Initial value problem for maximally nonlocal actions

We study the initial value problem for actions whose non-locality is “maximal” in the sense that there is no dependence upon the separation between points. In contrast with many other non-local actions, the classical solution set of these systems is at most discretely enlarged, and may even be restricted, with respect to that of a local theory. We show that the solutions are those of a local theory whose (spacetime constant) parameters vary with the initial value data according to algebraic equations. The various roots of these algebraic equations can be plausibly interpreted in quantum mechanics as different components of a multi-component wave function. It is also possible that the consistency of these algebraic equations imposes constraints upon the initial value data which appear miraculous from the context of a local theory. Although the discussion and examples are given in the context of simple mechanical systems the results should apply as well to field theory.