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

TL;DR

This paper re-examines the Hopf term, fractional spin, and soliton operators in the 2+1 dimensional O(3) nonlinear sigma model, clarifying their topological and quantum properties using the adjoint orbit parameterization.

Contribution

It demonstrates the well-defined nature of the Hopf term for all soliton charges and develops a Hamiltonian formulation that clarifies the fractional spin formula's applicability.

Findings

Hopf term is well-defined for any soliton charge with time-independent boundary conditions.

The Q^2 fractional spin formula applies only to a restricted class of configurations.

Constructed operators create states with specific soliton configurations in the Hilbert space.

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.