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REVIEW 3 major objections 6 minor 59 references

Lorentzian OPE Inversion Formula: A Geometric Perspective

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Mellin transforms of horocyclic Radon slices equal the partial wave amplitudes of the original CFT four-point function.

desk verdict Genuinely useful geometric repackaging of the Lorentzian OPE inversion, but the inverse step has an unproven analyticity assumption that keeps the advertised 'solution' from being a theorem. read the letter →

arxiv 2501.04621 v1 pith:6EA5WU3M submitted 2025-01-08 hep-th

Pulkit Agarwal , Richard Brower , Timothy Raben , Chung-I Tan This is my paper · ORCID
classification hep-th MSC 81T4022E7043A8544A12 PACS 11.25.Hf
keywords LorentzianOPEinversionformulaconformalpartialwavesRadontransformMellindoublediscontinuitycausalsphericalfunctionsfieldtheoryharmonicanalysis
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

The paper claims that the standard Lorentzian OPE inversion formula—the procedure that extracts the operator spectrum (dimensions and spins) of a conformal field theory from the double discontinuity of a four-point function—can be reorganized as a two-step tomographic procedure. First one integrates the reduced four-point function along horocycles, a Radon/Abel transform; then one takes a Mellin transform of the resulting auxiliary function. The paper's central equation (4.6) says this composition returns exactly the partial wave amplitudes of the original correlator. This matters because Minkowski-space conformal partial waves are distributions and do not form a complete orthonormal basis, which previously blocked a clean inversion; the Radon-then-Mellin route bypasses that obstruction. The claim is established explicitly for spacetime dimensions $d=1$ and $d=2$, with a framework laid out for general $d$.

What carries the argument

The load-bearing object is the Master Equation (Eq. 3.2), $N A_I P A_I^T N^T = H A P A^T H^T$, which relates the Iwasawa decomposition of the coset $G/H$ to the Cartan decomposition. It lets integrals over the subgroup $H$ be re-expressed as integrals over the nilpotent subgroup $N_+$ and the maximal abelian subgroup $A_I$, giving the horocyclic Radon transform $Ff(a_I)=\int_{N_+} f(n_+ a_I)\,dn_+$. The accompanying Mellin transform $\widetilde{Ff}(\lambda)=\int_{A_I} Ff(a_I)e^{\lambda^* a_I}\alpha\,da_I$ converts the Radon-transformed auxiliary four-point function into the partial wave amplitudes of the original correlator.

What would settle it

Take a known causal four-point function whose partial wave amplitudes have a singularity or branch structure reaching into the right half-plane $\mathrm{Re}\,\lambda>0$ (for instance a theory with an accumulation of Regge poles), and check whether the contour-closing reconstruction of Eqs. 4.14--4.20 reproduces the original double discontinuity; any pole or cut to the right makes the dropped boundary term nonzero and the reconstruction fails. A simpler version: compute the $d=1$ Abel and Laplace transforms for a compactly supported distributional $f$ whose transform has a right-half-plane singularity and verify that the two-sided inversion no longer returns $f$.

Watch

Extended reading notes

Core claim

The central claim is that $\hat{f}(\lambda)=\int Ff(a_I)\, e^{\lambda^* a_I}\,\alpha\, da_I$ (Eq. 4.6): the spherical transform of an $H$-bi-invariant reduced four-point function equals the Mellin transform of its horocyclic Radon transform, where $Ff(a_I)=\int_{N_+} f(n_+ a_I)\,dn_+$ and $\lambda$ labels the boost and dilatation eigenvalues. Since the $\hat{f}(\lambda)$ so obtained are the partial wave amplitudes of the original correlator, the Lorentzian inversion problem becomes a composition of two invertible transforms rather than a search for a complete orthonormal set of blocks. In $d=1$ the authors carry this out explicitly for $SO(1,2)/SO(1,1)$, where the Radon kernel is $(\cosh\eta_I-\cosh\eta)^{1/2}$, the partial wave amplitude is a Laplace transform of the Radon transform, and the inverse reproduces the four-point function through Legendre functions, recovering both principal and discrete series contributions. In $d=2$, the factorization $SO(2,2)\cong SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ makes the Radon transform factorize, and the authors compute the full generalized-free-field amplitude (Eq. 4.29), which is symmetric under the scale-shadow flip $\Delta\to -\Delta$. Geometrically, the statement is a generalization of the Projection-Slice Theorem: in the Iwasawa decomposition the transverse integrals are horocycles, and the remaining integration is a Laplace/Mellin transform.

