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A wavepacket (electromagnetic or otherwise) within an isotropic andhomogeneous space can be quantized on a regular lattice of discrete k-vectors.Each k-vector is associated with a temporal frequency omega; together, k andomega represent a propagating plane-wave. While the total energy and totallinear momentum of the packet can be readily apportioned among its individualplane-wave constituents, the same cannot be said about the packet's totalangular momentum. One can show, in the case of a reasonably smooth (i.e.,continuous and differentiable) wave packet, that the overall angular momentumis expressible as an integral over the k-space continuum involving only theFourier transform of the field and its k-space gradients. In this sense, theangular momentum is a property not of individual plane-waves, but of plane-wavepairs that are adjacent neighbors in the space inhabited by the k-vectors, andcan be said to be localized in the k-space. Strange as it might seem, thishallmark property of angular momentum does not automatically emerge from ananalysis of a discretized k-space. In fact, the discrete analysis shows theangular momentum to be distributed among k-vectors that pair not only withnearby k-vectors but also with those that are far away. The goal of the presentpaper is to resolve the discrepancy between the discrete calculations and thoseperformed on the continuum, by establishing the conditions under which thehighly non-local sum over plane-wave pairs in the discrete k-space wouldapproach the localized distribution of the angular momentum across thecontinuum of the k-space.

The continuum limit of k-space cavity angular momentum is controlled by an infinite range difference operator