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Star Product for Second Class Constraint Systems from a BRST Theory

I. A. Batalin, M. A. Grigoriev, S. L. Lyakhovich

TL;DR

This work tackles covariant deformation quantization of general second-class constraint systems on arbitrary symplectic manifolds by embedding the system into an extended phase space and converting the second-class constraints into an effective first-class, BRST-compatible set. The authors construct a Dirac-connected Fedosov-type geometry on the extended space and develop a BFV-BRST quantization that yields a star-product for the Dirac bracket, with a precise reduction to the constraint surface that recovers the Fedosov star-product. A key result is that, using a special Dirac connection, the original constraints become central in the star-commutator algebra, facilitating a clean reduction and a consistent quantum observables cohomology where $H^0$ corresponds to functions on the constraint surface $\Sigma$. The framework is explicitly phase-space covariant and avoids solving the constraints, providing a rigorous bridge between Dirac-bracket quantization and Fedosov quantization on $\Sigma$, along with an alternative formulation via first-class constraints on ${\bf V}(\mathcal M)$ and a total BRST charge.

Abstract

We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class one and can also be understood as a far-going generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV-BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.

Star Product for Second Class Constraint Systems from a BRST Theory

TL;DR

This work tackles covariant deformation quantization of general second-class constraint systems on arbitrary symplectic manifolds by embedding the system into an extended phase space and converting the second-class constraints into an effective first-class, BRST-compatible set. The authors construct a Dirac-connected Fedosov-type geometry on the extended space and develop a BFV-BRST quantization that yields a star-product for the Dirac bracket, with a precise reduction to the constraint surface that recovers the Fedosov star-product. A key result is that, using a special Dirac connection, the original constraints become central in the star-commutator algebra, facilitating a clean reduction and a consistent quantum observables cohomology where corresponds to functions on the constraint surface . The framework is explicitly phase-space covariant and avoids solving the constraints, providing a rigorous bridge between Dirac-bracket quantization and Fedosov quantization on , along with an alternative formulation via first-class constraints on and a total BRST charge.

Abstract

We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class one and can also be understood as a far-going generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV-BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.

Paper Structure

This paper contains 30 sections, 17 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General phase space
  4. Second-class constraint system
  5. Dirac bracket
  6. Dirac connection
  7. Modified Poisson bracket on the cotangent bundle over a symplectic manifold
  8. Symplectic structure
  9. Conversion -- classical description
  10. Unification of constraints $\theta_\alpha$ and $\theta_i$
  11. Symplectic connection
  12. Extended phase space
  13. Conversion -- classical description
  14. BRST cohomology
  15. Quantum description and star-product
  16. ...and 15 more sections

Key Result

Proposition 3.1

Let $f$ and $g$ be arbitrary functions on ${\mathcal{M}}$. Let $\pi^*$ be the pullback associated with the bundle projection $\pi {\rm\, :\,} {\mathcal{T}}^*_\omega{\mathcal{M}}\to {\mathcal{M}}$. Then where the Dirac bracket in the L.H.S. is taken w.r.t. the constraints $\Theta_A$ and in the R.H.S. w.r.t. $\theta_\alpha$ only. In particular, these Dirac brackets are identical on $\Sigma \subset

Theorems & Definitions (29)

  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Theorem 4.1
  • proof
  • ...and 19 more