anticanonical transformations, we consider the following anticanonically generalized action S ′′ where δF = δF (ϕ, ϕ * ) is an infinitesimal functional. The following representation holds In Appendix C, a proof of relations (4.25), (4.26) is given. Due to (4.25), we can present the action S ′′ in the form Although we need here of infinitesimal functional δF (ϕ, ϕ * ) but the representation (4.27) by itself is valid for arbitrary functional δF . In turn, the representation (4.27) allows us for the use of previous reasons concerning the case of infinitesimal anticanonical transformations and for the statement that the generating functionals Z ′′ and Z ′ constructed with the help of the actions S ′′ and S ′ , respectively, give the same physical results.
The next point of our study is connected with the behavior of generating functionals subjected to an arbitrary anticanonical transformation. Consider an one-parameter family of functionals Z ′ (J, ϕ * ; a), Taking into account (3.17) and (4.29), we derive the relation . (4.33) We will refer to (4.32) as the basic equation describing dependence of effective action on an anticanonical transformation in the field-antifield formalism. In Section 5, we present a solution to this equation.

Solution to the basic equation
In what follows below, we will use a short notation for all quantities depending on the variables ϕ, ϕ * , We will study solutions to (5.2) in the class of regular functionals in a, by using a power series expansion in this parameter. In the beginning, let us find a solution to this equation to the first order in a, presenting Γ(a) and F (a) in the form The dependence of Γ 2 (a) on a is described by the equation where ad(F n+1 (a)) = − exp{−ad(U n (a))}∂ a exp{ad(U n (a))} + + exp{−ad(U n (a))}ad(F 1 (a)) exp{ad(U n (a))}. (5.30) By the same reasons used at the previous stages, we conclude that ad(F n+1 (a)) = − n a ad(F n|n (a)) + exp{−ad(F n|n (a))}ad(F n (a)) exp{ad(F n|n (a))} + O(a n ) = Let us state again that the dependence of the effective action on a finite anticanonical transformation with a generating functional Y (ϕ, Φ * ; a) is really described in terms of anticanonical transformation with a generator functional U(ϕ, ϕ * ; a). As an anticanonical transformation is a change of variables in Γ, in particular, it means that, on-shell, the effective action does not depend on gauges introducing with the help of anticanonical transformations.

Discussions
In the present article, we have explored a conception of gauge fixing procedure in the fieldantifield formalism [1,2], based on the use of anticanonical transformations of general type. The approach includes an action (master-transformed action) constructed with the help of anticanonical master transformation and being non-degenerate. The master-transformed action is a sum of two terms: one is an action subjected to an anticanonical transformation and the other is a term connecting with a logarithm of a superdeterminant of this anticanonical transformation. This action satisfies the quantum master equation [19,13] (see, also Appendix D). Generating functionals of Green functions constructed via the master-transformed action obey the important property of the gauge independence of physical quantities on-shell, and satisfy the Ward identity. We have found that any (finite) anticanonical master transformation of an action leads to the corresponding anticanonical transformation of effective action (generating functional of vertex functions) provided the generating functional of Green functions is constructed with the help of anticanonical master action. We have proved the existence of a generator functional of an anticanonical transformation of the effective action. This result is essential when proving the independence of the effective action of anticanonical transformations on-shell and, on the other hand, it may supplement in a non-trivial manner the representation of anticanonical transformations in the form of a path-ordered exponential [13].

Appendix A Infinitesimal variation of effective action
Here we prove the possibility to present the equation (3.21) in the form (3.23). To do this, we introduce the matrix of second derivatives of Γ, Γ ij , and its inverse, M ij , From the Ward identity (2.20) written in the form it follows that the relations hold. By taking these relations into account, we have and Substituting (A.7) in (3.21), we have derived the equation (3.23) for variation of Γ.

Appendix B Generator of anticanonical transformation
Here we give a proof that any anticanonical transformation can be described by the corresponding generator ad(F ) in the sense of (4.18). Firstly, we note that ifZ A l (Z), l = 1, 2, are anticanonical variables, then the compositions of these variables,Z A 12 (Z) =Z A 1 (Z 2 (Z)) andZ A 21 (Z) =Z A 2 (Z 1 (Z)), are anticanonical as well. Indeed, we have (B.2) In particular, the variablesZ A 12 (Z) = exp{adF (Z)}Z A 1 (Z) are anticanonical ifZ A 1 (A) are anticanonical variables. Indeed, we havē Secondly, the next remark is obvious . Taking into account (4.2) -(4.4) and (4.6) -(4.8), we have Then, we introduce (anticanonical) variablesZ A 2 (Z; a), with the corresponding generating functional As a result, we have The corresponding generating functional Y n+1 (ϕ, Φ * ; a) has the form (B.14) By usual manipulations, we find Applying the induction method, we have proved that an arbitrary set of anticanonical variables Z A (Z) can be really represented in the form (4.18).

