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We couple the Hopf term to the relativistic $CP^1$ model and carry out theHamiltonian analysis at the classical level. The symplectic structure of themodel given by the set of Dirac Brackets among the phase space variables isfound to be the same as that of the pure $CP^1$ model. This symplecticstructure 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 theappearance of an additional first class constraint generating U(1) gaugetransformation. We then address the question of fractional spin as imparted bythe Hopf term at the classical level. For that we construct the expression ofangular momentum through both symmetric energy-momentum tensor as well asthrough Noether's prescription. Both the expressions agree for the modelindicating no fractional spin is imparted by this term at the classical level-aresult which is at variance with what has been claimed in the literature. Weprovide an argument to explain the discrepancy and corroborate our argument byconsidering a radiation gauge fixed Hopf term coupled to $CP^1$ model, wherethe desired fractional spin is reproduced and is given in terms of the solitonnumber. Finally, by making the gauge field of the $CP^1$ model dynamical byadding the Chern-Simons term, the model ceases to be a $CP^1$ model, as is thecase with its nonrelativistic counterpart. This model is also shown to revealsthe existence of `anomalous' spin. This is however given in terms of the totalcharge of the system, rather than any soliton number.Comment: 29 pages, Late

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

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