Strings in an electric field, and the Milne Universe

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.