Non-Abelian Proca model based on the improved BFT formalism

TL;DR

This paper introduces an improved BFT formalism for non-Abelian Proca models, simplifying the derivation of first class constraints and Hamiltonians, and establishing a connection with the generalized St"uckelberg Lagrangian.

Contribution

The authors develop a more transparent BFT method that simplifies handling infinite correction terms and provides a direct link between Hamiltonian and Lagrangian formulations for non-Abelian Proca models.

Findings

Infinite correction terms expressed as exponential functions

Poisson brackets match Dirac brackets in the extended phase space

Derived classical Lagrangian reduces to generalized St"uckelberg Lagrangian

Abstract

We present the newly improved Batalin-Fradkin-Tyutin (BFT) Hamiltonian formalism and the generalization to the Lagrangian formulation, which provide the much more simple and transparent insight to the usual BFT method, with application to the non-Abelian Proca model which has been an difficult problem in the usual BFT method. The infinite terms of the effectively first class constraints can be made to be the regular power series forms by ingenious choice of $X_{\alpha \beta}$ and $\omega^{\alpha \beta}$-matrices. In this new method, the first class Hamiltonian, which also needs infinite correction terms is obtained simply by replacing the original variables in the original Hamiltonian with the BFT physical variables. Remarkably all the infinite correction terms can be expressed in the compact exponential form. We also show that in our model the Poisson brackets of the BFT physical variables in the extended phase space are the same structure as the Dirac brackets of the original phase space variables. With the help of both our newly developed Lagrangian formulation and Hamilton's equations of motion, we obtain the desired classical Lagrangian corresponding to the first class Hamiltonian which can be reduced to the generalized St\"uckelberg Lagrangian which is non-trivial conjecture in our infinitely many terms involved in Hamiltonian and Lagrangian.