Physical unitarity in the Lagrangian Sp(2)-symmetric formalism

P.M. Lavrov? Tomsk State Pedagogical University, Tomsk 634041, Russia P.Yu. Moshin Tomsk State University, Tomsk 634050, Russia Abstract

The structure of state vector space for a general (non-anomalous) gauge theory is studied within the Lagrangian version of the Sp(2)-symmetric quantization method. The physical S-matrix unitarity conditions are formulated. The general results are illustrated on the basis of simple gauge theory models.

arXiv:hep-th/9604113v1 19 Apr 1996

1.

Introduction

The majority of advanced ?eld theory models are formulated in terms of gauge theories. The manifestly covariant quantization of gauge theories is carried out in the Lagrangian formalism with the use of functional integration. These methods can be divided into two groups depending on whether gauge theories quantization is based on the principle of invariance under the BRST (Becchi–Rouet–Stora–Tyutin) symmetry [1, 2] or the quantization rules are underlied by a realization of the principle of invariance under the extended BRST symmetry transformations including, along with the BRST transformations, also the so-called antiBRST transformations [3, 4]. For arbitrary gauge theories (general gauge theories), the BRST symmetry principle has ?rst been realized within the BV (Batalin–Vilkovisky) quantization scheme [5, 6] well-known at present. The same principle provides the basis of the method [7] of super?eld quantization discovered for general gauge theories quite recently. The studies of Refs. [8–10] by Batalin, Lavrov and Tyutin have suggested a quantization scheme (the Lagrangian Sp(2)-symmetric formalism), in which, for general gauge theories, the extended BRST symmetry has been manifestly realized. Note that the extended BRST symmetry principle also underlies a super?eld quantization method recently proposed for general gauge theories in Ref. [11]. Among the number of questions arising in connection with the Lagrangian quantization of gauge theories, two problems are of great importance. This is ?rst of all the unitarity problem of a theory. Next comes the issue of the dependence of a theory upon the gauge. These long-standing problems have been explicitly formulated by Feynman [12] (the unitarity problem) and Jackiw [13] (the issue of gauge dependence). For the Yang–Mills type theories, the problem of gauge dependence has been thoroughly studied in Refs. [12–19] (for more references, see Ref. [20]), and for general gauge theories, in Refs. [21, 22]. The studies of Refs. [21, 22] have proved general theorems on the gauge dependence of both the non-renormalized and renormalized Green’s functions and the Smatrix for general gauge theories in arbitrary gauges. The theorems themselves have been

2 proved on general assumptions of the absence of anomalies, the use of loop expansions and the existence of regularization respecting the Ward identities. The latest outburst of interest in the gauge dependence within gauge theories has been caused by Ref. [23], in which the author calculated the one-loop e?ective action for Einstein gravity within a special class of background gauges and found the e?ective action to depend upon the gauge on-shell which con?icts with the statements of the general theorems on gauge dependence [21, 22]. There have been papers either maintaining the result [23] and giving reasons for possible gauge dependence in arbitrary non-renormalizable gauge theories [24] or expressing a doubt [25] about the applicability of the general statements [21, 22], as formal ones, to speci?c theories (Einstein gravity, in particular). The study of Ref. [26] carried out the calculation of the one-loop e?ective action for Einstein gravity within the class of gauges suggested in Ref. [23]; it was shown, ?rstly, that the general assumptions applied for the proof of the theorems [21, 22] are valid in the particular case, secondly, that the one-loop e?ective action for Einstein gravity does not depend upon the gauge on-shell in exact conformity with the theorems [21, 22] and, ?nally, that the gauge dependence asserted in Ref. [23] results from incorrect calculations (this observation is also valid as regards Ref. [24]). For the non-renormalized Green’s functions and the S-matrix, the gauge dependence within the Lagrangian Sp(2)-symmetric quantization scheme was studied in Ref. [27]. Turning again to the unitarity problem in quantum gauge theories within the Lagrangian formalism, note that for the Yang–Mills type theories it was e?ciently analyzed in Ref. [28] by Kugo and Ojima in the framework of a formalism discovered by them and based on the study of the physical subspace Vphys of the total state vector space V with inde?nite inner product < | > (note that vector spaces having inde?nite inner product are also commonly referred to as vector spaces with inde?nite metric; see, for instance, Ref. [29]). ? ? ? The subspace Vphys ≡ {|phys >} is speci?ed by the operator QBRST (Q? BRST = QBRST ) ? QBRST |phys >= 0 (1.1)

being the generator of the BRST symmetry transformations and possessing an important nilpotency property ? Q2 BRST = 0. (1.2)

? In the Yang–Mills type theories, the nilpotency of the operator QBRST follows immediately from the nilpotency of the BRST transformations. Even though in arbitrary gauge theories the algebra of the BRST transformations is generally open (o?-shell), one can still prove (on the assumption of the absence of anomalies) ? that within such theories, for the corresponding operator QBRST the nilpotency property ? holds [30]. Thus, one can assume that the Noether charge operator QBRST in the BV quantization scheme satis?es Eq. (1.2) and that the Kugo–Ojima formalism, discovered for the Yang–Mills type theories, applies to the analysis of the unitarity problem for general gauge theories. In discussing the property (1.2), it is important to bear in mind that the widespread ? opinion that the nilpotency of the operator QBRST guarantees the unitarity of a theory (see, for example, Ref. [31]) proves to be incorrect [32], and that a more accurate examination of physicality conditions ful?lment ensuring the unitarity of a theory is then required. To this end, we shall now recall the main results of analysis of the unitarity problem within the framework of the formalism proposed by Kugo and Ojima. In Ref. [28] it was shown that if a theory satis?es the following conditions (physicality ? criteria) for the Hamiltonian H and the physical subspace Vphys in the total state vector space V with inde?nite inner product < | > ? ? (i) hermiticity of the Hamiltonian H = H ? (or (pseudo-)unitarity of the total

3 (iii) positive semi-de?neteness of inner product < | > in Vphys (Vphys ? |ψ >: < ψ|ψ >≥ 0), then the physical S-matrix Sphys is consistently de?ned in a Hilbert space Hphys equipped with positive de?nite inner product (the probabilistic interpretation of the quantum theory thus secured). Namely, Hphys can be identi?ed with a (completed) quotient space Vphys /V0 ? |Φ >, |Φ >= |Φ > +V0 , |Φ >∈ Vphys of Vphys with respect to the zero-norm subspace V0 V0 = {|χ >∈ Vphys :< χ|χ >= 0}, Vphys ⊥ V0 , where positive de?nite inner product in Vphys /V0 is de?ned by < Φ|Ψ >=< Φ|Ψ >. Given this, for the physical S-matrix in Hphys Hphys = Vphys /V0 , Sphys |Φ >= S|Φ > the unitarity property holds

? ? Sphys Sphys = Sphys Sphys = 1.

In this connection, note ?rst of all that the subsidiary condition (1.1) ensures, on the assumption of hermiticity of the Hamiltonian, the ful?lment of the condition (1.3), (ii) of in out invariance Vphys under the time development (Vphys = Vphys ). In Ref. [27], the analysis of the condition (1.3), (iii) for an arbitrary theory (1.2) was based on the study of representation ? ? ? ? of the algebra of the operator QBRST and the ghost charge operator iQC ([QC , H] = 0) ? ? ? [iQC , QBRST ] = QBRST (the other commutators trivially vanish) in the one-particle subspace of the total Fock space V. The one-particle subspace of the theory generally consists of the so-called genuine BRSTsinglets, singlet pairs, and quartets [28]. By de?nition, the BRST-singlets are introduced ? as state vectors |k, N > (iQC |k, N >= N|k, N >) from the physical subspace Vphys which ? cannot be represented in the form |k, N >= QBRST |? > for any state |? >. Here, k stands for all the quantum numbers (exept the ghost one) which specify the state. At that, if N = 0, then these BRST-singlets are called genuine ones and identi?ed with physical states having positive norm. Meanwhile, if N = 0, then state vectors (|k, ?N >, |k, N >) from the physical subspace (1.1) possess zero norm and form a singlet pair with non-vanishing inner product < k, ?N|k, N >= 1. Finally, the states (|k, N >, |k, ?N >, |k, N + 1 >, |k, ?(N + 1) >) such that ? ? |k, N + 1 >= QBRST |k, N >, |k, ?N >= QBRST |k, ?(N + 1) >, < k, ?(N + 1)|k, N + 1 >=< k, ?N|k, N >= 1 form a quartet. The study of Ref. [28] discovered a general mechanism, called the quartet one, by virtue of which any state that belongs to the physical subspace Vphys of the total Fock space and contains quartet particles has vanishing norm. At the same time, the condition (1.3), (iii) of positive semi-de?niteness of inner product < | > in Vphys is taken over by a requirement [28] of the absence of singlet pairs, which thus guarantees the physical S-matrix unitarity in the Hilbert space Hphys = Vphys /V0 .

