Spin 3/2 particle: Fradkin theory, nonrelativistic approximation

Abstract

The well-known relativistic wave equation for a spin 3/2 particle proposed by Pauli and Fierz is based on the use of the wave function with the transformation properties of vector-bispinor. Less known is the Fradkin theory based on the vector-bispinor wave function as well. At the vanishing Fradkin parameter Λ, this equation reduces to the Pauli – Fierz equation. To clarify the physical meaning of the additional parameter, in the present paper the nonrelativistic approximation in the Fradkin equation is studied, at this we take into account the presence of external electromagnetic fields. With the use of the technique of projective operators, we decompose the wave function into big and small constituents, and then derive a generalized nonrelativistic equation for a 16-component wave function. It is shown that when preserving only the terms of first order in the Fradkin parameter Λ after transition to 4 independent components of the nonrelativistic wave function there arises the ordinary nonrelativistic equation for the Pauli – Fierz theory without any additional interaction with electromagnetic fields. When preserving the terms of second order in parameter Λ, we obtain a 4-component nonrelativistic equation with additional interaction; however, only with the magnetic field. This interaction is quadratic in magnetic field components and governed by six 4-dimensional matrices. So the Fradkin theory may be understood as relevant to a particle with magnetic quadrupole moment.