V.V. Kisel¹, V.A. Pletyukhov², E.M. Ovsiyuk³, Ya.A. Voynova⁴, V.M. Redkov^4
1 Belarus State University of Informatics and Radio-electronics, Belarus
² Brest State University, Belarus
³ Mozyr State Pedagogical University, Belarus
⁴ Institute of Physics, National Academy of Sciences of Belarus, Belarus

Part of the book: Future Relativity, Gravitation, Cosmology

Abstract

In the chapter, starting from a 20-component wave function in general Gel’fandYaglom approach a new wave equation for spin 1/2 fermion, which is characterized by three mass parameters, is derived. In absence of external field, three involved bispinors obey three separate Dirac-like equations with different masses M1, M2, M3. In presence of external electromagnetic fields or non-Euclidean background with non-vanishing Ricci scalar curvature, the main equation is not split into separated equations, instead a quite definite mixing of three Dirac-like equations arises. It is shown that a generalized equation for Majorana particle with three mass parameters exists as well, such a generalized Majorana equation is not split into three separated equations in curved space-time if the Ricci scalar of the space-time model does not vanish.

Keywords: Lorentz group, extended sets of representations, generalized wave equation,
fermion, mass parameters, electromagnetic field, gravitational field.

References

[1] V. L. Ginzburg, Ya. A. Smorodinsky. On wave equations for particles with variable
spin. Zh. Eksp. Teor. Fiz. 1943. 13. 274 (in Russian).
[2] V. L. Ginzburg, To the theory of exited states of elementary particles. Zh. Eksp. Teor.
Fiz. 1943. 13. 33–58 (in Russian).
[3] A. S. Davydov. Wave equations of a particle having spin 3/2 in absence of field. Zh.
Eksp. Teor. Fiz. 1943. 13. no 9-10. 313–319 (in Russian).
[4] H. J. Bhabha, Harish-Chandra. On the theory of point particles.
Proc. Roy. Soc. Lon don. A. 1944. 183. 134–141.
[5] H. J. Bhabha. Relativistic wave equations for the proton. Proc. Indian Acad. Sci. A.
1945. 21. 241–264.
[6] H. J. Bhabha. Relativistic wave equations for elementary particles. Rev. Mod. Phys.
1945. 17. no 2-3. 200–215.
[7] H. J. Bhabha. The theory of the elementary particles.
Rep. Progr. Phys. 1946. 10. 253–271.
[8] Harish-Chandra. On the equations of motion of point particles.
Proc. Roy. Soc. Lon don. A. 1946. 185. 269–287.
[9] Harish-Chandra, Relativistic equations for elementary particles. Proc. Roy. Soc. Lon don. A. 1948. 192. 195–218.
[10] I. M. Gel’fand, A. M. Yaglom. General relativistic invariant equations snd infinitely
dimensional representation of the Lorentz group. Zh. Eksp. Teor. Fiz. 1948. 18. no 8.
703–733 (in Russian).
[11] I. M. Gel’fand, A.M. Yaglom. Pauli theorem for general relativistic invariant wave
equations. Zh. Eksp. Teor. Fiz. 1948. 18. no 12. 1096–1104 (in Russian).
[12] I. M. Gel’fand, A. M. Yaglom. Charge conjugation for general relativistic invariant
wave equations. Zh. Eksp. Teor. Fiz. 1948. 18. no 12. 1105–1111.
[13] H. J. Bhabha. Theory of elementary particles-fields. Lectures Delivered at 2nd Sum mer Seminar Canadian Math. Congress: held at University of British Columbia, Aug 1949. 1–103.
[14] F. I. Fedorov. To the problem of solving relativistic wave equations. Doklady AN
USSR. 1949. 65. no 6. 813–814 (in Russian).
[15] E. E. Fradkin. To the theory of particles with higher spins. Zh. Eksp. Teor. Fiz. 1950.
20. no 1. 27–38 (in Russian).
[16] F. I. Fedorov. On minimal polynomials of matrices of relativistic wave equations.
Dok lady AN USSR. 1951. 79. no 5. 787–790 (in Russian).
[17] F. I. Fedorov. To the theory of a spin 2 particle. Uchionye Zapiski Beloruaasian State
University. ser. Phys.-mat. 1951. no 12. 156–173 (in Russian).
