Third order wave equation in Duffin-Kemmer-Petiau theory: Massive case

Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism a more consistent approach to the derivation of the third order wave equation obtained earlier by M. Nowakowski [1] on the basis of introduction heuristic considerations 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 [eta] mu instead of the original matrices beta mu of the DKP algebra are introduced. It is shown that in terms of these [eta] mu matrices we have succeeded in reducing a 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 the passage to the limit z arrow right q, where z is some complex deformation parameter entering into the definition of the [eta]-matrices. A corresponding generalization of the result obtained to the case of the interaction with an external electromagnetic field introduced through the minimal coupling scheme is carried out and a comparison with M. Nowakowski result is performed. A detailed analysis of the general structure for a solution of the first order differential equation for the wave function [Psi](x; z) is performed and it is shown that the solution is singular in the z arrow right q limit. 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.