On the quantum Coulomb field

TL;DR

This paper extends the quantum Coulomb field theory by defining an energy operator, analyzing its eigenstates, and proposing a way to test quantum fluctuations' effects on atomic energy levels, linking theory with recent experiments.

Contribution

It introduces a classical energy operator in the quantum Coulomb field theory and suggests experimental tests using hydrogen spectroscopy.

Findings

Defined a quantum energy operator for the Coulomb field

Identified eigenstates and energy levels of the excitations

Proposed experimental verification through hydrogen and muonic-hydrogen spectroscopy

Abstract

The quantum theory of the Coulomb field has been developed by Staruszkiewicz in the long series of papers. This theory explains the universality and quantization of the electric charge observed in Nature. Moreover, the efforts have been made to determine the value of the elementary charge from its mathematical structure. Nonetheless, no other immediate applications of this theory have been proposed. We make such an attempt by (i) considering the classical energy operator and defining its counterpart in the quantum theory of the Coulomb field; (ii) determining the eigenstates of the energy operator and assigning energy to the excitations of the theory; and (iii) proposing a simple theoretical scheme to estimate the effect of the quantum fluctuations of the Coulomb field on the energy levels of hydrogen-like atoms. We argue that the recent experimental advances in hydrogen and muonic-hydrogen spectroscopy may provide the unique window of opportunity for the verification of the Staruszkiewicz's theory.