Initial value problem in string-inspired nonlocal field theory

TL;DR

This paper studies a nonlocal scalar field theory inspired by string theory, showing how to reformulate it locally in time for certain configurations and analyzing its causality and initial value problem.

Contribution

It introduces a perturbative field redefinition that maps the nonlocal theory to a local one for time-dependent configurations and explores local formulations and causality.

Findings

Time-dependent solutions map to local scalar potential

Local in time formulation is achievable despite nonlocality

Causality is consistent with no superluminal behavior

Abstract

We consider a nonlocal scalar field theory inspired by the tachyon action in open string field theory. The Lorentz-covariant action is characterized by a parameter $\xi^2$ that quantifies the amount of nonlocality. Restricting to purely time-dependent configurations, we show that a field redefinition perturbative in $\xi^2$ reduces the action to a local two-derivative theory with a $\xi^2$-dependent potential. This picture is supported by evidence that the redefinition maps the wildly oscillating rolling tachyon solutions of the nonlocal theory to conventional rolling in the new scalar potential. For general field configurations we exhibit an obstruction to a local Lorentz-covariant formulation, but we can still achieve a formulation local in time, as well as a light-cone formulation. These constructions provide an initial value formulation and a Hamiltonian. Their causality is consistent with a lack of superluminal behavior in the nonlocal theory.