Quantum Fluctuations of a Coulomb potential

TL;DR

This paper investigates the quantum fluctuations of the Coulomb potential, demonstrating gauge independence and finite correlations outside the particle localization region, with implications for the measurability of electromagnetic fields.

Contribution

It provides an explicit calculation of the long-range correlation function of the electromagnetic field using the Schwinger-Keldysh formalism, showing gauge independence and zero-order Planck constant contributions.

Findings

Root mean square fluctuation of Coulomb potential is 1/√2.

Correlation function is finite outside particle localization.

Quantum fluctuations are suppressed in macroscopic bodies by 1/N.

Abstract

Long-range properties of the two-point correlation function of the electromagnetic field produced by an elementary particle are investigated. Using the Schwinger-Keldysh formalism it is shown that this function is finite in the coincidence limit outside the region of particle localization. In this limit, the leading term in the long-range expansion of the correlation function is calculated explicitly, and its gauge independence is proved. The leading contribution turns out to be of zero order in the Planck constant, and the relative value of the root mean square fluctuation of the Coulomb potential is found to be 1/\sqrt{2}, confirming the result obtained previously within the S-matrix approach. It is shown also that in the case of a macroscopic body, the \hbar^0 part of the correlation function is suppressed by a factor 1/N, where N is the number of particles in the body. Relation of the obtained results to the problem of measurability of the electromagnetic field is mentioned.