4 quantization. With that, the structure of asymptotic space is analysed on the basis of rep? ? ? resentation of the algebra of operators Qa , iQC . Here, Qa is an Sp(2)-doublet of scalar Noether charge operators being the generators of the extended BRST symmetry transformations. The physical unitarity analysis implies such general assumptions [28] as the non-degeneracy of inde?nite inner product < | >, the absence of spontaneous symmetry ? ? breaking (Qa |0 >= QC |0 >= 0), the asymptotic completeness of state vector space V. The paper is organized as follows. In Section 2 we summarize the key points of the Lagrangian Sp(2)-symmetric method and discuss the algebraic properties of the extended BRST symmetry transformations (as well as their generators) for general gauge theories; in this section we use the condensed notations suggested by De Witt [33] and the designations adopted in Refs. [8–10]. Section 3 is devoted to the study of the general structure of state vector space and to the formulation of the physical unitarity conditions. In Section 4 we illustrate the application of the proposed physical unitarity analysis on the basis of the study of state vector spaces in concrete gauge theory models [32, 34] within the Lagrangian Sp(2) quantization method.

2.

Sp(2)-symmetric Lagrangian quantization

Let us now bring to mind the key points of the Lagrangian Sp(2)-symmmetric method [8–10]. To this end, note ?rst of all that the quantization of an arbitrary gauge theory within the formalism [8–10] involves introducing a complete set of ?elds φA and the set of ? the corresponding anti?elds φ? (a=1, 2), φA (the doublets of anti?elds φ? play the role of Aa Aa ? sources of the BRST and antiBRST transformations while the anti?elds φA are the sources of the mixed BRST and antiBRST transformations) with the following distribution of the Grassmann parity ? ε(φA ) ≡ εA , ε(φ? ) = εA + 1, ε(φA ) = εA Aa and the ghost number ? gh(φ? ) = (?1)a ? gh(φA ), gh(φA ) = ?gh(φA ). Aa The speci?c structure of con?guration space of the ?elds φA (including the initial classical ?elds, the ghosts, the antighosts and the Lagrangian multipliers) is determined by the properties of original classical theory, i.e. by the linear dependence (reducible theories) or independence (irreducible theories) of generators of gauge transformations. Namely, the studies of Refs. [8, 9] have shown that the ?elds φA form components of irreducible completely symmetric Sp(2)-tensors. The basic object of the Lagrangian Sp(2)-symmmetric scheme is the ? bosonic functional S = S(φ, φ? , φ), which enables one to construct the generating functional a of Green’s functions and satis?es the equations [8–10] 1 (S, S)a + V a S = i? ?a S h 2 with the boundary condition S|φ? =φ=? =0 = S, ? h a (2.2) (2.1)

where S is the initial gauge invariant classical action. In Eq. (2.1), h is the Planck constant; ? ? ( , )a is the extended antibracket introduced for two arbitrary functionals F = F (φ, φ? , φ) a ? and G = G(φ, φ? , φ) by the rule a (F, G)a = δG δF δF δG ? A ? (?1)(ε(F )+1)(ε(G)+1) ? δφA δφ δφ δφ

5 (derivatives with respect to the ?elds are understood as the right-hand, and those to the anti?elds as the left-hand ones), while V a and ?a are operators of the form ?a = (?1)εA δl δ δ , V a = εab φ? ? , Ab A δφ? δφ δ φA Aa

where δl /δφA denotes the left-hand derivatives with respect to the ?lds φA , and εab is a constant antisymmetric second rank tensor of the group Sp(2) subject to the normalization condition ε12 = 1. Note that by virtue of the explicit form of the operators ?a , V a , the generating equations (2.1) can be represented in the form of linear di?erential equations ? ?a exp i i ? S = 0, ?a = ?a + V a , h ? h ?

which enables one, in particular, to establish the compatibility [8] of Eq. (2.1). The algebra of operators ?a , V a and the properties of the extended antibracket were studied in detail in Ref. [8], and we omit here the corresponding discussion. The study of Ref. [10] proved the existence theorem for solutions of Eq. (2.1) with the boundary condition (2.2) in the form of expansions in h powers and described the charac? teristic arbitrariness of solutions. The generating functional Z(J) of Green’s functions for the ?elds of complete con?guration space is constructed, within the Lagrangian Sp(2)-symmetric quantization scheme, by the rule ? Z(J) = Z(J, φ? , φ)|φ? =φ=0 , ? a a (2.3)

? where the extended functional Z(J, φ? , φ) of Green’s functions is de?ned in the form of a a functional integral [8] ? Z(J, φ? , φ) = a dφ exp i ? Sext (φ, φ? , φ) + JA φA a h ? . (2.4)

In Eq. (2.3), (2.4) JA are the conventional sourses to the ?elds φA (ε(JA ) = εA , gh(JA ) = ? ?gh(φ)), while Sext = Sext (φ, φ? , φ) is a bosonic functional given by a exp i Sext = exp h ? ? i? T (F ) exp h? i S , h ? (2.5)

? ? where S = S(φ, φ? , φ) is a solution of Eq. (2.1) with the boundary condition (2.2), and T (F ) a is an operator de?ned as 1 ? ? ? T (F ) = εab [?b , [?a , F ]? ]+ . 2 (2.6)

Here, F = F (φ) is a bosonic functional ?xing a concrete choice of admissible gauge, i.e. ? chosen so as the functional Sext = Sext (φ, φ? , φ) be non-degenerate in φ (examples of such a functionals F have been given in Refs. [8, 9]). It follows from the algebraic properties of the ? ? ? operators ?a ([?a , ?b ]+ = 0) that the functional Sext (2.5) satis?es Eq. (2.1). Note that the ? gauge ?xing (2.5) is in fact a particular case of the transformation generated by T (F ), with any bosonic operator chosen for F , and describing the arbitrariness of solutions of Eq. (2.1). It should also be pointed out that the transformations (2.5) provide a basis for the proof [10] of physical equivalence between the quantization of a general gauge theory in the BV formalism and the one in the Lagrangian Sp(2)-symmetric formalism. ? By virtue of the expicit form of the operator T (F ) (2.6) with the functional F depending on the ?elds of complete con?guration space only

6 ? the generating functional Z(J, φ? , φ) of Green’s functions (2.3), (2.4) is representable in the a form [8] Z(J) = ? dφ dφ? dφ dλ dπ a exp a δF ? + φA ? A δφ where π Aa , λA ε(π Aa ) = εA + 1, gh(π Aa ) = ?(?1)a + gh(φA ), ε(λA ) = εA , gh(λA ) = gh(φA ) are auxiliary variables introducing the gauge. ? The validity of Eq. (2.1) for the functional S = S(φ, φ? , φ) enables one, ?rstly, to establish a an important fact that the integrand in Eqs. (2.7) is invariant for J = 0 under the following global supersymmetry transformations δφA = π Aa ?a , δφ? = ?a Aa δS ? , δ φA = εab ?a φ? , Ab δφA (2.8) δπ Aa = ?εab λA ?b , δλA = 0 , where ?a is an Sp(2)-doublet of constant anticommuting in?nitesimal parameters. The transformations (2.8) realize the extended BRST symmetry transformations in terms of the ? variables φ, φ? , φ, π a , λ and permit establishing the independence [8] of the S-matrix on a a choice of the gauge within the Lagrangian Sp(2)-symmetric quantization scheme. Namely, let us denote the vacuum functional as Z(0) ≡ ZF and change the gauge F → F + ?F . Then, making in the functional integral for ZF +?F the change of variables (2.8) and choosing for the parameters ?a ?a = i δ(?F ) Ab εab π , 2? h δφA i ? S(φ, φ? , φ) + φ? π Aa + a Aa h ? 1 δ2F λA ? εab π Aa A B π Bb + JA φA , 2 δφ δφ

(2.7)

we ?nd that ZF +?F = ZF and conclude that the S-matrix is in fact gauge-invariant. ? Secondly, by virtue of Eqs. (2.1), (2.5), the extended generating functional Z(J, φ? , φ) of a Green’s functions satis?es the Ward identities of the form [8] JA δ δ ? ? εab φ? ? Z(J, φ? , φ) = 0. a Ab ? δφAa φA (2.9)

The study of Ref. [27] showed, with the help of Eq. (2.9), that the generating functional ? Γ = Γ(φ, φ? , φ) of vertex functions (derivatives with respect to the sources J are understood a as the left-hand ones) ? h δ ln Z(J, φ? , φ) ? h ? a ? ? , Γ(φ, φ? , φ) = ln Z(J, φ? , φ) ? JA φA , φA = a a i i δJA calculated on its extremals δΓ/δφA = 0, does not depend upon the gauge on the hypersurface φ? = 0. ? At the same time, with allowance for Eq. (2.9), one derives for Γ = Γ(φ, φ? , φ) the Ward a identities 1 (Γ, Γ)a + V a Γ = 0, (2.10) 2

7 ? In particular, Eq. (2.10), considered at φ? = φ = 0, results in the invariance of the a ? = Γ(φ) ? e?ective action Γ ? Γ = Γ|φ? =φ=0 ? a of the ?elds φA under the following transformations δφA = δΓ δφ? Aa ?a

? φ? =φ=0 a

(2.11)

(2.12)

(we shall refer to Eq. (2.12) as quantum extended BRST symmetry transformations); namely, ? δΓ = δΓ δΓ δφA δφ? Aa δΓ ?a = ?εab φ? ? Ab δ φA ? φ? =φ=0

a

?a = 0.

