Non-Abelian Conversion and Quantization of Non-scalar Second-Class Constraints

TL;DR

This paper introduces a covariant method for deformation quantization of systems with non-scalar second-class constraints, converting them into non-Abelian first-class constraints within the BRST framework, and ensuring a consistent quantum structure.

Contribution

It develops a systematic, covariant procedure for converting non-scalar second-class constraints into non-Abelian first-class constraints using a connection in the constraint bundle.

Findings

Provides a systematic conversion procedure within the BRST framework.

Ensures covariant quantization of systems with non-scalar constraints.

Results in a weakly associative star-product on the phase space.

Abstract

We propose a general method for deformation quantization of any second-class constrained system on a symplectic manifold. The constraints determining an arbitrary constraint surface are in general defined only locally and can be components of a section of a non-trivial vector bundle over the phase-space manifold. The covariance of the construction with respect to the change of the constraint basis is provided by introducing a connection in the ``constraint bundle'', which becomes a key ingredient of the conversion procedure for the non-scalar constraints. Unlike in the case of scalar second-class constraints, no Abelian conversion is possible in general. Within the BRST framework, a systematic procedure is worked out for converting non-scalar second-class constraints into non-Abelian first-class ones. The BRST-extended system is quantized, yielding an explicitly covariant quantization of the original system. An important feature of second-class systems with non-scalar constraints is that the appropriately generalized Dirac bracket satisfies the Jacobi identity only on the constraint surface. At the quantum level, this results in a weakly associative star-product on the phase space.