Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of SU(N) Gauge Symmetry

Eckart Marsch,Yasuhito Narita

doi:10.3390/sym13030475

Eckart Marsch, Yasuhito Narita

Open Access

https://doi.org/10.3390/sym13030475

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Abstract

In this paper, we study possible mathematical connections of the Clifford algebra with the su(N)-Lie algebra, or in more physical terms the links between space-time symmetry (Lorentz invariance) and internal SU(N) gauge-symmetry for a massive spin one-half fermion described by the Dirac equation. The related matrix algebra is worked out in particular for the SU(2) symmetry and outlined as well for the color gauge group SU(3). Possible perspectives of this approach to unification of symmetries are briefly discussed. The calculations make extensive use of tensor multiplication of the matrices involved, whereby our focus is on revisiting the Coleman–Mandula theorem. This permits us to construct unified symmetries between Lorentz invariance and gauge symmetry in a direct product sense.

Highlights

Modern non-abelian gauge field theory started with the seminal paper by Yang andMills [1] in 1954, when they studied the conservation of isotopic spin and the associatedSU (2) gauge invariance
Their emphasis was on the gauge field equations, which resembled those of the electromagnetic field yet revealed new non-linear couplings due to the algebraic properties of the symmetry group involved
We will start from scratch concerning the fermion sector of the Standard Model (SM) and determine various links of the general SU ( N ) symmetry with the properties of the fermion Clifford algebra, as it is induced by the Lorentz transformation in Minkowski space-time

Summary

Introduction

Modern non-abelian gauge field theory started with the seminal paper by Yang and. Mills [1] in 1954, when they studied the conservation of isotopic spin and the associated. This subject was investigated in connection with the idea to assemble the proton and neutron in a doublet to describe the nuclear force Their emphasis was on the gauge field equations, which resembled those of the electromagnetic field yet revealed new non-linear couplings due to the algebraic properties of the symmetry group involved. An explicit example is given in a subsequent section illustrating the mathematical approach by which the symmetry group SU (2) can be incorporated in the standard Dirac equation in the Weyl basis They key point is that, instead of rotating the multiplet of Dirac spinors under SU ( N ), one can well tensor-multiply the gamma matrices from the right side by the related unit matrix 1N to accommodate the associated N-fold multiplet, expanding the Dirac spinor to a 4N-component spinor. Lorentz invariance and gauge symmetry in a direct product sense in compliance with the Coleman–Mandula theorem [5]

The Standard Dirac Equation in the Weyl Basis

Partial Unification of Symmetries

Summary and Conclusions