On a possible quantum manifestation of the perihelion advance in a hydrogen-like atom

Abstract

In a static and spherically symmetric case, by following an analog procedure for obtaining a Reissner–Nordstrom solution, thanks to the use of both the Einstein-type equation for an electromagnetic field as well as the Maxwell equation in curved space–time, we managed to find the Schwarzschild-like solution outside a charged sphere Q in interaction with a test charged particle q at an atomic scale where the gravitational force is negligibly small compared to the electrostatic force. Then, in the context of an extension of the equivalence principle to the electromagnetic interaction, we have studied the hydrogen atom. At the atomic scale, where quantum effects can no longer be ignored and the concept of trajectory is abandoned, the corrective terms derived from the geodesic equation for the electron are used within a perturbation theory. At the first order, these radial corrections of the Hamiltonian can partially remove the degeneracy of Bohr’s levels through a Lamb-like shift of the energy levels (2s) and (2p), estimated to E(2s) – E(2p) = 2.4595087102 × 10−4 eV. To match the experimental value of the Lamb shift, 4.372 × 10−6 eV, we have postulated the existence of an electromagnetic background that would describe statistically the vacuum energy by the introduction of a “cosmological”-like constant in the Einstein-like equation for electromagnetism. We have pointed out a divergence when n → +∞ in energy corrections due to a “cosmological” constant that technically could be overcome by the use of a cutoff as well as the renormalization of the electron’s mass.