Keldysh rotation in the large- N expansion and string theory out of equilibrium

Abstract

We extend our study of the large-N expansion of general nonequilibrium 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± - defined as the values of M on the forward and backward segments of the closed time contour - for their linear combinations Mcl and Mqu, known as the "classical"and "quantum"fields. First we develop a novel "signpost"notation for nonequilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each world-sheet surface ς in the dual string theory expansion was found to exhibit a triple decomposition into the parts ς± corresponding to the forward and backward segments of the closed time contour, and ς∧ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the world-sheet surface ς of the dual string theory undergoes a very different natural decomposition: ς consists of a "classical"part ςcl and a "quantum embellishment"part ςqu. We show that both parts of ς carry their own independent genus expansion. The nonequilibrium sum over world-sheet topologies is naturally refined into a sum over the double decomposition of each ς into its classical and quantum part. We apply this picture to the classical limits of the quantum nonequilibrium 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.