Celestial Bases and Boost Eigenstates as Boundary-Dictionary Inputs

Celestial conformal-primary wavefunctions are ordinary on-shell scattering solutions expressed in a boost eigenbasis. For massless particles, a Mellin transform in the energy produces the basis; on the principal series it has a Plancherel measure and an inverse. Moving away from that contour, changing spin, or adding mass changes the normalization and completeness statement.

Required background. Celestial CFT and Scattering Interfaces supplies the transformed observable, while Lorentz Field Representations and Poincaré Particle Representations supplies the one-particle representation theory.

Helpful background. Conformal Partial Waves and the Shadow Formalism explains shadow equivalence, and Bilinear and Hermitian Forms, Adjoints, and Isometries fixes what a unitary change of basis must preserve.

Scalar conformal-primary wavefunctions

For the null direction $q(z,\bar z)$, start with positive- or negative-frequency plane waves and define

$$\phi_\Delta^\pm(X;z,\bar z) =\int_0^\infty d\omega\,\omega^{\Delta-1} e^{\pm i\omega q\cdot X-\epsilon\omega} =\frac{\Gamma(\Delta)}{(\epsilon\mp iq\cdot X)^\Delta}.$$

They solve the massless wave equation. Under a Lorentz transformation, $z$ undergoes a Möbius map and $\phi_\Delta$ transforms as a scalar conformal primary of weights $(\Delta/2,\Delta/2)$. Spin adds the appropriate polarization and shifts $h-\bar h$.

For four-dimensional massless scattering, the delta-normalizable unitary principal series is

$$\Delta=1+i\nu, \qquad \nu\in\mathbb R.$$

With the Plancherel normalization fixed, Klein–Gordon inner products give delta functions in $\nu$ and the celestial point. The shadow transform relates $\Delta$ to $2-\Delta$ and must be handled to avoid double counting.

Mellin inversion

Because the energy transform is Mellin, the inverse has the schematic form

$$e^{\pm i\omega q\cdot X} =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} d\Delta\,\omega^{-\Delta}\phi_\Delta^\pm(X;q),$$

with the same regulator and normalization used in the forward transform. Choosing $c=1$ gives the principal-series contour. Poles crossed when the contour is deformed represent soft or distributional contributions; they cannot be dropped without changing the state.

First application. Construct normalized massless spin-zero conformal-primary wavefunctions, derive their Lorentz transformation, and invert the Mellin transform back to momentum space. Verify the Klein–Gordon delta normalization and recover a test wavepacket, not just a formal plane wave, so convergence and the regulator removal are explicit.

Massive, spinning, and off-contour states

Massive momenta lie on a hyperboloid rather than a null cone. Their celestial basis uses harmonic analysis on that hyperboloid and a bulk-to-boundary kernel; it is not the same one-dimensional energy Mellin transform. Spinning wavefunctions require gauge-equivalent polarizations and physical inner products. Complex weights away from $1+i\mathbb R$ can be useful analytic continuations or residues but are not automatically normalizable states.

Adversarial control. Move $\Delta$ off the principal series and apply the delta-normalization formula unchanged; the inner product ceases to be the claimed unitary measure. Then replace a massless momentum by a massive one while retaining $p=\omega q$; on-shellness fails. These tests isolate the exact basis theorem from a proposed interacting operator spectrum.

Evidence ceiling

The conformal-primary basis is complete and invertible for specified free one-particle sectors and contours. This establishes a representation of asymptotic states, not the completeness of an interacting celestial operator algebra or of a flat-space holographic dual.

The scalar conformal-primary basis and its principal-series inversion are constructed by Pasterski and Shao 2017; extending the basis off that contour or to additional sectors requires the extra completeness data stated above.

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, and Shu-Heng Shao. “Conformal Basis for Flat Space Amplitudes.” Physical Review D 96 (2017): 065022. 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.
• Pasterski, Sabrina. “Lectures on Celestial Amplitudes.” The European Physical Journal C 81 (2021): 1062. DOI; Open PDF.