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

Abstract

One of the outstanding problems in the holographic approach to many-body physics is the explicit computation of correlation functions of non-equilibrium states. In this work, we develop the method to compute the Schwinger-Keldysh correlation functions in a holographic theory in the limit in which the state hydrodynamizes, i.e. when the stress tensor is well described by the hydrodynamic effective theory, in the context of the Bjorken flow. Firstly, 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 KMS 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 constant surface gravity and area at late time, implying constant temperature and entropy density for the dual state (instead of perfect fluid expansion) at large proper time although the directions longitudinal and transverse to the flow expands and contract respectively implying eternal absence of time-translation symmetry. Undoing the Weyl rescaling, the correlation functions can be computed systematically in the large proper time expansion. Remarkably, the horizon cap has to be pinned to the non-equilibrium event horizon so that regularity and consistency conditions are satisfied. This mirrors the causal nature of Schwinger-Dyson equations in field theory. One of our key results is that in the limit of perfect fluid expansion, the Schwinger-Keldysh correlation functions are simply thermal at an appropriate temperature when expressed in terms of reparametrized spacetime arguments. A generalized bi-local 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 details of the initial state.