Subluminal and superluminal velocities of free-space photons

TL;DR

This paper analyzes the propagation velocities of free-space photons, demonstrating that wavepackets have inherently subluminal group velocities and superluminal phase velocities, with implications for electromagnetic and quantum descriptions.

Contribution

It provides a comprehensive analysis of photon velocities in free space, clarifying the relationship between group and phase velocities using multiple theoretical frameworks.

Findings

Wavepackets exhibit subluminal group velocities and superluminal phase velocities.

Explicit calculations for Gaussian and higher-order beams support the theoretical results.

Highlights subtleties in the quantum-mechanical photon wavefunction description.

Abstract

We consider rectilinear free-space propagation of electromagnetic wavepackets using electromagnetic field theory, scalar wavepacket propagation, and quantum-mechanical formalism. We demonstrate that spatially localized wavepackets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals $c^2$. These velocities are also known as the 'energy' and 'momentum' velocities, introduced by K. Milton and J. Schwinger. We illustrate general conclusions by explicit calculations for Gaussian and higher-order beams and wavepackets, and also highlight subtleties of the quantum-mechanical description based on the 'photon wavefunction'.