Appendix C Some useful formulae
Consider a set of differential operators ad(A(Z)), ad(B(Z)), ..., ε(A(Z)) = 1, ε(B(Z)) = 1, ... applied to any functional M(Z) of anticanonical variables Z = (ϕ, ϕ * ) as the left adjoint of the antibracket. If a multiplication operation is introduced as the commutator, then this set can be considered as a Lie superalgebra. Indeed, due to the symmetry properties and the Jacobi identity for the antibracket, we have In deriving (C.13) and (C.14), the identities (C.7) and (C.8) are used. Using initial data for X(t), it follows from (C.13) X(t) = −tad(D ∂aA (Z; a; t)), D ∂aA (Z; a; t) = f (tad(A(Z; a)))∂ a A(Z; a).

(C.15)
We will use the following convention and notation for applying operators R andR, where F (R) = F (x)| x=R , A(Z) is a function and (...) means an arbitrary quantity.
Consider a first-order differential operator where N A (Z) are some functionals of Z. Let where the relation

Appendix D Master-transformed actions
Here we present a set of properties concerning master-transformed actions.
Firstly, we prove that an action S ′ constructed by the rule (4.19) from S, being a solution to the quantum master equation, satisfies the quantum master equation, as well. To do this, we consider a functional X(Z) and the transformation X(Z) → X ′ (Z) = X(Z, 1) of the form Then, due to (D.11), we havē For a given action S(Z), the relations S 1 (Z) = exp{ad(F 1 (Z))}S(Z) = S(Z 1 (Z)), (D.15) S 2 (Z) = exp{[ad(F 2 (Z)) + ad(F 1 (Z))]}S(Z) = S(Z 2 (Z)) = S(Z 1 (Z 2 (Z)) = = exp{ad(F 2 (Z))}S 1 (Z) (D. 16) hold. Using the chain rule and multiplication rule for superdeterminants, one obtains for logarithm of superdeterminant of anticanonical transformation (D.14) Consider the action S ′ 2 constructed from an action S with the help of anticanonical master transformation with the generator functional F 1 + F 2 (D.12). We obtain where S 2 (Z) is defined by the first equality in (D.16), andZ A 2 is given by the second equality in (D.12). It follows from (D.17) and (D.18) that where S ′ 1 is master transformed action S under anticanonical transformation of variables Z with the generator functional F 1 (Z), and, as a result, S ′ 2 is presented as master transformed action S ′ corresponding to the anticanonical master transformation of Z with generator functional Finally, we give a proof of the relation For the sake of completeness of our study of anticanonical transformations, let us present here a simple proof of the factorization property of the Grand Jacobian of an anticanonical transformation within the field-antifield formalism [1,2]. The result was known at least since the article [18] of Batalin and Vilkovisky, although the proof was omitted therein.
We will proceed with the use of antisymplectic Darboux coordinates Z A in the form of an explicit splitting into fields φ i and antifields φ * i , where E AB is a constant invertible antisymplectic metric with the following block structure and antisymmetry property Let F = F (Z) be a Fermion generator of an anticanonical transformation, Z A satisfies the Lie equation Let us consider the (Grand) Jacobian, J(t), of the anticanonical transformation (E.5) together with its logarithm By using (E.6) and the relations which are valid for any invertible transformation Z A →Z A , together with the formula for a δ-variation, we derive the equation for ln J d dt where the operators ∆,∆ are defined 7 by Here ∂ A and ∂Ā denotes partial Z A -andZ A -derivative, respectively. Now, let J φ be the Jacobian in the sector of fields, together with its logarithm, where ∂ i denotes partial φ i -derivative. In what follows below, the symbols ∂k and ∂ * k , with barred indices, will be used to denote partial Φ k -and Φ * k -derivatives, respectively. To get the t-derivative of ln J φ , one needs of an inverse to the matrix Φ i ← − ∂ j .
Let us consider an anticanonical transformation in the sector of fields, Let us resolve that equation for initial fields φ i , with t and φ * i kept fixed, 16) so that φ i (t, Φ(t, φ, φ * ), φ * ) ≡ φ i . (E.17) 7 Notice that in (E.11) we mean just the second equality (E.12) as to define the transformed operator∆.
That definition is maintained by the two following motivations : it respects both the nilpotency property, and the multiplicative composition ∆G =∆Ḡ,Ḡ = G(Z), with arbitrary function G = G(Z).
It follows from (E.17) that the relation 18) holds, because of the initial fields φ i are inverse functions to Φ i (t, φ, φ * ) at the fixed values of t and φ * i . From now on, the variables Φ i , Φ * i are considered as functions of t, φ i , φ * i , while the fields φ i are functions of t, Φ i , φ * i , so that the short notation will be used naturally, Now, we derive the following equation for ln J φ In turn, let us consider the quantity, Then, by multiplying (E.21) subsequently from the right by the two Jacobi matrices accompanied with a special sign factor, we have, The latter can be rewritten in the form, Now, let us introduce the quantity, where E AB is an inverse to E AB , with the following block structure, and antisymmetry property Notice that the field-field components of (E.24), do coincide with (E.23). By taking the relation into account, we have for the Jacobian J φ * in the sector of antifields, That is just the antisymplectic counterpart to the Hamiltonian Liouville Theorem [18,13].