A General Prescription for Semi-Classical Holography

Budhaditya Bhattacharjee; Chethan Krishnan

REVIEW 4 major objections 5 minor 25 references

A General Prescription for Semi-Classical Holography

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A screen-source prescription for semi-classical holography beyond AdS.

desk verdict Source-on-screen holography is a genuinely different and concretely worked-out prescription, but the advertised generality rests on PDE well-posedness that the authors explicitly leave unproved. read the letter →

arxiv 1908.04786 v2 pith:GTM4NESN submitted 2019-08-13 hep-th

Budhaditya Bhattacharjee , Chethan Krishnan This is my paper · ORCID

classification hep-th

keywords semi-classicalholographyholographicscreensourceprescriptionFeynmanpropagatorhomogeneousmodebulkreconstructionflatspaceoperatordictionary

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a general recipe for semi-classical holography in spacetime regions that are not anti-de Sitter space, with flat space as a primary example. Instead of fixing the boundary values of bulk fields on the screen, the recipe puts localized sources on the screen and solves the bulk wave equation with those sources, using the Euclidean Green function that dies at infinity. The paper argues that this source prescription reproduces standard AdS/CFT when the screen sits at the boundary of AdS, while in more general settings it has two advantages: the analytic continuation of the Euclidean Green function is the Feynman propagator, and a regular homogeneous mode appears that plays the role of the normalizable mode and enables bulk reconstruction. The claims are demonstrated for a scalar field in 3+1 dimensional flat space with a timelike $\mathbb{R}\times S^2$ screen.

What carries the argument

The load-bearing object is the Euclidean bulk-to-bulk Green function $G_E$ that dies at infinity, used as a bulk-to-screen propagator for the source $J_0$, along with the regular homogeneous mode $\varphi_h$ that appears in Lorentzian signature. The former replaces the Dirichlet Green function, which is defined by vanishing on the screen and whose analytic continuation is not generally the Feynman propagator; the latter is the analogue of the AdS normalizable mode. The smearing kernel for bulk reconstruction is obtained from the spacelike version of this Green function via a Green's theorem identity.

What would settle it

Take a specific spacetime region with a timelike screen that meets the paper's stated sufficient conditions (e.g., a tube in Schwarzschild exterior whose constant-time intersections are $S^{d-1}$) and solve the Lorentzian scalar wave equation with a screen-localized source. If the regular homogeneous mode does not exist, or if the analytic continuation of the Euclidean Green function is not the Feynman propagator, the prescription's reconstruction and two-point function formulas fail. In the flat-space example, the correlator (3.5) can be computed directly; comparing it with the result of the alternative boundary-value prescription would test whether the two approaches are genuinely inequivalent at finite screen radius.

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Extended reading notes

Core claim

The central claim is that semi-classical holographic correlators can be computed from a bulk source problem on the holographic screen, rather than a boundary value problem. Concretely, the bulk field satisfies (2.2) with a source $J_0$ localized at $r=R$, the solution is (2.3) built from the bulk-to-bulk Green function $G_E$ that vanishes at Euclidean infinity, and the on-shell action equals $\int \sqrt{\gamma}\,\varphi(R,x)J_0(R,x)$. In Lorentzian signature, the Green function analytically continues to the Feynman propagator, and an additional regular homogeneous solution $\varphi_h$ appears; its on-screen value is the one-point function, and the two-point function is the restriction of the Green function. The extrapolate and differential dictionaries for computing screen correlators coincide, and the prescription reduces to the standard boundary-value AdS/CFT dictionary at the AdS boundary via the scaling map (2.13). Bulk reconstruction proceeds through a smearing kernel derived from the spacelike version of the Green function.

Load-bearing premise

The prescription assumes that for the spacetime regions in question the Euclidean source problem (2.2)-(2.4) has a unique solution with a Green function that dies at infinity, that its analytic continuation is the Feynman propagator, and that a regular homogeneous mode $\varphi_h$ exists in Lorentzian signature; the paper states these conditions with 'we expect' language and leaves the class of spacetimes imprecise.

