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

Authors

  • A. V. Hradyskyi V.N. Karazin National University of Kharkiv
  • Yu. P. Stepanoskiy National Science Center Kharkiv Institute of Physics and Technology

DOI:

https://doi.org/10.15407/ujpe63.7.584

Keywords:

Bargman–Wigner equation, massive graviton, wave equations

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.

References

<ol>
<li>A.S. Goldhaber, M.M. Nieto. Photon and graviton mass limits. Rev. Mod. Phys. 82, 939 (2010).
<a href="https://doi.org/10.1103/RevModPhys.82.939">https://doi.org/10.1103/RevModPhys.82.939</a>
</li>
<li>C. de Rham, J.T. Deskins, A.J. Tolley, S.-Y. Zhou. Gravi- ton mass bounds. Rev. Mod. Phys. 89, 025004 (2017).
<a href="https://doi.org/10.1103/RevModPhys.89.025004">https://doi.org/10.1103/RevModPhys.89.025004</a>
</li>
<li>E.P.Wigner. Relativistic invariance and quantum phenomena. Rev. Mod. Phys. 29, 255 (1957).
<a href="https://doi.org/10.1103/RevModPhys.29.255">https://doi.org/10.1103/RevModPhys.29.255</a>
</li>
<li>L. Landau, R. Peierls. Quantum electrodynamics in the configuration space. Zs. Phys. 62, 188 (1930).
<a href="https://doi.org/10.1007/BF01339793">https://doi.org/10.1007/BF01339793</a>
</li>
<li>I. Bialynicki-Birula. Photon wave function. Progress in Optics 36, edited by E. Wolf (Elsevier, 1996), p. ????????? [arXiv: quant-ph/0508202].
</li>
<li>I. Bialynicki-Birula. Photon as a quantum particle. Acta Phys. Pol. B 37, 935 (2006).
</li>
<li>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).
<a href="https://doi.org/10.1088/1751-8113/46/5/053001">https://doi.org/10.1088/1751-8113/46/5/053001</a>
</li>
<li>Yu.P. Stepanovskyi. The little Lorentz group and wave equations of free massless fields with arbitrary spin. Ukr. J. Phys. 9, 1165 (1964).
</li>
<li>H.Weyl. Electron and gravitation. Z. Phys. 56, 330 (1929).
<a href="https://doi.org/10.1007/BF01339504">https://doi.org/10.1007/BF01339504</a>
</li>
<li> M.P. Bronstein. Quantization of gravitational waves. J. Exper. Theor. Phys. 6, 195 (1936).
</li>
<li> Yu.P. Stepanovsky. On wave equations of massless fields. Teor. Matem. Fiz. 47, 343 (1981).
<a href="https://doi.org/10.1007/BF01019301">https://doi.org/10.1007/BF01019301</a>
</li>
<li> 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).
</li>
<li> Yu.P. Stepanovsky. On massless field and infinite component relativistic wave equations. Nucl. Phys. B. Proc. Suppl. 102 (1), 407 (2001).
<a href="https://doi.org/10.1016/S0920-5632(01)01587-0">https://doi.org/10.1016/S0920-5632(01)01587-0</a>
</li>
<li> 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].
<a href="https://doi.org/10.1007/978-94-010-0347-6_18">https://doi.org/10.1007/978-94-010-0347-6_18</a>
</li>
<li> Yu.P. Stepanovsky. ????? In: Problems in Contemporary Physics (KIPT, 2008), p. ?????????? (in Russian) [ISBN: 978-966-2136-15-9].
</li>
<li> B.L. Van der Waerden. Spinor analysis. Nachr. Ges. Wiss. G?ottingen, Math.-Phys. 100 (1929).
</li>
<li> O. Laporte, G.E. Uhlenbeck. Application of spinor analysis to the Maxwell and Dirac equations. Phys. Rev. 37, 1380 (1931).
