Foundations of Physics

, Volume 16, Issue 8, pp 701–719 | Cite as

Relativistic algebraic spinors and quantum motions in phase space

  • P. R. Holland


Following suggestions of Schönberg and Bohm, we study the tensorial phase space representation of the Dirac and Feynman-Gell-Mann equations in terms of the complex Dirac algebra C4, a Jordan-Wigner algebra G4, and Wigner transformations. To do this we solve the problem of the conditions under which elements in C4 generate minimal ideals, and extend this to G4. This yields the linear theory of Dirac spin spaces and tensor representations of Dirac spinors, and the spin-1/2 wave equations are represented through fermionic state vectors in a higher space as a set of interconnected tensor relations.


Phase Space Wave Equation State Vector Linear Theory Space Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Schönberg,An. Acad. Bras. Cienc. 28, 11 (1956);29, 473 (1957);30, 1, 117, 259, 429 (1958);Suppl. Nuovo Cimento (X) 6, 356 (1957); inMax-Planck-Festchrift 1958 (veb Deutscher Verlag der Wissenschaften, Berlin).Google Scholar
  2. 2.
    D. Bohm and B. J. Hiley, inOld and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology, A. van der Merwe, ed. (Plenum, New York, 1983), p. 67.Google Scholar
  3. 3.
    E. Wigner,Phys. Rev. 40, 749 (1932).Google Scholar
  4. 4.
    D. Bohm and B. J. Hiley,Found. Phys. 11, 179 (1981).Google Scholar
  5. 5.
    R. P. Feynman and M. Gell-Mann,Phys. Rev. 109, 193 (1958).Google Scholar
  6. 6.
    Ph. Gueret, P. R. Holland, A. Kyprianidis, and J.-P. Vigier,Phys. Lett. 107A, 379 (1985).Google Scholar
  7. 7.
    D. Hesteness,Space-Time Algebra (Gordon and Breach, New York, 1966).Google Scholar
  8. 8.
    P. Lounesto,Found. Phys. 11, 721 (1981); K. R. Greider,Found. Phys. 14, 467 (1984).Google Scholar
  9. 9.
    N. Salingaros,J. Math. Phys. 23, 1 (1982).Google Scholar
  10. 10.
    M. Riesz, inComptes Rendus 12me Cong. Math. Scand. (Lund, 1953), p. 241.Google Scholar
  11. 11.
    P. R. Holland,J. Phys. A: Math. Gen. 16, 2363 (1983).Google Scholar
  12. 12.
    E. Kähler,Rend. Mat. Roma 21, 425 (1962).Google Scholar
  13. 13.
    W. Graf,Ann. Inst. H. Poincaré, A29, 85 (1978).Google Scholar
  14. 14.
    P. Becher and H. Joos,Z. Phys. C 15, 343 (1982).Google Scholar
  15. 15.
    I. M. Benn and R. W. Tucker,Commun. Math. Phys. 89, 341 (1983).Google Scholar
  16. 16.
    S. Teitler,J. Math. Phys. 7, 1730, 1739 (1966).Google Scholar
  17. 17.
    A. S. Eddington,Fundamental Theory (Cambridge University Press, Cambridge, 1946), p. 144.Google Scholar
  18. 18.
    T. Takabayasi,Prog. Theor. Phys. Suppl., No. 4 (1957).Google Scholar
  19. 19.
    L. de Broglie,Théorie Générale des Particules à Spin (Gauthier-Villars, Paris, 1954).Google Scholar
  20. 20.
    P. R. Holland, Ph. D. Thesis, University of London, 1981.Google Scholar
  21. 21.
    J. M. Rabin,Nucl. Phys. B 201, 315 (1982).Google Scholar
  22. 22.
    C. W. Misner and J. A. Wheeler,Ann. Phys. 22, 525 (1957).Google Scholar
  23. 23.
    D. Bohm,Phys. Rev. 85, 166 (1952).Google Scholar
  24. 24.
    L. de Broglie,Non-Linear Wave Mechanics (Elsevier, Amsterdam, 1960).Google Scholar
  25. 25.
    S. R. de Groot, W. A. van Leeuwen, and Ch. G. van Weert,Relativistic Kinetic Theory (North-Holland, Amsterdam, 1980).Google Scholar
  26. 26.
    P. R. Holland, A. Kyprianidis, Z. Marić, and J. P. Vigier,Phys. Rev. A 33, 4350 (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • P. R. Holland
    • 1
  1. 1.Laboratoire de Physique ThéoriqueInstitut Henri PoincaréParis Cedex 05France

Personalised recommendations