[18] H. J. Bhabha. An equation for a particle with two mass states and positive charge
density. Phil. Mag. 1952. Ser VII. 43. 33–47.
[19] F. I. Fedorov. Generalized relativistic wave equations. Doklady AN USSR. 1952. 82.
no 1. 37–40 (in Russian).
[20] M. Petras. A contribution of the theory of the Pauli–Fierz’s equations a particle with
spin 3/2. Czech. J. Phys. 1955. 5. no 2. 169–170.
[21] M. Petras. A note to Bhabha’s equation for a particle with maximum spin 3/2. Czech.
J. Phys. 1955. 5. no 3. 418–419.
[22] V. Ya. Fainberg, To the interactionb theory of the particles of the higher spins with
electromagnetic and meson fields. Trudy FIAN USSR. 1955. 6. 269–332 (in Russian).
[23] V. L. Ginzburg. On relativistic wave equations with a mass spectrum. Acta Phys. Pol.
1956. 15. 163–175.
[24] H. Shimazu. A relativistic wave equation for a particle with two mass states of spin 1
and 0. Progress of Theoretical Physics. 1956. 16. no 4. 285–298.
[25] T. Regge. On properties of the particle with spin 2. Nuovo Cimento. 1957. 5. no 2.
325–326.
[26] P. G. Bergmann, A. I. Janis. Subsidiary conditions in covariant theories. Phys. Rev.
1958. 111. no 4. 1191–1200.
[27] H. A. Buchdahl. On the compatibility of relativistic wave equations for particles of
higher spin in the presence of a gravitational field. Nuovo Cim. 1958. 10. 96–103.
[28] L. A. Shelepin. Covariant theory of relativictic wave equations. Nucl. Phys. 1962. 33.
no 4. 580–593.
[29] A. Z. Capri. First order wave equations for multi-mass fermions. Nuovo Cim. B. 1969.
64. no 1. 151–158.
[30] A. Aurilia, H. Umezawa. Theory of high spin fields. Phys. Rev. 1969. 182. no 5. 1682–1694.
[31] F.I. Fedorov, V.A. Pletyukhov. Wave equations with repeated representations of the
Lorentz group. Proceedings of the National academy of sciences of Belarus. Phys.-
Math. series. 1969. no 6. 81–88 (in Russian).
[32] V. A. Pletyukhov, F. I. Fedorov. The wave equation with repeated representations for
spin 0 particle. Proceedings of the National academy of sciences of Belarus. Phys.-
Math. series. 1970. no 2. 79–85 (in Russian).
[33] F. I. Fedorov, V. A. Pletyukhov. Wave equations with repeated representations of the
Lorentz group. Half-integer spin. Proceedings of the National academy of sciences of
Belarus. Phys.-Math. series. 1970. no 3. 78–83 (in Russian).
[34] V. A. Pletyukhov, F. I. Fedorov. Wave equation with repeated reprentations for a spin
1 particle. Proceedings of the National academy of sciences of Belarus. Phys.-Math.
series. 1970. no 3. 84–92 (in Russian).
[35] A. Shamaly, A. Z. Capri. First-order wave equations for integral spin. Nuovo Cim. B.
1971. 2. no 2. 235–253.
[36] V. Amar, U. Dozzio. Finite dimensional Gel’fand–Yaglom equations for arbitrary in tegral spin. Nuovo Cim. B. 1972. 9. 53–63.
[37] A. Z. Capri. Electromagnetic properties of a new spin-1/2 field. Progr. Theor. Phys.
1972. 48. no 4. 1364–1374.
[38] A. Shamaly, A.Z. Capri. Unified theories for massive spin 1 fields. Can. J. Phys. 1973.
51. no 14. 1467–1470.
[39] M. A. K. Khalil. Properties of a 20-component spin 1/2 relativistic wave equation.
Phys. Rev. D. 1977. 15. no 6. 1532–1539.
[40] A. S. Wightman. Invariant wave equations: general theory and applications to the
external field problem. Lecture Notes in Physics. 1978. 73. 1–101.
[41] M. A. K. Khalil. An equivalence of relativistic field equations. Nuovo Cimento. A.
1978. 45. no 3. 389–404.
[42] L. Garding. Mathematics of invariant wave equations. Lect. Notes in Physics. 1978.
73. 102–164.
[43] J. P. Gazeau. L’equation de Dirac avec masse et spin arbitrares: une construction sim ple et naturelle
[The Dirac equation with arbitrary mass and spin: a simple and natural construction].