Load-bearing premise

The argument requires the partial wave amplitudes $a(\lambda)$ to be analytic in the right half-plane, with all physical singularities to the left, so that a boundary term at infinity (Eq. 4.15) vanishes and a zero can be inserted in Eq. 4.20; this analyticity is assumed, not proven, for the distributional correlators that occur in Minkowski CFT.

Editorial extensions

If this is right

  • For $d=1$ CFTs, the inversion automatically produces both the principal and discrete series, with the discrete series cancelling spurious principal-series poles exactly as in the SYK conformal limit.
  • For $d=2$, the generalized free field partial wave amplitude (Eq. 4.29) is recovered, and its scale-shadow symmetry $\Delta\to -\Delta$ reflects the semigroup invariance under $D\leftrightarrow -D$; it matches the bosonic two-dimensional SYK amplitude under the stated identifications.
  • Eq. 4.13 gives a new integral representation of the zonal spherical functions (the principal-series continuation of Minkowski conformal blocks) as a Radon transform followed by an exponential integral.
  • The inversion can be run in reverse: inverse Mellin transform of the amplitude, then inverse Radon transform, reconstructs the double discontinuity and, by contour closure, the conformal block expansion with all physical residues.
  • For $d>2$ the restricted root system is of type $B_2$ and the positive roots have no reflection symmetry, so the paper expects a full partial wave basis to require eight symmetry-related terms, one per Weyl chamber.

Reading between the lines

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

  • The Radon-then-Mellin factorization suggests a directly implementable numerical pipeline: compute or model the double discontinuity on the causal wedge, apply a numerical Radon transform along horocycles, and Mellin transform to read off OPE data—this would bypass constructing conformal blocks entirely, though the paper offers no numerical demonstration.
  • If the analyticity assumption on $a(\lambda)$ fails for realistic Minkowski correlators, the contour-closing steps (Eqs. 4.15 and 4.20) would acquire boundary contributions; reformulating the inversion distributionally, using wave-front sets rather than pointwise analyticity, is a natural extension the paper leaves open.
  • The projection-slice analogy may carry over to crossing: if the Radon transform intertwines the $s$, $t$, and $u$ channels in a simple way, the auxiliary four-point function could make crossing symmetry manifest in Mellin space, which would simplify bootstrap constraints.
  • The eight-chamber structure predicted for $d>2$ is testable already at the level of generalized free fields: computing the $d=3$ amplitude and checking whether it decomposes into the predicted Weyl-chamber sum would confirm or falsify the group-theoretic picture before any full inversion is attempted.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper proposes a geometric reinterpretation of the Lorentzian OPE inversion formula of Caron-Huot and of the authors' earlier work [3]. The reduced four-point function f(u,v), or its double discontinuity, is treated as an H-bi-invariant function on the conformal group, and a new "auxiliary" four-point function Ff is defined as its horocyclic (Radon/Abel) transform, Eq. (4.5). The main claim, Eq. (4.6), is that the Mellin/Laplace transform of Ff equals the conventional partial wave amplitudes a(λ) of f, presented as a CFT analogue of the projection-slice theorem (Fig. 1.1). The forward equivalence is verified explicitly for d = 1 in Eq. (4.12), and the generalized free-field example is computed for d = 2 in Eq. (4.29), where the SL(2,R) × SL(2,R) structure and the root-system symmetry of the partial waves are discussed (Sec. 4.2.3). The paper then attempts the inverse step, reconstructing f from a(λ), in Sec. 4.1.3 via an inverse Laplace transform followed by an inverse Abel transform, and connects the result to the conformal block expansion (Eqs. (4.20)–(4.22)).

