|^Ystf£P-W-~^Ü5 1 Introduction A common feature of most physical systems is that when passing to the Hamiltonian formulation of the theory the natural phase space appears to be too large, i.e. it contains unphysical degrees of freedom which are eliminated by constraints. This has to be taken into account in the quantization procedure. There are two kinds of constraints which are distinguished by their Poisson brackets. For the treatment of one kind of constraints, the first class-constraints, a quite general method has been developed by a group of authors [1,2,3], which is based on the BRST-symmetry [4, 5]. For a detailed review see [6]. Although the BRST-method is a powerful tool in quantizing systems with first class constraints it does not apply to systems with second-class constraints, the second kind of constraints. Since one would like to have one procedure which can handle both kind of constraints, several attempts have been made to generalise the BRST-method. There are two lines of approach which have been followed in order to solve that problem. One approach is to enlarge the phase by new variables (in addition to those variables which have to be introduced to construct the BRST-symmetry) such that the second class constraints become effectively first-class constraints. Then the usual BRST-machinery applies. This has been proposed in [7, 8, 9]. In the other line of approach, one does not introduce new variables except for the usual ghosts, but certain structures on the second-class constraints are assumed, which are used to change the ghost structure. There are several approaches to handle second- class constraints in this way [10]-[15]. In this paper we will investigate in more detail the method proposed in [13,14,15]. It is based on the assumption, that complexified second-class constraints can be split into a holomorphic and an anti-holomorphic part, such that the Poisson bracket of the holomorphic resp. anti-holomorphic constraints are in involution. If this is the case, one can impose the holomophic constraints on the ket-states, which automaticly imposes the anti-holomorphic constraints on the bra- states via hermitian conjugation. Such a procedure is known as the Gupta- Bleuler method. In section 2 we briefly review of the Gupta-Bleuler method in order to define the problem of the existence of a split in holomorphic and anti-holomorphic constraints. Section 3 contains the main result of this paper, namely the proof of the existence of a complex structure on the constraints which allows for a holomorphic split. In section 4 we use this result and apply the BRST-method on the holomorphic constraints. Section 5 contains the quantization of the 4-d superparticle as an example of this method and we end this paper with some conclusions in section 6. 1 2 Quantization of Holomorphic Constraints In this section we review the principal ideas of splitting second-class constraints in holomorphic and antiholomorphic first-class constraints and realizing them on the bra and ket states of the quantum theory. This will enable us to define the problem of the existence of snch a split. For a more detailed presentation we refer to [13, 14]. Let us consider a classical system given by a symplectic manifold M with a symplectic form UTM which is restricted by a set of constraints given as some real functions GA, A = 1,..,/ on M. The se. of solutions of the constraints: G,i(x) = 0 x€M defines a submanifold N C M. The pullback of UM to N (via the imbedding * : N —• M ) defines a pre-symplectic form ufjt on N (i.e. uqy = I'WM)- In case the Poisson brackets of the constraints vanish weakly, i.e. c {GA,GB} = f ABGc , (2.1) they are called first-class constraints and the Hamiltonian vector fields XA of GA are tangent to N. The Hamiltonian vector fields are the degenerate directions of u^ and the constraints generate symmetry transformations of some symmetry group Q on N. The physical submanifold T is therefore given by r = N/a . In this case one can apply the Dirac quantization method, i.e. one first constructs a Hubert space % on M. Of course, this Hubert space H is too large. However, it contains the physical Hilbert space as a subspace. The physical states in W are given as solutions of GA\9> = 0 *€W (2.2) where GA are now the hermitian operators corresponding to the constraint functions. Consistency of this method requires, at least, that the constraint algebra is of the form (2.1). However, the Poisson brackets of the constraints do not vanish on the constraint surface in general, i.e. we have {GA,GB} = ^BGC + ZAB • (2.3) In this case one cannot impose the constraints directly on H because this would lead to the condition ZXB|*> = 0 , 2 where the space of solutions in general is too small to be physically interesting even if it contains more than the trivial solution. In order to analyse the problem further, we first seperate the first-class constraints. This leads to a split into two sets of constraints G*, a = 1,.., / — 2k and $ol a = 1,.., 2k, where the Ga are first-class constraints and the $a are second-class constraints with the following Poisson brackets1: {G„Gh} = fJSc (2-4) {**,<?.