Soliton operators in the quantum equivalence of the $CP_1$ and $O(3)-\sigma$ models

Abstract

We discuss some interesting aspects of the well known quantum equivalence between the $O(3)-\sigma$ and $CP_1$ models in $3D$, working in the canonical and in the path integral formulations. We show first that the canonical quantization in the hamiltonian formulation is free of ordering ambiguities for both models. We use the canonical map between the fields and momenta of the two models and compute the relevant functional determinant to verify the equivalence between the phase-space partition functions and the quantum equivalence in all the topological sectors. We also use the explicit form of the map to construct the soliton operator of the $O(3)-\sigma$ model starting from the representation of the operator in the $CP_1$ model, and discuss their properties