Significance. If fully established, the framework provides a genuinely different and geometrically transparent view of the known Lorentzian inversion formula: partial wave amplitudes are Mellin transforms of the horocyclic Radon transform of the double discontinuity, and the Minkowski conformal blocks are recovered from the Radon kernel (Eq. (4.13)). The paper contains several concrete and verifiable computations, notably the d = 1 equivalence (Eq. (4.12)), the Radon transform of the d = 2 generalized free field (Eq. (4.26)), and the partial wave amplitude (Eq. (4.29)); its identification of the discrete symmetry of the d = 2 partial waves with the D2 root-system symmetry (Sec. 4.2.3) is a nice structural insight, and the connection to the SYK and 2d SYK analyses [40, 43] anchors the construction. These are real strengths. However, the paper is explicitly a proof of concept (Sec. 5): the general-d statement is acknowledged to be open, the d = 2 example is restricted to Re(∆) < 1, and the inverse reconstruction of Sec. 4.1.3 relies on an analyticity assumption not derived from CFT properties.

major comments (3)
  1. [Sec. 4.1.3, Eqs. (4.14)–(4.20)] The reconstruction of the reduced four-point function from the partial wave amplitudes is load-bearing for the paper's claim that the framework "allows one to address the issue of inversion for causal CFT partial wave amplitudes" (Sec. 1). The argument drops Eq. (4.15) and inserts the symmetric cosh term in Eq. (4.20) solely on the basis of the sentence after Eq. (4.14): "if we assume that the partial wave amplitudes a(λ) are analytic to the right...". No argument is given that the partial wave amplitudes of a Lorentzian CFT satisfy this analyticity or the needed decay at infinity for closing the contour, and the paper itself notes in Sec. 3 that Minkowski four-point functions are distributions on causal semigroups. The assumption does not follow from the group-theoretic construction of Sec. 3 and is not cosmetic: if a(λ) has any singularity for Re(eλ) > 1/2 (Re(λ) > 0), the boundary term in Eq. (4.15) contributes and Eq. (4.20) is not equivalent to Eq. (4.18). Note also that the identity operator (∆ = 0) sits exactly on the contour Re(eλ) = 1/2, so the contour itself needs a defining prescription. The authors should either prove the required analyticity and decay from the Regge growth and OPE convergence of the double discontinuity, or state them explicitly as hypotheses and delimit the class of theories for which the inversion is claimed; as written, the inversion result is conditional on an unverified property.
  2. [Sec. 4.2.2, Eqs. (4.25)–(4.29)] The d = 2 example, one of the two advertised explicit cases, has two gaps. First, Eq. (4.26) is stated to hold "only if R(∆) < 1" (i.e., Re(∆) < 1), which is a genuine convergence bound for the Abel integral; however, the physically relevant generalized free-field double-trace operators of an external primary of dimension Δ_φ ≥ 1/2 have ∆ = 2Δ_φ ≥ 1, so the example does not cover typical unitary cases. The paper does not give an analytic continuation in ∆ (e.g., a meromorphic continuation of Eq. (4.26)) and does not state what happens in the regime Re(∆) ≥ 1. Second, the steps from the Laplace-transform integral (4.27) to the final amplitude (4.29) are not shown: the integral evaluation, the convergence conditions, and the resulting pole structure — which underlies the symmetry discussion in Sec. 4.2.3 — are simply asserted. Since Eq. (4.29) is the main quantitative output for d = 2, the derivation should be presented, or at minimum its convergence domain and analytic structure should be stated explicitly.
  3. [Sec. 4, Eqs. (4.4)–(4.6)] The main equivalence (4.6) is derived by interchanging the N+ integral and the AI integral in Eq. (4.4) and identifying the result with the spherical transform (4.2), with the paper itself noting after Eq. (4.2) that this "assumes square integrability of the basis functions". For the double-discontinuity functions of interest, which are distributions with power-law singularities on the lightcones (e.g., Eq. (4.25) behaves as (cosh(y ± η) − 1)^{−∆} near contact) and Regge growth at large rapidity, neither the existence of the integrals nor the Fubini step is automatic; indeed, the same convergence issue forces the restriction Re(∆) < 1 in the d = 2 example (Eq. (4.26)). The explicit d = 1 verification in Eq. (4.12) supports the equivalence at a formal level for that case, and the paper is admirably explicit that the general-d problem is open (Secs. 1 and 5), but the statement "This is our main result" (Sec. 1) for Eq. (4.6) should be accompanied by a precise statement of the function space or growth conditions under which the Radon and Mellin transforms and their interchange are defined.
minor comments (6)
  1. [Sec. 4.1.2, Eqs. (4.8)–(4.9)] The exponent on (cosh ηI − cosh η) is printed as +1/2 in Eqs. (4.8) and (4.9); consistency with the Radon transform in Eq. (4.10) and with App. C, Eq. (C.8), requires −1/2. As printed, the measure contradicts the transform that immediately follows.
  2. [Sec. 4.1, Eqs. (4.11)–(4.15)] The notation is very hard to follow: λ, λ*, eλ, and e∆ are used with overlapping meanings, with "λ = −1/2 − e∆", "λ* = −1/2 + e∆", and "∆ = 1/2 − eλ" all in play within a few equations. A summary table of transform variables and conjugate integration contours would remove persistent sign ambiguities.
  3. [Sec. 2, Eq. (2.2)] The definition of (w, σ) is typeset without parentheses and is ambiguous as printed; presumably the intended definitions are w = (1 − √v)/√u and σ = (1 + √v)/√u.
  4. [Sec. 1 and References] On p. 2, "the well known F HAcycle" should be "the well-known FHA cycle" (Fourier–Hankel–Abel cycle), and reference [17] for the projection-slice theorem is a Wikipedia article; a monograph (e.g., Helgason's work, already cited as [41]) would be more appropriate.
  5. [Sec. 4.2.2, Eq. (4.26)] The subscript ℓ in F_{∆,ℓ}(yI, ηI) is introduced without definition; the generalized free-field double discontinuity is ℓ-independent, and ℓ enters only through the Laplace labels in Eq. (4.28), so the notation is misleading. Also, "R(∆)" in the sentence after Eq. (4.26) should be "Re(∆)".
  6. [Sec. 4.1.3, after Eq. (4.22)] The sentence "When the contour is closed in the above representation, one recovers the conformal block expansion of the fourpoint function" is not demonstrated; in light of the analyticity assumption discussed in the major comments, this step deserves a full contour-closing argument with the pole locations and the discrete-series contributions spelled out.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular step found: the central equivalence is derived from classical harmonic analysis, not assumed; the paper's reliance on prior self-citation [3] is load-bearing but real evidence, with an unproven analyticity assumption that is a rigor gap, not a circularity.

