Reducible gauge theories in local superfield Lagrangian BRST quantization

Abstract

The construction of theta-local superfield Lagrangian BRST quantization in non-Abelian hypergauges for generic gauge theories based on the action principle is examined in the case of reducible local superfield models (LSM) on the basis of embedding a gauge theory into a special theta-local superfield model with antisymplectic constraints and a Grassmann-odd time parameter theta. We examine the problem of establishing a new correspondence between the odd-Lagrangian and odd-Hamiltonian formulations of a local LSM in the case of degeneracy of the Lagrangian description with respect to derivatives over theta of generalized classical superfields A I(theta). We also reveal the role of the nilpotent BRST-BFV charge for a formal dynamical system corresponding to the BV-BFV dual description of an LSM.

Lagrangian (superfield) BV quantization; Odd Lagrangian and Hamiltonian formulations; Second-class constraints; Reducible multi-level non-Abelian hypergauges

REGULAR ARTICLES

Reducible gauge theories in local superfield Lagrangian BRST quantization

D. M. Gitman^I; P.Yu. Moshin^{I, II}; A.A. Reshetnyak^III

^IInstituto de Física, Universidade de São Paulo, Caixa Postal 66318-CEP, 05315-970 São Paulo, S.P., Brazil

^IITomsk State Pedagogical University, 634041 Tomsk, Russia

^IIILaboratory of Non-equilibrium State Theory, Institute of Strength Physics and Materials Science, 634021 Tomsk, Russia

ABSTRACT

The construction of q-local superfield Lagrangian BRST quantization in non-Abelian hypergauges for generic gauge theories based on the action principle is examined in the case of reducible local superfield models (LSM) on the basis of embedding a gauge theory into a special q-local superfield model with antisymplectic constraints and a Grassmann-odd time parameter q. We examine the problem of establishing a new correspondence between the odd-Lagrangian and odd-Hamiltonian formulations of a local LSM in the case of degeneracy of the Lagrangian description with respect to derivatives over q of generalized classical superfields A^I(q). We also reveal the role of the nilpotent BRST-BFV charge for a formal dynamical system corresponding to the BV-BFV dual description of an LSM.

Keywords: Lagrangian (superfield) BV quantization; Odd Lagrangian and Hamiltonian formulations; Second-class constraints; Reducible multi-level non-Abelian hypergauges

I. INTRODUCTION

Local superfield Lagrangian BRST quantization [1] was originally proposed for irreducible gauge theories and Abelian hypergauges [2], which introduce the gauge-fixing procedure following the BV method [3], and then extended to arbitrary gauge models in reducible non-Abelian hypergauges of finite stage of reducibility [4]. The quantization rules [1] combine, in terms of superfields, a generalization of the "first-level" Batalin-Tyutin formalism [5] (the case of reducible hypergauges is examined in [6]) and a geometric realization of BRST transformations [7, 8] in the particular case of q-local superfield models (LSM) of Yang-Mills-type. The concept of an LSM [1, 2, 4], which realizes a trivial relation between the even t and odd q components of the object c = (t,q) called supertime [9], unlike the nontrivial interrelation realized by the operator D = ¶_q + q¶_t in the Hamiltonian superfield N = 1 formalism [10] of the BFV quantization [11], provides the basis for the method of local quantization [1, 2, 4] and proves to be fruitful in solving a number problems that restrict the applicability of the functional superfield Lagrangian method [12] to specific gauge theories. The idea of an LSM makes it possible to obtain an odd-Lagrangian and odd-Hamiltonian form of the classical master equation as a condition that preserves a q-local analogue of the energy by virtue of Noether's first theorem with respect to the evolution along the variable q, defined by superfield extensions of the extremals for an initial gauge model, i.e., by odd-Lagrangian (LS) and odd-Hamiltonian (HS) systems. The concept of an LSM provides an inclusion of the dual BV-BFV description [13, 14] of a reducible gauge theory in terms of a BRST charge for formal topological dynamical systems (i.e., systems without a definite time parameter) subject to first-class constraints of higher-stage reducibility in the problem of embedding a gauge algebra of a special reducible LSM into that of a general [25] LSM. Finally, the idea of an LSM proves to be an adequate extension of a usual gauge model in a superfield construction of the quantum action as a superfield analogue of the Koszul-Tate complex resolution [15, 16] in Lagrangian formalism on the basis of interpreting the reducibility relations as special gauge transformations of ghosts transformed into a unique q-integrable odd HS.

