Open quantum systems and Schwinger-Keldysh holograms

TL;DR

This paper develops a framework for open quantum field theories where the environment is a strongly coupled holographic bath described by AdS/CFT, and derives an open effective action for a system scalar by integrating out the bath. Central to the construction is the gravitational Schwinger-Keldysh (grSK) saddle, a complex two-sheeted bulk geometry that fills the Schwinger-Keldysh contour and yields real-time bath correlators via boundary-to-bulk propagators, including ingoing (retarded) and outgoing (advanced) modes. The authors compute quadratic and nonlinear influence functionals in various dimensions, provide explicit 2d results, and show how the holographic bath induces both dissipation through quasinormal modes and fluctuations through Hawking radiation, with a stochastic open EFT exhibiting nonlinear fluctuation-dissipation relations. They also establish how to renormalize the open system's sources to obtain a finite, local effective theory and outline extensions to gauge fields, conserved currents, OTO observables, and gravity backreaction, highlighting the broader significance for holographic real-time observables and open quantum dynamics. Overall, the work offers a principled holographic route to open quantum dynamics, linking black hole physics, SK causality, and stochastic descriptions in a unified framework.

Abstract

We initiate the study of open quantum field theories using holographic methods. Specifically, we consider a quantum field theory (the system) coupled to a holographic field theory at finite temperature (the environment). We investigate the effects of integrating out the holographic environment with an aim of obtaining an effective dynamics for the resulting open quantum field theory. The influence functionals which enter this open effective action are determined by the real-time (Schwinger-Keldysh) correlation functions of the holographic thermal environment. To evaluate the latter, we exploit recent developments, wherein the semiclassical gravitational Schwinger-Keldysh saddle geometries were identified as complexified black hole spacetimes. We compute real-time correlation functions using holographic methods in these geometries, and argue that they lead to a sensible open effective quantum dynamics for the system in question, a question that hitherto had been left unanswered. In addition to shedding light on open quantum systems coupled to strongly correlated thermal environments, our results also provide a principled computation of Schwinger-Keldysh observables in gravity and holography. In particular, these influence functionals we compute capture both the dissipative physics of black hole quasinormal modes, as well as that of the fluctuations encoded in outgoing Hawking quanta, and interactions between them. We obtain results for these observables at leading order in a low frequency and momentum expansion in general dimensions, in addition to determining explicit results for two dimensional holographic CFT environments.

Figures (6)

• Figure 1: A comparison of the (a) thermofield double and (b) Schwinger-Keldysh complex time contours for a system prepared in a thermal state. The starting and end points of the contour are identified. The associated Euclidean (imaginary time) periodicity is set by the inverse temperature $\beta$.
• Figure 2: The two-sheeted complex grSK geometry shown from two different perspectives. On the top left we display the boundary thermal SK contour which is filled in the Euclidean portion by the Euclidean black hole geometry (the cigar) and in the Lorentzian section by two copies of the domain of outer communication of the Lorentzian black hole spacetime. The top right panel displays the bulk perspective to emphasize the smooth join of the two sheets of the Lorentzian section. On the bottom panel we illustrate the Lorentzian sections of the geometry on the Schwarzschild-AdSd+1 Penrose diagram. with the regions pertaining to the $L$ and $R$ sheets of the grSK spacetime shaded.
• Figure 3: The complex $r$ plane with the locations of the two boundaries and the horizon marked. The grSK contour is a codimension-1 surface in this plane (drawn at fixed $v$). The direction of the contour is as indicated counter-clockwise encircling the branch point at the horizon.
• Figure 4: The numerical values of the quantities $\mathfrak{g}_{0,2}$ and $i\, \mathfrak{g}_{2,0}$ as a function of the conformal dimension $\Delta$ in dimensions $2$, $3$, and $4$, respectively. We have confined attention to the case of relevant operators $\Delta \in (\frac{d}{2},\frac{d}{2}+1)$ when the integrals are convergent without need of additional counterterms.
• Figure 5: Illustration of Witten diagrams on the grSK geometry computing 3 and 4-point influence functions of the boundary field theory. The boundary operator insertions (blue) lie on the thermal SK contour. The bulk field is constructed using the boundary-bulk propagators, and the bulk vertex is integrated over the Lorentzian section of the grSK geometry.