Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 333–344 | Cite as

From Clifford Algebra of Nonrelativistic Phase Space to Quarks and Leptons of the Standard Model

Open Access


We review a recently proposed Clifford-algebra approach to elementary particles. We start with: (1) a philosophical background that motivates a maximally symmetric treatment of position and momentum variables, and: (2) an analysis of the minimal conceptual assumptions needed in quark mass extraction procedures. With these points in mind, a variation on Born’s reciprocity argument provides us with an unorthodox view on the problem of mass. The idea of space quantization suggests then the linearization of the nonrelativistic quadratic form \({{\mathbf{p}}^2 +{\mathbf{x}}^2}\) with position and momentum satisfying standard commutation relations. This leads to the 64-dimensional Clifford algebra \({{Cl}_{6,0}}\) of nonrelativistic phase space within which one identifies the internal quantum numbers of a single Standard Model generation of elementary particles (i.e. weak isospin, hypercharge, and color). The relevant quantum numbers are naturally linked to the symmetries of macroscopic phase space. It is shown that the obtained phase-space-based description of elementary particles gives a subquark-less explanation of the celebrated Harari–Shupe rishon model. Finally, the concept of additivity is used to form novel suggestions as to how hadrons are constructed out of quarks and how macroscopically motivated invariances may be restored at the hadron level.


Clifford algebra Phase space Elementary particles Harari–Shupe model 

Mathematics Subject Classification

Primary 81R05 Secondary 81P99 Tertiary 15A66 


  1. 1.
    Żenczykowski P.: Space, phase space and quantum numbers of elementary particles. Acta Phys. Pol. B 38, 2053 (2007)ADSMathSciNetGoogle Scholar
  2. 2.
    Żenczykowski P.: The Harari–Shupe preon model and nonrelativistic quantum phase space. Phys. Lett. B 660, 567 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Żenczykowski P.: Clifford algebra of nonrelativistic phase space and the concept of mass. J. Phys. A 42, 045204 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Żenczykowski P.: Leptons, quarks, and their antiparticles: a phase-space view. Int. J. Theor. Phys. 49, 2246 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Żenczykowski P.: Elementary Particles and Emergent Phase Space. World Scientific, Singapore (2014)MATHGoogle Scholar
  6. 6.
    Barbour, J.: The nature of time. Lect. Notes Phys. 633, 15 (2003). arXiv:0903.3489
  7. 7.
    Whitehead A.N.: Science and the Modern World. Cambridge Univ. Press, Cambridge (1926)MATHGoogle Scholar
  8. 8.
    Whitehead A.N.: Process and Reality. Free Press, NY (1979)Google Scholar
  9. 9.
    Born M.: Reciprocity theory of elementary particles. Rev. Mod. Phys. 21, 463 (1949)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Wigner E.P.: Relativistic invariance and quantum phenomena. Rev. Mod. Phys. 29, 255 (1957)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gell-Mann M., Oakes R.J., Renner B.: Behavior of current divergences under SU(3)\({\times}\)SU(3). Phys. Rev. Lett. 175, 2195 (1968)ADSGoogle Scholar
  12. 12.
    Beringer, J. et al. (Particle Data Group): Review of particle physics. Phys. Rev. D 86, 010001 (2012)Google Scholar
  13. 13.
    Capstick S., Roberts W.: Quark models of baryon masses and decays. Progr. Theor. Part. Nucl. Phys. 45, S241 (2000)ADSCrossRefGoogle Scholar
  14. 14.
    Harari H.: A schematic model of quarks and leptons. Phys. Lett. B 86, 83 (1979)ADSCrossRefGoogle Scholar
  15. 15.
    Shupe M.: A composite model of leptons and quarks. Phys. Lett. B 86, 87 (1979)ADSCrossRefGoogle Scholar
  16. 16.
    Żenczykowski, P.: J. Phys. Conf. Ser. (2015). (to appear) Google Scholar

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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.Institute of Nuclear PhysicsKrakówPoland

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