Spin 3/2 massless field, solutions with cylindrical symmetry, eliminating the gauge degrees of freedom

A. V. Ivashkevich

Problemy fiziki, matematiki i tehniki, no. 1 (2022), pp. 19-27.

Résumé

The system of equations for the vector-bispinor describing a spin 3/2 massless particle, is solved in cylindric coordinates of the Minkowski space. The covariant tetrad formalism is applied. Three operators, $i\partial_t$, $J_3$, $i\partial_z$ are diagonalized on the constructed solutions. After separating the variables, the system of 16 functions in the variable $r$ is derived. It is shown that the gauge solutions with cylindric symmetry, which are specified according to the Pauli–Fierz theory, satisfy these 16 equations identically. There are constructed 6 linearly independent solutions. Four of them coincide with the gauge ones, and two remain solutions do not contain the gauge degrees of freedom. These two solutions are expressed in terms of Bessel functions.

Détail

Keywords: spin 3/2, cylindric symmetry, massless field, gauge degrees of freedom.
Mots-clés : exact solutions

Bibliographie

[1] W. Pauli, M. Fierz, “Über relativistische Feldleichungen von Teilchen mit beliebigem Spin im elektromagnetishen Feld”, Helvetica Physica Acta, 12 (1939), 297–300 | Zbl

[2] M. Fierz, W. Pauli, “On relativistic wave equations for particles of arbitrary spin in an electromagnetic field”, Proc. Roy. Soc. London A, 173 (1939), 211–232 | DOI | MR

[3] W. Rarita, J. Schwinger, “On a theory of particles with halfintegral spin”, Phys. Rev., 60:1 (1941), 61–64 | DOI

[4] V.L. Ginzburg, “To the theory of particles of spin 3/2”, Journal of Experimental and Theoretical Physics, 12 (1942), 425–442 (in Russian) | MR | Zbl

[5] A.S. Davydov, “Wave equation for a particle with spin 3/2, in absence of external field”, Journal of Experimental and Theoretical Physics, 13 (1943), 313–319 | MR

[6] H.J. Bhabha, “Relativistic Wave Equations for the Elementary Particles”, Reviews of Modern Physics, 17:2–3 (1945), 200–216 | DOI | MR

[7] I.M. Gelfand, A.M. Yaglom, “General relativistically invariant equations and infinite-dimensional representations of the Lorentz group”, Journal of Experimental and Theoretical Physics, 18:8 (1948), 703–733 (in Russian) | MR

[8] E.S. Fradkin, “To the theory of particles with higher spins”, Journal of Experimental and Theoretical Physics, 20:1 (1950), 27–38 (in Russian) | MR

[9] F.I. Fedorov, “Generalized relativistic wave equations”, Proceedings of the Academy of Sciences of the USSR, 82:1 (1952), 37–40 (in Russian) | MR | Zbl

[10] V.Ya. Feinberg, “On the theory of interaction of particles with higher spins with electromagnetic and meson fields”, Proceedings of the Lebedev Physics Institute of the Academy of Sciences of the USSR, 6 (1955), 269–332 (in Russian)

[11] M.A. Petras, “Note to Bhabha's equation for a particle with maximum spin 3/2”, Czechoslovak Journal of Physics, 5:3 (1955), 418–419 | DOI | MR

[12] A.A. Bogush, “Equation for a 3/2 particle with anomalous magnetic moment”, Russian Physics Journal, 1 (1984), 23–27 (in Russian) | MR

[13] V.A. Pletyukhov, V.I. Strazhev, “To the theory of particles of spin 3/2”, Russian Physics Journal, 28:1 (1985), 91–95 (in Russian) | MR

[14] V.A. Pletyukhov, V.I. Strazhev, “On the relationship between various formulations of particle theory with spin 3/2”, Proceedings of the Academy of Sciences of the BSSR. Physics and Mathematics series, 5 (1985), 90–95 (in Russian) | MR

[15] K. Johnson, E.C.G. Sudarshan, “Inconsistency of the local field theory of charged spin 3/2 particles”, Ann. Phys. N.Y., 13 (1961), 121–145 | DOI | MR

[16] C.M. Bender, M. McCoy Barry, “Peculiarities of a free massless spin 3/2 field theory”, Phys. Rev., 148 (1966), 1375–1380 | DOI

[17] C.R. Hagen, L.P.S. Singh, “Search for consistent interactions of the Rarita-Schwinger field”, Phys. Rev. D, 26 (1982), 393–398 | DOI

[18] R.K. Loide, “Equations for a vector-bispinor”, J. Phys. A, 17 (1984), 2535–2550 | DOI | MR | Zbl

[19] V.M. Red'kov, Particle fields in the Riemann space and the Lorentz group, Belorusian science Publ., Minsk, 2009, 486 pp. (in Russian)

[20] V.A. Pletyukhov, V.M. Red'kov, V.I. Strazhev, Relativistic wave equations and internal degrees of freedom, Belaruskaya Navuka Publ., Minsk, 2015, 328 pp. (in Russian)

[21] V.V. Kisel et al., Elementary particles with internal structure in external fields, v. I, General theory, Nova Science Publishers Inc., New York, 2018, 404 pp.; v. II, Physical problems, 402 pp.

[22] V.V. Kisel et al., “Fradkin Equation for a Spin 3/2 Particle in Presence of External Electromagnetic and Gravitational Fields”, Ukr. J. Phys., 64:12 (2019), 1112–1117 | DOI

[23] A.V. Ivashkevich, E.M. Ovsiyuk, V.M. Red'kov, “Zero mass field with the spin 3/2: solutions of the wave equation and the helicity operator”, Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 55:3 (2019), 338–354 (in Russian) | DOI | MR

[24] A.V. Ivashkevich, E.M. Ovsiyuk, V.V. Kisel, V.M. Red'kov, “Spherical solutions of the wave equation for a spin 3/2 particle”, Doklady of the National Academy of Sciences of Belarus, 63:3 (2019), 282–290 | DOI | MR

[25] A.V. Ivashkevich, Ya.A. Voynova, E.M. Ovsiyuk, V.V. Kisel, V.M. Red'kov, “Spin 3/2 particle: Pauli-Fierz theory, nonrelativistic approximation”, Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 56:3 (2020), 335–349 | DOI | MR