BRST invariant $CP^{1}$ model through improved Dirac quantization

TL;DR

This paper applies an improved Dirac quantization scheme to the $CP^1$ model, deriving a BRST-invariant Lagrangian, and demonstrates its equivalence to the O(3) nonlinear sigma model with quantized energy spectra.

Contribution

It introduces a BRST-invariant formulation of the $CP^1$ model using the BFT scheme and establishes its exact equivalence to the O(3) nonlinear sigma model.

Findings

Derived a nontrivial first-class Hamiltonian for the $CP^1$ model.

Obtained a BRST-invariant gauge fixed Lagrangian.

Established the $CP^1$ model's equivalence to the O(3) nonlinear sigma model.

Abstract

The Batalin-Fradkin-Tyutin (BFT) scheme, which is an improved version of Dirac quantization, is applied to the $CP^1$ model, and the compact form of a nontrivial first-class Hamiltonian is directly obtained by introducing the BFT physical fields. We also derive a BRST-invariant gauge fixed Lagrangian through the standard path-integral procedure. Furthermore, performing collective coordinate quantization we obtain energy spectrum of rigid rotator in the $CP^1$ model. Exploiting the Hopf bundle, we also show that the $CP^1$ model is exactly equivalent to the O(3) nonlinear sigma model at the canonical level.