Heisenberg uncertainty relation for photons

TL;DR

This paper derives a new uncertainty relation for photons based on electromagnetic energy distribution, introducing a center of energy operator that modifies the traditional Heisenberg bound, especially at finite energies.

Contribution

It presents an alternative photon uncertainty relation using the center of energy operator, revealing a higher bound that approaches the nonrelativistic limit at high energies.

Findings

The uncertainty bound is $3/2 \, \hbar$ in three dimensions.

The bound decreases to the nonrelativistic value at infinite momentum.

The relation is closer to the original Heisenberg principle than previous formulations.

Abstract

The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. {\bf 108}, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of energy operator. The noncommutativity of the components of the center of energy operator results in the increase of the bound $3/2\,\hbar$ in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the limiting case of infinite momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics.