#### Publish with Nova Science Publishers

We publish over 800 titles annually by leading researchers from around the world. Submit a Book Proposal Now!

$39.50

V.V. Kisel^{1}, V.A. Pletyukhov^{2}, E.M. Ovsiyuk^{3}, Ya.A. Voynova^{4}, V.M. Redkov^{4}

^{1}Â Belarus State University of Informatics and Radio-electronics, Belarus

^{2}Â Brest State University, Belarus

^{3Â }Mozyr State Pedagogical University, Belarus

^{4}Â Institute of Physics, National Academy of Sciences of Belarus, Belarus

**Part of the book: **Future Relativity, Gravitation, Cosmology

In the chapter, starting from a 20-component wave function in general Gelâ€™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.

[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. 33â€“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. 313â€“319 (in Russian).

[4] H. J. Bhabha, Harish-Chandra. On the theory of point particles.

Proc. Roy. Soc. Lon don. A. 1944. 183. 134â€“141.

[5] H. J. Bhabha. Relativistic wave equations for the proton. Proc. Indian Acad. Sci. A.

1945. 21. 241â€“264.

[6] H. J. Bhabha. Relativistic wave equations for elementary particles. Rev. Mod. Phys.

1945. 17. no 2-3. 200â€“215.

[7] H. J. Bhabha. The theory of the elementary particles.

Rep. Progr. Phys. 1946. 10. 253â€“271.

[8] Harish-Chandra. On the equations of motion of point particles.

Proc. Roy. Soc. Lon don. A. 1946. 185. 269â€“287.

[9] Harish-Chandra, Relativistic equations for elementary particles. Proc. Roy. Soc. Lon don. A. 1948. 192. 195â€“218.

[10] I. M. Gelâ€™fand, A. M. Yaglom. General relativistic invariant equations snd infinitely

dimensional representation of the Lorentz group. Zh. Eksp. Teor. Fiz. 1948. 18. no 8.

703â€“733 (in Russian).

[11] I. M. Gelâ€™fand, A.M. Yaglom. Pauli theorem for general relativistic invariant wave

equations. Zh. Eksp. Teor. Fiz. 1948. 18. no 12. 1096â€“1104 (in Russian).

[12] I. M. Gelâ€™fand, A. M. Yaglom. Charge conjugation for general relativistic invariant

wave equations. Zh. Eksp. Teor. Fiz. 1948. 18. no 12. 1105â€“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. 1â€“103.

[14] F. I. Fedorov. To the problem of solving relativistic wave equations. Doklady AN

USSR. 1949. 65. no 6. 813â€“814 (in Russian).

[15] E. E. Fradkin. To the theory of particles with higher spins. Zh. Eksp. Teor. Fiz. 1950.

20. no 1. 27â€“38 (in Russian).

[16] F. I. Fedorov. On minimal polynomials of matrices of relativistic wave equations.

Dok lady AN USSR. 1951. 79. no 5. 787â€“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. 156â€“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. 33â€“47.

[19] F. I. Fedorov. Generalized relativistic wave equations. Doklady AN USSR. 1952. 82.

no 1. 37â€“40 (in Russian).

[20] M. Petras. A contribution of the theory of the Pauliâ€“Fierzâ€™s equations a particle with

spin 3/2. Czech. J. Phys. 1955. 5. no 2. 169â€“170.

[21] M. Petras. A note to Bhabhaâ€™s equation for a particle with maximum spin 3/2. Czech.

J. Phys. 1955. 5. no 3. 418â€“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. 269â€“332 (in Russian).

[23] V. L. Ginzburg. On relativistic wave equations with a mass spectrum. Acta Phys. Pol.

1956. 15. 163â€“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. 285â€“298.

[25] T. Regge. On properties of the particle with spin 2. Nuovo Cimento. 1957. 5. no 2.

325â€“326.

[26] P. G. Bergmann, A. I. Janis. Subsidiary conditions in covariant theories. Phys. Rev.

1958. 111. no 4. 1191â€“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. 96â€“103.

[28] L. A. Shelepin. Covariant theory of relativictic wave equations. Nucl. Phys. 1962. 33.

no 4. 580â€“593.

[29] A. Z. Capri. First order wave equations for multi-mass fermions. Nuovo Cim. B. 1969.

64. no 1. 151â€“158.

