Gauge group of the standard model in $Cl_{1,5}$
Résumé
Describing a wave with spin 1/2, the Dirac equation is form invariant under a group which is not the Lorentz group. This $SL(2,\mathbb{C})$ group is a subgroup of $Cl_3^*=GL(2,\mathbb{C})$ which is the true group of form invariance of the Dirac equation. Firstly we use the $Cl_3$ algebra to read all features of the Dirac equation for a wave with spin 1/2. We extend this to electromagnetic laws. Next we use both the $Cl_3$ algebra and the space-time algebra to get the gauge group of the electro-weak interactions, first in the leptonic case, electron+neutrino, next in the quark case. The complete wave for all objects of the first generation uses two supplementary dimensions of space and the Clifford algebra $Cl_{1,5}$. It is a function of the usual space-time with value into this enlarged algebra. The gauge group is then enlarged into a $U(1)\times SU(2)\times SU(3)$ Lie group in a way which gives automatically the insensitivity of electrons and neutrinos to strong interactions. This study gives new insights for many features of the standard model. It explains also how to get three generations and four kinds of neutrinos. We encounter not only two remarkable identities, we are able to explain several enigmas, like the existence of the Planck constant or why the great unification based on $SU(5)$ could not be successful. We consolidate both the standard model and the use of Clifford algebras as the true mathematical frame of quantum physics. We present in concluding remarks a simple solution to integrate together gravitation and quantum physics. Then the only great domain of physics which remains to study is the electromagnetism with magnetic monopoles.
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