Lorentz invariance and the spinor-helicity formalism yield the U(1) and SU(3) fermion symmetry

Abstract

AbstractIn this paper we use the so-called spinor-helicity formalism to represent three-vectors in terms of the Pauli matrices and derive a generalized relativistic wave equation for a massive fermion of spin one-half. We thus extend the Dirac equation by making use of the Pauli-Lubański operator that includes isospin explicitly. As a consequence, we get new degrees of freedom related to isospin helicity, in addition to the two standard ones of the Dirac equation that are associated with the kinetic spin-helicity doublet and the particle-antiparticle pair. Formally, isospin helicity has $$2(2s+1)$$ 2 ( 2 s + 1 ) degrees of freedom for an arbitrary general isospin s and has the eigenvalues s and $$-(s+1)$$ - ( s + 1 ) , and thus it reveals a kind of hidden symmetry in any isospin field. The resulting four degrees of freedom for isospin 1/2 are interpreted as being associated with two independent subspaces of dimension 1 related to the U(1) and 3 related to SU(3) symmetry, i.e. to the leptons and quarks.