full rationale

The central claim is that Mellin transforms of the Radon-transformed reduced four-point function equal the partial wave amplitudes (Eq. 4.6). This is not assumed: it is obtained by starting from the standard spherical transform definition (Eq. 4.2), substituting the integral representation of the zonal spherical functions (Eq. 4.1), interchanging integrations, and defining the Abel/Radon transform (Eq. 4.5). The explicit check in Eq. 4.12 evaluates the Radon-plus-Laplace object and shows it equals the spherical transform f-hat(lambda), i.e., the conventional partial wave amplitude, so the identification is demonstrated rather than imposed. The inverse reconstruction in Sec. 4.1.3 rests on an explicitly stated hypothesis, 'if we assume that the partial wave amplitudes a(lambda) are analytic to the right' (after Eq. 4.14), used to drop Eq. 4.15 and add the symmetric term in Eq. 4.20; this is a substantive analyticity assumption for distributional Lorentzian correlators, but it is an open condition, not an input renamed as an output. The paper leans heavily on the authors' prior work [3] for the Master Equation 3.2 and the causal zonal spherical functions, so the derivation is not fully self-contained; however, [3] is a separate published construction with stated assumptions and explicit d=1,2 examples, and it does not simply assert the present target result. External support is provided by the generalized-free-field example in Eq. 4.29 and its SYK comparison. Overall circularity is minimal; score 2 reflects the substantial but legitimate reliance on prior work and the unproven analyticity step, not a definitional or fitted reduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces an 'auxiliary four-point function' Ff as a Radon transform of the original correlator, but this is a mathematical construct defined from existing data, not a new physical entity. No free parameters are fitted; the computations are symbolic, and the only external parameters are the operator dimensions and spin, which are inputs. The derivation relies on the axioms listed above, several of which are imported from the authors' prior work [3].

