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

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.

arXiv commentLaTeX, 21 pages

References in corpus (8)

• Parent field theory and unfolding in BRST first-quantized terms
• BRST theory without Hamiltonian and Lagrangian
• Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model
• Characteristic classes of gauge systems
• Star Product for Second Class Constraint Systems from a BRST Theory
• Fedosov Deformation Quantization as a BRST Theory
• Generalized Poisson sigma models
• Weak Dirac bracket construction and the superparticle covariant quantization problem

Cited by in corpus (10)

• Fedosov Quantization of Lagrange-Finsler and Hamilton-Cartan Spaces and Einstein Gravity Lifts on (Co) Tangent Bundles
• BFT embedding of the Green-Schwarz superstring and the pure spinor formalism
• Unfree gauge symmetry in the BV formalism
• General method for including Stueckelberg fields
• Reducible Stueckelberg symmetry and dualities
• General conversion method for constrained systems
• Dualisation of free fields
• Hamiltonian and Lagrangian BRST quantization in Riemann Manifold
• Hamiltonian and Lagrangian BRST quantization in Riemann Manifold II
• Conversion of second-class constraints and resolving the zero curvature conditions in the geometric quantization theory