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

Abstract

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 [Phys.Lett.A {\bf 244} (1998) 329] on the basis of 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 $z\to q$, 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's 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 $z\to q$ 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.

arXiv comment39 pages

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• On the electromagnetic interaction and the anomalous term in the Duffin-Kemmer-Petiau theory
• Unitary quantization and para-Fermi statistics of order two
• Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. I