Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 279–290 | Cite as

Gauge Group of the Standard Model in Cl 1,5

Open Access


Describing a wave with spin 1/2, the Dirac equation is form invariant under \({SL(2,\mathbb{C})}\), 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 get the gauge group of 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 the Clifford algebra Cl 1,5. The gauge group is then enlarged into a \({U(1)\times SU(2)\times SU(3)}\) Lie group. We consolidate both the standard model and the use of Clifford algebras, true mathematical frame of quantum physics.

Mathematics Subject Classification

15A66 35Q41 81T13 83E15 


Invariance group Dirac equation Electromagnetism Weak interactions Strong interactions Standard model Clifford algebra 


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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Le Moulin de la LandePouillé-les-coteauxFrance

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