Nonrelativistic Approximation for a Spin-3/2 Particle in Combined Electromagnetic and Gravitational Fields

This paper investigates the nonrelativistic approximation in the first-order 39-component theory for a spin-3/2 particle in curved spacetime under external electromagnetic fields. Starting from the generally covariant matrix equations, generalized via the Weyl-Fock-Ivanenko tetrad method, we employ explicit forms of the four main $${{{\Gamma }}^a}$$ matrices of dimension $$16 \times 16$$ within the corresponding first-order system. The analysis is carried out for spacetime metrics that permit the existence of nonrelativistic equations. To separate the large and small components of the complete wave function, three projective operators are constructed from the fourth-order minimal polynomial of the $$\Gamma _{16 \times 16}^0$$ matrix. The explicit forms of these components are obtained, and the set of independent variables is identified; in particular, only four large components are independent. Following the standard procedure, we derive the nonrelativistic system of equations for a 4-component wave function. The resulting Hamiltonian includes contributions from the electromagnetic field and additional geometric terms expressed through the Ricci rotation coefficients, Ricci scalar R and Ricci tensor $${R_{ab}}$$ . We also isolate the term describing the interaction between the magnetic moment of the spin-3/2 particle and the external magnetic field. This interaction term is expressed via the spin matrices $${S_i}$$ and the components of the magnetic field vector B.