Fourth order wave equation in Bhabha-Madhavarao spin-$\frac{3}{2}$ theory

Within the framework of the Bhabha-Madhavarao formalism, a consistent approach to the derivation of a system of the fourth order wave equations for the description of a spin-$\frac{3}{2}$ particle is suggested. For this purpose an additional algebraic object, the so-called $q$-commutator ($q$ is a primitive fourth root of unity) and a new set of matrices $\eta_{\mu}$, instead of the original matrices $\beta_{\mu}$ of the Bhabha-Madhavarao algebra, are introduced. It is shown that in terms of the $\eta_{\mu}$ matrices we have succeeded in reducing a procedure of the construction of fourth root of the fourth order wave operator to a few simple algebraic transformations and to some operation of the passage to the limit $z \rightarrow q$, where $z$ is some (complex) deformation parameter entering into the definition of the $\eta$-matrices. In addition, a set of the matrices ${\cal P}_{1/2}$ and ${\cal P}_{3/2}^{(\pm)}(q)$ possessing the properties of projectors is introduced. These operators project the matrices $\eta_{\mu}$ onto the spins 1/2- and 3/2-sectors in the theory under consideration. A corresponding generalization of the obtained results to the case of the interaction with an external electromagnetic field introduced through the minimal coupling scheme is carried out. The application to the problem of construction of the path integral representation in parasuperspace for the propagator of a massive spin-$\frac{3}{2}$ particle in a background gauge field within the Bhabha-Madhavarao approach is discussed.