EFFECTS OF GAUGING ON SYMPLECTIC STRUCTURE, THE HOPF TERM COUPLED TO THE CP1 MODEL, AND FRACTIONAL SPIN

Abstract

We couple the Hopf term to the relativistic CP 1 model and carry out the Hamiltonian analysis at the classical level in a gauge-independent manner. 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 S3 model, and undergoes no modification upon gauging the U(1) subgroup, except for 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 the symmetric energy–momentum tensor and Noether's prescription. The two expressions agree for the model, indicating that no fractional spin is imparted by this term at the classical level in an approach which is different from the earlier analyses carried out in the literature. We next show that the model has to be altered à la Bowick et al., by using an identity (which is not a constraint as it involves a time derivative and thus changes the basic dynamical structure of the model) valid in the radiation gauge in order to yield the fractional spin 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, we find that the model ceases to be a CP 1 model, as is the case with its nonrelativistic counterpart. This model is also shown to reveal the existence of "anomalous" spin. This is, however, given in terms of the total charge of the system, rather than any soliton number.