Along the lines of our previous works, we consider it an interesting task to solve two of the problems mentioned in the conclusion of the first paper of Ref. [1]. The first problem is that of establishing a different (from the one given by a Legendre transformation [1, 2, 4]) correspondence between the odd-Lagrangian and odd-Hamiltonian formulations of an LSM in the case of degeneracy of the Lagrangian description with respect to derivatives over q of generalized classical superfields

^I(q), i.e., _q

^I(q). The second problem is to reveal the role of the nilpotent BRST-BFV charge for a formal dynamical system corresponding to the BV-BFV dual description of an LSM.

We devote this paper to the solution of the following problems:

1. A construction of an odd-Hamiltonian formulation for an LSM starting from an odd Lagrangian in the case of a degenerate Hessian supermatrix (q) of Refs. [1, 2, 4] as the supermatrix of second derivatives of the Lagrangian classical action, S_L(q) = S_L((q), ¶_q(q),q), with respect to odd velocities (¶_q

^I, ¶_q

^J) (q) on the basis of Dirac's algorithm in terms of a q-local antibracket.

2. An application of the BFV method [11] to a construction of formal counterparts of the BFV-BRST charge, gauge fermion and unitarizing Hamiltonian of a t-local field theory in terms of a q-local Dirac's antibracket, reflecting, in view of general gauge invariance, the presence of a subsystem of second-class constraints among all of the antisymplectic constraints.

3. Establishing a correspondence between the resulting odd-Hamiltonian formulation of an LSM with the BV quantum action for the gauge model corresponding to an LSM.

The paper is organized as follows. In Section 2, we apply Dirac's algorithm [17] to realize an odd Hamiltonization of a Lagrangian degenerate LSM, being an extension of a usual model of classical fields Aⁱ, i = 1,...,n = n₊ + n_-, on a configuration space M_cl, to a q-local theory defined on an odd tangent bundle T_odd

_CLº PT

_CL = {^I, ¶_q

^I} , I = 1,...,N = N₊+N_-, (n₊,n_-) < (N₊,N_-) with N₊,(N_-) bosonic (fermionic) superfields. The generalized classical superfields ^I(q), ^I(q) = A^I + l^Iq, parameterize the base _CL (_clÌ _CL) of the bundle PT

_CL and transform with respect to a J-superfield representation T of the direct product of supergroups , P: J = × P, P = exp(iµp_q) [1, 2, 4], with chosen as a spacetime SUSY group[26], and µ, p_q being the respective nilpotent parameter and generator of q-shifts. The non-Lorentz [18] character of superfields ^I(q) defined on = {(z^M, q)} = {z^K} , z^M Ì i Ì I, is reflected by a possible inclusion in their spectrum of additional (besides ⁱ(q)) superfields corresponding to the ghosts of the minimal sector in the BV quantization scheme [3]. In Section 3, following the BFV prescription, we construct q-local counterparts of the ghosts on the basis of a complete system of antisymplectic constraints and a Hamiltonian action S_H0(q) defined on PT*_CL = , a bosonic BRST charge S_W(q), a unitarizing Hamiltonian action (q), and a gauge fermion F_Y(q). We specify to the case of a singular LSM a derivation of Lagrangian and Hamiltonian master equations from Noether's first theorem [19] applied to q-shifts, and establish a relation between the complete Hamiltonian action, S_H(q) = (q) + (S_W(q), F_Y(q))_qD, constructed via q-local Dirac's antibracket, and the quantum action of the BV method [3].

We mostly follow the conventions of Refs. [1, 4] based on DeWitt's condensed notation [20] and distinguish between two types of superfield derivatives: the right (left) derivative

(q)/¶G^p(q) ((q)/¶G^p(q)) of a function (q) for a fixed q, and the right (left) variational derivative F/dF^A(q) (F/dF^A(q)) of a functional F. In the same manner, superfield right (left) covariant derivatives with respect to a superfield G^p(q) are denoted by _p(q)F (_p(q)F) for a fixed q, and variational derivatives are denoted by _p(q)F (_p(q)F). Derivatives with respect to super(anti)fields and their components are understood as acting from the right (left), for instance, or d/dF_A(q); in the opposite case we use arrows "®" ("¬") for left (right) differentiation. For right-hand derivatives with respect to ^I(q) for a fixed q, we use the notation ,_I(q) º ¶(q)/¶^I(q). As in [1, 4], following the definitions of Refs. [21, 22], a smooth supersurface [27]S, is parameterized by local coordinates zⁱ(q), and the rank of an even q-local supermatrix ||G(q)|| with Z₂-grading e is characterized by a pair of numbers = (m₊, m_-): rank||G(q)|| = rank||G(0)||, dimS = dimS|_{q = 0} with S| _{q = 0} parameterized by zⁱ(0). We characterize the property of a quantity F to be bosonic or fermionic by a triplet of Z₂-gradings, , so that the basic Grassmann parity e, according to [1, 2, 4], is given by the sum,, of Z₂-gradings , being the Grassmann parities of coordinates of the corresponding representation spaces of supergroups , P.