Editorial extensions

If this is right

The source prescription gives explicit holographic correlators for flat space with a timelike $\mathbb{R}\times S^2$ screen, and perturbative bulk interactions can be added using standard bulk perturbation diagrams.
Bulk reconstruction works in non-AdS regions: the homogeneous mode's boundary value determines the bulk field through a smearing kernel, extending the AdS reconstruction program.
The extrapolate and differential dictionaries coincide in the source prescription, a property that does not hold for the Dirichlet prescription in general.
The prescription reframes mechanics: instead of boundary conditions at a submanifold, one may localize sources on that submanifold and impose fall-off at infinity, a viewpoint with potential applications beyond holography.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the homogeneous mode exists generically, it could serve as a probe of bulk geometry in settings where the screen is not at infinity, potentially giving a handle on interior physics in black-hole or cosmological spacetimes; the paper only sketches this possibility.
The equivalence of source and boundary-value prescriptions at the AdS boundary suggests there may be a larger class of geometries where a rescaling of the source mimics a boundary value problem; searching for such geometries could sharpen the boundary of where the new prescription is needed.
In flat space, taking the large-$R$ limit of the screen correlators may yield flat-space S-matrix elements in the regime where $E R \gg 1$, mirroring the AdS/CFT relation to scattering; the paper raises this question but does not establish it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Buddy, here's the take.

The paper is a programmatic proposal for doing semi-classical holography with sources on the screen instead of Dirichlet boundary values. The genuinely new piece is the source prescription: you solve (2.2) with a delta-function source at r=R, use the bulk-to-bulk Euclidean Green function that dies at infinity, and define correlators by varying the on-shell action (2.6). That choice gives you, in Lorentzian signature, a homogeneous mode φ_h that behaves like the AdS normalizable mode, and it opens the door to HKLL-like reconstruction. The explicit flat-space example with an R×S^2 screen is concrete: the two-point function, the mode expansion (4.3), and the kernel (4.4) are all checkable, and the AdS limit correctly reduces to the standard dictionary with the scaling map (2.13). The authors are also honest that the match between extrapolate and differential dictionaries is basically built into the definition of the on-shell action—that's not a hidden circularity, it's just a feature of their setup.

The soft spot is the one the stress test flags, and the paper itself concedes it. The general prescription requires existence and uniqueness of the Euclidean Green function with decay at infinity, its analytic continuation as the Feynman propagator, and completeness of the homogeneous modes. None of that is proved for general screens. Footnote 7 says 'we expect' and footnote 15 worries about leakage through the top/bottom of a tube-like region. So the central claim is conditional. That's a real gap, but it's an addressable one—it's a matter of PDE analysis, not a contradiction. The appendix's alternate prescription (integrating only to the screen) is less compelling and the resulting correlator (A.5) looks messy, but it's peripheral.

This is a paper for people working on flat-space holography and bulk reconstruction. It gives a different route from Solodukhin and Li-Takayanagi, and the concrete example makes it worth engaging with. I would not cite it as a proof of anything, but I'd cite it as a formulation.

My recommendation: send it to peer review. A good referee should push for a sharper statement of the class of spacetimes and at least a plausibility argument for the homogeneous mode, but the paper is honest, the example is real, and the idea deserves to be in the literature. I'd bring it to the reading group.

Referee Report

4 major / 5 minor

Summary. The paper proposes a source-based reformulation of semi-classical holography. Instead of solving a Dirichlet boundary-value problem with boundary data on the holographic screen, the authors solve a bulk wave equation with a source localized on the screen, using the Euclidean Green function that vanishes at infinity. The on-shell action is then written as the integral of the bulk field times the source on the screen, and correlators are obtained by functional differentiation with respect to the source. In Lorentzian signature the authors add a regular homogeneous mode to the sourced solution, interpret it as a state-dependent one-point function, and construct an HKLL-like smearing kernel for bulk reconstruction. They claim the prescription reduces to standard AdS/CFT at the boundary of AdS, works for flat space with an R times S^2 timelike screen, and matches the extrapolate and differential dictionaries. The paper also sketches perturbative interactions and a broader philosophy of describing dynamics via sources on submanifolds.

