  The Laguerre-Gauss modes are a class of fundamental and well-studied optical
fields. These stable, shape-invariant photons - exhibiting circular-cylindrical
symmetry - are familiar from laser optics, micro-mechanical manipulation,
quantum optics, communication, and foundational studies in both classical
optics and quantum physics. They are characterized, chiefly, by two modes
numbers: the azimuthal index indicating the orbital angular momentum of the
beam - which itself has spawned a burgeoning and vibrant sub-field - and the
radial index, which up until recently, has largely been ignored. In this
manuscript we develop a differential operator formalism for dealing with the
radial modes in both the position and momentum representations, and - more
importantly - give for the first time the meaning of this quantum number in
terms of a well-defined physical parameter: the "intrinsic hyperbolic momentum
charge".
