A New Formulation of the Initial Value Problem for Nonlocal Theories

TL;DR

This paper proposes a novel way to formulate the initial value problem in nonlocal theories, using an initial function over a smeared time interval, which improves understanding of stability and initial conditions in string-inspired models.

Contribution

It introduces a new formulation of the initial value problem for nonlocal theories, replacing initial data with an initial function over a time interval, with implications for stability analysis.

Findings

Unstable directions in phase space are inaccessible from physical initial functions.

Previous unstable solutions correspond to unphysical initial conditions.

The new formulation clarifies stability issues in nonlocal cosmological models.

Abstract

There are a number of reasons to entertain the possibility that locality is violated on microscopic scales, for example through the presence of an infinite series of higher derivatives in the fundamental equations of motion. This type of nonlocality leads to improved UV behaviour, novel cosmological dynamics and is a generic prediction of string theory. On the other hand, fundamentally nonlocal models are fraught with complications, including instabilities and complications in setting up the initial value problem. We study the structure of the initial value problem in an interesting class of nonlocal models. We advocate a novel new formulation wherein the Cauchy surface is "smeared out" over the underlying scale of nonlocality, so that the the usual notion of initial data at t=0 is replaced with an "initial function" defined over -M^{-1} \leq t \leq 0 where M is the underlying scale of nonlocality. Focusing on some specific examples from string theory and cosmology, we show that this mathematical re-formulation has surprising implications for the well-known stability problem. For D-brane decay in a linear dilaton background, we are able to show that the unstable directions in phase space cannot be accessed starting from a physically sensible initial function. Previous examples of unstable solutions in this model therefore correspond to unphysical initial conditions, an observation which is obfuscated in the old formulation of the initial value problem. We also discuss implication of this approach for nonlocal cosmological models.