Ghosts of ghosts for second class constraints

TL;DR

This paper explores the incorporation of ghosts of ghosts in the BRST formalism for second class constraints, revealing an infinite hierarchy of ghosts necessary for correct quantum spectrum identification.

Contribution

It introduces a method to include ghosts of ghosts in the BRST treatment of second class constraints, addressing quantum reducibility issues.

Findings

Infinite tower of ghosts of ghosts is required.

Modified brackets offer an alternative approach.

Ensures correct physical spectrum in quantum theory.

Abstract

When one uses the Dirac bracket, second class constraints become first class. Hence, they are amenable to the BRST treatment characteristic of ordinary first class constraints. This observation is the starting point of a recent investigation by Batalin and Tyutin, in which all the constraints are put on the same footing. However, because second class constraints identically vanish as operators in the quantum theory, they are quantum-mechanically reducible and require therefore ghosts of ghosts. Otherwise, the BRST cohomology would not yield the correct physical spectrum. We discuss how to incorporate this feature in the formalism and show that it leads to an infinite tower of ghosts of ghosts. An alternative treatment, in which the brackets of the ghosts are modified, is also mentioned.