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

TL;DR

This paper develops a consistent fourth order wave equation for spin-3/2 particles within the Bhabha-Madhavarao formalism, introducing new algebraic tools and projectors, and extends the approach to include electromagnetic interactions and path integral formulation.

Contribution

It introduces a novel algebraic framework with $ ext{q}$-commutators and $ ext{eta}$-matrices to derive fourth order wave equations for spin-3/2 particles, advancing the Bhabha-Madhavarao formalism.

Findings

Derived a fourth order wave operator using new algebraic objects.

Constructed projectors for spin-1/2 and spin-3/2 sectors.

Extended the formalism to include electromagnetic interactions.

Abstract

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.