On effects of gauging on symplectic structure, the Hopf term coupled to CP^1 model, and fractional spin

TL;DR

This paper investigates the effects of gauging on the symplectic structure and fractional spin in the coupled Hopf term and CP^1 model, revealing no classical fractional spin but identifying anomalous spin when the gauge field is dynamical.

Contribution

It provides a Hamiltonian analysis of the coupled Hopf term and CP^1 model, clarifies the classical fractional spin issue, and explores the impact of dynamical gauge fields on spin properties.

Findings

Symplectic structure remains unchanged upon gauging U(1).

Classical analysis shows no fractional spin from the Hopf term.

Dynamical gauge fields introduce anomalous spin related to total charge.

Abstract

We couple the Hopf term to the relativistic $CP^1$ model and carry out the Hamiltonian analysis at the classical level. The symplectic structure of the model given by the set of Dirac Brackets among the phase space variables is found to be the same as that of the pure $CP^1$ model. This symplectic structure is shown to be inherited from the global SU(2) invariant $S^3$ model, and undergoes no modification upon gauging the U(1) subgroup, except the appearance of an additional first class constraint generating U(1) gauge transformation. We then address the question of fractional spin as imparted by the Hopf term at the classical level. For that we construct the expression of angular momentum through both symmetric energy-momentum tensor as well as through Noether's prescription. Both the expressions agree for the model indicating no fractional spin is imparted by this term at the classical level-a result which is at variance with what has been claimed in the literature. We provide an argument to explain the discrepancy and corroborate our argument by considering a radiation gauge fixed Hopf term coupled to $CP^1$ model, where the desired fractional spin is reproduced and is given in terms of the soliton number. Finally, by making the gauge field of the $CP^1$ model dynamical by adding the Chern-Simons term, the model ceases to be a $CP^1$ model, as is the case with its nonrelativistic counterpart. This model is also shown to reveals the existence of `anomalous' spin. This is however given in terms of the total charge of the system, rather than any soliton number.