Dynamics with Infinitely Many Time Derivatives and Rolling Tachyons

TL;DR

This paper explores the unique initial value problem in theories with infinite derivatives, analyzing time-dependent tachyon solutions in p-adic string theory and their implications for string dynamics and tachyon matter.

Contribution

It demonstrates how infinite derivatives constrain initial conditions and provides detailed solutions for rolling tachyons, revealing unexpected oscillatory behaviors.

Findings

Small oscillations at the tachyon vacuum for even potentials

Unbounded growth of oscillations for non-even potentials

Implications for closed string emergence and tachyon matter

Abstract

Both in string field theory and in p-adic string theory the equations of motion involve infinite number of time derivatives. We argue that the initial value problem is qualitatively different from that obtained in the limit of many time derivatives in that the space of initial conditions becomes strongly constrained. We calculate the energy-momentum tensor and study in detail time dependent solutions representing tachyons rolling on the p-adic string theory potentials. For even potentials we find surprising small oscillations at the tachyon vacuum. These are not conventional physical states but rather anharmonic oscillations with a nontrivial frequency--amplitude relation. When the potentials are not even, small oscillatory solutions around the bottom must grow in amplitude without a bound. Open string field theory resembles this latter case, the tachyon rolls to the bottom and ever growing oscillations ensue. We discuss the significance of these results for the issues of emerging closed strings and tachyon matter.