Dirac equation based on the vector representation of the Lorentz group

Eckart Marsch,Yasuhito Narita

doi:10.1140/epjp/s13360-020-00798-7

Eckart Marsch, Yasuhito Narita

Open Access

https://doi.org/10.1140/epjp/s13360-020-00798-7

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Abstract

In this paper, we derive an expanded Dirac equation for a massive fermion doublet, which has in addition to the particle/antiparticle and spin-up/spin-down degrees of freedom explicity an isospin-type degree of freedom. We begin with revisiting the four-vector Lorentz group generators, define the corresponding gamma matrices and then write a Dirac equation for the fermion doublet with eight spinor components. The appropriate Lagrangian density is established, and the related chiral and SU(2) symmetry is discussed in detail, as well as applications to an electroweak-style gauge theory. In “Appendix,” we present some of the relevant matrices.

Highlights

The goal of this study is to reconsider an old subject, namely the Lorentz group and its representations [1,2,3,4], yet here with emphasis on the four-vector generators of the Lorentz group
The new result will be an extended Dirac equation [5] for a massive fermion doublet which has another intrinsic degree of freedom related to SU (2) symmetry
Extensive use was made of differential forms and calculus and of the related advanced mathematics which even today is not a standard tool of theoretical physics or modern quantum field theory, as it is presented in text books [9,10,11] on the standard model (SM) of elementary particle physics

Summary

Introduction

The goal of this study is to reconsider an old subject, namely the Lorentz group and its representations [1,2,3,4], yet here with emphasis on the four-vector generators of the Lorentz group. The new result will be an extended Dirac equation [5] for a massive fermion doublet which has another intrinsic degree of freedom related to SU (2) symmetry. The outline of our paper is as follows: We start with revisiting the Lorentz transformation in Minkowski space and consider the related generators of the Lorentz group. These generators obey the Lorentz algebra, which in its simplest representation yields the Dirac equation. We will instead consider the four-vector generators of the Lorentz group They act instead of Pauli spinors on complex four-component vectors which describe two more degrees of freedom corresponding to an isospin doublet. Some relevant matrices are presented in “Appendix.” we present our conclusions

The four-vector Lorentz group generators

Dirac equation for an eight-component spinorial wave function

The Lagrangian density in the Weyl basis

Discussion and conclusion