Keldysh Rotation in the Large-N Expansion and String Theory Out of Equilibrium

TL;DR

This paper extends the large-N expansion and dual string theory description of non-equilibrium systems using the Keldysh rotation, revealing a new decomposition of worldsheet surfaces into classical and quantum parts with independent genus expansions.

Contribution

It introduces a novel Keldysh-rotated formalism and a signpost notation for non-equilibrium Feynman diagrams, leading to a new decomposition of dual string worldsheets into classical and quantum components.

Findings

Worldsheet surfaces decompose into classical and quantum parts after Keldysh rotation

Both parts have independent genus expansions in the dual string theory

Classical limits simplify the string perturbation theory to its classical form

Abstract

We extend our study of the large-$N$ expansion of general non-equilibrium many-body systems with matrix degrees of freedom $M$, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields $M_\pm$ -- defined as the values of $M$ on the forward and backward segments of the closed time contour -- for their linear combinations $M_{\textrm{cl}}$ and $M_{\textrm{qu}}$, known as the "classical" and "quantum" fields. First we develop a novel "signpost" notation for non-equilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each worldsheet surface $\Sigma$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts $\Sigma^\pm$ corresponding to the forward and backward segments of the closed time contour, and $\Sigma^\wedge$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the worldsheet surface $\Sigma$ of the dual string theory undergoes a very different natural decomposition: $\Sigma$ consists of a "classical" part $\Sigma^{\textrm{cl}}$, and a "quantum embellishment" part $\Sigma^{\textrm{qu}}$. We show that both parts of $\Sigma$ carry their own independent genus expansion. The non-equilibrium sum over worldsheet topologies is naturally refined into a sum over the double decomposition of each $\Sigma$ into its classical and quantum part. We apply this picture to the classical limits of the quantum non-equilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.