Non-relativistic Approximation for a Spin 3/2 Particle in Presence of Electromagnetic and Gravitational Fields

Abstract

The goal of the present paper is investigation of the non-relativistic approximation in the first order 39-component theory for a spin 2 particle, in curved space-time, and in presence of external electromagnetic fields. We start with the generally covariant matrix equations generalized according to Weyl–Fock–Ivanenko tetrad method. We apply explicit expressions for four main matrices Γa with dimension 16 16 in the relevant first order system of equations, for space-time metrics allowing for existence of the non-relativistic equations. For distinguishing the large and small constituents in the complete wave function, we use three projective operators constructed on the base of the minimal polynomial of the 4-th order for the matrix Γ0 16×16. The relevant large and small components are found in explicit form. Among them we have found independent variables; in particular, among the large components there exist only four independent ones. Acting in accordance with the known general procedure, we have derived the non-relativistic system of equations for a 4-component wave function; the relevant Hamiltonian depends on electromagnetic field and additional geometrical terms are determined by the Ricci rotation coefficients (these terms should be determined by Ricci scalar R and Ricci tensor Rab in tetrad form. The terms describing interaction of the magnetic moment of the spin 3/2 particle with the external magnetic field is separated< this additional term is constructed with the use of the spin matrices Si and the components of the magnetic field ⃗B .