Dynamics with Infinitely Many Derivatives: The Initial Value Problem

TL;DR

This paper investigates the initial value problem for infinite order differential equations relevant in string theory and cosmology, revealing conditions for well-posedness and the number of initial data needed, with implications for nonlocal theories.

Contribution

It provides a formal operator calculus framework for analyzing the initial value problem in infinite order differential equations, clarifying the number of initial data required and its physical interpretation.

Findings

Infinite order differential equations typically require two initial data per pole of the propagator.

Certain infinite order equations admit well-posed initial value problems with only two initial data.

String field theory and p-adic string theory are not among the equations with minimal initial data, but can be made ghost-free through operator definitions.

Abstract

Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists. Though these equations have been studied in the classical mathematical literature, it appears that the physics community is largely unaware of the relevant formalism. Of particular importance is the fate of the initial value problem. Under what circumstances do infinite order differential equations possess a well-defined initial value problem and how many initial data are required? In this paper we study the initial value problem for infinite order differential equations in the mathematical framework of the formal operator calculus, with analytic initial data. This formalism allows us to handle simultaneously a wide array of different nonlocal equations within a single framework and also admits a transparent physical interpretation. We show that differential equations of infinite order do not generically admit infinitely many initial data. Rather, each pole of the propagator contributes two initial data to the final solution. Though it is possible to find differential equations of infinite order which admit well-defined initial value problem with only two initial data, neither the dynamical equations of p-adic string theory nor string field theory seem to belong to this class. However, both theories can be rendered ghost-free by suitable definition of the action of the formal pseudo-differential operator. This prescription restricts the theory to frequencies within some contour in the complex plane and hence may be thought of as a sort of ultra-violet cut-off.