Consistent formulation of relativistic dynamics for massive spin-zero particles in external fields

It is shown that in the context of stochastic-phase-space formulations of quantum mechanics a nonlocal type of nonrelativistic as well as relativistic dynamics for a massive spin-zero particle moving in an external electromagnetic field can be formulated in $L^2\left(\Gamma \right)$ space. In a sharp-point limit $s\to +0\mathrm{}$ the nonrelativistic dynamics merges into the conventional local model. The relativistic dynamics is gauge invariant, it can be formulated covariantly in the relativistic phase space $M_m$, it possesses a covariant propagator, it leaves the space of positive-energy solutions of the Klein-Gordon equation invariant, and it gives rise to a covariant and conserved probability current on stochastic phase space. Furthermore, for states describing a particle moving at nonrelativistic speeds in relation to a given inertial frame, the relativistic propagation contracts into its nonrelativistic counterpart. The sharp-point limit $s\to +0\mathrm{}$ of the relativistic $S$ matrix also exists in each such inertial frame. The necessity of interactions acting at nonsharp configuration points for a consistent formulation of covariant relativistic one-particle dynamics is traced to the question of the nonexistence of covariant position operators in relativistic theories. The mandatory character of this feature is shown to be firmly rooted in the theory of quantum measurement.