Uncertainty relation for photons

TL;DR

This paper derives an uncertainty relation for photons within quantum electrodynamics, overcoming the challenge of defining a photon position operator, and reveals a Schrödinger-like equation for the photon wave function in momentum space.

Contribution

It introduces a photon uncertainty relation based on energy density and momentum spread, providing a novel approach to photon localization issues.

Findings

Photon energy density acts as a probability density in configuration space.

Derived an uncertainty relation for photons analogous to Heisenberg's.

Photon wave function in momentum space satisfies a Schrödinger-like equation.

Abstract

Uncertainty relation for photons that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space combined with a measure of the spread in the photon momentum leads to an inequality that is a natural counterpart of the standard Heisenberg relation. Unexpectedly, the equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schr\"odinger equation in coordinate space in the presence of electric and magnetic charges.