$CP^{1}$ model with Hopf interaction: the quantum theory

TL;DR

This paper quantizes the $CP^1$ model with Hopf interaction using the Batalin-Tyutin method, revealing quantum corrections to energy and angular momentum, with finite topological contributions and no fractional spin at the quantum level.

Contribution

It provides a detailed quantum analysis of the $CP^1$ model with Hopf interaction, including explicit quantum corrections and the extended Lagrangian in BRST formalism.

Findings

Quantum correction to energy density has a divergent ${ m O}( abla^2)$ contribution.

Hopf term can have a finite ${ m O}( abla)$ contribution in topologically nontrivial sectors.

Angular momentum shows no quantum correction, indicating no fractional spin.

Abstract

The $CP^1$ model with Hopf interaction is quantised following the Batalin-Tyutin (BT) prescription. In this scheme, extra BT fields are introduced which allow for the existence of only commuting first-class constraints. Explicit expression for the quantum correction to the expectation value of the energy density and angular momentum in the physical sector of this model is derived. The result shows, in the particular operator ordering that we have chosen to work with, that the quantum effect has a divergent contribution of $ {\cal O} (\hbar^2)$ in the energy expectation value. But, interestingly the Hopf term, though topological in nature, can have a finite ${\cal O} (\hbar)$ contribution to energy density in the homotopically nontrivial topological sector. The angular momentum operator, however, is found to have no quantum correction, indicating the absence of any fractional spin even at this quantum level. Finally, the extended Lagrangian incorporating the BT auxiliary fields is computed in the conventional framework of BRST formalism exploiting Faddeev-Popov technique of path integral method.