Fedosov Deformation Quantization as a BRST Theory

TL;DR

This paper links Fedosov deformation quantization of symplectic manifolds with BFV-BRST quantization of constrained systems, showing their equivalence and providing a quantization framework using BRST methods.

Contribution

It establishes a novel connection between Fedosov geometry and BRST quantization, formulating a consistent quantization scheme for symplectic manifolds via BRST theory.

Findings

Proves existence of quantum BRST charge and observables.

Identifies BRST charge with Fedosov connection.

Shows Fedosov star product as BRST invariant observable multiplication.

Abstract

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold $\mathcal M$ is presented as a second class constrained surface in the fibre bundle ${{\mathcal T}^*_\rho}{\mathcal M}$ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ${{\mathcal T}^*_\rho}{\mathcal M}$ and the tangent bundle $T {\mathcal M}$. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables.