The Massless Limit of Bargmann–Wigner Equations for a Massive Graviton

  • A. V. Hradyskyi V.N. Karazin National University of Kharkiv
  • Yu. P. Stepanoskiy National Science Center Kharkiv Institute of Physics and Technology
Keywords: Bargman–Wigner equation, massive graviton, wave equations


Information about the discovery of gravity waves attract attention to the graviton’s mass problem. The massive graviton is a spin-2 particle with a non-zero mass. In this work, relativistic wave equations for a massive graviton have been studied in the limiting case of zero particle mass. The equations for the non-zero-mass graviton are based on the Bargman–Wigner equations in the five-dimensional space-time with the (++++−) signature. In the massless limit of massive graviton, all states with possible helicity values –0 (LL-graviton), ±1 (TL-graviton), and ±2 (TT-graviton) –are preserved.


  1. A.S. Goldhaber, M.M. Nieto. Photon and graviton mass limits. Rev. Mod. Phys. 82, 939 (2010).

  2. C. de Rham, J.T. Deskins, A.J. Tolley, S.-Y. Zhou. Gravi- ton mass bounds. Rev. Mod. Phys. 89, 025004 (2017).

  3. E.P.Wigner. Relativistic invariance and quantum phenomena. Rev. Mod. Phys. 29, 255 (1957).

  4. L. Landau, R. Peierls. Quantum electrodynamics in the configuration space. Zs. Phys. 62, 188 (1930).

  5. I. Bialynicki-Birula. Photon wave function. Progress in Optics 36, edited by E. Wolf (Elsevier, 1996), p. ????????? [arXiv: quant-ph/0508202].

  6. I. Bialynicki-Birula. Photon as a quantum particle. Acta Phys. Pol. B 37, 935 (2006).

  7. I. Bialynicki-Birula, Z. Bialynicka-Birula. The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism. J. Phys. A 46, 053001 (2013).

  8. Yu.P. Stepanovskyi. The little Lorentz group and wave equations of free massless fields with arbitrary spin. Ukr. J. Phys. 9, 1165 (1964).

  9. H.Weyl. Electron and gravitation. Z. Phys. 56, 330 (1929).

  10. M.P. Bronstein. Quantization of gravitational waves. J. Exper. Theor. Phys. 6, 195 (1936).

  11. Yu.P. Stepanovsky. On wave equations of massless fields. Teor. Matem. Fiz. 47, 343 (1981).

  12. Yu.P. Stepanovsky. From Maxwell equation to Berry's phase and sonoluminescence: Problems of theory of electromagnetic and other massless fields. Electromag. Phenom. 1, 180 (1998).

  13. Yu.P. Stepanovsky. On massless field and infinite component relativistic wave equations. Nucl. Phys. B. Proc. Suppl. 102 (1), 407 (2001).

  14. Yu.P. Stepanovsky. Ettore Majorana and Matvei Bronstein (1906–1938): Men and scientists. In: Advances in the Interplay between Quantum and Gravity Physics (Kluwer Academic Publishers, 2002), p. 435 [ISBN: 978-94-010-0347-6].

  15. Yu.P. Stepanovsky. ????? In: Problems in Contemporary Physics (KIPT, 2008), p. ?????????? (in Russian) [ISBN: 978-966-2136-15-9].

  16. B.L. Van der Waerden. Spinor analysis. Nachr. Ges. Wiss. G?ottingen, Math.-Phys. 100 (1929).

  17. O. Laporte, G.E. Uhlenbeck. Application of spinor analysis to the Maxwell and Dirac equations. Phys. Rev. 37, 1380 (1931).

  18. V.I. Ogievetskii, I.V. Polubarinov. Notoph and its possible interactions. Yader. Fiz. 4, 216 (1966).

  19. Yu.B. Rumer. Spinor Analysis (Librocom, 2010) (in Russian) [ISBN: 978-5-397-01381-9].

  20. Yu. B. Rumer, A.I. Fet. Group Theory and Quantum Fields (Nauka, 1977) (in Russian).

  21. R. Penrose, W. Rindler. Spinors and Space-Time (Cambrige Univ. Press, 1984) [ISBN: 0521337070].

  22. A. Proca. Wave theory of positive and negative electrons. J. Phys. Radium 7, 347 (1936).

  23. V.V. Dvoeglazov. Photon-notoph equations. Phys. Scripta 64, 201 (2001).

  24. V. Bargmann, E.P. Wigner. Group theoretical discussion of relativistic wave equations. Proc. Nat. Acad. Sci. USA 34, 211 (1948).

  25. L. Bass, E. Schr?odinger. Must the photon mass be zero? Proc. R. Soc. London A 232, 1 (1955).

  26. J.K. Luba’nski. Sur la theorie des particules elementaires de spin quelconque. I. Physica 9, 310 (1942).

  27. T. Kaluza. Zum unitatsproblem der physik. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) 966 (1921).

  28. O. Klein. Quantentheorie und f?unfdimensionale relativit?atstheorie. Zeit. Phys. 37, 895 (1926).

  29. E. Majorana. Atomi orientati in campo magnetico variabile. Nuovo Cimento 9, 43 (1932).

  30. R. Penrose. A spinor approach to general relativity. Ann. Phys. 10, 171 (1960).

  31. A.Z. Petrov. Clasification of spaces defining gravity fields. Uchen. Zapisk. Kazan. Gosud. Univer. 114, 55 (1954).

  32. Yu.P. Stepanovsky. Complete set of Fierz's relations in six-dimensional form. Ukr. J. Phys. 11, 1191 (1966).

  33. H. Weyl. Reine infinitesimalgeometrie. Mat. Zeit. 2, 384 (1918).
How to Cite
Hradyskyi, A., & Stepanoskiy, Y. (2018). The Massless Limit of Bargmann–Wigner Equations for a Massive Graviton. Ukrainian Journal of Physics, 63(7), 584.
Fields and elementary particles