<a href="https://doi.org/10.1103/PhysRev.37.1380">https://doi.org/10.1103/PhysRev.37.1380</a>
</li>
<li> V.I. Ogievetskii, I.V. Polubarinov. Notoph and its possible interactions. Yader. Fiz. 4, 216 (1966).
</li>
<li> Yu.B. Rumer. Spinor Analysis (Librocom, 2010) (in Russian) [ISBN: 978-5-397-01381-9].
</li>
<li> Yu. B. Rumer, A.I. Fet. Group Theory and Quantum Fields (Nauka, 1977) (in Russian).
</li>
<li> R. Penrose, W. Rindler. Spinors and Space-Time (Cambrige Univ. Press, 1984) [ISBN: 0521337070].
<a href="https://doi.org/10.1017/CBO9780511564048">https://doi.org/10.1017/CBO9780511564048</a>
</li>
<li> A. Proca. Wave theory of positive and negative electrons. J. Phys. Radium 7, 347 (1936).
<a href="https://doi.org/10.1051/jphysrad:0193600708034700">https://doi.org/10.1051/jphysrad:0193600708034700</a>
</li>
<li> V.V. Dvoeglazov. Photon-notoph equations. Phys. Scripta 64, 201 (2001).
<a href="https://doi.org/10.1238/Physica.Regular.064a00201">https://doi.org/10.1238/Physica.Regular.064a00201</a>
</li>
<li> V. Bargmann, E.P. Wigner. Group theoretical discussion of relativistic wave equations. Proc. Nat. Acad. Sci. USA 34, 211 (1948).
<a href="https://doi.org/10.1073/pnas.34.5.211">https://doi.org/10.1073/pnas.34.5.211</a>
</li>
<li> L. Bass, E. Schr?odinger. Must the photon mass be zero? Proc. R. Soc. London A 232, 1 (1955).
<a href="https://doi.org/10.1098/rspa.1955.0197">https://doi.org/10.1098/rspa.1955.0197</a>
</li>
<li> J.K. Luba’nski. Sur la theorie des particules elementaires de spin quelconque. I. Physica 9, 310 (1942).
<a href="https://doi.org/10.1016/S0031-8914(42)90113-7">https://doi.org/10.1016/S0031-8914(42)90113-7</a>
</li>
<li> T. Kaluza. Zum unitatsproblem der physik. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) 966 (1921).
</li>
<li> O. Klein. Quantentheorie und f?unfdimensionale relativit?atstheorie. Zeit. Phys. 37, 895 (1926).
<a href="https://doi.org/10.1007/BF01397481">https://doi.org/10.1007/BF01397481</a>
</li>
<li> E. Majorana. Atomi orientati in campo magnetico variabile. Nuovo Cimento 9, 43 (1932).
<a href="https://doi.org/10.1007/BF02960953">https://doi.org/10.1007/BF02960953</a>
</li>
<li> R. Penrose. A spinor approach to general relativity. Ann. Phys. 10, 171 (1960).
<a href="https://doi.org/10.1016/0003-4916(60)90021-X">https://doi.org/10.1016/0003-4916(60)90021-X</a>
</li>
<li> A.Z. Petrov. Clasification of spaces defining gravity fields. Uchen. Zapisk. Kazan. Gosud. Univer. 114, 55 (1954).
</li>
<li> Yu.P. Stepanovsky. Complete set of Fierz's relations in six-dimensional form. Ukr. J. Phys. 11, 1191 (1966).
</li>
<li> H. Weyl. Reine infinitesimalgeometrie. Mat. Zeit. 2, 384 (1918).
<a href="https://doi.org/10.1007/BF01199420">https://doi.org/10.1007/BF01199420</a></li>

Published

2018-08-02

How to Cite

Hradyskyi, A. V., & Stepanoskiy, Y. P. (2018). The Massless Limit of Bargmann–Wigner Equations for a Massive Graviton. Ukrainian Journal of Physics, 63(7), 584. https://doi.org/10.15407/ujpe63.7.584

Issue

Section

Fields and elementary particles