assumptions (6)
  • domain assumption The Master Equation N AI · P · AT I N T = HA · P · AT H T (Eq. 3.2) correctly relates Iwasawa and Cartan decompositions on G/H.
    Taken from the authors' prior work [3]; it is the basis for deriving all Radon transform measures in Sec. 4 and is not re-derived here.
  • domain assumption The H-bi-invariant zonal spherical functions φλ(a) form a complete basis (with appropriate distributional interpretation) for the double-discontinuity reduced four-point function.
    Eq. 4.2 assumes square integrability of the basis; the paper notes that Lorentzian CFT functions are distributions, so completeness is nontrivial and is not proven in this paper.
  • domain assumption The partial wave amplitudes a(λ) are analytic in the right half-plane.
    Assumed in Sec. 4.1.3 (after Eq. 4.14) to justify contour deformation in Eqs. 4.15 and 4.20; this is explicitly stated as an assumption.
  • domain assumption The double discontinuity dDisc F is H-bi-invariant, f(hgh') = f(g) (Eq. 3.1).
    Established in the authors' earlier paper [3] and used throughout; not re-derived.
  • domain assumption The Radon transform integrals over the nilpotent subgroup N+ are convergent and the transforms are well-defined on the relevant function space.
    The paper does not give general convergence criteria; for the d=2 example it notes the result holds only for Re(Δ)<1 (Sec. 4.2.2).
  • standard math For d=2, SO(2,2) ≅ SL(2,R) × SL(2,R), so the problem factorizes into two copies of the d=1 case.
    Used in Sec. 4.2 to define the Radon transform and compute the generalized free field example; standard Lie algebra fact.

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Pith. "Pith review of Lorentzian OPE Inversion Formula: A Geometric Perspective." pith.science (2026). https://pith.science/paper/6EA5WU3M

@misc{pith2026250104621,
  author       = {Pith},
  title        = {Pith review of: Lorentzian OPE Inversion Formula: A Geometric Perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6EA5WU3M}},
  note         = {Machine review of arXiv:2501.04621}
}
read the original abstract

We give a new perspective on the Lorentzian OPE inversion formula of arXiv:1703.00278, building on arXiv:2302.06469. We introduce an ``auxiliary'' fourpoint function that can be related to the traditionally defined ones via a Radon transform. The Mellin amplitudes associated with this auxiliary function can be shown to be equivalent to the conventional partial wave amplitudes. This has the intuitive geometrical meaning of a generalization of the Projection-Slice Theorem.

Figures

Figures reproduced from arXiv: 2501.04621 by the authors.

Figure 1.1
Figure 1.1. Tomography of the conformal fourpoint function. [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Orbits on the Poincar´e disc model of H2 in two different group parametrizations. Fig. (a) corresponds to the Cartan decomposition where the manifold is identified with coset elements of the form K(φ)A(η). Lines of constant η are shown in green whereas lines of constant φ are shown in red. Fig. (b) is in the Iwasawa decomposition where the coset is parametrized by elements of the form N(c)AI (ηI ). Lines of constant… view at source ↗
Figure 2.1
Figure 2.1. The first quadrant of the (u, v) plane divides into four regions, E, Ms, Mu Mt , which are multi-sheeted. In terms of new variables (w, σ), Eq. (2.2), this is resolved into multple copies. The first quadrant of the √ u, √ v plane divides into four regions (shown in solid colors) with striaghtline segments. In terms of new variables (w, σ), the corresponding 4 regions are shown in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figures from the paper (3 more)
Figure 2.2
Figure 2.2. Figure 2.2: Illustration of the w, σ variables on the Minkowski spacetime with coordinates (z, t). Dashed lines illustrate causal scattering configurations where x 2 14 < 0 and x 2 23 < 0. The variable e η gives the radius of the hyperbole whereas w parametrizes the position of …
Figure 1
Figure 1. Figure 1: is standard. It essentially states that the spherical transform of a function is equivalent [PITH_FULL_IMAGE:figures/full_fig_p012_1.png]
Figure 4.1
Figure 4.1. Figure 4.1: The restricted root space diagram for SO(d, 2) is of type B2. The green shaded region contains the set of positive roots. Figure (a) corresponds to ordering (L, D) whereas figure (b) corresponds to (D, L). The set of positive roots do not possess any discrete or refl…

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Works this paper leans on

59 extracted references · 23 canonical work pages

  1. [3]

