The Initial Value Problem For Maximally Non-Local Actions

TL;DR

This paper investigates the initial value problem for a class of non-local actions, revealing that solutions are closely related to local theories with parameters governed by algebraic equations, potentially linking to quantum mechanics interpretations.

Contribution

It introduces a new class of non-local actions with solutions connected to local theories and explores their implications for initial data constraints and quantum interpretations.

Findings

Solutions are at most discretely enlarged compared to local theories.

Parameters vary with initial data according to algebraic equations.

Roots of algebraic equations may represent quantum wave components.

Abstract

We study the initial value problem for actions which contain non-trivial functions of integrals of local functions of the dynamical variable. In contrast to 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.