Advances in Applied Clifford Algebras

, Volume 15, Issue 2, pp 271–290 | Cite as

Algebra of matter

  • Bernd Schmeikal
Original Paper


Here it is shown that the known forces of nature unfold in parallel with an exact decomposition of the geometric algebra Cl3,1 of spacetime. Up to an important common scalar this decomposition is a partition into a positive definite commutative graded space of strong forces and two negative non-commutative spaces with a quaternion structure for weak and other fields. The 6 fundamental spaces of strong forces are acted on by a rank 2 Lie algebra ℒ = sl Cl (2, ℝ) × so Cl (3, ℝ) of dimension 8 which brings in an isotropy group of the neutrinos, the flavour and colour symmetries and an isotropy of the weaker forces. The standard model, in an improved form, is a feature of the Clifford algebra of spacetime, and relativity as Lorentz invariance reduced to dimension (2, 1) is compatible with quantum theory. The whole Lorentz group cannot be, however, a property of physical motion. Due to the total exploitation of the whole geometric space and its convincing logical structure, the author believes there is at present no better algebraic model for the totality of known symmetries of physical dynamics.


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Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  • Bernd Schmeikal
    • 1
  1. 1.GarstenAustria

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