On exact solutions of the Dirac equation in a homogeneous magnetic field in the Lobachevsky space

TL;DR

This paper derives exact solutions to the Dirac equation for a spin-1/2 particle in a hyperbolic Lobachevsky space with a magnetic field, revealing how curvature influences quantum energy levels.

Contribution

It provides the first exact solutions of the Dirac equation in Lobachevsky space with a magnetic field, extending quantum mechanics to curved geometries.

Findings

Derived explicit solutions to the Dirac equation in hyperbolic space.

Obtained a generalized formula for energy quantization in curved space.

Showed the influence of negative curvature on particle energy levels.

Abstract

There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in Riemannian space of constant negative curvature, hyperbolic Lobachevsky space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the motion of the particle in magnetic field on the background of the Lobachevsky space geometry, has been obtained.