} = flfic + rjt-, (2.5) {*.,**} = /y?c + £A + £* (2.6) Since all first class constraints are given by the Ga the anti-symmetric matrix Z^ is now invertible and therefore the number of second-class constraints must be even. In contrast to the Hamiltonian vector fields Xa of the first class constraints the Hamilto- nian vector fields Xa of the second class constraints $„ are skew orthogonal to N. The idea of Gupta-Bleuler quantization of second-class constraints is to relax the strong condition (2.2) and impose only half of the second-class constraints on the states such that the expectation-values of all constraints still vanish. In order to accomplish this, one first has to find a complex structure Jf compatible with Zafiy i.e.: JiZfh + ZafiJ! = 0 , ra J* = -&*a , (2.7) and than use this complex structure to split the (complexified) second-class constraints into holomorphic and anti-holomorphic contraints. This can be done with the two projection operators P+% = \(% + iJfi) , P-0 = \(&t - iJt) • (2-8) Let us denote the k holomorphic contraints by Gr. They obey Gr = P+',G, . (2.9) The anti-holomorphic constraints are then given by Gl = &tP.'r . (2.10) It is easy to check, that the bracket of two holomorphic or two anti-holomorphic con­ straints vanishes weakly, i.e.: {Gr,G.}*0 , {Gl,G\}*0 (2.11) {G„G\) *\zr.. (2.12) 'Actually second-class constraint* may also appear on the right hand side of (2.4), but those terms have to be at least qnadratic in #„ 3 Together with the Ga, we now have two sets of first-class constraints, (Ga,Gr) and (G«,(?J), which are related by hermitian conjugation. However, the bracket of two constraints of one set is not necessarily in involution, i.e. in general we have {Gr,G.} = f:.G. + n,Gt + K.G\ (2.13) b t {Gr,Ga} = f nG> + fr.Gt + h r.G\. (2.14) l The requirement n', = h ra = 0 leads to additional conditions on the complex structure •CA-/LJf-{<?««£> = 0 (2.15) fr-£WZ-J&h + 4M>*4 + ^W-W,4l** = °- (2-16) Only if one can find a complex structure which solves eq. (2.15,2.16) the split in holomorphic and anti-holomorphic contraints is such that the bracket on each set is in involution. Supposing we have found a solution we can then proceed and implement the holomorphic contraints on the Hilbert-space % on M, i.e. one defines the physical subspace "Hphy, of H as the set of solutions of G«|¥> = 0 (2.17) Gr\9> = 0. (2.18) Hermitian conjugation of eq. (2.18) leads to <9\Gl = 0 (2.19) and the eqs (2.17) and (2.18) are therefore sufficient conditions for the vanishing of the expectation-values of all constraints: <*|Ga|*> =<*|Gr|*> =<f|Gj|*> = 0 (2.20) In contrast to the Dirac quantization method, in this approach the physical bra and ket states are different due to the fact that GT and Gr* are independent constraints. The crucial point in this quantization scheme is to find a complex structure which is compatible with Zap and solves the eqs (2.15,2.16). In fact, the second condition is the hardest, since it is in some is sense the Poisson-bracket analog of the Nijenhuis-tensor of a complex structure on a manifold. However in the next section, when we consider the question of existence of a complex structure, we shall also discuss to what extent the condition of global existence may be relaxed. 3 The Existence of a Complex Structure We now turn to the question whether there exists a complex structure on second-class constraints in classical systems. This allows for a geometrical treatment of the problem. 4 Since we have for the brackets of the 2Jb second-class constraints with Zap invertible, we can use the constraint-functions as part of a local coordinate system of M. We further assume that the first-class constraints, if there are any, fulfil the regularity condition i.e. the matrix dGm/dzj has maximum rank, where the zj are some local coordinates. This guarantees that we can use them also as part of a local coordinate system.