Field Theory of the Spinning Electron: II — The New Non-Linear Field Equations

  • Erasmo Recami
  • Giovanni Salesi
Part of the NATO ASI Series book series (NSSB, volume 358)


One of the most satisfactory picture of spinning particles is the Barut-Zanghi (BZ) classical theory for the relativistic electron, that relates the electron spin to the so-called zitterbewegung (zbw). The BZ motion equations constituted the starting point for two recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. Here, employing on the contrary the tensorial language, more common in the (first quantization) field theories, we “quantize” the BZ theory and derive for the electron field a non-linear Dirac equation (NDE), of which the ordinary Dirac equation represents a particular case.

We then find out the general solution of the NDE. Our NDE does imply a new probability current J μ , that is shown to be a conserved quantity, endowed (in the center-of-mass frame) with the zbw frequency ω = 2m, where m is the electron mass. Because of the conservation of J μ , we are able to adopt the ordinary probabilistic interpretation for the fields entering the NDE.

At last we propose a natural generalization of our approach, for the case in which an external electromagnetic potential A μ is present; it happens to be based on a new system of five first-order differential field equations.


General Solution Electron Spin Dirac Equation Classical Analogue Spinorial Field 
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]
    A.H. Compton: Phys. Rev. 14 (1919) 20, 247, and refs. therein. See also W.H. Bostick: “Hydromagnetic model of an elementary particle”, in Gravity Res. Found. Essay Contest (1958 and 1961).CrossRefGoogle Scholar
  2. J. Frenkel: Z. Phys. 37 (1926) 243.zbMATHCrossRefGoogle Scholar
  3. M. Mathisson: Acta Phys. Pol. 6 (1937) 163.Google Scholar
  4. H. Hönl and A. Papapetrou: Z. Phys. 112 (1939) 512.CrossRefGoogle Scholar
  5. H. Hönl and A. Papapetrou: Z. Phys. 116 (1940) 153.CrossRefGoogle Scholar
  6. M.J. Bhabha and H.C. Corben: Proc. Roy. Soc. (London) A178 (1941) 273.MathSciNetGoogle Scholar
  7. K. Huang: Am. J. Phys. 20 (1952) 479.zbMATHCrossRefGoogle Scholar
  8. H. Hönl: Ergeb. Exacten Naturwiss. 26 (1952) 29.Google Scholar
  9. A. Proca: J. Phys. Radium 15 (1954) 5.MathSciNetCrossRefGoogle Scholar
  10. M. Bunge: Nuovo Cimento 1 (1955) 977.CrossRefGoogle Scholar
  11. F. Gursey: Nuovo Cimento 5 (1957) 784.MathSciNetCrossRefGoogle Scholar
  12. B. Liebowitz: Nuovo Cimento A63 (1969) 1235.Google Scholar
  13. H. Jehle: Phys. Rev. D3 (1971) 306.Google Scholar
  14. F. Riewe: Lett. Nuovo Cim. 1 (1971) 807.CrossRefGoogle Scholar
  15. G.A. Perkins: Found. Phys. 6 (1976) 237.CrossRefGoogle Scholar
  16. D. Gutkowski, M. Moles and J.P. Vigier: Nuovo Cim. B39 (1977) 193.CrossRefGoogle Scholar
  17. A.O. Barut: Z. Naturforsch. A33 (1978) 993.Google Scholar
  18. J.A. Lock: Am. J. Phys. 47 (1979) 797.MathSciNetCrossRefGoogle Scholar
  19. M.H. McGregor: The enigmatic electron (Kluwer; Dordrecht, 1992).Google Scholar
  20. W.A. Rodrigues, J. Vaz and E. Recami: Found. Phys. 23 (1993) 459.MathSciNetGoogle Scholar
  21. [2]
    E. Schrödinger: Sitzunger. Preuss. Akad. Wiss. Phys. Math. Kl. 24 (1930) 418.Google Scholar
  22. See also P.A.M. Dirac: The principles of quantum mechanics (Claredon; Oxford, 1958), 4th edition, p. 262.zbMATHGoogle Scholar
  23. J. Maddox: “Where Zitterbewegung may lead”, Nature 325 (1987) 306.Google Scholar
  24. [3]
    H.C. Corben: Phys. Rev. 121 (1961) 1833.zbMATHCrossRefGoogle Scholar
  25. H.C. Corben: Nuovo Cim. 20 (1961) 529.MathSciNetzbMATHCrossRefGoogle Scholar
  26. H.C. Corben: Phys. Rev. D30 (1984) 2683.MathSciNetGoogle Scholar
