Quaternion symmetry of the Dirac equation

  • T. Saue
  • H. J. Aa. Jensen
Part of the Lecture Notes in Chemistry book series (LNC, volume 74)


Following van der Waerden, the Dirac equation is derived from linearization of the Klein-Gordon equation using the algebraic properties of the Pauli spin matrices. As the algebra of these matrices is identical to that of quaternions, the Dirac equation can be reformulated in terms of quaternion algebra and therefore without reference to a specific spin quantization axis. In this paper we consider the symmetry content of the Dirac equation. It is found that the basic binary symmetry operations in spin space map onto the unit vectors of complex quaternions. We argue that a consistent choice of the inversion operator in spin space is of order four. We furthermore show that quaternion algebra is the natural language for time reversal symmetry. These considerations lead to the formulation of a symmetry scheme that automatically provides maximum point group and time reversal symmetry reduction in the solution of the Dirac equation in the finite basis approximation.


Dirac Equation Dirac Operator Symmetry Operation Spin Space Quaternion Algebra 
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]
    B. N. Kursunoglu and E. P. Wigner, editors. Reminiscences about a great physicist: Paul Adrien Maurice Dimc. Cambridge University Press, 1987.Google Scholar
  2. [2]
    F. G. Frobenius. Grelle84:59, 1878.Google Scholar
  3. [3]
    C. S. Peirce. Amer.Jour.Math., 4:225, 1881.Google Scholar
  4. [4]
    S. L. Altmann. Rotations, Quatemions, and Double Groups. Clarendon Press, Oxford, 1986.Google Scholar
  5. [5]
    G. Aucar, T. Saue, H. LÃa. Jensen, and L. Visscher. J.Ghem.Phys., 110:6208–6218, 1999.CrossRefGoogle Scholar
  6. [6]
    B. L. van der Waerden. Group Theory and Quantum Mechanics. Springer, Berlin, 1974.(Translation of the German Original Edition: Die Grundlehren der mathematischen Wissenschaften Band 37, Die Gruppentheoretische Methode in der Quantenmechanik. Publisher: Verlag von Julius Springer, Berlin 1932).CrossRefGoogle Scholar
  7. [7]
    H. Weyl. Z.Phys., 56:330–352, 1929.CrossRefGoogle Scholar
  8. [8]
    C. S. Wu, E. Ambler, W. Hayward, D. D. Hoppes, and R. P. Hudson. Phys.Rev., 1413:1413–1414,1957.CrossRefGoogle Scholar
  9. [9]
    E. P. Wigner. Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New York, 1959.Google Scholar
  10. [10]
    M. Tinkham. Group theory and Quantum Mechanics. McGraw-Hill, New York, 1964.Google Scholar
  11. [11]
    M. Lax. Symmetry Principles in Solid State and Molecular Physics. Wiley and Sons, New York, 1974.Google Scholar
  12. [12]
    T. Saue and H. J. Aa Jensen. J.Ghem.Phys., 111:6211–6222, 1999.CrossRefGoogle Scholar
  13. [13]
    H. Bethe. Ann.Phys., 3:133–208, 1929.CrossRefGoogle Scholar
  14. [14]
    J. G. Snijders. Relativity and pseudopotentials in the HFS method. PhD thesisVrije Universiteit, Amsterdam, 1979.Google Scholar
  15. [15]
    S. A. Werner. Phys.Rev.Lett., 35:1053–1055, 1975.CrossRefGoogle Scholar
  16. [16]
    H. Rauch, A. Zeilinger, G. Badurek, A. Wilfing, W. Bauspiess, and U. Bonse. Phys.Lett., 54:425–427, 1975.CrossRefGoogle Scholar
  17. [17]
    A. G. Klein and G.l. Opat. Phys.Rev.Lett., 37:238–240, 1976.CrossRefGoogle Scholar
  18. [18]
    M. E. Stoll. Phys.Rev. A16:1521–1524, 1977.CrossRefGoogle Scholar
  19. [19]
    C. P. Poole and H.A. Farach. Found.Phys., 12:719–738, 1982.CrossRefGoogle Scholar
  20. [20]
    G. Artken. Mathematical Methods for Physicists. Academic Press, San Diego, 1985.Google Scholar
  21. [21]
    S. L. Altmann and P. Herzig. Mol.Phys., 45:585–604, 1982.CrossRefGoogle Scholar
  22. [22]
    E. Wigner. Nachrichten der Akad. der Wissensch. zu Göttingen,II, pages 546–559, 1932.Google Scholar
  23. [23]
    H. J. Aa Jensen, KG. Dyall, T. Saue, and K Fægri. J.Chem.Phys., 104:4083–4097, 1996.CrossRefGoogle Scholar
  24. [24]
    Max Jammer. The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York, 1966.Google Scholar
  25. [25]
    G. M. Dixon. Division Algebras. Kluwer Academic, 1994.Google Scholar
  26. [26]
    Footnote 2 on page 607 in [38].Google Scholar
  27. [27]
    L. E. Dickson. Linear Algebras. Cambridge University Press, London, 1914.Google Scholar
  28. [28]
    C. Lanczos. Z.Phys57:474–483, 1929.CrossRefGoogle Scholar
  29. [29]
    C. Lanczos. Z.Phys57:484–493, 1929.CrossRefGoogle Scholar
  30. [30]
    A. W. Conway. Proc. Roy. Soc. (London) A162:145–154, 1937.CrossRefGoogle Scholar
  31. [31]
    P. Rastall. Rev. Mod.Phys., 36:820–832, 1964.CrossRefGoogle Scholar
  32. [32]
    K. Morita. Prog. Theor.Phys., 70:1648–1665, 1983.CrossRefGoogle Scholar
  33. [33]
    A. D. McLean and Y.S. Lee. In R. Carbo, editor, Current Aspects of Quantum Chemistry 1981, volume 21, pages 219–238, Amsterdam, 1982. Elsevier.Google Scholar
  34. [34]
    R. E. Stanton and S. Havriliak. J.Chem.Phys., 81:1910–1918, 1984.CrossRefGoogle Scholar
  35. [35]
    F. J. Dyson. J.Math. Phys., 3:1199–1215, 1962.CrossRefGoogle Scholar
  36. [36]
    Dirac, a relativistic ab initio electronic structure program, Release 3.1 (1998), written by T. Saue, T. Enevoldsen, T. Helgaker, H.J. Aa. Jensen, J. Laerdahl, K Ruud, J. Thyssen, and L. Visscher. See Scholar
  37. [37]
    T. Saue, K. Fægri, T. Helgaker, and O. Gropen. Mol.Phys., 91, 1997.Google Scholar
  38. [38]
    W. Pauli. Z.Phys., 43:601–623, 1927.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • T. Saue
    • 1
  • H. J. Aa. Jensen
    • 2
  1. 1.The Institute of ChemistryUniversity of TromsøTromsøNorway
  2. 2.Department of ChemistryUniversity of Southern Denmark - Main campus: Odense UniversityOdense MDenmark

Personalised recommendations