Quantum Limit for the Emittance of Dirac Particles Carrying Orbital Angular Momentum

In this article, we highlight that the interaction potential confining Dirac particles in a box must be invariant under the charge conjugation to avoid the Klein paradox, in which an infinite number of negative-energy particles are excited. Furthermore, we derive the quantization rules for a relativistic particle in a cylindrical box, which emulates the volume occupied by a beam of particles with a non-trivial aspect ratio. We apply our results to the evaluation of the quantum limit for emittance in particle accelerators. The developed theory allows the description of quantum beams carrying Orbital Angular Momentum (OAM). We demonstrate how the degeneracy pressure is such to increase the phase-space area of Dirac particles carrying OAM. The results dramatically differ from the classical evaluation of phase-space areas, that would foresee no increase in emittance for beams in a coherent state of OAM. We discuss the quantization of the phase-space cell's area for single Dirac particles carrying OAM, and, finally, provide an interpretation of the beam entropy as the measure of how much the phase-space area occupied by the beam deviates from its quantum limit.