Ghosts of ghosts for second class constraints

When one uses the Dirac bracket, second class constraints become first class. Hence, they are amenable to the BRST treatment characteristic of ordinary first class constraints. This observation is the starting point of a recent investigation by Batalin and Tyutin, in which all the constraints are put on the same footing. However, because second class constraints identically vanish as operators in the quantum theory, they are quantummechanically reducible and require therefore ghosts of ghosts. Otherwise, the BRST cohomology would not yield the correct physical spectrum. We discuss how to incorporate this feature in the formalism and show that it leads to an infinite tower of ghosts of ghosts. An alternative treatment, in which the brackets of the ghosts are modified, is also mentioned. Constraints in constrained Hamiltonian systems can arise for different physical reasons [1]. They may originate from a gauge invariance of the theory, in which case they are first class. Or they may indicate that the q − p commutation rules must be modified, in which case they are second class. First class constraints kill twice as many degrees of freedom as second class constraints do. Geometrically, first class constraints define co-isotropic submanifolds, while second class constraints define regular submanifolds (i.e. submanifolds with an invertible induced two-form for a recent review, see [2]). The standard quantization rules implement first class and second class constraints quite differently. This is natural in view of the different physical meanings of the constraints. However, this feature is somewhat unfortunate because it requires an explicit separation of the constraints into first and second classes before going to the quantum theory. This is always possible in principle, but in practice, the split may be cumbersome or may spoil manifest Lorentz invariance. For this reason, various efforts have been devoted to the question of quantizing the constraints in a more uniform manner. Recently, Batalin and Tyutin have constructed a very interesting algebraic scheme in which the constraints are formally kept on the same footing [3, 4, 5]. Their starting point is the observation that all the constraints are first class if one uses the Dirac bracket. Hence, one may define a BRST generator in which one associates a conjugate pair of ghosts to all (first and second class) constraints. In that approach, the ghost spectrum appears to make no distinction between the constraints. Batalin and Tyutin then go on to develop a powerful formalism that incorporates the covariance properties of the Dirac bracket under redefinitions of the constraints, but this analysis will not be needed for our discussion, which focuses on a much more elementary point. We shall only investigate here the extent to which one can introduce a ghost spectrum uniformly for all the constraints. The main result of this letter is to indicate that the ghost spectrum must be augmented by ghosts of ghosts for the second class constraints. These ghosts of ghosts destroy the symmetry between first class and second class constraints. A uniform treatment may be recovered if one introduces further an infinite tower of additional ghosts of ghosts. Let z (A = 1, ..., n) be the phase space coordinates and let φa(z ) (a = 1, ..., m) be the constraint functions. We assume the constraints to be irreducible for simplicity. We denote the Dirac bracket by [, ] and will make no mention of the original Poisson bracket thereafter. One important feature of the Dirac bracket is that it is degenerate in the algebra C∞(P ) of smooth phase space functions f(z). Indeed, if χα are the second class constraints that have been used in the definition of the Dirac bracket, then one has [f, χα] = 0. (1) It is the degeneracy of the Dirac bracket that enables one to distinguish between first class and second class constraints when one has eliminated the Poisson bracket. Although degenerate in the algebra C∞(P ) of smooth functions defined in the entire phase space, the Dirac bracket is regular in the algebra C∞(Σ2) of functions defined on the surface of the second class constraints. But since we want to maintain a symmetric treatment of the constraints, we do not reduce the system to Σ2 at this stage and work instead with the full algebra C∞(P ). In terms of the Dirac bracket, all the constraints are first class [φa, φb] = C c ab φc (2) and we shall denote by Ω(z, ηa,Pa) the corresponding BRST generator . There is one pair of conjugate ghosts (ηa,Pa) for each constraint φa. The BRST generator fulfills [Ω,Ω] = 0 (3) (in the Dirac bracket) everywhere in phase space and not just on the surface of the second class constraints. The fact that the Dirac bracket is degenerate does not prevent one from obtaining the BRST generator in the usual manner. The Koszul-Tate differential used in the standard construction explained in [2] is acyclic at positive resolution degree independently of what the bracket is, and the first class property (2) guarantees that one can complete the φaη -term in Ω = φaη a + ... (4) It is denoted by Ω0 in [3]. Note that once the solution of (3) is found, the general solution of the equations of [3, 4] reads, in the notations of [3, 4], Ω0 + Γ ∗ A(Γ A +O(η)) +O(Γ∗AJΓ ∗ B) (and ∆ is defined by Eq. (3.5) of [4]). Thus, one may say that the heart of the construction of [3, 4] is contained in the equation [Ω0,Ω0] = 0.