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

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.

Keywords

• extended Dirac equation
• isospin
• Clifford algebra
• <i>SU</i>(<i>N</i>) symmetry