II. CLASSICAL FORMULATION OF A DEGENERATE q-LOCAL SUPERFIELD MODEL

In order to provide an equivalent description of a general LSM degenerate with respect to superfields ( ¶_q)^I(q) in odd-Lagrangian and odd-Hamiltonian formulations, we consider a procedure of the odd Hamiltonization of a Lagrangian LSM that preserves -covariance. Using these results, in the next section we will apply the BFV-BRST approach in order to construct from the odd Hamiltonian formulation of an LSM a complete Hamiltonian action S_H(q) and extend the BV-BFV dual description inherent in the q-local approach to the case of a degenerate LSM. We will also establish a relation with the quantum action of the BV method for the physical gauge model contained in an LSM.

2.1 Odd-Lagrangian Formulation

Let us recall that the Lagrangian formulation (based on the variational principle) of an L_g-stage reducible LSM of generalized classical superfields , is defined by a Lagrangian action S_L, PT

_CL× {q} ® , being a C^¥(PT

_CL)-function taking its values in a real Grassmann algebra , and (independently) by a functional Z_L[], whose q-density is defined with accuracy up to an arbitrary -bosonic function f((, ¶_q)(q),q) Î ker{¶_q},

invariant with respect to general gauge transformations,

with a superfield Euler-Lagrange derivative that determines LSM dynamics and (on the assumption of locality and -covariance) with functionally dependent generators (q; q₀)

The dependence of implies an existence (on solutions of the odd LS (2.3)) of proper zero-eigenvalue eigenvectors, , with a structure similar to in (2.4), which exhaust the zero-modes of the generators and are dependent in case rank . Thus, a general L_g-reducible LSM is defined by the reducibility relations, for s = 1,...,L_g, _s = 1,..., M_s = M_s++M_s-,

For L_g = 0, the LSM is an irreducible general gauge theory.

Due to the J-scalar nature of Z_L[] it is only S_L(q), among the objects S_L(q) and Z_L[] invariant under the action of a J-superfield representation T restricted to , that transforms nontrivially with respect to the total representation T under

Eq. (2.6) is written [1, 2, 4] in terms of the nilpotent operator (¶_qU)(q) = ¶_q

^I(q)/¶^I(q) = [¶_q,U(q)]_-, U(q) = P₁

^I(q)/¶^I(q), by means of projectors onto C^¥(PT^(*)

_CL) × {q}: {P_a(q) = d_a0(1 - q¶_q) + d_a1q¶_q, a = 0,1}. The superfield Euler-Lagrange equations (2.3) are equivalent (in view of to an odd LS characterized by 2N formally second-order differential equations in q,

so that the subset of formal equations (2.7) does not affect LSM dynamics, in particular, identities (2.2) and relations (2.5); however, it determines, depending on the (non)degeneracy of the supermatrix , a possibility of presenting LS (2.3) in the normal form. The Lagrangian constraints (2.8), Q_I(q) = Q_I((,¶_q)(q),q) are functionally dependent as first-order equations in q, restricting the setting of the Cauchy problem for the LS, and thus determine the general gauge algebra of an LSM by Eqs. (2.2)-(2.5).

For an LSM which enables one to represent S_L(q) in the form of a natural system in usual classical mechanics, S_L(q) = T( ¶_q(q)) - S((q),q) , the functions Q_I(q), Q_I(q) Î C^¥(_CL×{q}) are given by the relations

being, for q = 0, the extremals of the functional S₀(A) = S((0),0) , S₀(A) Î C^¥(M_CL) , M_CL = _{CL|q = 0}. Relations (2.2) and (2.5) assume the standard (in case q = 0) form of the relations of a special gauge algebra with linearly dependent generators of special gauge transformations, , that leave invariant only S(q), in contrast to T(q),

in the rank conditions for the special zero-eigenvalue eigenvectors (q):

In case rank an LSM defined by relations (2.10), (2.11) is called a special gauge theory of L_g-stage reducibility.

The special gauge algebra of this LSM is q-locally embedded into the gauge algebra of a general gauge theory with the functional Z_L[] = (¶_qT-S)(q), which leads to a relation between the eigenvectors,

and to a possible parametric dependence of the structure functions of a special gauge theory on ¶_q

^I(q) . As noted in [1, 4], an extended (in comparison with {P_a(q) }, a = 0,1) system of projectors onto C^¥(PT

_CL × {q}), {P₀(q),q¶/¶q,U(q)}, selects in (2.10), (2.11) two kinds of gauge algebra: one is selected by means of the subsystem {P₀(q),U(q)}, with the corresponding structure equations and functions [S,]((q)) not depending explicitly on q, and another with the help of P₀(q) with the standard (in case q = 0, (e_P)_I = = 0, s = 1,...,L_g) relations for the gauge algebra of a reducible model with quantities (A) under the assumption of the completeness of reduced generators and eigenvectors .

