Celestial CFT and Scattering Interfaces

Celestial amplitudes express four-dimensional scattering in a basis that diagonalizes boosts rather than energies. The Mellin transform makes Lorentz covariance look like two-dimensional conformal covariance on the celestial sphere. This is an exact representation-theoretic statement for its domain; a local operator algebra, positive Hilbert space, intrinsic dynamics, and complete reconstruction require further evidence.

Required background. Celestial Amplitudes supplies the transformed scattering object, while Flat-Space Holography and Asymptotic Observables defines the stronger duality target.

Helpful background. Celestial and Cosmological Correlators: Axiom and Handoff Audit compares the relevant axioms, and Conjugation and Reflection Positivity explains why conformal covariance alone does not give a Euclidean CFT inner product.

From momentum to celestial variables

A future-directed massless momentum can be written

$$p^\mu=\omega q^\mu(z,\bar z), \qquad q^\mu=(1+|z|^2,\ z+\bar z,\ -i(z-\bar z),\ 1-|z|^2),$$

up to an overall normalization absorbed into $\omega$. The celestial transform of an $n$-point amplitude is

$$\widetilde{\mathcal A}_{\Delta_i,J_i}(z_i,\bar z_i) =\prod_{i=1}^n\int_0^\infty d\omega_i\, \omega_i^{\Delta_i-1} \mathcal A_{J_i}(\epsilon_i\omega_iq_i),$$

where $\epsilon_i=+1$ for outgoing and $-1$ for incoming momenta. Lorentz transformations act on $z$ by $SL(2,\mathbb C)$ Möbius maps, and each external state transforms as a two-dimensional primary with

$$h_i=\frac{\Delta_i+J_i}{2}, \qquad \bar h_i=\frac{\Delta_i-J_i}{2}.$$

The momentum-conserving factor $\delta^{(4)}(\sum_i\epsilon_i\omega_iq_i)$ remains. For four massless particles it gives distributional support on the physical real-cross-ratio locus, unlike a generic Euclidean CFT correlator.

What the transform preserves

On a valid inversion contour the transform preserves the S-matrix information. Crossing changes the signs $\epsilon_i$ and analytic region; unitarity remains the four-dimensional optical theorem rewritten in the boost basis. Soft limits become poles in $\Delta$, and collinear factorization becomes a short-distance expansion on the sphere. These are substantive structural simplifications, not new independent dynamics by themselves.

First application. Transform a simple massless four-point amplitude into a celestial correlator and identify its conformal covariance, support distributions, and retained scattering normalization. A constant scalar contact amplitude is sufficient: perform three energy integrals against momentum conservation, display the remaining Mellin-scale distribution, and keep the delta-function support rather than treating it as a smooth four-point function.

Axioms still to be supplied

A complete celestial CFT interpretation needs an infrared-finite state space, inner product and conjugation, a spectrum including massive states, an OPE with a controlled convergence domain, loop-level operator mixing, and a way to reconstruct all desired bulk observables. Principal-series boost eigenstates are delta-normalizable scattering states, not ordinary radial-quantization states. Reflection positivity is therefore not inherited from Lorentz covariance.

Adversarial control. Demand a reflection-positive Euclidean four-point function away from the momentum-support locus, or ask the transform to predict an amplitude not already supplied. Covariance alone cannot answer. The correct surviving claim is that celestial amplitudes give a conformal representation of scattering data, with additional soft and collinear structures that may guide a boundary theory.

Evidence ceiling

The celestial transform and its Lorentz covariance are exact for appropriately regulated amplitudes and valid contours. They do not alone establish an interacting celestial CFT, fixed-theory unitarity, or a complete quantum-gravity dictionary.

The celestial transform’s conformal covariance for flat-space amplitudes is established by Pasterski, Shao, and Strominger 2017; covariance alone does not supply a Euclidean CFT spectrum, reflection positivity, or a local operator algebra.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Pasterski, Sabrina. “Lectures on Celestial Amplitudes.” The European Physical Journal C 81 (2021): 1062. DOI; Open PDF.
• Pasterski, Sabrina, Shu-Heng Shao, and Andrew Strominger. “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere.” Physical Review D 96 (2017): 065026. DOI; Open PDF.
• Stieberger, Stephan, and Tomasz R. Taylor. “Strings on Celestial Sphere.” Nuclear Physics B 935 (2018): 388–411. DOI; Open PDF.