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Strings in an electric field, and the Milne Universe
Open accessAbstract
Arguably the simplest model of a cosmological singularity in string theory, the Lorentzian orbifold ℝ 1,1 /boost is known to lead to severe divergences in scattering amplitudes of untwisted states, indicating a large backreaction towards 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 string scattering states of the Milne orbifold continue to non-normalizable modes in an unusual Euclidean orbifold of ℝ 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.
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TL;DR
The paper argues that twisted sectors of the Lorentzian orbifold \(R^{1,1}/\text{boost}\) (the Milne Universe) do contain physical scattering states, contrary to earlier claims, by analogy with charged open strings in a constant electric field. It proposes a zero-mode quantization yielding a continuous spectrum of delta-normalizable states and Schwinger pair production, and gives a Euclidean continuation to non-normalizable modes and an unusual rotation orbifold.
Key findings
- Twisted-sector zero modes behave as charged particles in an electric field: the inverted harmonic oscillator has a continuum of delta-normalizable states with real positive \(M^2\), with reflection and transmission coefficients \(R=-1/(1+e^{-\pi M^2/\nu})\), \(T=1/(1+e^{\pi M^2/\nu})\), satisfying \(R^2=1+T^2\); the stimulated pair-creation rate is \(\eta=R^2-1\).
- The vacuum persistence amplitude is \(|0,\text{in}|0,\text{out}|^2=\exp[-\int dp\,\ln(1+e^{-\pi M^2/\nu})]\), agreeing with the two-dimensional Schwinger creation rate. In Milne space, the semi-classical pair-production distribution is \(w(X^+,X^-)=1/[(j-\nu X^+X^-)^2+2M^2 X^+X^-]\), diverging on the light-cone and in the whiskers; for \(M^2=0\) the total number of produced pairs is logarithmically divergent.
- Analytic continuation maps the electric field to a Euclidean magnetic field and the Milne orbifold to a Euclidean rotation orbifold by angle \(\beta\), with non-normalizable modes rather than the usual localized twisted states; for irrational \(\beta\), smoothness requires winding/spectrally flowed states.
- Low-lying physical states include a bosonic open-string tachyon with \(M^2=k_i^2+\nu^2-2\), a first-excited gauge boson with \(D-2\) transverse degrees of freedom despite a \(\nu^2\) mass-squared gap, and closed-string twisted sectors with scalar tachyon at level \((0,0)\), massless graviton with \((D-2)^2\) transverse degrees at \((1,1)\), and gauge bosons at \((0,1)/(1,0)\) with boost momentum \(\pm 2\pi/(w\beta)\).
Methods
- First-quantized string theory: charged open strings in a constant electric field and closed strings in the \(w\)-twisted sector of \(R^{1,1}/\text{boost}\), with the identification \(w\beta=-2\pi\nu=-2\,\text{arcth}(\pi e_0)\) for \(e_1=0\).
- Zero-mode quantization via the charged Klein-Gordon equation in static coordinates (inverted harmonic oscillator), Rindler coordinates (Whittaker functions), and light-cone coordinates; old covariant quantization is used for levels 0 and 1, and one-loop amplitudes are compared.
- Analytic continuation to Euclidean signature: rotate \(t=-i\tau\), \(\nu=i b\) for the electric-to-magnetic case, and \(\beta=i\theta\) for the Milne-to-rotation-orbifold case; non-normalizable vertex operators and Wilson-line observables are outlined.
Limitations
- The paper states that the dynamics and backreaction are not solved: whether twisted-state condensation resolves the singularity or prevents the circle from reaching zero size cannot be decided by the analogy alone; the D-brane collision outcome depends on production versus recombination and bound-state formation.
- It states that the physical interpretation of the Milne one-loop amplitude is obscured by poles in the fundamental domain, twisted vertex operators are left for future work, and the Euclidean continuation proposal “clearly more work is needed to establish its terms of usage.”
- Evident from the text: the analysis is largely kinematic/first-quantized, relies on a formal analogy between open strings in an electric field and closed strings in the Milne orbifold, and does not compute the full backreacted S-matrix or resolve the non-perturbative multi-particle condensation problem.
