Photon Momentum in Linear Dielectric Media

Abstract

According to the scientific literature, the momentum of a photon in a simple linear dielectric is either $\hslash \omega /(nc)$ or $n\hslash \omega /c$ with a unit vector ${\hat{\mathbf{e}}}_k$ in the direction of propagation. These momentums are typically used to argue the century-old Abraham--Minkowski controversy in which the momentum density of the electromagnetic field in a dielectric is either the Abraham momentum density, ${\mathbf{g}}_A=\mathbf{E}\times \mathbf{H}/c$, or the Minkowski momentum density, ${\mathbf{g}}_M=\mathbf{D}\times \mathbf{B}/c$. The elementary optical excitations, photons, are typically known as polaritions in the particular case of light traveling in a dielectric medium. Applying the relativistic energy formula, we find that the total momentum that is attributable to a polariton in a dielectric is $\hslash \omega {\hat{\mathbf{e}}}_k/c$ corresponding to a total momentum density ${\mathbf{g}}_T=n\mathbf{E}\times \mathbf{B}/c$.