Description of Lepton and Baryon Phases in Skyrme-Faddeev Spinor Model

8-spinor field is suggested to unify Skyrme and Faddeev models describing baryons and leptons as topological solitons. In the Skyrme model the particles-solitons possess the topological charge of the degree type, which is interpreted as the baryon number. In the Faddeev model the particles are endowed with the topological invariant of the Hopf type, which is interpreted as the Lepton number. The special 8-spinor Brioschi identity is used to include leptons and baryons as two possible phases of the effective spinor field model with Higgs potential depending on the jμjμ being added to the Lagrangian. In the present paper the generalization of the Mie electrodynamics within the scope of the effective 8-spinor field model is suggested. For this field model the quadratic spinor quantities entering the Brioschi identity are constructed. Also we find the symmetry groups, which generate S2- and S3-submanifolds in general S8 biquadratic spinor manifold. As a result we have the homotopy groups π3(S2) and π3(S3), which describes lepton and baryon phases. To unifying these phases, we find common vacuum state which conserves only one component in two cases. Finally, we obtain the resulting 8-spinor model permitting unified description of baryon and lepton.

8-spinor, topological charge, solitons, homotopy groups, Brioshi identity, Skyrme-Faddeev model, Hopf index