J. Phys, G. Nucl. Phys. 1980. 6. no 12. 1459–1475.
[44] W. Cox. Higher-rank representations for zero-spin filds theories. J. Phys. A. 1982. 15.
627–635.
[45] W. Cox. First-order formulation of massive spin-2 field theories. J. Phys. A. 1982. 15.
253–268.
[46] P. M. Mathews, B. Vijayalakshmi, M.Sivakuma. On the admissible Lorentz group rep resentations
in unique-mass, unique-spin relativistic wave equations. Phys. A. 1982.
15. no 11. 1579–1582.
[47] P. M. Mathews, B. Vijayalakshmi. On inequivalent classes unique-mass-spin relativis tic wave equations involving repeated irreducible representations with arbitrary mul tiplicities. J. Math. Phys. 1984. 25. no 4. 1080–1087.
[48] W. Cox. On the Lagrangian and Hamiltonian constraint algorithms for the Rarita Schwinger field coupled to an external electromagnetic field. J. Phys. A. 1989. 22. no 10. 1599–1608.
[49] S. Deser, A. Waldron. Inconsistencies of massive charged gravitating higher
spins.Nucl. Phys. B. 2002. 631. 369–387.
[50] V. Simulik. Relativistic wave equations of arbitrary spin in quantum mechanics and
field theory, example spin S = 2. J. of Phys. Conference Series, 2017. 804. no 1. paper 012040.
[51] E. M. Ovsiyuk, V. V. Kisel, Y. A. Voynova, O. V. Veko, V. M. Red’kov Spin 1/2
particle with anomalous magnetic moment in a uniform magnetic field, exact solutions
Nonlinear Phenomena in Complex Systems. 2016. 19. no 2. 153–165.
[52] V. Kisel, Ya. Voynova, E. Ovsiyuk, V. Balan, V. Red’kov. Spin 1 Particle with Anoma lous Magnetic Moment in the External Uniform Magnetic Field. Nonlinear Phenom ena in Complex Systems. 2017. 20. no 1. 21–39.
[53] E. M. Ovsiyuk, Ya. A. Voynova, V. V. Kisel, V. Balan, V. M. Red’kov Spin 1 Particle
with Anomalous Magnetic Moment in the External Uniform Electric Field Chapter
in: Quaternions: Theory and Applications. Editor: Sandra Griffin. – Nova Science
Publishers, Inc. USA, 2017. – P. 47–84.
[54] E. M. Ovsiyuk, Ya. A. Voynova, V. V. Kisel, V. Balan, V. M. Red’kov Spin 1 Particle
with Anomalous Magnetic Moment in the External Uniform Electric Field Chapter
in: Quaternions: Theory and Applications. Editor: Sandra Griffin. – Nova Science
Publishers, Inc. USA, 2017. – P. 47–84.
[55] V. V. Kisel, E. M. Ovsiyuk, Ya. A. Voynova, V. M. Red’kov. Quantum mechanics of
spin 1 particle with quadrupole moment in external uniform magetic field. Problems
of Physics, Mathemativs, and Thechnics. 2017. no 3(32). P. 18–27.
[56] V. V. Kisel, V. A. Pletyukhov, V. V. Gilewsky, E. M. Ovsiyuk, O. V. Veko, V. M.
Red’kov. Spin 1/2 particle with two mass states. interaction with external fields.
Non linear Phenomena in Complex Systems. 2017. 20. 4. 404–423.
[57] E. M. Ovsiyuk, O. V. Veko, Ya. A. Voynova, V. V. Kisel, V. Balan, V. M. Red’kov. Spin
1/2 particle with two masses in magnetic field. Applied Sciences. 2018. 20. 148–166.
[58] E. M. Ovsiyuk, O. V. Veko, Ya. A. Voynova, V. M. Red’kov, V. V. Kisel. N. V. Sam sonenko.
Spin 1/2 particle with two masses in external magnetic field. J. Mech. Cont.
and Math. Sci. Special Issue. 2019. no 1. P. 651–660
[59] I. M. Gel’fand, R. A. Minlos, Z. Ya. Shapiro. Representations of the rotation and
Lorentz groups and their applications. Translated from the Russian edition (1958) by
G. Cummins and T. Boddington. Pergamon, London; Macmillan, New York, 1963.
[60] V. M. Red’kov. Fields in Riemannian space and the Lorentz group. Publishing House:
Belarusian Science, Minsk, 2009.
[61] V. V. Kisel, E. M. Ovsiyuk, V. Balan, O. V. Veko, V. M. Red’kov. Elementary particles
with internal structure in external field. Vol. I. General theory. – Nova Science Pub lishers, Inc. USA, 2018;
Vol. II. Physical problems. – Nova Science Publishers, Inc.
USA, 2018.