Schr\"odinger representation for topological solitons

Abstract

Exploiting the SU(2) Skyrmion Lagrangian with second-class constraints associated with Lagrange multiplier and collective coordinates, we convert the second-class system into the first-class one in the Batalin-Fradkin-Tyutin embedding through introduction of the St\"uckelberg coordinates. In this extended phase space we construct the "canonical" quantum operator commutators of the collective coordinates and their conjugate momenta to describe the Schr\"odinger representation of the SU(2) Skyrmion, so that we can define isospin operators and their Casimir quantum operator and the corresponding eigenvalue equation possessing integer quantum numbers, and we can also assign via the homotopy class $\pi_{4}(SU(2))=Z_{2}$ half integers to the isospin quantum number for the solitons in baryon phenomenology. Different from the semiclassical quantization previously performed, we exploit the "canonical" quantization scheme in the enlarged phase space by introducing the St\"uckelberg coordinates, to evaluate the baryon mass spectrum having global mass shift originated from geometrical corrections due to the $S^{3}$ compact manifold involved in the topological Skyrmion. Including ghosts and anti-ghosts, we also construct Becci-Rouet-Stora-Tyutin invariant effective Lagrangian.