? φ? =φ=0 a

(2.13)

By virtue of Eq. (2.10), one readily ?nds that the algebra of the symmetry transformations (2.12), (2.13) is open o?-shell δ(1) δ(2) φA ? δ(2) δ(1) φA = ? δΓ δ2Γ = (?1)εA δφB δφ? δφ? Bb Aa

?(1){a ?(2)b}

? φ? =φ=0 a

(2.14)

(here, the symbol { } denotes the symmetrization with respect to the Sp(2) indices: A{ab} = Aab + Aba ). In this connection, note that the study of Ref. [30] investigated the properties of the symmetry transformations δα which form an open algebra

γ δα (δβ q i ) ? δβ (δα q i ) = fαβ δγ q i + ?i αβ

(2.15)

within the Lagrangian formulation of an arbitrary non-degenerate theory. Here, q i are conγ ?guration space variables, fαβ are some structure coe?cients (depending generally on q i ) and ?i are some functions vanishing on-shell. In Ref. [30] it was shown, on the assumption αβ of the absence of anomalies, that within the quantum theory constructed in accordance with the Dirac procedure, the following relations hold

γ ? ? ? ? ? [Qα , H] = 0, [Qα , Qβ ] = fαβ Qγ ,

(2.16)

? ? where H is the Hamiltonian operator and Qα are the Noether charge operators generating, on the quantum level, the symmetry transformations δα . The comparison of Eq. (2.14) with Eqs. (2.15), (2.16) yields the algebra of the operators ? ? ? ? ? of Hamiltonian H and Noether charges Q(1) ≡ Qa ?(1)a , Q(2) ≡ Qa ?(2)a corresponding to the transformations δ(1) , δ(2) (2.12), (2.14) ? ? ? ? [Q(1,2) , H] = 0, [Q(1) , Q(2) ] = 0. (2.17)

By virtue of the arbitrariness of parameters ?(1)a , ?(2)a , Eq. (2.17) implies the relations ? ? [Qa , H] = 0, ? ? [Qa , Qb ]+ = 0.

Hence it follows that within a general gauge theory (the anomalies out of account) there exists ? a doublet of nilpotent anticommuting operators Qa generating the quantum transformations of the extended BRST symmetry.

8

3.

Representation of the algebra of Qa, QC

Let us consider the representation of the algebra ? ? [Qa , Qb ]+ = 0, (3.1) ? [iQC , Q ] = ?(?1) Q

a

?a

?a

? ? ? of the operators L = (Qa , iQC ) in the one-particle subspace V (1) of the total Fock space V with inde?nite inner product < | > ? ? ? LV (1) ? V (1) , < Ψ|LΦ >=< L? Ψ|Φ >, |Ψ >, |Φ >∈ V (1) , (3.2) ? ? ? ? (Qa )? = ?(?1)a Qa , (QC )? = QC . We shall demonstrate it here that the space V (1) of representation of the algebra (3.1) is generally a direct sum V (1) =

n (1) (1) ? (1) Vn , LVn ? Vn ,

(3.3)

(1) Vn

Vn′ = ?, n = n′ ,

(1)

(1) where subspaces Vn include the following one-particle state complexes (i) genuine BRST–antiBRST-singlets (physical particles), (ii) pairs of BRST–antiBRST-singlets, (iii) BRST-quartets, (iv) antiBRST-quartets, (3.4) (v) BRST–antiBRST-quartets, (vi) BRST–antiBRST-sextets, (vii) BRST–antiBRST-octets. In order to construct the basis of representation (3.3), (3.4) explicitly, note that for an arbitrary state |Φ > one of the following conditions holds

1 ?a ?b εab Q Q |Φ > = 0, 2 1 ?a ?b εab Q Q |Φ > = 0. 2

(3.5) (3.6)

(1) ? If a state |φ(k,N ) >∈ Vn (iQC |φ(k,N ) >= N|φ(k,N ) >) satis?es the condition (3.5), then, by virtue of Eq. (3.1), there exists a set of linearly independent states

1 ? ? ? |φ(k,N ) >, Qa |φ(k,N ) >, εab Qa Qb |φ(k,N ) >, 2

(3.7)

which form a basis of a four-dimensional representation of the algebra (3.1). Given this, owing to the properties (3.1), (3.2), the states 1 ? ? ? Qa |φ(k,N ) >, εab Qa Qb |φ(k,N ) > 2

1 ? ? have vanishing norm, in particular, |k, N >≡ 2 εab Qa Qb |φ(k,N ) >

< k, N|k, N >= 0.

(3.8)

In accordance with Ref. [28], for an arbitrary one-particle zero-norm (3.8) state |k, N > there

9 (by virtue of Eq. (3.2), any states |k, N >, |k ′ , N ′ > can only have a non-vanishing inner product < k ′ , N ′ |k, N > when N = ?N ′ ). At the same time, among all the basis states, the vector |k, ?N > subject to the normalization (3.9) is without the loss of generality unique [28]. In fact, in the subspace of linearly independent states (|k, ?N >, {|l, ?N >}) with the properties < k, ?N|k, N >=< l, ?N|k, N >= 1, one can always choose a basis (|k, ?N >, {|l, ?N >≡ |l, ?N > ?|k, ?N >}) such that < l, ?N |k, N >= 0. Note that, owing to Eqs. (3.8), (3.9), the basis in the subspace of states |Ψ >= {|l, N >, l = k}, < k, N|Ψ >= 0 can always be chosen so as < k, ?N|l, N >= 0. Indeed, in order to go over from the basis states |k, N >, {|l, N >} < k, N|k, N >= 0, < k, ?N|k, N >= 1, < k, N|l, N >= 0, < k, ?N|l, N >= 1, ?l to an equivalent linearly independent set |k, N >, {|l, N >} < k, N |k, N >= 0, < k, ?N|k, N >= 1, < k, N |l, N >= 0, < k, ?N|l, N >= 0, ?l it is su?cient, for example, to identify |k, N >= |k, N >, |l, N >= |l, N > ?|k, N >, ?l. From Eqs. (3.8), (3.9) and the hermiticity assignment (3.2) it follows that there exists a set of four states 1 ? ? ? ? ? ? (3.10) |φ(k,?N ) >, Qa |φ(k,?N ) >, εab Qa Qb |φ(k,?N ) >, 2 which are also linearly independent and form a basis of representation of the algebra (3.1). ? Here, |φ(k,?N ) > is a state (3.5) chosen from the condition 1 ? ? ? εab < φ(k,?N ) |Qa Qb |φ(k,N ) >= 1. 2 (3.11)

? ? ? By virtue of Eq. (3.11), the state vectors |φa >≡ (|φ1 >, |φ2 >) satisfying the normalization 1 2 ? ? ? ? ? < φ1 |Q φ >=< φ2 |Q φ >= 1 that corresponds to the zero-norm states Qa |φ > can be chosen ?a >= εba (Qb )? |φ >. Note that without the loss of generality, one can assume ? ? in the form |φ ? < φ(k,?N ) |φ(k,N ) >= 0, since if there does exists such α = 0 that ? < φ(k,?N ) |φ(k,N ) >= α, then one can choose the basis in the subspace (3.7) so as 1 ? ? ? |φ′ (k,N ) >, Qa |φ′(k,N ) >, εab Qa Qb |φ′ (k,N ) >, 2 1 ? ? ? ? εab < φ(k,?N ) |Qa Qb |φ′ (k,N ) >= 1, < φ(k,?N ) |φ′(k,N ) >= 0, 2

1 ? ? where |φ′ (k,N ) >≡ α?1 |φ(k,N ) >? 2 εab Qa Qb |φ(k,N ) >. Let us consider the conditions of the linear dependence of the whole set of states (3.7), ? (3.10). In fact, let there among the numbers (β, βa , β, γ, γa , γ ) be a non-zero one and let ? ?(k,?N ) >≡ |φ >) ? (|φ(k,N ) >≡ |φ >, |φ

10 Eq. (3.12), in turn, implies, by virtue of Eq. (3.1), that there exists such α = 0 that 1 α ? ? ? ? ? εab Qa Qb |φ(k,N ) >= εab Qa Qb |φ(k,?N ) > . 2 2 (3.13)

