Communications in Mathematical Physics

, Volume 6, Issue 4, pp 286–311 | Cite as

Nonrelativistic particles and wave equations

  • Jean-Marc Lévy-Leblond


This paper is devoted to a detailed study of nonrelativistic particles and their properties, as described by Galilei invariant wave equations, in order to obtain a precise distinction between the specifically relativistic properties of elementary quantum mechanical systems and those which are also shared by nonrelativistic systems. After having emphasized that spin, for instance, is not such a specifically relativistic effect, we construct wave equations for nonrelativistic particles with any spin. Our derivation is based upon the theory of representations of the Galilei group, which define nonrelativistic particles. We particularly study the spin 1/2 case where we introduce a four-component wave equation, the nonrelativistic analogue of the Dirac equation. It leads to the conclusion that the spin magnetic moment, with its Landé factorg=2, is not a relativistic property. More generally, nonrelativistic particles seem to possess intrinsic moments with the same values as their relativistic counterparts, but are found to possess no higher electromagnetic multipole moments. Studying “galilean electromagnetism” (i.e. the theory of spin 1 massless particles), we show that only the displacement current is responsible for the breakdown of galilean invariance in Maxwell equations, and we make some comments about such a “nonrelativistic electromagnetism”. Comparing the connection between wave equations and the invariance group in both the relativistic and the nonrelativistic case, we are finally led to some vexing questions about the very concept of wave equations.


Wave Equation Dirac Equation Relativistic Property Massless Particle Multipole Moment 
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  1. 1.
    Lévy-Leblond, J.-M.: Thèse de Doctorat, Orsay, 1965.Google Scholar
  2. 2.
    Wigner, E. P.: Ann. Math.40, 149 (1939).Google Scholar
  3. 3.
    Jauch, J. M.: Helv. Phys. Acta37, 284 (1964).Google Scholar
  4. 4.
    Bargmann, V.: Ann. Math.59, 1 (1954).Google Scholar
  5. 5.
    Inönu, E., andE. P. Wigner: Nuovo Cimento9, 705 (1952).Google Scholar
  6. 6.
    Hamermesh, H.: Ann. Phys. (N.Y.)9, 518 (1960).Google Scholar
  7. 7.
    Lévy-Leblond, J.-M.: J. Math. Phys.4, 776 (1963).Google Scholar
  8. 8.
    —— Commun. Math. Phys.4, 157 (1967).Google Scholar
  9. 9.
    Newton, T. D., andE. P. Wigner: Rev. Mod. Phys.21, 400 (1956)Google Scholar
  10. 9a.
    Wightman, A. S.: Rev. Mod. Phys.34, 845 (1962).Google Scholar
  11. 10.
    Dirac, P. A. M.: Proc. Roy. Soc. A117, 610 (1928).Google Scholar
  12. 11.
    Pauli, W.: Z. Physik43, 601 (1927).Google Scholar
  13. 12.
    Galindo, A., andC. Sanchez del Rio: Am. J. Phys.29, 582 (1961).Google Scholar
  14. 13.
    Eberlein, W. F.: Am. Math. Monthly69, 587 (1962).Google Scholar
  15. 14.
    Pursey, D. L.: Ann. Phys. (N.Y.)32, 157 (1965).Google Scholar
  16. 15.
    Gel'fand, I. M., R. A. Minlos, andZ. Y. Shapiro: Representations of the rotation and Lorentz groups. Oxford: Pergamon Press 1963.Google Scholar
  17. 16.
    Georges, C., andM. Lévy-Nahas: J. Math. Phys.7, 980 (1966).Google Scholar
  18. 17.
    Inönu, E., andE. P. Wigner: Proc. Nat. Acad. Sci.39, 510 (1953);40, 119 (1954).Google Scholar
  19. 17a.
    Saletan, E.: J. Math. Phys.2, 1 (1961).Google Scholar
  20. 18.
    Bargmann, V., andE. P. Wigner: Proc. Nat. Acad. Sci.34, 211 (1948).Google Scholar
  21. 19.
    Moldauer, P. A., andK. M. Case: Phys. Rev.102, 279 (1956).Google Scholar
  22. 20.
    Proca, A.: C. R. Acad. Sci.202, 1420 (1936); J. Phys. Radium9, 61 (1938).Google Scholar
  23. 21.
    Tumanov, V. S.: Soviet Physics JETP19, 1182 (1964).Google Scholar
  24. 22.
    Lévy-Leblond, J.-M.: Communication at the Second Rochester Conference on Coherence and Quantum Optics. Rochester: 1966.Google Scholar
  25. 23.
    Kibble, T. W.: Phys. Rev.150, 1060 (1966).Google Scholar
  26. 24.
    Havas, P.: Rev. Mod. Phys.36, 938 (1964).Google Scholar

Copyright information

© Springer-Verlag 1967

Authors and Affiliations

  • Jean-Marc Lévy-Leblond
    • 1
  1. 1.Laboratoire de Physique ThéoriqueFaculté des SciencesNiceFrance

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