Why photons cannot be sharply localized

TL;DR

This paper demonstrates the fundamental limitations on localizing photons sharply in space, showing that only electric or magnetic aspects can be momentarily localized, while coherent states can be localized without such restrictions.

Contribution

It introduces a novel approach using space-dependent photon operators to analyze localization, revealing intrinsic quantum constraints on photon localization.

Findings

Photon energy and number densities cannot be sharply localized, only approximately.

Sharp localization of electric or magnetic footprints is possible only momentarily.

Coherent states can be localized without limitations.

Abstract

Photons cannot be localized in a sharply defined region. The expectation value of their energy density and the photon number density can only be approximately localized, leaving an exponential tail. We show that one may sharply localize either electric or magnetic (but not both) footprints of photons, and only momentarily. In the course of time evolution this localization is immediately destroyed. However, the coherent states, like their classical counterparts, can be localized without any limitations. The main tool in our analysis is a set of space-dependent photon creation and annihilation operators defined without any reference to the mode decomposition.