Geodesic
Algebraic approach to the construction of the wave equation for particles with spin 3/2
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 50-65.

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Within the framework of the Bhabha–Madhava Rao formalism, we propose a self-consistent approach to a system of fourth-order wave equations for describing massive particles with spin $3/2$. For this purpose, we introduce a new set of matrices $\eta_{\mu}$ instead of the original matrices $\beta_{\mu}$ of the Bhabha–Madhava Rao algebra. We prove that, in terms of the matrices $\eta_{\mu}$, the procedure for constructing the fourth root of the fourth-order wave operator can be reduced to some simple algebraic transformations and passing to the limit as $z\to q$, where $z$ is a complex deformation parameter and $q$ is a primitive fourth root of unity, which is included in the definition of the $\eta$-matrices. Also, we introduce a set of three operators $P_{1/2}$ and $P_{3/2}^{(\pm)}(q)$, which possess the properties of projectors. These operators project the matrices $\eta_{\mu}$ onto sectors with $1/2$- and $3/2$-spins. We generalize the results obtained to the case of interaction with an external electromagnetic field introduced by means of a minimal substitution. We discuss the corresponding applications of the results obtained to the problem of constructing a representation of the path integral in para-superspace for the propagator of a massive particle with spin $3/2$ in an external gauge field within the framework of the Bhabha–Madhava Rao approach.
Keywords: fourth-order wave operator, Bhabha–Madhava Rao algebra, particles with spin $3/2$, deformation parameter. 
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Yu. A. Markov; M. A. Markova; A. I. Bondarenko. Algebraic approach to the construction of the wave equation for particles with spin 3/2. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 50-65. http://geodesic.mathdoc.fr/item/INTO_2021_196_a5/

[1] Baisya H. L., “An equation for particles with spin 5/2”, Prog. Theor. Phys., 94 (1995), 271–292 | DOI | MR

[2] Bhabha H. J., “Relativistic wave equations for the elementary particles”, Rev. Mod. Phys., 17 (1945), 200–216 | DOI | MR

[3] Dirac P. A. M., “Relativistic wave equations”, Proc. Roy. Soc. London A., 155 (1936), 447–459 | DOI

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

[5] Kamefuchi S. and Takahashi Y., “A generalization of field quantization and statistics”, Nucl. Phys., 36 (1962), 177–206 | DOI | MR | Zbl

[6] Madhava Rao B. S., “Generalised algebra of elementary particles”, Proc. Indian Acad. Sci. A., 26 (1947), 221–233 | DOI | MR

[7] Madhava Rao B. S., Thiruvenkatachar V. R., Venkatachaliengar K., “Algebra related to elementary particles of spin 3/2”, Proc. Roy. Soc. London A., 187 (1946), 385–397 | DOI | MR | Zbl

[8] Markov Yu. A., Markova M. A., Bondarenko A. I., “Third-order wave equation in Duffin–Kemmer–Petiau theory. Massive case”, Phys. Rev. D., 92 (2015), 105017 (24) | DOI | MR

[9] Ohnuki Y., Kamefuchi S., “Para-Grassmann algebras with applications to para-Fermi systems”, J. Math. Phys., 21 (1980), 609–616 | DOI | MR | Zbl

[10] Scharfstein H., “Trilinear commutation relations between fields”, Nuovo Cim., 30 (1963), 740–761 | DOI | MR