Topological solitons in the Skyrme–Faddeev spinor model and quantum mechanics

Abstract

We discuss the 16-spinor field realization of the Skyrme–Faddeev chiral model of baryons and leptons as topological solitons. The main idea behind this paper consists in unifying the approaches suggested by Skyrme and Faddeev for the description of baryons and leptons, respectively, by using the special 8-semispinor identity invented by the Italian geometrician F. Brioschi. A peculiar property of this unified model is the necessity of generalizing the Einstein gravitational theory by including the Kretschmann invariant (i.e., the Riemann curvature tensor squared) in the Lagrangian through a special structure of the Higgs potential, implying spontaneous symmetry breaking. This fact reveals two consequences. The first one concerns an essential role of higher derivatives of the metric tensor at small distances (strong gravity), and the second one concerns the behavior of the model at large distances implying a correspondence with quantum mechanics. We consider axially symmetric states in the lepton and baryon sectors and demonstrate a method of calculating topological charges. We also give a definition of the wave function for extended particles—solitons in a special stochastic representation, which is illustrated by T. Young’s famous experiment with n slits and also by the spin–statistics correlation as a natural consequence of this representation.