Dirac Equation with Space Contributions Embedded in a Quantum-Corrected Gravitational Field

TL;DR

This paper explores the Dirac equation within a quantum-corrected gravitational field incorporating relativistic and quantum effects, analyzing multipole expansions and proposing the Bethe-ansatz method for solutions.

Contribution

It introduces a framework for including space contributions in the Dirac equation with quantum and relativistic corrections, and discusses solution strategies for complex potentials.

Findings

Formulation of the Dirac equation with generalized gravitational and electromagnetic potentials.

Analysis of multipole expansions for scalar and vector potentials.

Proposal of the Bethe-ansatz approach for solving the quantum-corrected Coulomb problem.

Abstract

The Dirac equation is considered with the recently proposed generalized gravitational interaction (Kepler or Coulomb), which includes post-Newtonian (relativistic) and quantum corrections to the classical potential. The general idea in choosing the metric is that the spacetime contributions are contained in an external potential or in an electromagnetic potential which can be considered as a good basis for future studies of quantum physics in space. The forms considered for the scalar potential and the so-called vector (magnetic) potential, can be viewed as the multipole expansion of these terms and therefore the approach includes a simultaneous study of multipole expansions to both fields. We also discuss several known generalizations of the Coulomb potential within this formulation in terms of certain Heun functions. The impossibility of solving our equation for the quantum-corrected Coulomb terms using known exact or quasi-exact nonperturbative analytical techniques is discussed, and finally the Bethe-ansatz approach is proposed to overcome this challenging problem.