Geometric quantization of the BRST charge

In the first half of this paper (Sects. 1–4) we generalise the standard geometric quantization procedure to symplectic supermanifolds. In the second half (Sects. 5, 6) we apply this to two examples that exhibit classical BRST symmetry, i.e., we quantize the BRST charge and the ghost number. More precisely, in the first example we consider the reduced symplectic manifold obtained by symplectic reduction from a free group action with Ad^*-equivariant moment map; in the second example we consider a foliated configuration space, whose cotangent bundle admits the construction of a BRST charge associated to this foliation. We show that the classical BRST symmetry can be described in terms of a hamiltonian supergroup action on the extended phase space, and that geometric quantization gives us a super-unitary representation of this supergroup. Finally we point out how these results are related to reduction at the quantum level, as compared with the reduction at the classical level.Research supported by the Dutch Organization for Scientific Research (NWO)     

Geometric BRST quantization,I: Prequantization

This is the first part of a two-part paper dedicated to the definition of BRST quantization in the framework of geometric quantization. After recognizing prequantization as a manifestation of the Poisson module structure of the sections of the prequantum line bundle, we define BRST prequantization and show that it is the homological analog of the symplectic reduction of prequantum data. We define a prequantum BRST cohomology theory and interpret it in terms of geometric objects. We then show that all Poisson structures correspond under homological reduction. This allows to prove, in the BRST context, that prequantization and reduction commute.     

On BRST quantization of second class constraint algebras

A BRST quantization of second-class constraint algebras that avoids Dirac brackets is constructed, and the BRST operator is shown to be related to the BRST operator of first class algebra by a nonunitary canonical transformation. The transformation converts the second class algebra into an effective first class algebra with the help of an auxiliary second class algebra constructed from the dynamical Lagrange multipliers of the Dirac approach. The BRST invariant path integral for second class algebras is related to the path integral of the pertinent Dirac brackets, using the Parisi-Sourlas mechaism. As an application the possibility of string theories in subcritical dimensions is considered.     

Hamiltonian BRST quantization of the interacting antisymmetric tensor field

We study the hamiltonian BRST quantization of the non-abelian antisymmetric tensor field. We find the constrained system which arises from the standard action by Dirac's procedure, and eliminate the second-class constraints by introducing Dirac brackets. Having isolated the underlying first-class constrained system, we quantize it using the hamiltonian BRST techniques of Batalin and Fradkin. We study the Lorentz covariant gauge fixing of this system, and discuss the relationship between our results and other recent studies of the interacting antisymmetric tensor field.     

BRST quantization of the string model with extrinsic curvature

The string model proposed by Polyakov is investigated as a two-dimensional field theory with higher-order derivatives. We reduce the model lagrangian to a simple useful form and achieve the BRST quantization. We show the nilpotence of the BRST charge under certain conditions, and discuss the unitarity of the theory.     

Thermal conditions for scalar bosons in a curved spacetime

The conditions that allow us to consider the vacuum expectation value of the energy-momentum tensor as a statistical average, at some particular temperature, are given. When the mean value of created particles is stationary, a planckian distribution for the field modes is obtained. In the massless approximation, the temperature dependence is like that corresponding to a radiation-dominated Friedmann-like model.     

Nonlocal stochastic quantization of scalar electrodynamics

Quantization of the electromagnetic interactions of scalar charged particles is considered within the stochastic Langevin and Schwinger-Dyson equations with nonlocal white noise. Fulfillment of the gauge-invariant condition in such a scheme is studied in detail. Matrix elements of the vacuum polarization and self-energy diagrams of the scalar electrodynamics are calculated explicitly, which reduce to usual nonlocal scalar electrodynamic results.     

BRST quantization and computation of static potentials for bosonic membranes

The static potentials for both open and closed bosonic membranes are derived using the extended phase-space functional integral. It is shown that the BRST quantization scheme in the case of background gauge coincides with the ordinary phase-space quantization. The results for the mass of a rectangular pointlike membrane for the critical radius of the spherical membranes (under which appear tachyons) as well as for the tachyonic mass differ by numerical factors from those found using configuration-space functional methods. The latter is a consequence of the noncorrectness of the configuration-space quantization for the membrane theory.     

Nilpotency ofQ BRST and background field equations

Following the method of Fradkin and Vialkovsky, we study the BRST charge operator.Q _BRST , for the closed bosonic string coupled to the background of its massless states. We show that the requirement of nilpotency ofQ _BRST gives rise to the correct equations for the background fields.