Significance. If the prescription is correct for a wide class of screens, it provides a concrete semi-classical holographic dictionary beyond AdS, with the notable advantage that the bulk propagator is the standard Feynman propagator and it yields a natural analogue of the normalizable mode. The paper is clearly written and the basic construction is simple and explicit, especially in the flat-space example where the Green functions and mode expansion are given. The main value is as a proposal: it identifies a plausible generalization of the AdS/CFT dictionary and spells out the calculational scheme, including correlators, homogeneous modes, and bulk reconstruction. However, the claimed generality is not established, and some of the advertised checks are definitional rather than substantive.

major comments (4)

[Sec. 2, Eq. (2.4), and Sec. 3, Eq. (3.2)]
[Sec. 2, Eqs. (2.5)-(2.7) and Sec. 3, Eqs. (3.3)-(3.5)]
[Sec. 3, text before Eq. (3.2), and Sec. 4.3, Eq. (4.3)]
[Sec. 4.3 and Appendix A]

minor comments (5)

[Sec. 1, footnote 5]
[Sec. 2, Eq. (2.13)]
[Sec. 4.3, Eq. (4.4)]
[Sec. 5, paragraph after Eq. (5.2)]
[References]

Circularity Check

1 steps flagged · score 5.0 of 10

The claimed match between the extrapolate and differential dictionaries is built into the definition of the on-shell action; the AdS-limit check and flat-space Green-function computations are independent, so the circularity is partial.

self definitional [Section 2, Eq. (2.7); see also Section 3, Eqs. (3.3)-(3.5)]
"When evaluated at zero-source, this procedure leads to a vanishing 1-point function, and a two-point function of the form ⟨O(x′)O(x′′)⟩ = G(R,x′;R,x′′). In other words, the differential prescription yields an on-screen 2-point function that is just the restriction of the bulk 2-point function to the screen. In yet other words, what we have done here amounts to a demonstration of the equivalence between the differential and the extrapolate dictionaries!"

The on-shell action (2.6) was deliberately chosen, by integrating the action to infinity, to be S = ∫√γ φ(R,x) J0(R,x). Combining this with (2.3), φ(R,x) = ∫ G(R,x;R,x′) J0(R,x′), makes S a quadratic functional of J0 whose second functional derivative is identically the screen-restricted Green function G(R,x′;R,x′′). The 'extrapolate dictionary' value is the same screen-restricted bulk two-point function, so the match is true by construction rather than by independent derivation.

full rationale

The paper is a prescription paper: the central formulas (2.3), (2.7) and (3.2)-(3.5) define the holographic dictionary rather than deriving it from an external framework. The advertised consistency check that the extrapolate and differential dictionaries match is self-definitional, because S was chosen to make it true. However, there are independent, non-circular contents: the reduction to standard AdS/CFT is checked explicitly via the bulk-to-bulk Green function near the boundary and the map (2.13) using the standard AdS propagator; the flat-space example computes explicit Euclidean/Lorentzian/spacelike Green functions (4.2) and a momentum-space HKLL kernel (4.4); and no load-bearing claim rests on a same-author citation ([14] is a companion paper supplying further details, not the argument). The unproved existence/uniqueness of the Green function and homogeneous mode is a genuine gap in generality, but it is a correctness/rigor risk, not circularity. Overall, one central consistency claim reduces by construction; the rest has independent content. Hence a moderate partial-circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The prescription depends on PDE existence and uniqueness, on a specific Green function choice and its analytic continuation, and on the regular homogeneous mode structure. These are domain assumptions explicitly flagged by the authors through 'we expect' wording. No numerical data are fitted, and no new particles, forces, or dimensions are introduced.

