Levy flights and nonlocal quantum dynamics

We develop a fully fledged theory of quantum dynamical patterns of behavior that are nonlocally induced. To this end we generalize the standard Laplacian-based framework of the Schr"{o}dinger picture quantum evolution to that employing nonlocal (pseudodifferential) operators. Special attention is paid to the Salpeter (here, $m\ge 0$) quasirelativistic equation and the evolution of various wave packets, in particular to their radial expansion in 3D. Foldy’s synthesis of “covariant particle equations” is extended to encompass free Maxwell theory, which however is devoid of any “particle” content. Links with the photon wave mechanics are explored.