Lorentz-covariant quantum 4-potential and orbital angular momentum for the transverse confinement of matter waves

In two recent papers exact Hermite-Gaussian solutions to relativistic wave equations were obtained for both electromagnetic and particle beams. The solutions for particle beams correspond to those of the Schr\"odinger equation in the nonrelativistic limit. Here, it will be shown that each beam particle has additional 4-momentum resulting from transverse localization compared to a free particle traveling in the same direction as the beam with the same speed. This will be referred to as the quantum 4-potential term since it will be shown to play an analogous role in relativistic Hamiltonian quantum mechanics as the Bohm potential in the nonrelativistic quantum Hamilton-Jacobi equation. Low-order localization effects include orbital angular momentum, Gouy phase, and beam spreading. Toward a more systematic approach for calculating localization effects at all orders, it will be shown that both the electromagnetic and quantum 4-potentials couple into the canonical 4-momentum of a particle in a similar way. This offers the prospect that traditional methods used to calculate the affect of an electromagnetic field on a particle can now be adapted to take localization effects into account. The prospects for measuring higher order quantum 4-potential related effects experimentally are also discussed alongside some questions to challenge the quantum information and quantum field theorists.