    Agarwal, R.C

    P. Agarwal, R.C. Brower, T.G. Raben and C.-I. Tan, Embedding space approach to Lorentzian CFT amplitudes and causal spherical functions , Phys. Rev. D 110 (2024) 086019 [2302.06469]

    arXiv 2024

  2. [1]

    Caron-Huot, Analyticity in Spin in Conformal Theories , JHEP 09 (2017) 078 [1703.00278]

    S. Caron-Huot, Analyticity in Spin in Conformal Theories , JHEP 09 (2017) 078 [1703.00278]

    arXiv 2017

  3. [2]

    Simmons-Duffin, D

    D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula , JHEP 07 (2018) 085 [ 1711.03816]

    arXiv 2018

  4. [4]

    Dolan and H

    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nuclear Physics B599 (2001) 459 [ hep-th/0011040]

    arXiv 2001

  5. [5]

    Dolan and H

    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion , Nuclear Physics B678 (2004) 491 [ hep-th/0309180]

    arXiv 2004

  6. [6]

    Dolan and H

    F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results , [1108.6194]

    arXiv

  7. [7]

    Hogervorst and S

    M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks , Physical Review D87 (2013) 106004 [ 1303.1111]

    arXiv 2013

  8. [8]

    Qualls, Lectures on Conformal Field Theory , [1511.04074]

    J.D. Qualls, Lectures on Conformal Field Theory , [1511.04074]

    arXiv

Show all 59 references
  1. [9]

    D. Simmons-Duffin, The Conformal Bootstrap , in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015): Boulder, CO, USA, June 1-26, 2015 , pp. 1–74, 2017, DOI [ 1602.07982]

    2015 arXiv

  2. [10]

    Rychkov, EPFL Lectures on Conformal Field Theory in D >= 3 Dimensions , SpringerBriefs in Physics (1, 2016), 10.1007/978-3-319-43626-5, [ 1601.05000]

    S. Rychkov, EPFL Lectures on Conformal Field Theory in D >= 3 Dimensions , SpringerBriefs in Physics (1, 2016), 10.1007/978-3-319-43626-5, [ 1601.05000]

    2016 arXiv

  3. [11]

    Schomerus, E

    V. Schomerus, E. Sobko and M. Isachenkov, Harmony of Spinning Conformal Blocks , JHEP 03 (2017) 085 [ 1612.02479]

    2017 arXiv

  4. [12]

    Isachenkov and V

    M. Isachenkov and V. Schomerus, Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory, JHEP 07 (2018) 180 [ 1711.06609]

    2018 arXiv

  5. [13]

    Schomerus and E

    V. Schomerus and E. Sobko, From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models, JHEP 04 (2018) 052 [ 1711.02022]. 26

    2018 arXiv

  6. [14]

    Buri´ c, M

    I. Buri´ c, M. Isachenkov and V. Schomerus,Conformal Group Theory of Tensor Structures, JHEP 10 (2020) 004 [ 1910.08099]

    2020 arXiv

  7. [15]

    Buric and V

    I. Buric and V. Schomerus, Universal spinning Casimir equations and their solutions , JHEP 03 (2023) 133 [ 2211.14340]

    2023 arXiv

  8. [16]

    Agarwal, R.C

    P. Agarwal, R.C. Brower, T.G. Raben and C.-I. Tan, Application of Lorentzian CFT Principal Series Representation to Near Forward Scattering , SciPost Phys. Proc. 10 (2022) 026 [ 2110.15256]

    2022 arXiv

  9. [17]

    Projection-slice theorem — Wikipedia, the free encyclopedia

    Wikipedia contributors, “Projection-slice theorem — Wikipedia, the free encyclopedia.” https://en.wikipedia.org/w/index.php?title=Projection-slice_theorem&oldid= 1167004207, 2023

    2023

  10. [18]

    Carmi and S

    D. Carmi and S. Caron-Huot, A Conformal Dispersion Relation: Correlations from Absorption, JHEP 09 (2020) 009 [ 1910.12123]

    2020 arXiv

  11. [19]

    Costa, V

    M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, Journal of High Energy Physics 12 (2012) 091 [ 1209.4355]

    2012 arXiv

  12. [20]

