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A.V. Ivashkevich^{1}, E.M. Ovsiyuk^{2}, V.V. Kisel^{3}, V.M. Redkov^{1
1} Institute of Physics, National Academy of Sciences of Belarus

^{2}Mozyr State Pedagogical University, Belarus

^{3} Belarus State University of Informatics and Radio-Electronics, Belarus

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

Many year ago, a special generalized equation for a spin 3/2 particle, different from Pauli–Fierz and Rarita–Schwinger model, was proposed by Fradkin. To the present time it is not clear which additional structure underlies this extended wave equation. We investigate this model systematically, applying the general Gel’fand–Yaglom formalism. Having used a standard set of requirements: relativistic invariance, single nonzero mass and single spin S=3/2, P-symmetry, existence of Lagrangian formulation for the model, we derive a set of spinor equations. The 20-component wave function consists of bispinor and vector-bispinor. It is shown that in the free case the Fradkin equation reduces to minimal Pauli–Fierz and Rarita–Schwinger equation for a bisbinor. In presence of external electromagnetic fields, the minimal form the Fradkin equation for a bispinor function contains an additional interaction term governed by electromagnetic tensor Fαβ. Finally we take into account the external curved space-time background, in generally covariant case the Fradkin equation contains additional gravitational interaction through the Ricci tensor Rαβ. If the electric charge of the particle is zero, The Fradkin model remains correct and describes a neutral spin 3/2 particle of Majorana type interacting additionally with geometrical background by means of the Ricci tensor.

**Keywords:** Lorentz group, extended set of representations, generalized wave equation,

fermion, intrinsic electromagnetic structure, external fields, curved space-time.

[1] P. A. M. Dirac. Relativistic wave equations. Proc. R. Soc. London. A. 1936. 155.

447–459.

[2] E. Majorana. Teoria simmetrica dell electrone e dell positrone. Nuovo Cim. 1937. 14.

171–186.

[3] M. Fierz. Uber die relativistische theorie KraftefreierTeilchen mit beliebigem Spin.

Helv. Phys. Acta. 1939. 12. 3–37.

[4] W. Pauli, M. Fierz. Uber relativistische Feldleichungen von Teilchen mit beliebigem

Spin im elektromagnetishen Feld. Helv. Phys. Acta. 1939. 12. 297–300.

[5] V. L. Ginzburg, Ya. A. Smorodinsky. On wave equations for particles with variable

spin. Zh. Eksp. Teor. Fiz. 1943. 13. 274 (in Russian).

[6] 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).

[7] H. J. Bhabha, Harish-Chandra. On the theory of point particles. Proc. Roy. Soc. Lon don. A. 1944. 183. 134–141.

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

1945. 21. 241–264.

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

1945. 17. no 2-3. 200–215.

[10] H. J. Bhabha. The theory of the elementary particles. Rep. Progr. Phys. 1946. 10.

253–271.

[11] I. M. Gel’fand, A. M. Yaglom. General relativistic invariant equations snd infinitely

dimensional representation of the Lorentz group. Zh. Eksp. Teor. Fiz. 1848. 18. no 8.

703–733 (in Russian).

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

20. no. 1. 27–38 (in Russian).

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

Doklady AN USSR. 1951. 79. no 5. 787–790 (in Russian).

[14] 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).

[15] H. J. Bhabha. An equation for a particle with two mass states and positive charge

density. Phil. Mag. 1952. 43. P. 33–47.

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

no 1. 37–40 (in Russian).

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

325–326.

[18] M. Petras. A contribution of the theory of the Pauli-Fierz’s equations a particle with

spin 3/2. Czech. J. Phys. 1955. 5. 169–170.

[19] M. Petras. A note to Bhabha’s equation for a particle with maximum spin 3/2. Czech.

J. Phys. 1955. 5. 418–419.

[20] V. Ya. Fainberg. To the interaction theory of the particles of the higher spins with

electromagnetic and meson fields. Trudy FIAN USSR. 1955. 6. 269–332 (in Russian).

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

1956. 15. P. 163–175.

[22] 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.

[23] 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.

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

33. no 4. 580–593.

[25] L. A. Shelepin. Covariant theory of relativictic wave equations for paricle of arbitrary

spins. Trudy FIAN USSR. 1964. 30. 253–321 (in Russian).

[26] A. Z. Capri. Non uniqueness of the spin 1/2 equation.

Phys. Rev. 1969. 178. 1811–1815.

[27] A. Z. Capri. First order wave equations for multimass fermions. Nuovo Cim. B. 1969.

64. 151–158.

[28] 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).

[29] 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).

[30] 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).

[31] 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).

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

1971. 2. no 2. 235–253.

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

1972. 48. 1364–1374.

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

1973. 51. no 14. 1467–1470.

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

Phys. Rev. D. 1977. 15. no 6. 1532–1539.

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

external field problem. Lecture Notes in Physics. 1978. 73. 1–101.

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

1978. 45. no 3. 389–404.

[38] L. Garding, L. Mathematics of invariant wave equations. Lect. Notes in Physics.

1978. 73. 102–164.

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

15. 627–635.

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

15. 253–268.

[41] P. M. Mathews, B. Vijayalakshmi, M. Sivakuma. On the admissible Lorentz group

representations in unique-mass, unique-spin relativistic wave equations. Phys. A.

1982. 15. no 11. 1579–1582.

[42] F. I. Fedorov. The Lorentz group. Moscow. Nauka. 1979.

[43] P. M. Mathews, B. Vijayalakshmi. On inequivalent classes unique-mass-spin relativistic wave equations involving repeated irreducible representations with arbitrary

multiplicities. J. Math. Phys. 1984. 25. no 4. 1080–1087.

[44] 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.

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

Nucl. Phys. B. 2002. 631. 369–387.

[46] 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.

[47] V. Kisel, Ya. Voynova, E. Ovsiyuk, V. Balan, V. Red’kov. Spin 1 Particle with

Anomalous Magnetic Moment in the External Uniform Magnetic Field. Nonlinear

Phenomena in Complex Systems. 2017. 20. no 1. 21–39.

[48] 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, Mathematics, and Thechnics. 2017. 32 no 3. 18–27.

[49] 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. no 4. 404–423.

[50] 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.

[51] V. M. Red’kov. Fields in Riemannian space and the Lorentz group. Publishing

House: Belarusian Science, Minsk, 2009.

[52] V. A. Pletjukhov, V. M. Red’kov, V. I. Strazhev. Relativistic wave equations and

intrinsic degrees of freedom. Belarusian Science. Minsk. 2015.

[53] V. V. Kisel, E. M. Ovsiyuk, O. V. Veko, Ya. A. Voynova, V. Balan, V. M. Red’kov.

Elementary particles with internal structure in external fields. I. General theory, II.

Physical problems. New York: Nova Science Publishers Inc. 2018.

[54] A. V. Ivashkevich, O. A. Vasiluyk, V. V. Kisel, V. M. Red’kov. Spin 3/2 particle:

the Fradkin theory, non-relativistic approximation. Nonlinear Dynamics and Applications. 2021. 27. 138–175.

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