As shown in Refs. [1, 2, 4], a characteristic feature of the q-local extension of a usual field theory to an LSM is the application of Noether's first theorem [19] to provide the invariance of the density dqS_L(q) under global q-translations as symmetry transformations of the superfields ^I(q) and coordinates (z^M,q), (^I,z^M,q) ® (^I, z^M,q + µ). It is easy to see that the function

is an LS integral of motion, namely, a quantity preserved by q-evolution, assuming the fulfillment of the generalized Lagrangian master equation

Eq. (2.15) follows from the principle of dynamical symmetry in contrast to the standard (Hamiltonian-like) master equation in the minimal sector of the BV method [3] which is based on differential-algebraic reasons as a generating equation encoding the standard relations of the gauge algebra and the structure functions (A). The function S_E(q) may also be an LS integral in the case of an explicit dependence on q, unlike its analogue in a t-local field theory, the energy E(t), in case S_L(q) admits the representation

If S_L(q) does not depend on q explicitly, Eq. (2.15) transforms into a Lagrangian master equation, . As was announced in Ref. [2], a sufficient condition for the existence of a proper solution _L(q) for the Lagrangian master equation is the presence of an independent special Lagrangian constraints for a certain division of the index I,

2.2 Odd Hamiltonization

A possibility of presenting the odd LS (2.3) in the normal form (which, in a t-local field theory, provides a basis for the generalized canonical quantization of a given dynamical system) depends on the existence of an inverse for the supermatrix in (2.7).

2.2.1 Nondegenerate Case

In this case, the LSM is reformulated in the odd-Hamiltonian description [1, 2, 4] on the odd phase space PT*_CL = = (q)} in terms of a Hamiltonian action being a C^¥(PT*_CL)-function, S_H: PT*_CL× {q} ® L₁(q;), constructed from the Lagrangian formulation through a Legendre transformation of S_L(q) with respect to _q

^I(q),

The action S_H(q) coincides with S_E(q) in terms of the PT*_CL-coordinates.

An odd HS equivalent to the LS follows from (2.3) due to transformations (2.18) and is implied by the condition of the existence of a critical superfield configuration for a fermionic functional Z_H[G_CL] identical to Z_L[],

it is written in the square brackets in Eq. (2.20) with the help of a q-local antibracket and an antisymplectic potential, V_P(G_k(q)) = 1/2(q), defined with respect to an odd Poisson bivector and a flat antisymplectic metric antidiag .

The equivalence of HS (2.20) and LS (2.7), (2.8) is provided by the fact that the Lagrangian constraints Q_I(q) transformed into the Hamiltonian constraints in terms of the PT*_CL-coordinates coincide with half of the HS equations due to transformations (2.18) and their consequences:

The equivalence between the LS and HS is guaranteed by the corresponding [formal, in view of the degeneracy conditions (2.2)] setting of the Cauchy problem (q = 0, k = CL) for integral curves modulo the continuous part of I,

The definition of a special gauge algebra (2.10), (2.11) remains the same in the Hamiltonian formulation, whereas the definition (2.2), (2.3) of a general L_g-stage reducible LSM is transformed by the rule

The fulfillment of the generalized Lagrangian master equation (2.15) for S_L(q) implies, in view of definition (2.14), transformations (2.18) and their consequence, that the Hamiltonian action is an HS integral of motion (i.e., a quantity invariant under q-shifts along arbitrary solutions by (e_P, e)-odd µ) due to a generalized Hamiltonian master equation,

The equation Q_compl(q)S_H(q) = 0, written in terms of an odd operator Q_compl(q), holds true also in the case of an explicit dependence of S_H(q) on q, according to (2.16),

where is a Legendre transform of , and If S_H(q) does not depend on q explicitly, (¶/¶q)S_H(q) = 0; then Eq. (2.25) transforms into the Hamiltonian master equation (S_H(q), S_H(q)) _q = 0. This imposes the condition [3] for S_H(q) to be proper, which has no counterpart in a t-local field theory. Sufficient conditions of the solvability of the latter equation is the presence of irreducible special Hamiltonian constraints j_a(G_k(q)) [2], equivalent to the constraints (2.17), being of first-class with respect to the antibracket:

As a consequence, the q-superfield integrability of HS (2.20) is guaranteed by the properties of the antibracket, including the Jacobi identity

and the q-translation formula written in terms of a nilpotent BRST-like generator of q-shifts along an (e_p,e)-odd vector field Q(q) º adS_H(q) = (S_H(q), ·)_q, acting on C^¥(PT*_CL×{q}), namely,

Following Refs. [1, 2, 4] and depending on the realization of additional properties of a gauge theory, we assume the fulfillment of the equation

for a trivial (in the case of a flat odd phase-space) choice of the density function r(G(q)), r = 1, which is equivalent to a vanishing antisymplectic divergence of Q(q), .

