Fourth-order wave equation in Bhabha–Madhavarao spin-3 2 theory

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-[Formula: see text] particle is suggested. For this purpose an additional algebraic object, the so-called [Formula: see text]-commutator ([Formula: see text] is a primitive fourth root of unity) and a new set of matrices [Formula: see text], instead of the original matrices [Formula: see text] of the Bhabha–Madhavarao algebra, are introduced. It is shown that in terms of the [Formula: see text] 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 [Formula: see text], where [Formula: see text] is some (complex) deformation parameter entering into the definition of the [Formula: see text]-matrices. In addition, a set of the matrices [Formula: see text] and [Formula: see text] possessing the properties of projectors is introduced. These operators project the matrices [Formula: see text] 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 para-superspace for the propagator of a massive spin-[Formula: see text] particle in a background gauge field within the Bhabha–Madhavarao approach is discussed.