Quantisation of second class systems in the Batalin-Tyutin formalism

TL;DR

This paper reviews and extends the Batalin-Tyutin formalism for quantising second class systems, illustrating the approach with the Proca model and its nonabelian extension, and clarifying the role of boundary terms and extra fields.

Contribution

It generalizes the Batalin-Tyutin quantisation method for nonabelian theories and explicitly constructs first class constraints for specific second class models.

Findings

Explicit construction of first class constraints for Proca and nonabelian models

Identification of boundary terms crucial for field correspondence

Connection established between Hamiltonian and Lagrangian formalisms

Abstract

We review the Batalin-Tyutin approach of quantising second class systems which consists in enlarging the phase space to convert such systems into first class. The quantisation of first class systems, it may be mentioned, is already well founded. We show how the usual analysis of Batalin-Tyutin may be generalised, particularly if one is dealing with nonabelian theories. In order to gain a deeper insight into the formalism we have considered two specific examples of second class theories-- the massive Maxwell theory (Proca model) and its nonabelian extension. The first class constraints and the involutive Hamiltonian are explicitly constructed. The connection of our Hamiltonian approach with the usual Lagrangian formalism is elucidated. For the Proca model we reveal the importance of a boundary term which plays a significant role in establishing an exact identification of the extra fields in the Batalin-Tyutin approach with the St\"uckelberg scalar. Some comments are also made concerning the corresponding identification in the nonabelian example.