On excited states in real-time AdS/CFT

TL;DR

The authors analyze the Skenderis–van Rees real-time holographic prescription and generalize it to excited states by turning on nonzero Euclidean boundary data, thereby preparing initial/final wave functionals for non-vacuum CFT states.They demonstrate that these SvR-excited states are coherent in the AdS Hilbert space, derive the corresponding boundary-mode eigenvalues, and compute the inner product, 1- and 2-point functions to confirm the coherent-state structure holographically.A detailed bulk analysis solves the Klein–Gordon equation on Lorentzian and Euclidean AdS regions, matches solutions across gluing surfaces, and shows the on-shell action encodes the excited-state data in a Gaussian inner product and standard AdS/CFT correlators, independent of the specific excitations for certain observables.Overall, the work shows that Hartle–Hawking-type wavefunctionals can be generalized to excited gravitational states in AdS, offering a precise holographic realization of coherent excitations and guiding principles for semiclassical gravity in holographic settings.

Abstract

The Skenderis-van Rees prescription, which allows the calculation of time-ordered correlation functions of local operators in CFT's using holographic methods is studied and applied for excited states. Calculation of correlators and matrix elements of local CFT operators between generic in/out states are carried out in global Lorentzian AdS. We find the precise form of such states, obtain an holographic formula to compute the inner product between them, and using the consistency with other known prescriptions, we argue that the in/out excited states built according to the Skenderis-Van Rees prescription correspond to {\it coherent} states in the (large-$N$) AdS-Hilbert space. This is confirmed by explicit holographic computations. The outcome of this study has remarkable implications on generalizing the Hartle-Hawking construction for wave functionals of excited states in AdS quantum gravity.

Figures (6)

• Figure 1: (a) Euclidean AdS (hyperbolic space) and the insertion of a source at its conformal boundary. (b) Lorentzian AdS with a source at its boundary. We also depict the initial and final wave functions.
• Figure 2: (a) In-Out contour in complex t-plane appropriate for scattering problems showing a temporal evolution $\Delta T=T_+-T_-$. (b) SvR geometry dual to In-Out QFT contour obtained by gluing together Lorentzian ${\cal M}_L$ and Euclidean ${\cal M_\pm}$ AdS sections. We explicitly depict the gluing surfaces $\Sigma^\pm$. On the bottom we show the glued geometry and a generic smooth configuration on the spacetime.
• Figure 3: In standard global coordinates, the point $\tau = \pm \infty$ is mapped to the origin of the conformal boundary of $\mathcal{M}_\pm$.
• Figure 4: The manifold shown can be interpreted as taking figure \ref{['SvR']} and splitting it the Lorentzian piece, and also removing $\mathcal{M}_+$ since in principle, the final state $\Psi_f$ is not necessarily expressed as any (Euclidean) path integral.
• Figure 5: The configuration considered in figure \ref{['SvR']}b becomes an Euclidean AdS space when the Lorentzian cilinder ${\cal M}$ length is set to zero.