? Namely, if β = γ = βa = γa = 0, then from the condition β = 0 it follows that γ = 0 ? ? = 0) with α = β? ?1 . In the case ?a : βa = 0 the condition β = γ = 0 ?γ (reversely, γ = 0 ? β ? ?1 implies γa = 0 (similarly, γa = 0 ? βa = 0), here α = βa γa (no summation). Finally, if ?1 β = 0 (or, equivalently, γ = 0), then we have α = βγ . By virtue of Eq. (3.13), the state ? |φ(k,?N ) > can be chosen from the normalization condition (3.11) in the form ? |φ(k,?N ) >= α|φ(k,N ) > . (3.14)

Hence, evidently, N = 0, and the spaces of representations corresponding to the vector sets (3.7), (3.10) coincide. For a set of basis vectors we choose, say, (3.7), i.e. (|φ(k,N =0) >≡ |k, 0 >) 1 ? ? ? |k, 0 >, Qa |k, 0 >, εab Qa Qb |k, 0 > . 2 Given this, owing to Eq. (3.14), the relation holds α? ? ? εab < k, 0|Qa Qb |k, 0 >= 1. 2 (3.16) (3.15)

By virtue of Eq. (3.16), the set of states (3.15) can be represented in the form of both a ? ? BRST-quartet ((Q1 )? = Q1 ) |k, 0 >, |k, 0 >, |k, 1 >, |k, ?1 >, ? ? |k, 1 >= Q1 |k, 0 >, |k, 0 >= Q1 |k, ?1 >, < k, 0|k, 0 >=< k, ?1|k, 1 >= 1 ? ? ? (choosing for |k, ?1 >≡ ?αQ2 |k, 0 >), and an antiBRST-quartet ((iQ2 )? = iQ2 ) |k, 0 >, |k, 0 >, |k, ?1 >, |k, 1 >, ? ? |k, ?1 >= iQ2 |k, 0 >, |k, 0 >= iQ2 |k, 1 >, < k, 0|k, 0 >=< k, ?1|k, 1 >= 1 ? (choosing for |k, 1 >≡ ?iα? Q1 |k, 0 >). In what follows, we shall refer to the state complexes of the form (3.15), (3.16) as BRST–antiBRST-quartets (3.4), (v). In case of linear independence, the states (3.7), (3.10) form a BRST–antiBRST-octet (3.4), (vii) and can be represented as both a pair of BRST-quartets ? ? ? ? ? ? (|φ(k,N ) >, ?Q1 Q2 |φ(k,?N ) >, Q1 |φ(k,N ) >, ?Q2 |φ(k,?N ) >), (3.19) ? ? (|φ(k,?N ) >, ?Q Q |φ(k,N ) >, Q |φ(k,?N ) >, ?Q |φ(k,N ) >) ?1 ?2 ?1 ?2 and a pair of antiBRST-quartets ? ? ? ? ? ? (|φ(k,N ) >, Q2 Q1 |φ(k,?N ) >, iQ2 |φ(k,N ) >, ?iQ1 |φ(k,?N ) >), (3.18) (3.17)

11 Further, we shall consider the states (3.7), (3.10), (3.11) (i.e. BRST–antiBRST-quartets and octets), provided they do exist in a speci?c theory, as components of the basis in subspace V (1) . The state vectors (3.7), (3.10), (3.11) evidently exhaust all the states (3.5). At the same time, by construction, linear combinations of the vectors (3.7), (3.10) constitute, by construction, a subspace of states |Ψ >, having non-degenerate inner product (?|Ψ >= 0, ? ?|Ψ′ >: < Ψ|Ψ′ >= 0), which is invariant under the action of the operators L. (1) Consider now the states |Φ >∈ V which cannot be represented as linear combinations of the vectors (3.7), (3.10), (3.11) (i.e. those which do not belong to BRST–antiBRSTquartets or octets); from the previous treatment it follows immediately that the states |Φ > under consideration satisfy the condition (3.6). Making allowance for the properties of the zero-norm vectors from the set (3.7), (3.10), a basis in the subspace of states |Φ > can always be chosen so as < Ψ|Φ >= 0 (|Ψ > is an arbitrary linear combination of BRST–antiBRSTquartet or octet vectors), and hence, the states |Φ > ? ? < Ψ|LΦ >=< L? Ψ|Φ >= 0 ? form a space of representation of the algebra of the operators L. Given this, the following conditions generally hold ? ?a : Qa |Φ >= 0, ? ?a : Qa |Φ >= 0. (3.21) (3.22)

Let us ?rst turn ourselves to the states of the form (3.21). For such states the condition is valid (|? > implies arbitrary one-particle states) ? |Φ >= Qa |? >, (3.23)

since otherwise the states |Φ > under consideration would be some linear combinations of the states (3.7). An arbitrary state |Φ > (3.6), (3.21), (3.23), in its turn, satis?es one of the three conditions ? ? (i) Q1 |Φ >= 0, Q2 |Φ >= 0, ? 1 |Φ >= 0, Q2 |Φ >= 0, ? (ii) Q (3.24) ? 1 |Φ >= 0, Q2 |Φ >= 0. ? (iii) Q If a state |φ(k,N ) > satis?es the condition (3.24), (i), then, by virtue of Eq. (3.1), there exist linearly independent states ? |φ(k,N ) >, Qa |φ(k,N ) >, (3.25)

which form a basis of a three-dimensional representation of the algebra (3.1). At the same ? time, the states Qa |φ > (we omit, for the sake of brevity, the notations of the quantum numbers) have vanishing norm ? ? ? ? < Q1 φ|Q1 φ >=< Q2 φ|Q2 φ >= 0. From the above relations it follows, with allowance made for Eqs. (3.2), (3.8), (3.9), that there exist three linearly independent states 1 ?a ? (Q ) |φa >, (3.26) 2 ? where the states |φa >= Qa |? >, chosen without the loss of generality as eigenvectors for ? the ghost charge operator iQC , satisfy the normalization conditions |φa >,

a ? < φb |Qa φ >= δb

(3.27)

? ? ? (here, iQC |φa >= ?(N ? (?1)a )|φa > and the conditions < φ2 |Q1 φ >=< φ1 |Q2 φ >= 0 hold, therefore, automatically); at the same time, by virtue of Eqs. (3.2), (3.27), we have 1

a ?

12 ? ? < (Qb )? φb |Qa φa >= 0, < φa |φ >= 0 (the inequality < φa |φ >= 0 leads one to the condition ?a : N = N ? (?1)a and, therefore, does not hold for any N). Owing to Eqs. (3.27), (3.28), the bases (3.25), (3.26) (|φ >, ? ? Qa |φ >) ≡ |ei >, (|φa >, 1 (Qa )? |φa >) ≡ |fi > are dual with respect to each other 2 < fi |ej >= δij . Hence follows the non-degeneracy of bilinear form < | > de?ned on the pair X ≡ {|ei >}, Y ≡ {|fi >} of state spaces corresponding to the vector sets (3.25), (3.26). ? ? This fact implies that in the space Y exists the (unique) representation L? |fi >= (L? )ij |fj > ? i >= (L)ij |ej > de?ned in X, i.e. ? of the algebra (3.1) conjugate to the representation L|e ? ? ? (L? )ij = (L)ji. Namely, 1 a ? ? (Qa )? |φb >= δb (Qc )? |φc >, 2 (3.29) ? ? (Qa )? (Qb )? |φb >= 0 ? (for the ghost charge operator iQC , the basis states of the subspace Y are by construction eigenvectors). Let us show, with Eqs. (3.1), (3.2) taken into account, that the whole set of states (3.25), (3.26) 1 ? ? |φ >, Qa |φ >, |φa >, (Qa )? |φa > 2 is linearly independent. Indeed, assuming the reverse, i.e. γ ? ? β|φ > +βa Qa |φ > +γ a |φa > + (Qa )? |φa >= 0 2 (3.30)

(the numbers (β, βa , γ a , γ) not all vanishing), one arrives, by virtue of Eq. (3.6) and the normalization conditions (3.27), at the relation ? ?a : < (Qa )? φ|φ >≡ αa = 0 representable as β = 0 ? ?a : γ a = 0, αa = (?1)a γ a /β, β = γ = 0, γ = 0 ? ?a : βa = 0, αa = ?γ/βa . ? If we now suppose, for example, that a = 1, then, owing to Eq. (3.27) (< Q1 φ1 |φ >= 1), ? C that correspond to the states Q1 |φ > and ? the eigenvalues of the ghost charge operator iQ ? Q1 |φ1 > ? ? ? iQC |Q1 φ >= (N + 1)|Q1 φ >, ? ? ? iQC |Q1 φ1 >= ?N|Q1 φ1 > must coincide, i.e. N + 1 = ?N. In the case a = 2 we similarly have N ? 1 = ?N and ?nd that neither condition can be satis?ed for an integer N. Note that the states (3.30), (3.27), (3.29) are representable in the form of a BRST-quartet ? ? |φ >, Q1 |φ1 >, Q1 |φ >, |φ1 >, ? ? < Q1 φ1 |φ >=< φ1 |Q1 φ >= 1 ? and a pair of BRST-singlets (Q2 |φ >, |φ2 >)

