Equation for Doublets of Particles with Half-Integer Spin and Parasupersymmetry

It is well-known that equations for particles with spin more than 12 have a lot of defects such as: acausality [1], incompatibility of the theoretical value of gyromagnetic ratio g = 1s with experimental value g = 2 for any spin, complex value of energy of charged particle in constant magnetic field and others. In the papers [2, 3] tensor-bispinor equation was considered, which describes the motion of particle with half-integer spin. It was shown [2, 3], that tensor-bispinor equation remains causal after introducing the interaction with electromagnetic field. Moreover, this equation, in the case of minimal interaction, is equivalent to the equation in Dirac-like form, which was considered in the paper [4]. The aim of this paper is generalization of the result that was obtained in the papers [2,3], for the case of anomalous interaction quadratic in electromagnetic field. We note that there is a kind of anomalous interaction quadratic in electromagnetic field at which the extra degeneration of energy levels of particle appear. This degeneratior is caused by parasupersymmetry of considered equations. In Section 2 we consider tensor-bispinor equations for free particle with arbitrary half-integer spin. We also show the equivalence of such equations and the Dirac-like equations [4]. In Section 3 the minimal and anomalous interaction with electromagnetic field is discussed. Section 4 is dedicated to the problem of motion a particle in constant magnetic field. The condition is found at which the problem of complex energy levels does not arise. Finally, in Section 5, we retrieve the conditions the constants to which should satisfy of interactions in order that the equation for particle in constant magnetic field be parasupersymmetric (It is worth to note that parasupersymmetry of equations for spin-1 and spin-0 was considered by Beckers, Debergh, Nikitin [5] and Sergeyev [6]). We also obtain corresponding parasupercharges.