On hierarchy and equivalence of relativistic equations for massive fields

Vladimir V. Kassandrov1


A non-canonical correspondence of the complete sets of solutions to the Dirac and Klein-Gordon free equations in Minkowski space-time is established. This allows for a novel viewpoint on the relationship of relativistic equations for different spins and on the origin of spinor transformations. In particular, starting from a solution to the Dirac equation, one obtains a chain of other solutions to both Dirac and Klein-Gordon equations. A comparison with the massless case is performed, and examples of non-trivial singular solutions are presented. A generalization to Riemannian space-time and inclusion of interactions are also discussed.


  1. N.N. Bogolyubov and D.V. Shirkov, Quantum Fields (Nauka, Moscow, 1993, in Russian).
  2. A.I. Akhiezer and V.B. Berestetskii, Quantum Electrodynamics (John Wiley & Sons, New York, 1965).
  3. Yu.P. Rybakov, B. Saha and G.N. Shikin, Commun. Theor. Phys. 3, 199 (2004); V.A. Zhelnorovich, Zh. Eksp. Teor. Fiz. 125, 707 (2004), gr-qc/0010039.
  4. C. Armendriz-Picon and P.B. Greene, Gen. Rel. Grav. 35, 1637 (2003); hep-th/0301129.
  5. M.O. Ribas, F.P. Devecchi and G.M. Kremer, Phys. Rev. D 72 123502 (2005), gr-qc/0511099.
  6. J.R. Oppenheimer, Phys. Rev. 38, 725 (1931); E. Majorana, Nuovo Cim. 9, 335 (1932); M.S. Shneerson, Izvestiya VUZov, Fizika No. 9, 119 (1981); S. Esposito, Found. Phys. 28, 231 (1998).
  7. A.A. Campolattaro, Int. J. Theor. Phys. 29, 141; 477 (1990); J. Vaz and W.A. Rodriges Jr., Int. J. Theor. Phys. 32, 945 (1993); A. Gsponer, Int. J. Theor. Phys. 41, 689 (2002), math-ph/0201053.
  8. D.M. Gitman and A.L. Shelepin, Int. J. Theor. Phys. 40, 603 (2001), hep-th/0003146v2; V.V. Varlamov, Int. J. Theor. Phys. 42, 583 (2003), math-ph/0209036; 46, 741 (2007), math-ph/0503058v2.
  9. L.C. Biedenharn, H.W. Braden, P. Truini and H. vam Dam, J. Phys. A 21, 3593 (1988).
  10. V. Bargman and E.P. Wigner, Proc. Nat. Acad. USA 34, 211 (1948).
  11. Yu. B. Rumer and A. I. Fet, Group Theory and the Quantized Fields, (Nauka, Moscow, 1977, in Russian).
  12. R. Penrose and W. Rindler, Spinors and Space-Time. Vols. I, II (Cambridge Univ. Press, Cambridge, 1986).
  13. I. M. Gel'fand and A. M. Yaglom, Zh. Eksp. Teor. Fiz. 18, 703 (1948); I.M. Gel'fand, R.A. Minlos and Z.Ya. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon Press, Oxford, 1963).
  14. V.V. Kassandrov, Vestnik Univ. Druzhby Narodov, Fizika, 8 (1), 34 (2000); V.V. Kassandrov, in: Has the Last Word Been Said on Classical Electrodynamics?, ed. by A. Chubykalo et al. (Rinton Press, 2004), physics/0308045.
  15. V.V. Kassandrov, Algebraic Structure of Space-Time and the Algebrodynamics, Peopl. Fried. Univ. Press, 1992 (in Russian); V.V. Kassandrov, Grav. & Cosmol. 3 216 (1995); gr-qc/0007026.
  16. V.V. Kassandrov and J.A. Rizcallah, in: Recent Problems in Field Theory, ed. by A.V. Aminova (Kasan Univ. Press, Kasan, 1998), gr-qc/9809078.
  17. V.V. Kassandrov and J.A. Rizkallah, Twistor and "Weak" Gauge Structures in the Framework of Quaternionic Analysis, gr-qc/ 0012109; V.V. Kassandrov, in: Space-Time Structure. Algebra and Geometry, ed. by D.G. Pavlov et al. (Lilia-Print, Moscow, 2007); math-ph/0710.2895.
  18. V.V. Kassandrov, Grav. & Cosmol. 8 Suppl. 2, 57 (2002); math-ph/0311006.
  19. V.V. Kassandrov and V.N. Trishin, Gen. Rel. Grav. 36 1603, (2004); gr-qc/0411120.
For more information about this paper please visit Springer's Home Page of this paper.

Back to The Contents Page