The European Physical Journal Plus | VOL. 137
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Lorentz invariance and the spinor-helicity formalism yield the U(1) and SU(3) fermion symmetry

Publication Date Jul 15, 2022

Abstract

In 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-Luba\u0144ski 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) degrees of freedom for an arbitrary general isospin s and has the eigenvalues s and -(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.

Concepts

Spinor-helicity Formalism Particle-antiparticle Pair Fermion Symmetry Standard Ones Lorentz Invariance Dirac Equation Arbitrary Isospin Degrees Of Freedom Relativistic Equation Relativistic Wave Equation

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