[30] A. Aurilia, H. Umezawa. Theory of high spin fields. Phys. Rev. 1969. 182. no 5. 1682â€“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. 81â€“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. 79â€“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. 78â€“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. 84â€“92 (in Russian).

[35] A. Shamaly, A. Z. Capri. First-order wave equations for integral spin. Nuovo Cim. B.

1971. 2. no 2. 235â€“253.

[36] V. Amar, U. Dozzio. Finite dimensional Gelâ€™fandâ€“Yaglom equations for arbitrary in tegral spin. Nuovo Cim. B. 1972. 9. 53â€“63.

[37] A. Z. Capri. Electromagnetic properties of a new spin-1/2 field. Progr. Theor. Phys.

1972. 48. no 4. 1364â€“1374.

[38] A. Shamaly, A.Z. Capri. Unified theories for massive spin 1 fields. Can. J. Phys. 1973.

51. no 14. 1467â€“1470.

[39] M. A. K. Khalil. Properties of a 20-component spin 1/2 relativistic wave equation.

Phys. Rev. D. 1977. 15. no 6. 1532â€“1539.

[40] A. S. Wightman. Invariant wave equations: general theory and applications to the

external field problem. Lecture Notes in Physics. 1978. 73. 1â€“101.

[41] M. A. K. Khalil. An equivalence of relativistic field equations. Nuovo Cimento. A.

1978. 45. no 3. 389â€“404.

[42] L. Garding. Mathematics of invariant wave equations. Lect. Notes in Physics. 1978.

73. 102â€“164.

[43] J. P. Gazeau. Lâ€™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. 1459â€“1475.

[44] W. Cox. Higher-rank representations for zero-spin filds theories. J. Phys. A. 1982. 15.

627â€“635.

[45] W. Cox. First-order formulation of massive spin-2 field theories. J. Phys. A. 1982. 15.

253â€“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. 1579â€“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. 1080â€“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. 1599â€“1608.

[49] S. Deser, A. Waldron. Inconsistencies of massive charged gravitating higher

spins.Nucl. Phys. B. 2002. 631. 369â€“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. Redâ€™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. 153â€“165.

[52] V. Kisel, Ya. Voynova, E. Ovsiyuk, V. Balan, V. Redâ€™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. 21â€“39.

[53] E. M. Ovsiyuk, Ya. A. Voynova, V. V. Kisel, V. Balan, V. M. Redâ€™kov Spin 1 Particle

with Anomalous Magnetic Moment in the External Uniform Electric Field Chapter

in: Quaternions: Theory and Applications. Editor: Sandra Griffin. â€“ Nova Science

Publishers, Inc. USA, 2017. â€“ P. 47â€“84.

[54] E. M. Ovsiyuk, Ya. A. Voynova, V. V. Kisel, V. Balan, V. M. Redâ€™kov Spin 1 Particle

with Anomalous Magnetic Moment in the External Uniform Electric Field Chapter

in: Quaternions: Theory and Applications. Editor: Sandra Griffin. â€“ Nova Science

Publishers, Inc. USA, 2017. â€“ P. 47â€“84.

[55] V. V. Kisel, E. M. Ovsiyuk, Ya. A. Voynova, V. M. Redâ€™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. 18â€“27.

[56] V. V. Kisel, V. A. Pletyukhov, V. V. Gilewsky, E. M. Ovsiyuk, O. V. Veko, V. M.

Redâ€™kov. Spin 1/2 particle with two mass states. interaction with external fields.

Non linear Phenomena in Complex Systems. 2017. 20. 4. 404â€“423.

[57] E. M. Ovsiyuk, O. V. Veko, Ya. A. Voynova, V. V. Kisel, V. Balan, V. M. Redâ€™kov. Spin

1/2 particle with two masses in magnetic field. Applied Sciences. 2018. 20. 148â€“166.

[58] E. M. Ovsiyuk, O. V. Veko, Ya. A. Voynova, V. M. Redâ€™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. 651â€“660

[59] I. M. Gelâ€™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. Redâ€™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. Redâ€™kov. Elementary particles

with internal structure in external field. Vol. I. General theory. â€“ Nova Science Pub lishers, Inc. USA, 2018;

Vol. II. Physical problems. â€“ Nova Science Publishers, Inc.

USA, 2018.

We publish over 800 titles annually by leading researchers from around the world. Submit a Book Proposal Now!