2.2.2 Degenerate Case

The degeneracy of implies the impossibility of applying the above procedure of odd Hamiltonization and requires the use of an odd counterpart of Dirac's algorithm in order to reduce the odd LS (2.3) to a normal form (with only -covariance preservation). Let the degree of degeneracy of be given by the relation

valid almost everywhere in PT_CL. It means the impossibility to express each _q

^I(q) as a function of using equations in (2.18), which is equivalent to a functional dependence among the antifields in the form of primary antisymplectic constraints,

In view of the preservation of -covariance, the constraints may be dependent for and assuming the existence of an analog of the regularity conditions [16], these constraints define a (2-_-1)-dimensional submanifold smoothly embedded in . The above regularity conditions imply the existence of an open covering of the constraints surface, = 0, by open regions on each of which there exists a local separation of constraints:

where A₀ = (A'₀,a'₀) .

For an LSM with primary constraints, there hold the following statements:

Theorem 1 For any C^¥(PT*

_CL)-function (G_CL(q),q) vanishing on the constraint surface, = 0, there exist functions such that (q) is a linear combination of constraints: .

Theorem 2 Due to the solution of the equation L_P(G_CL(q),q) = 0 for arbitrary variations , the functions

are tangent to the constraints surface.

A sketch of a proof: the validity of Theorem 1 follows from the partition of unity on PT*_CL and from the fact that from the local validity of representation (2.33) follows the existence of an invertible change of variables, (G_k(q)), M₀ = 1,...,2N - R_-1, so that = 0. Therefore, in a regular coordinate system one has

The validity of Theorem 2 follows from the fact that the variations tangent to the surface = 0 at a certain point form a (2 - _-1)-dimensional superspace so that there exist R_-1 independent solutions of = 0. By virtue of the regularity conditions (2.33), the above solutions may be chosen as a linear combination (2.34) of the antisymplectic gradients, (q), with functions unambiguously defined for .

The Hamiltonian action S_H(G_k(q),q) in (2.18) is well-defined as a Legendre transformation of S_L(q) only on the surface The definition of the Hamiltonian action by the rule (2.18) on the entire PT*_k is possible due to an extension of S_H(q) by the addition of linear combination of constraints:

Actually, Eqs. (2.18) implies the following equations under a variation on the constraints surface:

which, in accordance with Theorem 2, admit the solution

In the case of independent constraints, the unknown functions can be uniquely obtained from the first equations for _q

^I in (2.36) as functions on PT

_CL, in contrast to dependent .

The invertible Legendre transformation, corresponding to Eqs. (2.36), of S_L(q) from PT

_CL to the surface = 0 of THE bundle and its inverse are given by the relations

We assume that the local consideration of the Legendre transformation made in Eqs. (2.36)-(2.38) holds globally.

The corresponding odd HS equivalent to LS (2.3) is implied by relations (2.18), (2.35)-(2.38) following from the variational principle for a fermionic functional extended, as compared with Z_H[G_k] in (2.19), by means of linear combinations of the constraints with superfields as Lagrangian multipliers,

A relation between the Cauchy problem (or the boundary problem) for an LS and HS should be specified in view of , whereas the restriction of the action remains an expression of the function S_E(q) in terms of PT*_CL-coordinates.

By the Legendre transformation, the consequence (2.37) is a change of S_H,_I(q) by (q) in expression (2.21) for the Hamiltonian constraints , preserving the equivalence of a generalized HS that consists of the constraints ^H(q) and HS (2.40) itself. In this case, the definition of the gauge algebra (2.23) in case of a nondegenerate supermatrix is not affected by the change by virtue of the identities dZ_H/d^I(q) = implied by Eqs. (2.37).

Locally, having assumed that condition (2.31) is fulfilled in a vicinity of the configurations = (0,0) the structure of the primary constraints can be specified. Now, as the regularity conditions (2.33), one can assume, while neglecting -covariance, that there exists a separation of the superfields compatible with (2.31),

In this case, the action depends neither on the antifields nor on the primary non-expressible superfields due to the direct Legendre transformation (2.37), in which, instead of , one writes , and due to the fact that the action S_L(q) is at most homogeneous of degree 1 with respect to . The action ((G_k,L)(q),q) has the form

where the odd velocities ¶_q

^I₀(q) are resolved from the relations . In turn, Eqs. (2.38) imply the coincidence of the Lagrangian multipliers and .

