2005/01/13 by Igor Batalin, I. Batalin, Maxim Grigoriev +3 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Homotopy and Cohomology in Algebraic Topology #Quantum Mechanics and Non-Hermitian Physics #hep-th #math.DG #math.QA #math.SG
paper · pdf · doi:10.1063/1.1935430
published as J.Math.Phys. 46 (2005) 072301 · LaTeX, 21 pages
arxiv created 2005/01/13 · openalex publication_date 2005/06/17 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We propose a general method for the 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 nontrivial 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 nonscalar 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 nonscalar 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 nonscalar 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.