13 ? ? ? ? ? ? Q1 |Q2 φ >= Q1 |φ2 >= 0, |φ2 >= Q1 |? >, |Q2 φ >= Q1 |? >, ? ? as well as in the form of an antiBRST-quartet ((Q1 )? |φ1 >= (Q2 )? |φ2 >) ? ? |φ >, ?Q2 |φ2 >, iQ2 |φ >, i|φ2 >, ? ? < iφ2 |iQ2 φ >= ? < Q2 φ2 |φ >= 1 ? and a pair of antiBRST-singlets (Q1 |φ >, |φ1 >) ? < φ1 |Q1 φ >= 1, (3.32) ? ? ? ? ? ? Q2 |Q1 φ >= Q2 |φ1 >= 0, |φ1 >= Q2 |? >, |Q1 φ >= Q2 |? > . We shall refer to the states (3.30), (3.27) as BRST–antiBRST-sextets (3.4), (vi) and consider them (supposing they generally exist in a theory) as a part of the basis state vectors in subspace V (1) . The above considerations imply that the variety of linear combinations of BRST–antiBRSTsextet states contain all the states of the form (3.24), (i); at the same time, the sextet representations (3.30), (3.29) partly include the states (3.24), (ii), (iii), that is to say ? (|φ1 >, |φ2 >) = Qa |? >, (3.33) ? ? ? ? Q1 |φ1 >= ?Q2 |φ2 >= 0, Q2 |φ1 >= Q1 |φ2 >= 0. Reversely, any states (3.33) belong to a BRST–antiBRST-sextet ? ? ? ? |φ >, Q1 |φ >, Q2 |φ >, |φ1 >, |φ2 >, Q1 |φ1 >= ?Q2 |φ2 >, where |φ > is chosen from the relations ? ? < Q1 φ1 |φ >= ? < Q2 φ2 |φ >= 1. Hence, for the further analysis of representations of the algebra (3.1) that contain the states speci?ed by the conditions (3.6), (3.21), (3.23), it is su?cient for us to con?ne ourselves to the states of the form (3.24), (ii), (iii) not repreresentable as linear combinations of BRST– antiBRST-sextet states (such states belong, without the loss of generality, to the subspace orthogonal to BRST–antiBRST-sextet states and, therefore, invariant under the action of ? the operators L). For the states ? ? ? |φ >= Qa |? >, Q1 |φ >= 0, Q2 |φ >= 0, ? ? ? ? ? ? |φ >= Qa |? >, Q2 |φ >= 0, Q1 |φ >= 0 under consideration, the following supplementary conditions hold ? ? Q1 |φ >= Q2 |? >, ? ? ? Q2 |φ >= Q1 |? > . (3.37) (3.38) (3.35) (3.36) (3.34)

Let us show that the violation, for instance, of the condition (3.37) leads one to a contradiction. In fact, |? > is not, by de?nition, representable as a linear combination of ? ? BRST–antiBRST-sextet states, and, consequently, the relation Q1 |φ >= Q2 |? > is only

14 From the above relation it follows, by virtue of Eqs. (3.33)–(3.36), that the states (|φ >, ? ? ? ? |φ >, Q1 |φ >= ?Q2 |φ >) belong to some BRST–antiBRST-sextet (3.34). The inequality (3.38) is proved in a similar way. Eqs. (3.37), (3.38) imply, in particular, that the spaces of representations (3.35) and (3.36) respectively cannot be transformed into each another by ? the action of the operators L. By repetition of the given above considerations with respect to Eqs. (3.35)–(3.38) we ?nd that the state complexes (3.35), (3.37) constitute some BRST-quartets (3.4), (iii) ? ? |φ >, |φ′ >, Q1 |φ >, Q1 |φ′ >, ? ? < φ′ |Q1 φ >=< Q1 φ′ |φ >= 1, |Φ >≡ (|φ >, |φ′ >), ? ? ? ? Q2 |Φ >= 0, |Φ >= Qa |? >, Q1 |Φ >= Q2 |? > (|φ > (3.39) is orthogonal to all the BRST–antiBRST-sextet states and, in particular, to ? any state |ψ >: ?a, Qa |ψ >= 0; hence, |φ′ > also satis?es Eqs. (3.35), (3.37)), representable as well in the form of two pair of antiBRST-singlets ? ? (|φ >, Q1 |φ′ >), (|φ′ >, Q1 |φ >). (3.40) Similarly, the states (3.36), (3.38) constitute antiBRST-quartets (3.4), (iv) ? ? ? ? ? ? |φ >, |φ′ >, iQ2 |φ >, iQ2 |φ′ >, ? ? ? ? ? ? < φ′ |iQ2 φ >=< iQ2 φ′ |φ >= 1, ? ? |Φ >≡ (|φ >, |φ′ >), ? ? ? ? Q1 |Φ >= 0, |Φ >= Qa |? >, Q2 |Φ >= Q1 |? > and BRST-singlet pairs ? ? ? ? ? ? (|φ >, iQ2 |φ′ >), (|φ′ >, iQ2 |φ >). (3.42) (3.41) (3.39)

Thus, with allowance for Eqs. (3.25)–(2.32), (3.35)–(3.42), we have described the structure of representations containing the states of the form (3.6), (3.21), (3.23). Finally, we turn to the states |Φ > not representable as linear combinations of the above considered BRST–antiBRST-quartets, sextets, octets and states (3.35)–(3.38) (the state vectors just mentioned are without the loss of generality all orthogonal to |Φ >). One readily ?nds that these restrictions can only be met by the states ? |Φ >≡ {|φ(k,N ) >} = Qa |? > ? of the form (3.32) Qa |Φ >= 0, which, for N = 0, we shall identify, following Ref. [28], with physical particles |φk >≡ |φ(k,N =0) > (genuine BRST–antiBRST-singlets (3.4), (i)) ? ? < φk |φk >= 1, Qa |φk >= 0, |φk >= Qa |? > . (3.43) Meanwhile, in the case N = 0 (< φ(k,N ) |φ(k,N ) >= 0) we shall refer to the states |Φ >≡ (|φ(k,?N ) >, |φ(k,N ) >) ? ? < φ(k,?N ) |φ(k,N ) >= 1, Qa |Φ >= 0, |Φ >= Qa |? > . (3.44) as BRST–antiBRST-singlet pairs (2.4), (ii); with that, the state complexes (3.43) and (3.44) are orthogonal to each other. Thus, taking Eqs. (3.7)–(3.44) into account, we have in a general case described the (1) (1) (1) structure (3.4) of the one-particle state subspace V (1) ? Vn , (Vn ⊥ Vn′ , n = n′ ) as a ? ? ? ? space of representation LV ? V, L = (Qa , iQC ) of the algebra (3.1) of the generators ? a of extended BRST symmetry transformations and the ghost charge operator iQC . By ? Q (1) construction, inde?nite inner product < | > is non-degenerate in each subspace Vn (see the normalization conditions (3.11), (3.27), (3.39), (3.41), (3.43), (3.44) for basis vectors),

15

4.

Physical unitarity conditions

We now consider, with allowance for Eqs. (3.3), (3.4), (3.7)–(3.44), the conditions of the physical S-matrix unitarity in the Hilbert space Hphys = Vphys /V0 , where the physical subspace Vphys ? |phys > is speci?ed by the Sp(2)-covariant subsidiary condition ? Qa |phys >= 0 (4.1)

(which evidently ensures the invariance of Vphys under the time development). By virtue of Eq. (4.1), the structure of Vphys has the form

1 Vphys = Vphys 2 Vphys ,

where

1 ? 1 V ? Vphys , Q1 Vphys = 0, 2 ? 2 V ? Vphys , Q2 Vphys = 0.

In particular, for the zero-norm subspace V0 ? V we have

1 V0 = V0 2 V0 ,

(4.2)

1 1 2 2 V0 ? Vphys , V0 ? Vphys .

The analysis of representations (3.3), (3.4), (3.7)–(3.44) on the basis of the quartet mechanism [28] shows that the state vectors from Vphys containing particles of BRST–antiBRSTquartets (3.4), (v) and octets (3.4), (vii) (i.e. state complexes simultaneously representable as BRST- (3.17), (3.19) and antiBRST- (3.18), (3.20) quartets) belong to the zero-norm subspace V0 (4.2). The remaining unphysical particles (3.4), (ii), (iii), (iv), (vi) generally contain BRST- (antiBRST-) singlet pairs (3.31), (3.32), (3.40), (3.42), (3.44) and are present in the physical subspace Vphys . In this connection, the physical S-matrix conditions within the suggested approach are the reqirements of absence of the pointed out unphysical particles, i.e. BRST–antiBRST-singlet pairs (3.4), (ii), BRST-quartets (3.4), (iii) (antiBRST-singlet pairs (3.40)), antiBRST-quartets (3.4), (iv) (BRST-singlet pairs (3.42)) and BRST–antiBRSTsextets (3.4), (vi).