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In the paper
“It is now possible to construct plane-wave combinations, such that only incoming or outgoing waves are present at u = ±∞:”
§ Strings in an electric field, and the Milne Universe > 3.2.3 In and out states
hedged- Asserted: passes — The authors put it forward as true.
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In the paper
“The Klein-Gordon equation now takes the form of an inverted harmonic oscillator in one variable, -∂ 2 u - 1 4 u 2 + M 2 2ν ψ p(u) = 0 (3.24)”
§ Strings in an electric field, and the Milne Universe > 3.2.1 The inverted harmonic oscillator
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“As we shall see, in both the electric and Milne case, the zero-mode degrees of freedom describe the motion of a charged particle (the string center of motion) in an electric field.”
§ Strings in an electric field, and the Milne Universe > 1. Introduction
plainstates an associationn = 2- Asserted: passes — The authors put it forward as true.
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In the paper
“The level matching condition L 0 -L0 = 0 relates the product of the boost momentum J by the winding w to the difference of occupation numbers of the excited levels,”
§ Strings in an electric field, and the Milne Universe > 3.1.2 Closed strings in the Milne Universe
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“In Rindler coordinates, best suited for the Lorentzian orbifold case where one needs to project on states with integer boost momentum J, they correspond to eigenstates of the Schrödinger equation with Liouville-like potential V (y) = M 2 e 2y -(J + 1 2 νe 2y ) 2 .”
§ Strings in an electric field, and the Milne Universe > 1. Introduction
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In the paper
§ Strings in an electric field, and the Milne Universe > 3.2.2 Parabolic cylinder functions
“φ(u) = 1 4 u 2 - M 2 ν log |u| + π 4 + 1 2 arg Γ 1 2 + i M 2 2ν (3.28)”
§ Strings in an electric field, and the Milne Universe > 3.2.1 The inverted harmonic oscillator
“-∂ 2 u - 1 4 u 2 + M 2 2ν ψ p(u) = 0 (3.24)”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Strings in an electric field, and the Milne Universe > 5.2 The electric field case
“[dφ][dφ * ] e -d d xdτ {|(∂τ -i b 2 x)φ| 2 +|(∂x+i ν 2 τ )φ| 2 } (5.6)”
§ Strings in an electric field, and the Milne Universe > 5.2 The electric field case
“t = e -iθ τ, ν = e iθ b (5.5)”
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In the paper
§ Strings in an electric field, and the Milne Universe > 2.3.3 One-loop amplitude
“The one-loop amplitude for the Milne orbifold has been computed in [28,29], using an Euclidean world-sheet but a Lorentzian target space.”
§ Strings in an electric field, and the Milne Universe > References
“28. T. N. Tomaras, N. C. Tsamis and R. P. Woodard, "Pair creation and axial anomaly in light-cone QED(2)," JHEP 0111 (2001) 008 [arXiv:hep-th/0108090].”
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(2002) · arXiv (Cornell University)
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Metadata
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- Berkooz, M., & Pioline, B.
- Affiliations
- Weizmann Institute of Science (IL)
- Laboratoire de Physique Théorique et Hautes Energies (FR)
- ORCID
- Pioline, B
- Venue
- Journal of Cosmology and Astroparticle Physics
- Publisher
- IOP Publishing
- Published
- 2003-11-17
- Volume
- 2003
- Issue
- 11
- Pages
- 007-007
- ISSN
- 1475-7516 (ISSN-L)
- OpenAlex
- W1976678443
- Semantic Scholar
- dfeca4474b8d9e26401ebdc508feb3affbb85fcb
- Cited by
- 88
- References
- 81
- Citations by year
- 2026: 1 · 2023: 1 · 2022: 1 · 2020: 3 · 2018: 1 · 2017: 1 · 2014: 1
- FWCI
- 9.02
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Cosmology and Astroparticle Physics (publishedVersion)
- HAL (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Minerva Foundation
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- Milne universe, cosmological singularity, twisted states, Schwinger mechanism, backreaction, string cosmology
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-11-14, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-11-14 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