    Raben and C.-I

    T.G. Raben and C.-I. Tan, Minkowski conformal blocks and the Regge limit for Sachdev-Ye-Kitaev-like models, Physical Review D98 (2018) 086009 [ 1801.04208]

    2018 arXiv

  13. [21]

    Lang, SL2(R), Springer-Verlag (1974)

    S. Lang, SL2(R), Springer-Verlag (1974)

    1974

  14. [22]

    Sugiura, Unitary Representation and Harmonic Analysis-An Introduction , North-Holland/Kodansha (1990)

    M. Sugiura, Unitary Representation and Harmonic Analysis-An Introduction , North-Holland/Kodansha (1990)

    1990

  15. [23]

    Beerends, An introduction to the abel transform , in Miniconference on Harmonic Analysis, vol

    R. Beerends, An introduction to the abel transform , in Miniconference on Harmonic Analysis, vol. 15, pp. 21–34, Australian National University, Mathematical Sciences Institute (1987)

    1987

  16. [24]

    Gangolli and V

    R. Gangolli and V. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, Springer Berlin Heidelberg (2012)

    2012

  17. [25]

    Shilin and J

    I.A. Shilin and J. Choi, On matrix elements of the SO(2,2)-representation in a space of functions on 2×4-matrices, Integral Transforms and Special Functions 29 (2018) 761

    2018

  18. [26]

    Kravchuk, J

    P. Kravchuk, J. Qiao and S. Rychkov, Distributions in CFT. Part I. Cross-ratio space , JHEP 05 (2020) 137 [ 2001.08778]

    2020 arXiv

  19. [27]

    Kravchuk, J

    P. Kravchuk, J. Qiao and S. Rychkov, Distributions in CFT. Part II. Minkowski space , JHEP 08 (2021) 094 [ 2104.02090]. 27

    2021 arXiv

  20. [28]

    Qiao, Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions, SciPost Phys

    J. Qiao, Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions, SciPost Phys. 13 (2022) 093 [ 2005.09105]

    2022 arXiv

  21. [29]

    Costa, V

    M.S. Costa, V. Goncalves, A. Salgarkar and J. Vilas Boas, Conformal multi-Regge theory, JHEP 09 (2023) 155 [ 2305.10394]

    2023 arXiv

  22. [30]

    Knapp, Representation Theory of Semisimple Groups: An overview based on examples , Princeton mathematical series, Princeton University Press, Princeton, New jersey (1986)

    A. Knapp, Representation Theory of Semisimple Groups: An overview based on examples , Princeton mathematical series, Princeton University Press, Princeton, New jersey (1986)

    1986

  23. [31]

    Knapp, Lie Groups Beyond an Introduction , Progress in Mathematics, Birkh¨ auser Boston (2002)

    A. Knapp, Lie Groups Beyond an Introduction , Progress in Mathematics, Birkh¨ auser Boston (2002)

    2002

  24. [32]

    Bargmann, Irreducible unitary representations of the lorentz group , Annals of Mathematics 48 (1947) 568

    V. Bargmann, Irreducible unitary representations of the lorentz group , Annals of Mathematics 48 (1947) 568

    1947

  25. [33]

    S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Mathematical Surveys and Monographs, American Mathematical Society (2022)

    2022

  26. [34]

    ´Olafsson and J

    G. ´Olafsson and J. Hilgert, Causal Symmetric Spaces , Elsevier Science & Techn. (1996)

    1996

  27. [35]

    Faraut and G.A

    J. Faraut and G.A. Viano, Volterra algebra and the bethe–salpeter equation , Journal of Mathematical Physics 27 (1986) 840

    1986

  28. [36]

    Bendokat, R

    T. Bendokat, R. Zimmermann and P.-A. Absil, A grassmann manifold handbook: Basic geometry and computational aspects , Advances in Computational Mathematics 50 (2024) 6

    2024

  29. [37]

    Maz´ aˇ c,A Crossing-Symmetric OPE Inversion Formula , JHEP 06 (2019) 082 [1812.02254]

    D. Maz´ aˇ c,A Crossing-Symmetric OPE Inversion Formula , JHEP 06 (2019) 082 [1812.02254]

    2019 arXiv

  30. [38]

    A simple model of quantum holography (part 1)