  27. H.C. Corben: Am. J. Phys. 61 (1993) 551.CrossRefGoogle Scholar
  28. “Primitive quantization of Zitterbewegung”, preprint (June 1994).Google Scholar
  29. [4]
    F.A. Berezin and M.S. Marinov: J.E.T.P. Lett. 21 (1975) 320.Google Scholar
  30. R. Casalbuoni: Nuovo Cimento A33 (1976) 389.MathSciNetGoogle Scholar
  31. [5]
    W.A. Rodrigues Jr., J. Vaz and E. Recami: Found. Phys. 23 (1993) 469.MathSciNetCrossRefGoogle Scholar
  32. [6]
    A.O. Barut and N. Zanghi: Phys. Rev. Lett. 52 (1984) 2009.MathSciNetCrossRefGoogle Scholar
  33. See also A.O. Barut and A.J. Bracken: Phys. Rev. D23 (1981) 2454.MathSciNetGoogle Scholar
  34. See also A.O. Barut and A.J. Bracken: Phys. Rev. D24 (1981) 3333.MathSciNetGoogle Scholar
  35. A.O. Barut and M. Pavsic: Class. Quantum Grav. 4 (1987) L131.MathSciNetCrossRefGoogle Scholar
  36. A.O. Barut: Phys. Lett. B237 (1990) 436.Google Scholar
  37. [7]
    G. Salesi and E. Recami: Phys. Lett. A190 (1994) 137.MathSciNetGoogle Scholar
  38. G. Salesi and E. Recami: Phys. Lett. A195 (1994) E389.MathSciNetGoogle Scholar
  39. E. Recami and G. Salesi: “Field theory of the electron: Spin and Zitterbewegung”, in Particles, Gravity and Space-Time, ed. by P.I. Pronin and G.A. Sardanashvily (World Scient.; Singapore, 1996), pp.345–368.CrossRefGoogle Scholar
  40. [8]
    M. Pavsic, E. Recami, W.A. Rodrigues, G.D. Maccarrone, F. Raciti and G. Salesi: Phys. Lett. B318 (1993) 481.MathSciNetGoogle Scholar
  41. [9]
    M. Pavsic: Phys. Lett. B205 (1988) 231.MathSciNetGoogle Scholar
  42. M. Pavsic: Phys. Lett. B221 (1989) 264.MathSciNetGoogle Scholar
  43. M. Pavsic: Class. Quant. Grav. 7 (1990) L187.MathSciNetCrossRefGoogle Scholar
  44. [10]
    A.O. Barut and M. Pavsic: Phys. Lett. B216 (1989) 297.MathSciNetGoogle Scholar
  45. F.A. Ikemori: Phys. Lett. B199 (1987) 239.MathSciNetGoogle Scholar
  46. See also D. Hestenes: Found. Phys. 20 (1990) 1213.MathSciNetCrossRefGoogle Scholar
  47. S. Gull, A. Lasenby and C. Doran: “Electron paths, tunneling and diffraction in the space-time algebra”, to appear in Found. Phys. (1993).Google Scholar
  48. D. Hestenes and A. Weingartshofer (eds.): The electron (Kluwer; Dordrecht, 1991).Google Scholar
  49. in particular the contributions by H. Krüger, by R. Boudet, and by S. Gull; A. Campolattaro: Int. J. Theor. Phys. 29 (1990) 141.MathSciNetzbMATHCrossRefGoogle Scholar
  50. D. Hestenes: Found. Phys. 15 (1985) 63.MathSciNetCrossRefGoogle Scholar
  51. [11]
    W.A. Rodrigues Jr., J. Vaz, E. Recami and G. Salesi: Phys. Lett. B318 (1993) 623.MathSciNetGoogle Scholar
  52. [12]
    V.I. Fushchich and R.Z. Zhdanov: Sov. J. Part. Nucl. 19 (1988) 498.MathSciNetGoogle Scholar
  53. [13]
    For the physical interpretation of the negative frequency waves, without any recourse to a “Dirac sea”, see e.g. E. Recami: Found. Phys. 8 (1978) 329.CrossRefGoogle Scholar
  54. E. Recami and W.A. Rodrigues: Found. Phys. 12 (1982) 709.MathSciNetCrossRefGoogle Scholar
  55. E. Recami and W.A. Rodrigues: Found. Phys. 13 (1983) 533.MathSciNetCrossRefGoogle Scholar
  56. M. Pavsic and E. Recami: Lett. Nuovo Cim. 34 (1982) 357.MathSciNetCrossRefGoogle Scholar
  57. See also R. Mignani and E. Recami: Lett. Nuovo Cim. 18 (1977) 5.CrossRefGoogle Scholar
  58. A. Garuccio et al: Lett. Nuovo Cim. 27 (1980) 60.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Erasmo Recami
    • 1
    • 2
    • 3
  • Giovanni Salesi
    • 4
  1. 1.Facoltà di IngegneriaUniversità Statale di BergamoDalmine (BG)Italy
  2. 2.Sezione di MilanoINFNMilanItaly
  3. 3.Dept. of Applied Math.State University at CampinasCampinasBrazil
  4. 4.Dipart. di FisicaUniversità Statale di CataniaCataniaItaly

Personalised recommendations