Having obtained an equivalent description of the LSM in terms of the Hamiltonian action , and following the q-local analog of Dirac's algorithm, we need to check the compatibility of the odd HS in (2.40), being the preservation of the constraints by the q-evolution generated by the vector field ,

Eqs. (2.43) may contain a subsystem with equations independent from with residual equations permitting to find a part of as functions of if . In the former case, if a subsystem of equations does not depend on , i.e., if there exists a set of {} such that

The above expression defines dependent (in general) secondary antisymplectic constraints Î C^¥(PT*_k),

The constraints , in contrast to , are not valid identically in the entire PT*_k, but only on solutions of the odd HS (2.40).

The consistency conditions for the secondary constraints of the form (2.43), lead to new (generally dependent) secondary antisymplectic constraints, = (q) = 0, A₂ = 1,..., = among equations not containing in the relations for the subset ,

The remaining part of equations in permits one to define, in the case of a part of the superfields as functions of . A proof of consistency conditions for the constraints permits one to obtain new secondary antisymplectic constraints and a part from , etc.

As a result of this algorithm, the complete set of

₀ dependent antisymplectic constraints consisting of the primary and the remaining , called secondary ones, is defined as

From the nonhomogeneous linear equations with unknowns (q), whose number is ,

with the symbol "»" for a weak equality in view of Theorem 1, the consistency of Dirac's algorithm implies the existence of a general solution in the form

Here, is a particular solution of the nonhomogeneous equations and the functions define linearly independent solutions of homogeneous equations associated to (2.46), with const[28]. The coefficients in Eqs. (2.47) are arbitrary.

Relations (2.47) allow one to rewrite HS (2.40) in terms of a total Hamiltonian action ,

with = and linearly dependent constraints and proper zero-eigenvalue eigenvectors, for a rigorous inequality being dependent as well.

Following Dirac's terminology, the concept of (q-local) quantities of first and second classes is defined by the fact that an arbitrary C^¥(PT*_k)-function (G_k(q),q) either obeys the relations

(and is said to be a first-class function) or is a second-class one if it does not obey Eqs. (2.49). Jacobi's identity for the antibracket (·,·)_q implies that the set of first-class functions forms a Lie algebra _I with a multiplication with respect to the antibracket, i.e., for any (q),(q) Î _I, ((q), (q))_q Î _I. By definition, it follows that the total action is the sum of certain quantities and in view of the non-uniqueness of the particular solution of Eqs. (2.46) which are given by the gauge algebra of reducible antisymplectic primary first-class constraints :

with -odd quantities (U, V) and -even .

The primary constraints may be related with the reducible antisymplectic gauge transformations, in C^¥(PT*_k) (different from general gauge transformations, which correspond to the difference between the values of at q and at (q + µ) under two different choices of the Lagrangian multipliers : with arbitrary functions (q) on ,

For the choice with arbitrary functions (q) on , being the Grassmann parities + (1,0,1) if , the above transformations vanish on the constraint surface, .

The functional , given by

is invariant not only with respect to the transformations , in view of the relation Z_H = and Eqs. (2.23), but also with respect to extended antisymplectic transformations in C^¥(PT*_CL Å _l),

Following an analog of "Dirac's conjecture" in the usual Hamiltonian treatment of a gauge model, one may suggest that the structure of S_L(q) for an LSM be such that secondary first-class antisymplectic constraints are generators of antisymplectic gauge transformations as well.

A characteristic feature of odd Hamiltonization is the possibility of an explicit identification of so-called antisymplectic gauge freedom in comparison with the odd-Lagrangian formulation of an LSM which is related to the construction of an extended Hamiltonian action and a functional , including all the constraints ,

The variational principle for encodes an HS, which is not entirely equivalent to LS (2.3),

The system of antisymplectic constraints {(q)} contains two subsystems {} = of first-class and second-class constraints extracted from the initial system by means of certain quantities = so that only the functions from the set of Lagrangian multipliers = , remain completely undetermined by the evolution of HS (2.54).

By definition, we consider that an -stage reducible system of antisymplectic constraints (q) = (q) is divided to a subsystem of -stage reducible first-class constraints and that of -stage reducible second-class ones if

where and

where we have used the standard distribution [1, 3] of ghost number in PT*_CL.

