Weak Dirac bracket construction and the superparticle covariant quantization problem

TL;DR

This paper develops a covariant Dirac-type bracket for models with mixed constraints, applies it to superparticles, and discusses its limitations for quantization, especially in D=9 massive superparticles.

Contribution

It introduces a weak Dirac bracket construction that handles reducible constraints and applies it to superparticle models, highlighting limitations for quantization.

Findings

The weak Dirac bracket satisfies Jacobi identity only on the constraint surface.

Application to superparticles demonstrates the method's practicality.

In D=9 massive superparticle, a strong Jacobi identity bracket cannot be constructed.

Abstract

The general procedure of constructing a consistent covariant Dirac-type bracket for models with mixed first and second class constraints is presented. The proposed scheme essentially relies upon explicit separation of the initial constraints into infinitely reducible first and second class ones (by making use of some appropriately constructed covariant projectors). Reducibility of the second class constraints involved manifests itself in weakening some properties of the bracket as compared to the standard Dirac one. In particular, a commutation of any quantity with the second class constraints and the Jacobi identity take place on the second class constraints surface only. The developed procedure is realized for N=1 Brink--Schwarz superparticle in arbitrary dimension and for N=1, D=9 massive superparticle with Wess--Zumino term. A possibility to apply the bracket for quantizing the superparticles within the framework of the recent unified algebra approach by Batalin and Tyutin [20--22] is examined. In particular, it is shown that for D=9 massive superparticle it is impossible to construct Dirac-type bracket possessing (strong) Jacobi identity in a full phase space.