Symplectic structure in open string field theory I: Rolling tachyons

TL;DR

The paper develops a covariant phase-space symplectic framework for open string field theory and applies it to rolling tachyon configurations, navigating transgressive locality by performing momentum-space calculations in Lorentzian signature. It validates the approach in a scalar EFT with stub operators and then implements it in Siegel-gauge open SFT, computing the energy of rolling tachyons up to quartic order and comparing with boundary-state data. The results reproduce the boundary-state coefficients with high accuracy and clarify the relation between the new formalism and the Witten/Cho–Mazel–Yin constructions, illuminating how different sigmoid choices affect the observable energy. Overall, the work strengthens the connection between covariant phase space methods and concrete tachyon dynamics in SFT, and provides practical computational tools via oscillator methods for higher-point amplitudes.

Abstract

We discuss a new formula for the symplectic structure on the phase space of open string field theory. Revisiting the setup of Cho, Mazel, and Yin, we use the formula to compute the energy of rolling tachyon solutions on unstable D-branes. An important aspect of the analysis is dealing with the singular ultraviolet behavior of string vertices in Lorentzian signature, a feature we refer to as transgressive locality. This forces us to carry out computations in momentum space, where time and causality are somewhat obscure. Nevertheless the symplectic structure appears to be sensible, giving results in agreement with boundary state computations. As further confirmation of our methods, we study the symplectic structure for rolling tachyons in scalar effective field theory, where vertices show similar high energy behavior to string field theory but the physics is that of local field theory. This model gives interesting insight into the runaway oscillations of the rolling tachyon.

Figures (6)

• Figure 3.1: The delta function $\delta(E-ib)$ can be realized as the limit (\ref{['eq:Gaussian']}) of Gaussian distributions. In the grey region the limit is divergent, while outside the limit vanishes. In this sense the delta function with imaginary argument vanishes for large real energies.
• Figure 3.2: The exponential rolling tachyon solution at $\Lambda\mu^2=1/2$, shown in red, compared to the solution of the original $\phi^3$ theory, shown in black. In the effective field theory the vacuum decay enters a phase of violent oscillation. The peaks and troughs of the oscillation have been cropped out of the figure after the first two turns. In the original $\phi^3$ theory, by contrast, the configuration returns to the unstable vacuum in the infinite future.
• Figure 3.3: Because the rolling tachyon solution in the effective field theory diverges faster than exponentially for large real energies, to describe the time evolution we must implement a Fourier transform over a contour in the quadrant above or below the real axis. Closing the contours at infinity we obtain an infinite sum of residues at positive or negative integer multiples of $i\mu$. This creates two solutions given respectively as an expansion in powers of $e^{-\mu t}$ or $e^{\mu t}$ with infinite radius of convergence. The former gives a good representation of the evolution of the background towards the future, while the later gives a better representation in the past.
• Figure 4.1: The correlation function (\ref{['eq:4tach']}) is defined by connecting the star product of two Fock vacua to the star product of another two Fock vacua by a Siegel gauge propagator strip. A plane wave insertion of momentum appears exactly half way along the length of the propagator strip.
• Figure 4.2: Eigenvalues of the quadratic form $Q^{AB}$ plotted as a function of the modulus $y$. The results are obtained by truncating the matrices at mode number $L=100$. The zero eigenvalue corresponds to the momentum $k^3$ inserted in the propagator, and the remaining eigenvalues are positive.