Non-relativistic description of the Dirac-KaЁhler particle on the background of curved space-time

Abstract

In 16-component relativistic wave equation for the Dirac–Kähler particle, the procedure of the non-relativistic approximation in presence of external electromagnetic field is performed. An eight-component quantum mechanical Pauli-like equation is constructed, the wave function includes scalar, pseudoscalar, 3-vector and 3-pseudovector. The Pauli equation is invariant with respect to spatial P -reflection4 it consists of two disconnected sub-systems for scalar-pseudovector and pseudoscalar-vector respectively. In presence of only electric field, the Pauli equation reduces to more simple form of four disconnected wave equations for scalar, pseudoscalar, vector, and pseudovector. These features are substantial for physical interpretation of the Dirac–Kähler particle: it interacts in very different manner with magnetic and electric fields. This approach is generalized to a Riemannian space-time structure. Curved geometry substantially influences the structure of the nonrelativistic equation. This theory is considered in more detail on the background of the spherical Riemann space.