assumptions (5)

domain assumption The bulk metric can be put in Gaussian normal form ds^2 = dr^2 + gamma_ab dx^a dx^b with the holographic screen at r = R, and the screen separates spacetime into two disconnected regions.
Used to set up the source problem (2.2); the paper notes that choosing the screen to be normal to the radial coordinate restricts generality (Sec. 2).
domain assumption The Euclidean bulk-to-bulk Green function that vanishes at spatial infinity exists, is unique, and analytically continues to the Feynman propagator in Lorentzian signature for the screens under consideration.
This is the core of the prescription; the paper states 'we expect' these solutions to exist and be regular and unique (Sec. 2), with no general proof.
domain assumption In Lorentzian signature, the homogeneous solution phi_h of the wave equation is regular everywhere and can be added to the sourced solution, providing state data phi_h(R,x).
The claim that the divergent Euclidean mode is replaced by regular homogeneous solutions, and that phi_h(R,x) and its radial derivative are not independent, is used for bulk reconstruction (Secs. 3 and 4.1); no general proof is given.
domain assumption The bulk theory is a free scalar with two-derivative action, and interactions can be added perturbatively so that the semiclassical on-shell action is a valid generating functional.
The entire dictionary is formulated for this scalar; gauge fields, gravitons, and nonlinear gravity are left as open questions (Sec. 5).
standard math Standard properties of Euclidean and Lorentzian Green functions for Laplace and wave operators, including integration by parts and Green's theorem, hold.
Used to derive S = integral phi J0 (Eq. 2.6) and the HKLL reconstruction formula (Eq. 4.1).

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Pith. "Pith review of A General Prescription for Semi-Classical Holography." pith.science (2026). https://pith.science/paper/GTM4NESN

@misc{pith2026190804786,
  author       = {Pith},
  title        = {Pith review of: A General Prescription for Semi-Classical Holography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GTM4NESN}},
  note         = {Machine review of arXiv:1908.04786}
}

read the original 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 spacetimes, including flat space with timelike screens. We find that our approach reduces to the standard Dirichlet-like approach, when restricted to the boundary of AdS. 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. We also find that the extrapolate and differential dictionaries match. 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.

Figures

Figures reproduced from arXiv: 1908.04786 by the authors.

**Figure 1.** The blue region denotes the Euclidean d+1-dimensional bulk M. The ddimensional boundary aka holographic screen is ∂M, the bulk field is φ(r, x), and the source on the screen is J0(R, x). The coordinate r represents the bulk foliation and r = R will be taken as the screen. The region M can be a sub-region of spacetime, and the x is d-dimensional. screens connected to the trivial (ie., non-existent) screen. Note that… view at source ↗

**Figure 2.** The blue region is bounded by a timelike tubular holographic screen in the exterior [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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J. M. Maldacena, “The Large N limit of superconformal ﬁeld theories and supergrav- ity,” Int. J. Theor. Phys.38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)] doi:10.1023/A:1026654312961, 10.4310/ATMP.1998.v2.n2.a1 [hep-th/9711200]

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S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from non- criticalstringtheory,” Phys.Lett.B 428, 105(1998)doi:10.1016/S0370-2693(98)00377-3 [hep-th/9802109]

arXiv 1998

[3] [3]

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E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys.2, 253 (1998) doi:10.4310/ATMP.1998.v2.n2.a2 [hep-th/9802150]

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[4] [4]

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V. Balasubramanian, P. Kraus and A. E. Lawrence, “Bulk versus bound- ary dynamics in anti-de Sitter space-time,” Phys. Rev. D 59, 046003 (1999) doi:10.1103/PhysRevD.59.046003 [hep-th/9805171]

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arXiv 1999

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D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, “Correlation functions in the CFT(d) / AdS(d+1) correspondence,” Nucl. Phys. B546, 96 (1999) doi:10.1016/S0550- 3213(99)00053-X [hep-th/9804058]. 22

arXiv 1999

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I. R. Klebanov and E. Witten, “AdS / CFT correspondence and symmetry breaking,” Nucl. Phys. B556, 89 (1999) doi:10.1016/S0550-3213(99)00387-9 [hep-th/9905104]

arXiv 1999

[8] [8]

Holographic representation of local bulk operators,

A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, “Holographic representation of local bulk operators,” Phys. Rev. D74, 066009 (2006) doi:10.1103/PhysRevD.74.066009 [hep-th/0606141]

arXiv 2006