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