The Jacobi identity for Dirac-like brackets

TL;DR

This paper proves that the Jacobi identity holds for new brackets introduced for systems with redundant second-class constraints, with explicit examples in particle physics.

Contribution

It establishes the Jacobi identity for these brackets on the constraint surface and demonstrates this with explicit physical models.

Findings

Jacobi identity holds on the constraint surface for the new brackets.

Explicit examples include fractional spin particles and superparticles.

The proof applies to systems with redundant second-class constraints.

Abstract

For redundant second-class constraints the Dirac brackets cannot be defined and new brackets must be introduced. We prove here that the Jacobi identity for the new brackets must hold on the surface of the second-class constraints. In order to illustrate our proof we work out explicitly the cases of a fractional spin particle in 2+1 dimensions and the original Brink-Schwarz massless superparticle in D=10 dimensions in a Lorentz covariant constraints separation.