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

TL;DR

This work studies the Dirac equation in curved spacetime with a generalized gravitational potential that includes post-Newtonian relativistic corrections and quantum corrections, embedding curvature effects into external scalar and vector potentials via a multipole expansion. Using z(r)=u/r+v/r^2+w/r^3 and a Bethe-ansatz approach, the authors transform the problem into a Schrödinger-like equation and obtain quasi-exact analytic solutions for the energy spectrum ε_n and corresponding wavefunctions ρ₁,n(r) under parameter constraints; explicit results are provided for the ground state (n=0) and the first excited state (n=1), as well as the special Coulomb limit, which recovers known results. The Bethe-ansatz equations govern the roots r_i and encode the necessary constraints among potential parameters, illustrating both the power and the limits of this method for such a nontrivial, Heun-type problem. The study offers a framework for quantum physics in spatially curved backgrounds and provides a nonperturbative handle on how multipole-space corrections to gravity influence relativistic quantum systems, with consistency checks against established Coulomb results. The findings have implications for theoretical explorations of quantum dynamics in curved space and related condensed-matter analogs where Dirac fermions experience curvature-like effects.

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.