Subluminal and superluminal velocities of free-space photons

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Abstract
We consider rectilinear free-space propagation of electromagnetic wave packets using electromagnetic field theory, scalar wave-packet propagation, and quantum-mechanical formalism. We demonstrate that spatially localized wave packets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals c2. 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 wave packets, and also highlight the subtleties of the quantum-mechanical description based on the “photon wavefunction”.

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