5.

State vector spaces in antisymmetric tensor ?eld models

In this section, in order to illustrate the general results of the paper, we shall study the physical unitarity conditions for two simple gauge theory models [32, 34] within the Sp(2)-symmetric Lagrangian formalism. p Consider the theory of a non-abelian antisymmetric ?eld B?ν suggested by Freedman and Townsend [34] and described by the action

p S = S(Ap , B?ν ) = ?

1 1 p d4 x{? ε?νρσ Gp Bρσ + Ap Ap? }, ?ν 4 2 ?

(5.1)

where Ap is a vector gauge ?eld with the strength Gp = ?? Ap ? ?ν Ap + f pqr Aq Ar (the ? ?ν ν ? ? ν coupling constant is absorbed into the structure coe?cients f pqr ), and ε?νρσ is a constant comlpetely antisymmetric four-rank tensor (ε0123 = 1).

16

p pq pq where ξ? are arbitrary parameters; D? is the covariant derivative with potential Ap (D? = ? δ pq ?? + f prq Ar ). The algebra of the gauge transformations is abelian, and the generators ? pq pq pq R?να have at the extremals of the action (5.1) the zero-eigenvectors Z? ≡ D? pr R?να Z rqα = ε?ναβ f prq

δS , r δBαβ

(5.3)

which, in their turn, are linearly independent. According to the generally accepted terminology, the model (5.1)–(5.3) is an abelian gauge theory of ?rst stage reducibility, and its quantization can be carried out, for example, within the BV scheme [6] for the theories with linearly dependent (reducible) generators. The study of Ref. [32] showed that the application of the rules [6] to the model (5.1)–(5.3) leads one to a physically unitary theory, equivalent to the principal chiral ?eld model (non-linear σ model for d = 4). We should also mention Refs. [35, 36] devoted to various aspects of quantization of the model (5.1)–(5.3) within the standard BRST symmetry. p Next, consider the gauge model [32], in which the set of ?elds (Ap , B?ν ) is extended on ? p account of a scalar gauge ?eld ω with the transformation rule

p δω p = ? ? ξ? , p and the initial action of the ?elds (Ap , B?ν , ω p ) is chosen as ? p p S(Ap , B?ν , ω p) = S(Ap , B?ν ). ? ?

(5.4)

(5.5)

p Here, S(Ap , B?ν ) is de?ned in Eq. (5.1). ? The action (5.5) is invariant under the gauge transformations (5.2), (5.4). The generators of these transformations are linearly independent (irreducible), and their algebra is abelian. At the same time, the Lagrangian quantization of the model (5.5), (5.2), (5.4) within the standard BRST symmetry fails [32] to provide physical unitarity. As mentioned above, the study of Ref. [10] proved the physical equivalence between the Lagrangian quantizations of a general gauge theory within the standard (BV formalism) and extended (Sp(2)-covariant formalism) BRST symmetries. In this connection, we shall reveal, with the help of the analysis of asimpotic states of the reducible (5.1)–(5.3) and irreducible (5.5), (5.2), (5.4) models within the Lagrangian Sp(2)-symmetric formalism, the reason for the physical unitarity in (5.1)–(5.3) and the origin of unitarity violation in (5.5), (5.2), (5.4) (see the proof of the physical S-matrix unitarity of the theory (5.1)–(5.3) within the Lagrangian Sp(2)-symmetric formalism in Ref. [37]). Consider the model (5.5), (5.2), (5.4) in the Lagrangian Sp(2)-symmetric quantization of irreducible gauge theories [8]. To this end, note that the manifest structure of complete con?guration space φA of the theory has the form

φA = (Ap? , B p?ν , ω p , B p? , C p?a ), where C p?a , B p? are the Sp(2)-doublets of the Faddeev–Popov ghosts and the auxiliary ?elds respectively, introduced according to the number of gauge parameters ξ p in Eqs. (5.2), (5.4). ? The set of anti?elds φ? , φA corresponding to the ?elds φA reads explicitly A

? ? ? ? φ? = (A? , Bp?νa , ωpa , Bp?a , Cp?a|b ), Aa p?a

φA = (Ap? , B p?ν , ω p , B p? , C p?a ). The Grassmann parities and the ghost numbers of the ?elds φA take on the values ε(Ap? ) = ε(B p?ν ) = ε(ω p) = ε(B p?) = 0,

17 gh(Ap? ) = gh(B p?ν ) = gh(ω p) = gh(B p? ) = 0, gh(C p?a ) = 3 ? 2a. The solution of the generating equations (2.1) with the boundary condition (2.2) for the model under consideration can be found in a closed form. In order to avoid the overloading of the following relations with an abundance of indices, we shall further omit the gauge ? indices p. Then the bosonic functional S = S(φ, φ? , φ) for the theory (5.5), (5.2), (5.4) can a be represented as S = S+

? ? a d4 x {B?νa (D ? C νa ? D ν C ?a ) + ωa ? ? C? ?

? ?εab C?a|b B ? + B ?ν (D ? B ν ? D ν B ? ) + ω? ? B? },

where S is the initial classical action de?ned in Eqs. (5.5), (5.1); besides, we have used for the ?elds Ap ≡ A, B p ≡ B the notations Ap B p ≡ AB, D? B ≡ ?? B + A? ∧ B, (A ∧ B)p = f pqr Aq B r . Consider the generating functional Z(J) of Green’s functions represented in the form of the functional integral (2.7) and choose for the gauge ?xing Boson F = F (φ) F = 1 1 1 a d4 x {? B?ν B ?ν + ω 2 ? εab C? C ?b }. 4 2 4

Integrating in Eq. (2.7) over the variables λA , π Aa , φA , φ? , we obtain, for the theory Aa concerned, the following representation of the generating functional Z(J) Z(J) = where SFP = ? 1 2

a a d4 x {εab ? ? C? ? ν + εab D? Cν (D ? C νb ? D ν C ?b )},

dφ exp

i S + SFP (φ) + SGF (φ) + JA φA h ?

,

(5.6) SGF = 1 d4 x {(D ν Bν? + ?? ω)B ? ? B? B ? }. 2

The application of the Dirac procedure [38] to the quantum action S + SFP + SGF (5.6) enables one to establish the fact that half the constraints of the theory (the constraints are all second-class ones) have the form π = 0. According to the theorem [39], these momenta and the corresponding conjugate coordinates can be eliminated with the help of the constraint equations; the remaining (physical) variables form canonical pairs 1 (Ai , π(A)i = ? εoijk B jk ), 2 (B oi , π(B)oi = Bi ), (ω, π(ω) = Bo ),

b (C ia , π(C)ia = εab (Do Cib ? Di Co )),

(5.7)

(C oa , π

= ε ? ? C b ),

18 The quantum action (5.6) of the theory is invariant under the following transformations of the extended BRST symmetry δB αβ = (D α C βa ? D β C αa )?a , δω = ?α C αa ?a , δC αa = ?εab B α ?b , δB α = 0. Speci?cally, the Lagrangian 1 1 1 1 a b L = ? ε?νρσ G?ν Bρσ + A? A? ? εab ? ? C? ? ν Cν ? B? B ? ? 4 2 2 2 1 a ? εab D? Cν (D ? C νb ? D ν C ?b ) + (D ν Bν? + ?? ω)B ? 2 corresponding to the quantum action (5.6) changes under the transformations (5.8) by the total derivative δL = ? ν Fν 1 a a a Fν = {? ενγρσ Gρσ C γa + (Dν Cγ ? Dγ Cν )B γ + Bν ? γ Cγ }?a . 2

a This implies the conserved Noether currents Jν a Jν ≡ Jν ? a =

δAα = 0, (5.8)

?L δφ ? Fν , ? ν Jν = 0 ?(? ν φ)

(the variations δφ of ?elds are given by Eq.(5.8)), and the corresponding Noether charges a Qa = d3 x J0 , expressed in terms of the physical variables (5.7), have the form Qa = 1 oi d3 x{ εoijk Gjk Cia + εab π(C)ib π(B) + εab π(C)ob π(ω) }. 2 (5.9)