    A. Kitaev, “A simple model of quantum holography (part 1).” Apr., 2015

    2015

  31. [39]

    A simple model of quantum holography (part 2)

    A. Kitaev, “A simple model of quantum holography (part 2).” May, 2015

    2015

  32. [40]

    Maldacena and D

    J. Maldacena and D. Stanford, Comments on the Sachdev-Ye-Kitaev model , Physical Review D94 (2016) 106002 [ 1604.07818]

    2016 arXiv

  33. [41]

    Helgason, The Radon Transform, Springer US (1999), 10.1007/978-1-4757-1463-0

    S. Helgason, The Radon Transform, Springer US (1999), 10.1007/978-1-4757-1463-0

    1999 doi

  34. [42]

    Samko, A

    S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Taylor & Francis (1993)

    1993

  35. [43]

    Murugan, D

    J. Murugan, D. Stanford and E. Witten, More on Supersymmetric and 2d Analogs of the SYK Model, Journal of High Energy Physics 08 (2017) 146 [ 1706.05362]. 28

    2017 arXiv

  36. [44]

    Caron-Huot, D

    S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Dispersive CFT Sum Rules , JHEP 05 (2021) 243 [ 2008.04931]

    2021 arXiv

  37. [45]

    Czech, L

    B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A Stereoscopic Look into the Bulk, JHEP 07 (2016) 129 [ 1604.03110]

    2016 arXiv

  38. [46]

    Bhowmick, K

    S. Bhowmick, K. Ray and S. Sen, Bulk reconstruction in AdS and Gel’fand-Graev-Radon transform, JHEP 10 (2017) 082 [ 1705.06985]

    2017 arXiv

  39. [47]

    S.R. Das, A. Ghosh, A. Jevicki and K. Suzuki, Space-Time in the SYK Model , JHEP 07 (2018) 184 [ 1712.02725]

    2018 arXiv

  40. [48]

    Antunes, M.S

    A. Antunes, M.S. Costa, V. Goncalves and J.V. Boas, Lightcone bootstrap at higher points, JHEP 03 (2022) 139 [ 2111.05453]

    2022 arXiv

  41. [49]

    Buric, S

    I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, From Gaudin Integrable Models to d-dimensional Multipoint Conformal Blocks , Phys. Rev. Lett. 126 (2021) 021602 [ 2009.11882]

    2021 arXiv

  42. [50]

    Buric, S

    I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, Gaudin models and multipoint conformal blocks: general theory , JHEP 10 (2021) 139 [ 2105.00021]

    2021 arXiv

  43. [51]

    Buric, S

    I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D , JHEP 11 (2021) 182 [ 2108.00023]

    2021 arXiv

  44. [52]

    Buric, S

    I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, Gaudin models and multipoint conformal blocks III: comb channel coordinates and OPE factorisation , JHEP 06 (2022) 144 [ 2112.10827]

    2022 arXiv

  45. [53]

    Fitzpatrick, J

    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [ 1212.3616]

    2013 arXiv

  46. [54]

    Komargodski and A

    Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin , Journal of High Energy Physics 2013 (2013) 140

    2013

  47. [55]

    Kaviraj, K

    A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin , Journal of High Energy Physics 11 (2015) 083 [ 1502.01437]

    2015 arXiv

  48. [56]

    Vilenkin and A.U

    N.J. Vilenkin and A.U. Klimyk, Representation of Lie Groups and Special Functions , vol. 1, Springer Netherlands (1991), 10.1007/978-94-011-3538-2

    1991 doi

  49. [57]

    Vilenkin and A.U

    N.J. Vilenkin and A.U. Klimyk, Representation of Lie Groups and Special Functions , vol. 2, Springer Netherlands (1993), 10.1007/978-94-017-2883-6. 29

    1993 doi

  50. [58]

    Vilenkin and A.U

    N.J. Vilenkin and A.U. Klimyk, Representation of Lie Groups and Special Functions , vol. 3, Springer Netherlands (1992), 10.1007/978-94-017-2881-2

    1992 doi

  51. [59]

    Dobrev, G

    V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, On the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory , Springer Berlin, Heidelberg (1977), https://doi.org/10.1007/BFb0009678. 30

    1977 doi

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Reviewed August 10, 2026 · model on record in the stance chip above.