Higher-Order Wave Equation Within the Duffin–Kemmer–Petiau Formalism

Within the framework of the Duffin–Kemmer–Petiau (DKP) formalism a consistent approach to derivation of the third-order wave equation is suggested. For this purpose, an additional algebraic object, the so-called q -commutator ( q is a primitive cubic root of unity) and a new set of matrices η μ instead of the original matrices β μ of the DKP algebra are introduced. It is shown that in terms of these η-matrices, we have succeeded to reduce the procedure of the construction of cubic root of the third-order wave operator to a few simple algebraic transformations and to a certain operation of passage to the limit z → q , where z is some complex deformation parameter entering into the definition of the η μ -matrices. A corresponding generalization of the result obtained to the case of interaction with an external electromagnetic field introduced through the minimal coupling scheme is performed. The application to the problem of construction within the DKP approach of the path integral representation in parasuperspace for the propagator of a massive vector particle in a background gauge field is discussed.