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

Abstract

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.

Keywords Bargman–Wigner equation, massive graviton, wave equations

References


  1. A.S. Goldhaber, M.M. Nieto. Photon and graviton mass limits. Rev. Mod. Phys. 82, 939 (2010).
    https://doi.org/10.1103/RevModPhys.82.939

  2. C. de Rham, J.T. Deskins, A.J. Tolley, S.-Y. Zhou. Gravi- ton mass bounds. Rev. Mod. Phys. 89, 025004 (2017).
    https://doi.org/10.1103/RevModPhys.89.025004

  3. E.P.Wigner. Relativistic invariance and quantum phenomena. Rev. Mod. Phys. 29, 255 (1957).
    https://doi.org/10.1103/RevModPhys.29.255

  4. L. Landau, R. Peierls. Quantum electrodynamics in the configuration space. Zs. Phys. 62, 188 (1930).
    https://doi.org/10.1007/BF01339793

  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).
    https://doi.org/10.1088/1751-8113/46/5/053001

  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).
    https://doi.org/10.1007/BF01339504

  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).
    https://doi.org/10.1007/BF01019301

  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).
    https://doi.org/10.1016/S0920-5632(01)01587-0

  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].
    https://doi.org/10.1007/978-94-010-0347-6_18

  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).
    https://doi.org/10.1103/PhysRev.37.1380

  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].
    https://doi.org/10.1017/CBO9780511564048

  22. A. Proca. Wave theory of positive and negative electrons. J. Phys. Radium 7, 347 (1936).
    https://doi.org/10.1051/jphysrad:0193600708034700

  23. V.V. Dvoeglazov. Photon-notoph equations. Phys. Scripta 64, 201 (2001).
    https://doi.org/10.1238/Physica.Regular.064a00201

  24. V. Bargmann, E.P. Wigner. Group theoretical discussion of relativistic wave equations. Proc. Nat. Acad. Sci. USA 34, 211 (1948).
    https://doi.org/10.1073/pnas.34.5.211

  25. L. Bass, E. Schr?odinger. Must the photon mass be zero? Proc. R. Soc. London A 232, 1 (1955).
    https://doi.org/10.1098/rspa.1955.0197

  26. J.K. Luba’nski. Sur la theorie des particules elementaires de spin quelconque. I. Physica 9, 310 (1942).
    https://doi.org/10.1016/S0031-8914(42)90113-7

  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).
    https://doi.org/10.1007/BF01397481

  29. E. Majorana. Atomi orientati in campo magnetico variabile. Nuovo Cimento 9, 43 (1932).
    https://doi.org/10.1007/BF02960953

  30. R. Penrose. A spinor approach to general relativity. Ann. Phys. 10, 171 (1960).
    https://doi.org/10.1016/0003-4916(60)90021-X

  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).
    https://doi.org/10.1007/BF01199420
Published
2018-08-02
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. doi:10.15407/ujpe63.7.584
Section
Fields and elementary particles