Localization of relativistic particles and uncertainty relations

TL;DR

This paper explores two photon localization schemes based on energy density, analyzing their properties and how they relate to the Heisenberg uncertainty principle for relativistic particles.

Contribution

It introduces and compares two novel localization schemes for relativistic particles, clarifying their mathematical structure and physical implications.

Findings

One scheme yields a positive operator-valued measure for localization.

Both schemes' uncertainty relations approach the Heisenberg bound.

The schemes connect localization with polarization density matrices.

Abstract

Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces the effective 3x3 polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound $\Delta p \Delta z= \hbar/2$