The algebra of the charges Qa with respect to the Poisson superbracket { , } constructed by the canonically conjugate variables (5.7) is abelian {Qa , Qb } = 0. As is well-known, within the canonical quantization (according to Dirac), classical variables correspond to operators subject to canonical (anti-)commutation relaitons resulting from the replacement of Poisson (super)brackets by (anti-)commutators (with respect to the Grassmann parities of the variables) in accordance with the rule [ , ]+ = i{ , }. In particular, the ? a ? a , generating the extended BRST Noether charges Q (5.9) correspond to the operators Q symmetry transformations in terms of the operators of physical variables ? ? ? ? ? ? [Ai , Qa ] = 0, [?(A)i , Qa ] = ?iε0ijk (? j C ka + Aj ∧ C ka ), π ? ? ? [B 0i , Qa ] = iεab π(C)b , [?(B)0i , Qa ] = 0, ?i π ? [? , Qa ] = iεab π(C)0b , [?(ω) , Qa ] = 0, ω ? ? π 1? a ? ? ? ? ? [C ib , Qa ]+ = iεab π(B) , [?(C)ib , Qa ]+ = iε0ijk δb (? j Ak + Aj ∧ Ak ), ? 0i π 2 ? ? ? [C 0b , Qa ]+ = iεab π(ω) , [?(C)0b , Qa ]+ = 0. ? π

19 Consider the asymptotic state space of the model (5.6), (5.8) and study its structure. To this end, we con?ne ourselves to the analysis of free in-?elds (assuming the existence of the corresponding in-limits). The quadratic approximation S (0) of the quantum action (5.6) reads (F?ν ≡ ?? Aν ? ?ν A? ) S (0) = 1 1 1 a d4 x{? ε?νρσ F?ν Bρσ + A? A? ? εab ?? Cν ? ? C νb ? 4 2 2 1 ? B? B ? + (? ν Bν? + ?? ω)B ? }. 2

(5.10)

From Eq. (5.10) follow the equations of motion for the in-?elds (the equations for the inoperators have the same form)

a 2B?ν = 0, 2ω = 0, 2C? = 0,

(5.11) 1 A? = ε?νρσ ? ν B ρσ , B? = ? ν Bν? + ?? ω 2 (we omit the symbol of in-limit). The solution of Eq. (5.11) for the operator-valued ?eld ? ? (?) ? (+) B?ν (x) is representable as ((B?ν )? = B?ν ) ? B?ν (x) = d3 k 2(2π)3 k

0

? (?) ? (+) B?ν (k)e?ikx + B?ν (k)eikx .

(5.12)

?a Similar decompositions are valid for the operators ω (x), C? (x) with allowance made for ? ? a(?) ? a(+) (? (?) )? = ω (+) , (C? )? = (?1)a+1 C? . ω ? At the same time, the analysis of equal-time (anti-)commutation relations yields ? (?) ? (+) [B?ν (k), Bρσ (k ′ )] = η?[ρ ησ]ν δ(k ? k ′ ), [? (?) (k), ω (+) (k ′ )] = δ(k ? k ′ ), ω ? ? a(?) ? b(+) [C? (k), Cν (k ′ )]+ = εab η?ν δ(k ? k ′ ). The action (5.10) of the in-?elds is invariant under the following (non-vanishing) transformations δB αβ = (? α C βa ? ? β C αa )?a , δω = ?α C αa ?a , δC αa = ?εab B α ?b . ? In terms of the creation operators, the corresponding transformations for B?ν , ω , C? have ? ?a the form ? (+) ? ? a(+) ? a(+) [B?ν (k), Qa ] = ? k? Cν (k) ? kν C? (k) , ? ? a(+) [? (+) (k), Qa ] = ?k ? C? (k), ω (5.14) (5.13)

20 ? In Eq. (5.14), Qa is a doublet of generators of the extended BRST symmetry transformations for the operatorial in-?elds, its normal form being ? Qa =

a(+) ? a(?) ? ω d3 k k ? ω (+) (k)C? (k) + C? (k)? (?) (k) +

? (+) ? ? ? (?) + B?ν (k)C νa(?) (k) + C νa(+) (k)B?ν )(k) . The analysis of asymptotic state space structure is conveniently carried out with the help of the following local basis 1 1 ? k = 2 (k 0 , k), 2 2k0 2k0 1 ?? 1 e? (k) = k = 2 (k 0 , ?k), T 2 2k0 2k0 e? (k) = L e? (k) = (0, ελ (k)), kελ (k) = 0, ελ (k)ελ′ (k) = δλλ′ , λ = 1, 2. λ Given this, the decomposition of any vector a? (k) in the basis vectors (5.15) have the form a? (k) = e? (k)aL (k) + e? (k)aT (k) + e? (k)aλ (k), L T λ where ? aL (k) = k ? a? (k), aT (k) = k ? a? (k), aλ (k) = ?e? (k)a? (k). λ Note that for the study of one-particle state space it is su?cient to analyze the structure of creation operators. We shall decompose all the vector creation operators in the basis ? (+) (5.15), representing the operator B?ν as ? ? (+) . ? (+) ? (+) 1 B?ν = Boi , εoijk B jk(+) ≡ Di 2 Namely, ? (+) ? (+) ? [DL (k) ? DT (k), Qa ] = 0, ? (+) ? ? a(+) [Dλ (k), Qa ] = ?k0 ελλ′ Cλ′ (k), ελλ′ = ?ελ′ λ , ε12 = ?1, ? (+) ? (+) ? ? a(+) [BL (k) ? BT (k), Qa ] = 2k0 CT (k), ? ? ? [Bλ (k), Qa ] = ?k0 Cλ

(+) a(+)

(5.15)

(5.16)

(k),

? ? a(+) [? (+) (k), Qa ] = ?CT (k), ω ? [CL ? [CT

b(+)

1 ? (+) ? ? (+) (k), Qa ]+ = ?2εab {? (+) (k) + ω (BL (k) ? BT (k))}, 2k0 ? (k), Qa ]+ = 0, ? ? (+) (k), Qa ]+ = ?εab k0 Bλ (k).

b(+)

? [Cλ

b(+)

Hence it follows that among the 15 (p, k ?xed) one-particle states there is only one genuine ? (+) ? (+) BRST–antiBRST-singlet: (DL ? DT )|0 >. There are two BRST–antiBRST-quartets (respectively, for λ = 1, 2)

21 and a BRST–antiBRST-sextet 1 ? ? b(+) ? ? ? a(+) ω (+) |0 >, Qa ω (+) |0 >, CL |0 >, εab Qa CL |0 > . ? 2 The presence of the BRST–antiBRST-sextet in the state space thus accounts for the physical S-matrix unitarity violation [32] in the theory concerned. Let us now turn ourselves to the quantizaiton of the reducible model (5.1)–(5.3) within the Lagrangian Sp(2)-symmetric scheme [8–10]. In accordance with the rules [9], we introduce the set of ?elds φA φA = (A? , B ?ν , B ? , B a , C ?a , C ab ) and the sets of the corresponding anti?elds φ? , φA Aa

? ? ? ? ? φ? = (A? , B?νa , B?a , Ba|b , C?a|b , Ca|bc ), Aa ?a

φA = (A? , B ?ν , B ? , B a , C ?a , C ab ). Note that C ab , B a are the ghost ?elds (symmetric second rank Sp(2)-tensors) and Sp(2)doublets of ?rst stage respectively, introduced in accordance with the number of gauge pq pr parameters ξ ≡ ξ p for the generators R1?ν ≡ R?να Z rqα . Given this, ε(C ab ) = 0, ε(B a ) = 1, gh(C ab ) = 6 ? 2(a + b), gh(B a ) = 3 ? 2a

(the remaining ?elds A? , B ?ν , B ? , C ?a have been described above). ? The solution S = S(φ, φ? , φ) of the generating equations (2.1) with the boundary condia tion (2.2) for the model (5.1)–(5.3) can be represented as S = S+

? ? d4 x {B?νa (D ? C νa ? D ν C ?a ) ? εab C?a|b B ? + B ?ν (D ? B ν ? D ν B ? ) +

? ? ? +C?a|b D ? C ab ? 2εab Ca|bc B c ? B?a D ? B a + 2C ?a D ? B a ? 1 ? ? ? ?ε?νρσ B?νa (B ρσ ∧ B a ) + ε?νρσ (B?νa ∧ Bρσb )C ab }, 2

where S is the classical action (5.1). Choosing for the gauge ?xing bosonic functional F = F (φ) F = 1 1 1 a d4 x {? B?ν B ?ν ? εab C? C ?b ? εab εcd C ac C bd } 4 4 12

and integrating in Eq. (2.7) over the variables λ, π a , φ, φ? , one arrives at the generating a functional Z(J) of Green’s functions of the form Z(J) = where SFP = SGF = 1 1 b[?ν][ρσ] c Gρσ ? εab εcd D? C ac D ? C bd }, d4 x{ Ga Mab Kc ?ν 4 4 1 1 d4 x{B? Dν B ν? + εab B a D? C ?b ? B? B ? ? εab B a B b }, 2 2

?

dφ ? exp

i S + SFP (φ) + SGF (φ) + JA φA h ?

