Schr\"odinger representation for topological solitons

TL;DR

This paper develops a Schr"odinger representation for topological solitons in the SU(2) Skyrmion model by converting second-class constraints into first-class ones using the Batalin-Fradkin-Tyutin method, enabling canonical quantization and detailed baryon spectrum analysis.

Contribution

It introduces a canonical quantization scheme in an extended phase space for SU(2) Skyrmions, incorporating St"uckelberg coordinates and topological considerations, advancing the quantization and spectrum calculation of topological solitons.

Findings

Constructed the Schr"odinger representation for SU(2) Skyrmions.

Derived the baryon mass spectrum with geometrical corrections.

Established a BRST-invariant effective Lagrangian including ghosts.

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.