Strings in an electric field, and the Milne Universe

TL;DR

The paper tackles the fate of spacelike singularities in the Milne orbifold by analyzing twisted closed-string sectors and their potential to regulate contraction via pair production. It develops a detailed first-principles treatment of zero-modes, drawing a precise analogy with charged open strings in a constant electric field, and demonstrates the existence of physical twisted-state scattering states and Schwinger-like production. By constructing in/out states in static and Rindler coordinates and performing analytic continuation to Euclidean space, the authors derive non-normalizable Euclidean observables and outline vertex operators suitable for S-matrix calculations in time-dependent backgrounds. The work provides a concrete framework to study time-dependence in string theory and suggests how twisted-state dynamics could influence singularity resolution, while highlighting the need for nonperturbative tools to capture backreaction and potential condensation effects.

Abstract

Arguably the simplest model of a cosmological singularity in string theory, the Lorentzian orbifold $\Real^{1,1}/boost$ is known to lead to severe divergences in scattering amplitudes of untwisted states, indicating a large backreaction toward the singularity. In this work we take a first step in investigating whether condensation of twisted states may remedy this problem and resolve the spacelike singularity. By using the formal analogy with charged open strings in an electric field, we argue that, contrary to earlier claims, twisted sectors do contain physical scattering states, which can be viewed as charged particles in an electric field. Correlated pairs of twisted states will therefore be produced, by the ordinary Schwinger mechanism. For open strings in an electric field, on-shell wave functions for the zero-modes are determined, and shown to analytically continue to non-normalizable modes of the usual Landau harmonic oscillator in Euclidean space. Closed strings scattering states of the Milne orbifold continue to non-normalizable modes in an unusual Euclidean orbifold of $\Real^2$ by a rotation by an irrational angle. Irrespective of the formal analogy with the Milne Universe, open strings in a constant electric field, or colliding D-branes, may also serve as a useful laboratory to study time-dependence in string theory.