A complete definition of the subsystem of second-class antisymplectic constraints may be obtained directly from Eqs. (2.57) under the change . For = 0 ( = 0), the subsystem , is referred to as irreducible first-class (second-class) antisymplectic constraints. The definitions in Eqs. (2.55)-(2.58) generalize, to the case of reducible mixed antisymplectic constraints in the context of a dynamical LSM, the formal definition [6, 23] of irreducible second-class antisymplectic constraints for q = 0 .

The presence of the reducible constraints {} permits one to construct a so-called weak Dirac's antibracket possessing all the properties of the odd Poisson bracket on the constraint surface = 0 by means of a degenerate odd Poisson-Dirac bivector:

with the quantities determined on the surface for any characterizing the Dirac antibracket, by the equations

The functions in (2.60) may be specified due to the consequence of relations (2.56), (2.58), applied to the constraints in the form .

The fact that Dirac's antibracket obeys the generalized Jacobi identity in the entire PT*_k or on the surface = 0 can be extended according to the construction of a weak even Dirac bracket for infinitely reducible second-class symplectic constraints [24] used to quantize the N = 1 Brink-Schwarz superparticle and the N = 1, d = 9 massive superparticle with the Wess-Zumino term.

As in the case of the initial antibracket, there exists an odd weak nilpotent (on = 0) Laplacian , corresponding to antibracket (·,·)_qD, such that Dirac's antibracket weakly equals to the failure of to act as a derivative on the product of two functions in C^¥(PT*_k),

with an -even density function r_D(G_k(q)). The operator generalizes the properties of its analogue for irreducible antisymplectic second-class constraints in Ref. [6], for in Eqs. (2.56).

From the definition of Dirac's antibracket and relations (2.56)-(2.59), there follows, for any first-class C^¥(PT*_k)-functions [, ](q) and arbitrary C^¥(PT*_k)-function (q), the validity of the weak equalities

By virtue of the fact that is a first-class quantity, the Dirac antibracket generates the odd HS (2.54), where it is possible to change the antibracket (·,·)_q by (·,·)_qD. In turn, the definition of the first-class constraint subsystem (2.55), with allowance for relations (ref2.61), may be rewritten in terms of Dirac's antibracket with a weak equality which means an equality modulo . Note that in view of the inequality the functional is not invariant under the general gauge transformations due to the presence of the secondary constraints .

III. ODD-LAGRANGIAN FORM OF BFV-BRST METHOD APPLICATION TO ODD-HAMILTONIAN LSM

Let us apply the BFV-BRST method [11] to an LSM in the odd Hamiltonian formulation which has been made more complex by the presence of the -reducible first- and second-class antisymplectic constraints. The construction of an analog S_W(q) of the BFV-BRST charge, encoding, in terms of Dirac's antibracket, the gauge algebra structure functions of the antisymplectic first-class constraints and the eigenvectors , as well as the enhanced antisymplectic gauge transformations, can be described by a superfield algorithm similar to the construction of the superfield BV action in Refs. [1, 4]. Let us consider the gauge transformations (2.52) restricted in the minimal sector of superfields for all first-class constraints ,

which, due to the definition and the substitution, instead of arbitrary , of the ghosts , are embedded into the odd HS in PT*_k with 2N equations (for µ = dq)

In view of (2.58), the function is invariant, modulo , with respect to antisymplectic gauge transformations of ghosts , with arbitrary functions Î C^¥, (,gh) ,

Making the substitution and extending first-order equations in q with respect to unknown in transformations (3.3) to an HS-like set of 2 equations with the even Hamiltonian for unknown , we obtain a system of the form (3.2). An extension of the union of the latter HS with Eqs. (3.2) is formally identical to the system (3.2) under the replacement

The iteration sequence related to a reformulation of the antisymplectic gauge transformations of ghost variables obtained from relations (2.58), leads, for an -stage reducible LSM at the s-th step, with 0 < s < L and , to invariance transformations for , modulo the constraints , namely,

The substitution transforms the antisymplectic gauge transformations (3.4) to r_s-1 equations for unknown , extended by the superantifields of odd momenta (q), into an HS on PT*_s-1 =

Having combined the system (3.5) with an HS in PT*_[s-1] = of the same form with and the even BFV-BRST charge (Hamiltonian) and having expressed the result for equations with , we obtain, by induction, the following HS:

The even function quadratic in the powers of with vanishing ghost number provided by the (, gh)-spectrum for PT*_k-coordinates

is a solution of the classical master equation written in terms of Dirac's antibracket trivially extended in PT*, k = MIN} with accuracy up to , modulo ,

Additionally, the function is subject to the BFV-like condition of properness in the sense that

where the codimension of the surface is calculated with respect to PT*_CL. The integrability of the HS in (3.8) is guaranteed by a double deformation of first in the powers of and then in the powers of , in the framework of the existence theorem [3] for the classical master equation in the minimal sector, in the case of a purely topological theory (i.e., one without the potential term S₀(A) = S((0),0) in Sec. 2.1 presenting the classical action of a standard gauge model for (e_P)_I = (e_P)_I = = 0):

Theorem 3: There exists a solution of equation (3.9), in PT^*

_MIN, as a power series in the ghost coordinates , subject to the boundary conditions

and to the condition of properness (3.8) written for S_W;k(q).