,

(5.17)

2i

4

?

bc

?

ij

22 In Eq. (5.17), we have used the following notations Kb

a[?ν][ρσ]

1 a a ≡ {δb (η ?ρ η νσ ? η ?σ η νρ ) + Xb ε?νρσ }, 2

1 a a Ga ≡ D? Cν ? Dν C? ? ε?νρσ B a ∧ B ρσ . ?ν 2 The matrix Mab is the inverse of M ab

a b M ab ≡ εab ? Xc Xd εcd , a M ac Mcb = δb ,

a while the action of the matrix Xb on the objects E ≡ E p carring the gauge indices p is de?ned by the rule a Xb E ≡ εbc (C ac ∧ E).

The functional ? (5.17) can be considered as a contribution to the integration measure (in φ space), invariant under the extended BRST symmetry transformations δB αβ = ?εab Mbc Kd

c[αβ][γδ]

Gd ?a , δAα = 0, γδ

δC αa = (D α C ab ? εab B α )?b , δB α = D α B a ?a , δC ab = B {a εb}c ?c , δB a = 0 for the quantum action S + SFP + SGF (5.17). The corresponding Noether charges Qa Qa = 1 d3 x{? εoijk Gjk Cia + π(C)ib D i C ab ? εab π(C)0b D iB0i + 2 0i +εab π(C)ib π(B) + 2εab εcd π(C)bc π(C)od },

expressed in terms of physical variables 1 (Ai , π(A)i = ? εoijk B jk ), 2 (B oi , π(B)oi = Bi ), 1 b (C ia , π(C)ia = ?Mab (Gb + εoi jk Xc Gc )) jk 0i 2 (C oa , π(C)oa = ?εab B b ), 1 (C ab , π(C)ab = ? εac εbd D0 C cd ), 2 have the algebraic properties {Qa , Qb } = 0 ? with respect to the Poisson bracket in phase space (5.18). The Noether charge operators Qa ? a , Qb ]+ = 0) generate the transformations ? ([Q ? ? ? ? ? ? ? ? [Ai , Qa ] = 0, [?(A)i , Qa ] = i{ε0ijk (? j C ka + Aj ∧ C ka ) + π(C)ib ∧ C ab ? εab π(C)ob ∧ Boi }, π ? ? ? ? ? ? [B 0i , Qa ] = iεab π(C)b , [?(B)0i , Qa ] = ?iεab (?i π(C)0b + Ai ∧ π(C)0b ), ?i π ? ? ? ? ? ? [C ib , Qa ]+ = i(? i C ab + Ai ∧ C ab + εab π(B) ), ? 0i (5.18)

23 ? ? ? ? ? ? [C 0b , Qa ]+ = ?i{εab (? i B0i + Ai ∧ B0i ) + 2εac εbd π(C)cd }, [?(C)0b , Qa ]+ = 0. ? π ? ? [C bc , Qa ] = iεa{b εc}d π(C)0d , ? i a a ? ? ? ? [?(C)bc , Qa ] = (? i π(C)i{b δc} + Ai ∧ π(C)i{b δc} ). π 2 Next, turn to the analysis of the in-?elds and consider the quadratic approximation S (0) for the quantum action (5.17) S (0) = 1 1 d4 x{? ε?νρσ F?ν Bρσ + A? A? + 4 2 1 1 a + εab ?? Cν (? ? C νb ? ? ν C ?b ) ? εab εcd ?? C ac ? ? C bd + 2 4 1 1 b + B? ?ν B ν? + εab B a ? ? C? ? B? B ? ? εab B a B b }. 2 2

(5.19)

The equations of motion for the in-?elds following from Eq. (5.19) are representable in the form

a 2B?ν = 0, 2C? = 0, 2C ab = 0,

1 a A? = ε?νρσ ? ν B ρσ , B? = ? ν Bν? , B a = ? ? C? . 2 ? ?a The decompositions of B?ν (x), C? (x) in the creation and annihilation operators are given by ? the relations (5.12), (5.13), while in the corresponding decomposition for C ab (x) one should make allowance for ? ? (C ab(?) )? = (?1)a+b+1 C ab(+) . The (anti-)commutation relations for the creation and annihilation operators read ? (?) ? (+) [B?ν (k), Bρσ (k ′ )] = η?[ρ ησ]ν δ(k ? k ′ ), ? a(?) ? b(+) [C? (k), Cν (k ′ )]+ = εab η?ν δ(k ? k ′ ), ? ? [C ab(?) (k), C cd(+) (k ′ )] = ?εa{c εd}b δ(k ? k ′ ). ? Owing to Eq. (5.20), the doublet of operators Qa ? Qa = ? (+) ? ? ? (?) d3 k k ? Bν? (k)C νa(?) (k) + C νa(+) (k)Bν? )(k) + ? ? c(?) ? c(+) ? + εcb(C ab(+) (k)C? (k) + C? (k)C ab(?) (k) generates for the in-operators the extended BRST symmetry transformations ? ? ? δ B αβ = ?(? α C βa ? ? β C αa )?a , ? ? ? ? ? δ C αa = (? α C ab ? εab B α )?b , δ B α = ? α B a ?a , ab {a b}c ? ? δC = B ε ?c , which, for the creation operators, take on the form ? (+) ? ? a(+) ? a(+) [B?ν (k), Qa ] = k? Cν (k) ? kν C? (k), (5.20)

24 ? ? ? c}(+) [C bc(+) (k), Qa ] = ?k ? εa{b C? (k). Making use of the decompositions of operators in the local basis (5.15) and taking the notation (5.16) into account, we ?nd, by virtue of Eq. (4.21), that the 17 one-particle states of the theory form a genuine BRST–antiBRST-singlet ? (+) ? (+) (DL ? DT )|0 >, two BRST–antiBRST-quartets ? (B1 1 ? ? ? (+) ? (+) ? (+) ? ? (+) ? (+) + D2 )|0 >, Qa (B1 + D2 )|0 >, εab Qa Qb (B1 + D2 )|0 >, 2 1 ? ? ? (+) ? (+) ? (+) ? (+) ? ? (+) ? (+) (B2 ? D1 )|0 >, Qa (B2 ? D1 )|0 >, εab Qa Qb (B2 ? D1 )|0 > 2

(+)

and a BRST–antiBRST-octet 1 ? 1(+) ? ? 1(+) ? ? ? 1(+) CL |0 >, Qa CL |0 >, εab Qa Qb CL |0 >, 2 1 2(+) 2(+) ? ? ? 2(+) ? ? ? CL |0 >, Qa CL |0 >, εab Qa Qb CL |0 > . 2 Given this, the absense of BRST–antiBRST-singlet pairs and BRST–antiBRST-sextets as well as BRST-quartets (antiBRST-singlet pairs) and antiBRST-quartets (BRST-singlet pairs) ensures the physical unitarity of the theory (see [37]).

6.

Conclusion

In this paper, we have studied the unitarity problem for general gauge theories within the Lagrangian Sp(2)-symmetric scheme proposed by Batalin, Lavrov and Tyutin in Refs. [8–10] and underlied by the principle of invariance under the extended BRST transformations. The present study is based on the investigation of asymptotic state space structure with the help ? of representations of the algebra of generators Qa (a = 1, 2) of the extended BRST symmetry ? transformations and the ghost charge operator iQC . It is shown that the space of represen? a , iQC can be described by 7 types of one-particle ? tation of the algebra of the operators Q state complexes referred to in the paper as genuine BRST–antiBRST-singlets (physical particles), pairs of BRST–antiBRST-singlets, BRST-quartets, antiBRST-quartets, BRST– antiBRST-sextets and BRST–antiBRST-octets. The conditions of the S-matrix unitarity are formulated. To provide the unitarity of a gauge theory in the Lagrangian Sp(2)-symmetric scheme, we require that in the theory be no BRST–antiBRST-singlet pairs, BRST-quartets, antiBRST-quartets and BRST–antiBRST-sextets. The general results are exempli?ed on the basis of the well-known Freedman–Townsend model [34] and the antisymmetric tensor ?eld model with auxiliary gauge ?elds, proposed in Ref. [32].

25

References

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26 [31] T.E. Clark, C.H. Lee and S.T. Love, Nucl. Phys. B 308 (1988) 379. [32] P.M. Lavrov and I.V. Tyutin, Yad. Fiz. 50 (1989) 1467. [33] B.S. De Witt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965). [34] D.Z. Freedman and P.K. Townsend, Nucl. Phys. B 177 (1981) 282. [35] S.P. De Alwis, M.T. Grisaru and L. Mezincescu, Phys. Lett. B 190 (1987) 122. [36] A.A. Slavnov and S.A. Frolov, Teor. Mat. Fiz. 75 (1988) 201. [37] P.M. Lavrov and P.Yu. Moshin, Teor. Mat. Fiz. 102 (1995) 60. [38] P.A.M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964). [39] D.M. Gitman and I.V. Tyutin, Quantization of Fields with Constraints (SpringerVerlag, Berlin, Heidelberg, 1990).

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# Physical unitarity in the Lagrangian Sp(2)-symmetric formalism

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