Algebraic approach to the construction of the wave equation for particles with spin 3/2

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.