Correlation functions of the Bjorken flow in the holographic Schwinger-Keldysh approach

TL;DR

This paper develops a systematic method to compute correlation functions in nonequilibrium holographic states, specifically for the Bjorken flow, by extending the Schwinger-Keldysh formalism and analyzing the dual black hole geometry.

Contribution

It generalizes the horizon cap prescription to the hydrodynamic Bjorken flow, enabling systematic calculation of correlation functions in expanding nonequilibrium states.

Findings

Correlation functions are thermal in the perfect fluid limit.

The method applies a Weyl rescaling to achieve constant temperature and entropy density.

Stokes data encode quantum fluctuations behind the evolving horizon.

Abstract

One of the outstanding problems in the holographic approach to many-body physics is the explicit computation of correlation functions in nonequilibrium states. We provide a new and simple proof that the horizon cap prescription of Crossley-Glorioso-Liu for implementing the thermal Schwinger-Keldysh contour in the bulk is consistent with the Kubo-Martin-Schwinger periodicity and the ingoing boundary condition for the retarded propagator at any arbitrary frequency and momentum. The generalization to the hydrodynamic Bjorken flow is achieved by a Weyl rescaling in which the dual black hole's event horizon attains a constant surface gravity and area at late time although the directions longitudinal and transverse to the flow expands and contract respectively. The dual state's temperature and entropy density thus become constants (instead of the perfect fluid expansion) although no time-translation symmetry emerges at late time. Undoing the Weyl rescaling, the correlation functions can be computed systematically in a large proper time expansion in inverse powers of the average of the two reparametrized proper time arguments. The horizon cap has to be pinned to the nonequilibrium event horizon so that regularity and consistency conditions are satisfied. Consequently, in the limit of perfect fluid expansion, the Schwinger-Keldysh correlation functions with space-time reparametrized arguments are simply thermal at an appropriate temperature. A generalized bilocal thermal structure holds to all orders. We argue that the Stokes data (which are functions rather than constants) for the hydrodynamic correlation functions can decode the quantum fluctuations behind the horizon cap pinned to the evolving event horizon, and thus the initial data.