Dirac Equation and Planck-Scale Quantities

TL;DR

This paper explores how Planck-scale quantities naturally appear in the Dirac equation within a gravitational context, proposing a gravitational atom model that becomes meaningful under certain quantum and mass conditions.

Contribution

It introduces a test theory linking Dirac equation solutions with Planck quantities and derives energy eigenvalues for a gravitational atom model based on Newtonian gravity.

Findings

Exact energy eigenvalues for gravitational atoms derived

A meaningful gravitational atom model requires test mass below Planck mass

Planck quantities naturally emerge in quantum theory with Lorentz symmetry

Abstract

This work investigates in which form quantities with Planck dimensions occur already in the common quantum theory with local Lorentz symmetry. Since such Planck quantities as Planck length or Planck mass involve the Planck constant h, the velocity of light c and the Newton gravitational constant G, the relativistic Dirac equation (h, c) in the Newtonian gravitational potential (G) can be considered as a test theory. The evaluation of the break-off condition of the power series of the radial energy eigenfunctions of a purely gravitational atom leads to exact terms for the energy eigenvalues E for various special cases of the quantum numbers N, k and n = N + |k|. It turns out that a meaningful atom model, based solely on Newtonian gravitational forces, can result if, inter alia, the test mass m in the gravitational field of the mass M is selected to be smaller than the Planck mass.