Relativistic Equations for Arbitrary Spin, Especially for the Spin s = 2


  • V. M. Simulik Institute of Electron Physics, Nat. Acad. of Sci. of Ukraine



Dirac equation, relativistic quantum mechanics, arbitrary spin, graviton, spin (2,2) particle-antiparticle doublet


The further approbation of the equation for the particles of arbitrary spin introduced recently in our papers is under consideration. The comparison with the known equations suggested by Bhabha, Pauli–Fierz, Bargmann–Wigner, Rarita–Schwinger (for spin s =3/2) and other authors is discussed. The advantages of the new equations are considered briefly. The advantage of the new equation is the absence of redundant components. The important partial case of spin s =2 is considered in details. The 10-component Dirac-like wave equation for the spin s =(2,2) particle-antiparticle doublet is suggested. The Poincar´e invariance is proved. The three-level consideration (relativistic canonical quantum mechanics, canonical Foldy–Wouthuysen-type field theory, and locally covariant field theory) is presented. The procedure of our synthesis of arbitrary spin covariant particle equations is demonstrated on the example of spin s =(2,2) doublet.


V.M. Simulik. Derivation of the Dirac and Dirac-like equations of arbitrary spin from the corresponding relativistic canonical quantum mechanics. Ukr. J. Phys. 60, 985 (2015).

V.M. Simulik. Link between the relativistic canonical quantum mechanics of arbitrary spin and the corresponding field theory. J. Phys: Conf. Ser. 670, 012047 (2016).

P.A.M. Dirac. Relativistic wave equations. Proc. Roy. Soc. Lond. A. 155, 447 (1936).

M. Fierz, W. Pauli. On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. R. Soc. Lond. A. 173, 211 (1939).

H.J. Bhabha. Relativistic wave equations for the elementary particles. Rev. Mod. Phys. 17, 200 (1945).

V. Bargmann, E.P. Wigner. Group theoretical discussion of relativistic wave equations. Proc. Nat Acad. Sci. USA. 34, 211 (1948).

A. Zecca. Massive field equations of arbitrary spin in Schwarzschild geometry: separation induced by spin-3/2 case. Int. J. Theor, Phys. 45, 2241 (2006).

V.M. Simulik. On the relativistic canonical quantum mechanics and field theory of arbitrary spin. Univ. J. Phys. Appl. 11, 202 (2017).

L.L. Foldy, S.A. Wouthuysen. On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Phys. Rev. 78, 29 (1950).

L.L. Foldy. Synthesis of covariant particle equations. Phys. Rev. 102, 568 (1956).

W. Rarita, J. Schwinger. On a theory of particles with half-integral spin. Phys. Rev. 60, 61 (1941).

A.S. Davydov. Wave equations of a particle having spin 3/2 in the absence of a field. J. Exper. Theor. Phys. 13, 313 (1943) (in Russian).

G. Velo, D. Zwanziger. Propagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential. Phys. Rev. 186, 1337 (1969).

J.D. Kimel, L.M. Nath. Quantization of the spin-3/2 field in the presence of interactions. Phys. Rev. D. 6, 2132 (1972).

A.E. Kaloshin, V.P. Lomov. Rarita-Schwinger field and multi-component wave equation. Phys. Part. Nucl. Lett. 8, 517 (2011).

A.Z. Capri, R.L. Kobes. Further problems in spin-3/2 field theories. Phys. Rev. D. 22, 1967 (1980).

V.M. Red'kov. Particle with spin S = 3/2 in Riemannian space-time. arXiv: 1109.3871v1.

V.S. Vladimirov. Methods of the Theory of Generalized Functions (Taylor and Francis, 2002) [ISBN:9780429153013].

B. Wybourne. Classical Groups for Physicists (Wiley, 1974) [ISBN-13: 978-0471965053].

J. Elliott, P. Dawber. Symmetry in Physics, Vol. 1 (Macmillan Press, 1979) [ISBN-13: 978-0333382707].

H.J. Bhabha. On a class of relativistic wave equations of spin 3/2. Proc. Indian Acad. Sci. A 34, 335 (1951).



How to Cite

Simulik, V. M. (2019). Relativistic Equations for Arbitrary Spin, Especially for the Spin s = 2. Ukrainian Journal of Physics, 64(11), 1064.



Fields and elementary particles