arxiv: 2509.04974 · v3 · submitted 2025-09-05 · ✦ hep-th

Minkowski Space holography and Radon transform

Samrat Bhowmick , Koushik Ray This is my paper

Pith reviewed 2026-05-18 19:09 UTC · model grok-4.3

classification ✦ hep-th

keywords Minkowski spaceRadon transformholographybulk reconstructionscalar fieldde SitterMellin modeshypergeometric functions

0 comments p. Extension

The pith

A free scalar field in Minkowski spacetime maps to a scalar field on a codimension-two sphere through Radon transform and bulk reconstruction.

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

The paper shows how a free scalar field living in flat Minkowski space can be connected to a field on a sphere that has two fewer dimensions. The connection works by taking the Radon transform of the Minkowski field, which produces a field on a hyperplane equivalent to a de Sitter or Euclidean anti-de Sitter space. That result is then matched to a field reconstructed from the codimension-two sphere using standard holographic bulk reconstruction methods. The authors also provide expressions for the Mellin modes of the original field as generalized hypergeometric functions. If correct, this gives a concrete dictionary between bulk flat-space data and lower-dimensional holographic data.

Core claim

We relate a free scalar field in the Minkowski spacetime with a scalar field with a certain scaling dimension on a sphere of codimension two. This is realised by first performing a Radon transform of the bulk field on the Minkowski space to a field on a hyperplane identified with a de Sitter or Euclidean anti-de Sitter slice. The Radon transform is identified in turn with a scalar field obtained from a sphere of one further lower dimension through the so-called bulk reconstruction programme. We write down the Mellin modes of the bulk field as generalised hypergeometric functions using the Lee-Pomeransky method developed for evaluation of Feynman loop diagrams.

What carries the argument

The Radon transform applied to the Minkowski bulk field, which maps it to a hyperplane field identified with the output of bulk reconstruction from the codimension-two sphere.

If this is right

This establishes a holographic-type relation for fields in flat spacetime.
The scaling dimension of the sphere field is determined by the matching to the transformed Minkowski field.
Computational tools from Feynman diagram evaluation can be used to find Mellin modes in this setup.
Similar mappings might apply to other fields or curved backgrounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could offer an alternative to AdS/CFT for understanding flat-space holography.
Extending this to gauge fields or gravity might reveal more about asymptotic symmetries in Minkowski space.
Testing the exact scaling dimension match numerically in low dimensions could verify the claim.

Load-bearing premise

That the field obtained after the Radon transform precisely equals the one from bulk reconstruction with the required scaling dimension.

What would settle it

Computing the Radon transform of a simple Minkowski scalar mode and comparing it to the reconstructed field from the sphere for the proposed scaling dimension; any mismatch would disprove the identification.

Figures

Figures reproduced from arXiv: 2509.04974 by Koushik Ray, Samrat Bhowmick.

**Figure 1.** Figure 1: Different regions of M1,d The inner product of two vectors X and Y in M1,d is X · Y = ηµνX µY ν = −X 0Y 0 + X 1Y 1 + · · · + X d−1Y d−1 + X dY d . (2) The spacetime is partitioned into different regions or patches, M+ = {X ∈ M1,d; |X| 2 > 0}, (3) M↑ − = {X ∈ M1,d; |X| 2 < 0, X0 > 0, }, M↓ − = {X ∈ M1,d; |X| 2 < 0, X0 < 0, }, M− = M↑ − ∪ M↓ −, (4) M↑ 0 = {X ∈ M1,d; |X| 2 = 0, X0 > 0}, M↓ 0 = {X ∈ M1,d; |X| … view at source ↗

read the original abstract

We relate a free scalar field in the Minkowski spacetime with a scalar field with a certain scaling dimension on a sphere of codimension two. This is realised by first performing a Radon transform of the ``bulk" field on the Minkowski space to a field on a hyperplane identified with a de Sitter or Euclidean anti-de Sitter slice. The Radon transform is identified in turn with a scalar field obtained from a sphere of one further lower dimension through the so-called bulk reconstruction programme. We write down the Mellin modes of the bulk field as generalised hypergeometric functions using the Lee-Pomeransky method developed for evaluation of Feynman loop diagrams.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a Radon-transform chain from Minkowski scalars to codim-two sphere data but leaves convergence of the transform unaddressed.

read the letter

The one thing to know is that this work tries to build a dictionary between a free scalar field in Minkowski space and a scalar with specific scaling dimension on a codimension-two sphere. It does this by applying the Radon transform to get a field on a hyperplane that they identify with a de Sitter or Euclidean anti-de Sitter slice, then matching that to the output of bulk reconstruction from the sphere. They also write the Mellin modes as generalized hypergeometric functions using the Lee-Pomeransky method. What stands out as new is the particular way they chain the Radon transform with the bulk reconstruction program and the codimension reduction. The reuse of Lee-Pomeransky for these modes is a legitimate technical step that gives explicit, calculable expressions others could test. The paper does a reasonable job of spelling out these steps in sequence and showing how the identification might work. If the central map holds, it supplies a practical way to relate flat-space data to lower-dimensional quantities, which fits into ongoing efforts on flat-space holography. A soft spot is the applicability of the Radon transform itself. Minkowski space solutions, including the plane-wave or Mellin-transformed modes, do not decay in every direction. This means the integral that defines the transform over hyperplanes could run into convergence problems. The abstract does not mention any regularization or iε prescription, so it is not obvious whether the exact correspondence with the required scaling dimension survives once that is handled. If the full paper does not demonstrate this explicitly, the identification might rest on an assumption that needs more justification. The math setup looks formally consistent on the surface, and the citation pattern seems appropriate for the topic without obvious omissions. There is no data or numerics here, just analytic expressions. This paper is for people already working on flat-space holography, bulk reconstruction techniques, or Mellin transforms in scattering amplitudes. Someone looking for explicit constructions in non-AdS settings could find the mode expressions useful to play with. I would recommend sending it for peer review. The construction is specific enough that referees can check the details on the transform and the mode matching, and the area is active enough that even a partial result is worth putting through the process.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to relate a free scalar field in Minkowski spacetime to a scalar field with a specific scaling dimension on a codimension-two sphere. This is realized by applying a Radon transform to map the Minkowski bulk field onto a hyperplane identified with a de Sitter or Euclidean anti-de Sitter slice, which is then identified with the scalar obtained from the sphere via the bulk reconstruction programme. The authors express the Mellin modes of the bulk field as generalized hypergeometric functions using the Lee-Pomeransky method.

