Classical angular momentum of light: A paradox and its resolution

Abstract

Angular momentum carried by a classical circularly polarized electromagnetic plane wave (light) appears to be identically zero inasmuch as its linear field-momentum density is directed along wave propagation, and, therefore, the angular momentum, being the integrated moment of the linear momentum density about an axis parallel to the direction of propagation, necessarily vanishes — in detail. This, however, contradicts the established fact that circularly polarized light does carry angular momentum that remains classically non-zero. The paradox is resolved in a physically transparent manner by treating this problem as that of a transversely bounded, and hence necessarily non-trans verse, electromagnetic wave propagating along a circular waveguide, in the limit as its radius tends to infinity. We get a non-zero angular momentum that bears the correct ratio to wave energy. This angular momentum derives essentially and exactly from the boundary conditions for the geometry considered. This is an interesting example of surface terms giving a volume (bulk) contribution, much as in the entirely different context of orbital diamagnetism, which was regarded as a surprise of theoretical physics by Rudolf Peierls.[1]