A General Prescription for Semi-Classical Holography

Abstract

We present a version of holographic correspondence where bulk solutions with sources localized on the holographic screen are the key objects of interest, and not bulk solutions defined by their boundary values on the screen. We can use this to calculate semi-classical holographic correlators in fairly general spacetime regions, including flat space with timelike boundaries. In AdS, the distinction between our approach and the standard Dirichlet-like approach is superficial. But in more general settings, the analytic continuation of the Dirichlet Green function does not lead to a Feynman propagator in the bulk. Our prescription avoids this problem. Furthermore, in Lorentzian signature we find an additional homogeneous mode. This is a natural proxy for the AdS normalizable mode and allows us to do bulk reconstruction. Perturbatively adding bulk interactions to these discussions is straightforward. We conclude by elevating some of these ideas into a general philosophy about mechanics and field theory. We argue that localizing sources on suitable submanifolds can be an instructive alternative formalism to treating these submanifolds as boundaries.

Page Curve and the Information Paradox in Flat Space

“…The implicit assumption here is that the bulk metric, though quantum corrected, is still a well-defined quantity. In passing, let us also note that the semi-classical holographic correspondence between the bulk and the screen can be developed quite a bit [21], with striking parallels to the conventional AdS/CFT correspondence [2,26]. The key difference of course in many of these discussions being that the on-screen correlation functions one finds are not those of a conventional local theory.…”

“…Along with the conformal boundary, we will also find it useful to have the notion of a holographic screen (of finite but large radius) in flat space, introduced previously in [19]. See also a closely related, but different aspect of the screen in [21]. We will present evidence that to describe the evolution of the black hole states that we are after, we can treat the Hilbert space of the full quantum gravity to be an approximate tensor factorization between semi-classical limit it contains black holes, the information paradox needs a resolution in flat space.…”

“…This is similar to the picture in [14,15] where the Hawking radiation left the AdS/CFT system via a coupling at the boundary. Though perhaps less familiar to the reader, a parallel construction can be done in asymptotically flat spaces where the source lives on a codimension-1 holographic screen [21] 8 . This leads again to the correct Page curve, but now we are extracting Hawking radiation out of the quantum gravity Hilbert space.…”

“…We will do this in four steps. First, we note that a spherical sub-region on a t = 0 slice of the Minkowski space at the asymptotic boundary of AdS can be defined via the waists of symmetric boundary causal diamonds 21 . Second, we note that an arbitrary sub-region can be defined in terms of a (possibly infinite) union of such solid spheres, and therefore in terms of symmetric causal diamonds.…”

Dirichlet Baths and the Not-so-Fine-Grained Page Curve

“…• Our Dirichlet prescription here can be viewed as the doubly holographic analogue to the Gibbons-Hawking idea of fixing field values at the cut-off. It would be interesting to see if a more natural formulation would be to work with sources at the boundary instead of boundary values, as suggested in [36].…”