A sketch of a proof: the proof of Theorem 3 repeats the Koszul-Tate construction in Hamiltonian formalism [15] for the antisymplectic case.

IV. CONCLUSION

Let us summarize the main results of the present work.

We have examined a q-local description of an arbitrary degenerate reducible superfield theory as a natural extension of a usual gauge theory, defined on a configuration space of classical fields Aⁱ, to a local superfield model. Namely, we apply Dirac's algorithm to realize an odd Hamiltonization of a Lagrangian degenerate local superfield model (LSM), being an extension of a usual gauge model of classical fields Aⁱ, i = 1,...,n = n₊+n_-, on a configuration space M_cl, to a q-local theory defined on an odd tangent bundle T_odd

_CLº PT

_CL {^I,¶_q

^I} , I = 1,...,N = N₊+N_-, (n₊,n_-) < (N₊,N_-) with N₊,( N_-) bosonic (fermionic) superfields. The generalized classical superfields ^I(q), ^I(q) = A^I + l^Iq, parameterize the base _CL (_clÌ _CL) of the bundle PT

_CL and transform with respect to a J-superfield representation T of the direct product of supergroups , P: J = × P, P = exp(iµp_q), with chosen as a spacetime SUSY group, and µ, p_q being the respective nilpotent parameter and generator of q-shifts. The non-Lorentz character of superfields ^I(q) defined on = {(z^M,q)} = {z^K} , z^M Ì i Ì I, is reflected by a possible inclusion in their spectrum of additional, besides ^I(q), superfields corresponding to the ghosts of the minimal sector in the BV quantization scheme. Following the BFV prescription, we construct their q-local counterparts on the basis of a complete system of antisymplectic constraints and a Hamiltonian action defined on PT*_CL = , a bosonic BRST charge S_W(q), a unitarizing Hamiltonian action (q), and a gauge fermion F_Y(q). We specify to the case of a singular LSM a derivation of Lagrangian and Hamiltonian master equations from Noether's first theorem applied to q-shifts, and establish a relation between the complete Hamiltonian action, S_H(q) = (q)+(S_W(q),F_Y(q))_qD, constructed via q-local Dirac's antibracket, and the quantum action of the BV method.

We have constructed an odd-Hamiltonian formulation for an LSM starting from an odd Lagrangian in the case of a degenerate Hessian supermatrix as the supermatrix of second derivatives of the Lagrangian classical action, S_L(q) = S_L((q),¶_q(q),q), with respect to odd velocities (¶_q

^I,¶_q

^J) (q) on the basis of Dirac's algorithm in terms of a q-local antibracket. We apply the BFV method to a construction of formal counterparts of the BFV-BRST charge, gauge fermion and unitarizing Hamiltonian of a t-local field theory in terms of a q-local Dirac's antibracket, reflecting, in view of general gauge invariance, the presence of a subsystem of second-class constraints among all of the antisymplectic constraints. We present a plan of establishing a correspondence between the resulting odd-Hamiltonian formulation of an LSM with the BV quantum action for a gauge model corresponding to an LSM.

Acknowledgments D.M.G. thanks the foundations FAPESP and CNPq for permanent support. P.Yu.M. is grateful to CNPq.

[25] An LSM of a general type, unlike an LSM of a special type, contains the operator ¶q º = d/dq in the structure functions of the gauge algebra.

[26] In the case of a nontrivial relation between the t- and q-components of supertime c by means of the operator D = ¶q + q¶_t, [D, D]₊ =2¶_t, as the generator of the supergroup J, an adequate realization of J, whose quotient is the superspace , is made possible in terms of (non(super)commutative with each other) and P_ntriv = exp(iµD) by analogy with the construction of a simple SUSY group.

[27] In the infinite-dimensional case, the concept of dimension has to be clarified; for a vector bundle ® , it is formally understood that dim is the dimension of the fiber over an arbitrary p Î .

[28] Here, the rank of a supermatrix is calculated with respect to the Grassmann parity

Received on 29 August, 2007

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Publication Dates

• Publication in this collection
  28 Jan 2008
• Date of issue
  Dec 2007

History

• Received
  29 Aug 2007