Hopf Term, Fractional Spin and Soliton Operators in the O(3) Nonlinear Sigma Model

Abstract

We re-examine three issues, the Hopf term, fractional spin and the soliton operators, in the 2+1 dimensional O(3) nonlinear sigma model based on the adjoint orbit parameterization (AOP) introduced earlier. It is shown that the Hopf Term is well-defined for configurations of any soliton charge $Q$ if we adopt a time independent boundary condition at spatial infinity. We then develop the Hamiltonian formulation of the model in the AOP and thereby argue that the well-known $Q^2$-formula for fractional spin holds only for a restricted class of configurations. Operators which create states of given classical configurations of any soliton number in the (physical) Hilbert space are constructed. Our results clarify some of the points which are crucial for the above three topological issues and yet have remained obscure in the literature.