Significance. If the central identification holds, the work would provide a concrete integral-transform realization of flat-space holography linking Minkowski fields to lower-dimensional sphere data. The explicit use of the Lee-Pomeransky method to write Mellin modes as generalized hypergeometric functions is a strength, supplying concrete expressions that could be checked for consistency with the claimed scaling dimension.

major comments (2)

The section on the Radon transform and subsequent identification: the central claim requires that the Radon transform of Minkowski plane-wave (or Mellin) modes yields a field on the hyperplane that exactly matches the bulk-reconstructed scalar with the stated scaling dimension. The manuscript does not demonstrate convergence of the transform for non-decaying modes or specify an iε prescription/cutoff that preserves the mode correspondence; this is load-bearing for the exact match asserted in the abstract.
The part deriving the scaling dimension from bulk reconstruction: it is unclear whether the dimension is fixed independently by the reconstruction programme or is adjusted to fit the Radon-transformed field; an explicit check that the dimension emerges without post-hoc choice would strengthen the result.

minor comments (2)

The abstract and introduction could state the numerical value of the scaling dimension obtained, rather than referring to it only as 'a certain scaling dimension'.
Notation for the hyperplane identification with dS/EAdS slices should be introduced with a clear diagram or coordinate chart to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

Referee: The section on the Radon transform and subsequent identification: the central claim requires that the Radon transform of Minkowski plane-wave (or Mellin) modes yields a field on the hyperplane that exactly matches the bulk-reconstructed scalar with the stated scaling dimension. The manuscript does not demonstrate convergence of the transform for non-decaying modes or specify an iε prescription/cutoff that preserves the mode correspondence; this is load-bearing for the exact match asserted in the abstract.

Authors: We agree that an explicit treatment of convergence is necessary for the claimed exact identification. In the revised manuscript we will add a dedicated paragraph (or short subsection) in the Radon-transform section that introduces a standard iε prescription for the integration contour, demonstrates distributional convergence for the Mellin modes, and verifies that the resulting hyperplane field coincides with the bulk-reconstructed scalar. The same regularization will be used consistently when expressing the modes via the Lee-Pomeransky representation. revision: yes
Referee: The part deriving the scaling dimension from bulk reconstruction: it is unclear whether the dimension is fixed independently by the reconstruction programme or is adjusted to fit the Radon-transformed field; an explicit check that the dimension emerges without post-hoc choice would strengthen the result.

Authors: The scaling dimension is fixed by the bulk-reconstruction programme itself: it is the unique value for which the reconstructed scalar on the dS/EAdS slice satisfies the correct wave equation and reproduces the known conformal dimension on the codimension-two sphere. This determination precedes and is independent of the subsequent Radon-transform step. To remove any ambiguity we will insert an explicit derivation (using the standard reconstruction formula) that obtains the dimension directly from the sphere data, followed by a consistency check confirming that the Radon-transformed Minkowski modes match this same dimension without adjustment. revision: yes

Circularity Check

0 steps flagged

No circularity: relation constructed via external transforms and standard bulk reconstruction

full rationale

The derivation applies the Radon transform (a standard integral transform) to map the Minkowski free scalar to a field on a hyperplane, then identifies this with the output of the bulk reconstruction programme on the codimension-two sphere. The Lee-Pomeransky method is invoked only for explicit mode computation and is an independent technique from Feynman integral evaluation. No equation reduces the target scaling dimension or the identification to a fitted parameter, self-definition, or prior self-citation chain. The central claim is an explicit construction relating two fields through these operations rather than a tautological renaming or forced prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard properties of the Radon transform and the bulk reconstruction framework from prior literature without introducing new free parameters or postulated entities.

axioms (2)

standard math The Radon transform maps the Minkowski scalar field to a well-defined field on a hyperplane that can be identified with a de Sitter or Euclidean anti-de Sitter slice.
This identification is the first step stated in the abstract.
domain assumption Bulk reconstruction relates the hyperplane field to a scalar field on a codimension-two sphere with a definite scaling dimension.
Invoked to complete the chain from Minkowski space to the sphere.

pith-pipeline@v0.9.0 · 5627 in / 1558 out tokens · 38352 ms · 2026-05-18T19:09:33.147111+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We relate a free scalar field in the Minkowski spacetime with a scalar field with a certain scaling dimension on a sphere of codimension two. This is realised by first performing a Radon transform of the “bulk” field on the Minkowski space to a field on a hyperplane identified with a de Sitter or Euclidean anti-de Sitter slice.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The Mellin modes of the bulk field as generalised hypergeometric functions using the Lee-Pomeransky method

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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