Fractional Orbital Angular Momentum as a Quantitative Measure of Wave-Particle Duality

In an earlier letter [Ducharme \textit{et al.} Phys. Rev. Lett. \textbf{126}, 134803 (2021)], a solution to the Dirac equation for a relativistic Gaussian electron beam showed that for a diverging beam the spin of each electron is the sum of fractional contributions from both the spin matrix and orbital angular momentum operators. It will be argued here that fractional orbital angular momentum (FOAM) is a measure of the wavelike character of a system such that fractional spin angular momentum (FSAM) as the complement of FOAM quantifies the system's particlelike character. Thus, diffraction splits the spin of a single particle into FOAM and FSAM as the double-slit experiment probabilistically splits it between distinguishable paths. Beyond diffraction it is shown Dirac equations for quantum bound systems have two alternate solutions - one wavelike and the other particlelike. The observable properties of the two solutions are identical except for the amount of FOAM in them. It is further shown that the quantity of FOAM is velocity dependent, indicating FOAM and FSAM are interrelated through Lorentz transformations. Overall, the implication being that the Dirac formalism has an intrinsic sense of the dualistic wave-particle nature of matter built into it.