{C ell{^ast_3}} Invariance of the Dirac Equation and Electromagnetism

In order to obtain a form invariance of the Dirac wave, the relativistic quantum theory uses ${{rm SL}(2, mathbb{C})}$ , a subset of the Clifford algebra C? ₃ of the 3-dimensional physical space. In Dirac theory and electromagnetism, we study a replacement of this invariance group by the full group ${C ell{^ast_3}}$ consisting of all invertible elements in C? ₃. We extend this invariance to systems of identical particles. We explain how geometry linked to the Dirac wave implies existence of a second space-time variety, and we review its consequences to particle physics.