Gauge dependence of spontaneous radiation spectrum in a time-dependent relativistic non-perturbative Coulomb field

TL;DR

This paper investigates how gauge choices affect the spectrum of spontaneous radiation in a time-dependent relativistic Coulomb field, revealing significant gauge dependence that can be experimentally measurable and offering insights into the physical meaning of gauge fields.

Contribution

It extends the gauge choice problem to relativistic non-perturbative Coulomb fields and demonstrates the measurable gauge dependence of radiation spectra in such scenarios.

Findings

Gauge dependence can be of 10 MHz or larger in the spectrum.

The gauge choice influences the effective external field used in modeling.

This gauge dependence can be exploited to understand the physical significance of gauge fields.

Abstract

We extend the "gauge choice" problem Lamb noticed to include a time-dependent relativistic non-perturbative Coulomb field, which can be produced by a cluster of relativistic charged particles. If adiabatic conditions are carefully maintained, such a field must be included along side the nuclear Coulomb potential when defining the atomic state. We reveal that when taking the external field approximation, the gauge choice for this time-dependent relativistic non-perturbative Coulomb field cannot be overcome by previous method, and leads to considerable gauge-dependence of the transient spontaneous radiation spectrum. We calculate explicitly with a simple one-dimensional charged harmonic oscillator that such a gauge-dependence can be of a measurable magnitude of 10 MHz or larger for the commonly used Coulomb, Lorentz, and multipolar gauges. Contrary to the popular view, we explain that this gauge dependence is not really a disaster, but actually an advantage here: The relativistic bound-state problem is so complicated that a fully quantum-field method is still lacking, thus the external field approximation cannot be derived and hence not guaranteed. However, by fitting to the experimental data, one may always define an effective external field, which may likely be parameterized with the gauge potential in a particular gauge. This effective external field would not only be of phenomenological use, but also shed light on the physical significance of the gauge field.