Celestial and Cosmological Correlators: Axiom and Handoff Audit

Celestial amplitudes and late-time cosmological correlators transform conformally, but conformal covariance alone does not make either object an ordinary reflection-positive Euclidean CFT correlator. Before importing a block expansion, positivity bound, inversion formula, or dispersion relation, one must identify the spectrum, inner product, ordering, distributional support, factorization rule, and map to the physical observable.

Required background. Lorentzian correlators and causal orderings fixes what a sheet and discontinuity mean in an ordinary CFT. Celestial amplitudes defines the transform and scattering observable; this page uses that interface without rederiving it.

Helpful background. Multipoint correlators and multilightcone limits helps separate global conformal kinematics from OPE convergence and positivity.

Evidence cutoff. Research-sensitive celestial and cosmological interface statements below reflect primary sources available through 2026-08-09. Later observable maps, infrared prescriptions, or bootstrap theorems require a renewed source check.

Celestial covariance comes from the Lorentz group

For a massless four-dimensional momentum, write

$$p_i^\mu=\epsilon_i\omega_i q^\mu(z_i,\bar z_i), \qquad \omega_i>0, \qquad \epsilon_i=\begin{cases} +1,&\text{outgoing},\\ -1,&\text{incoming}. \end{cases}$$

A celestial transform of an $n$-particle amplitude is schematically

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

with the momentum-conserving distribution and the infrared prescription retained. The Lorentz group $SL(2,\mathbb C)$ acts by Möbius transformations on $(z,\bar z)$, and boost eigenstates on the principal series have

$$\Delta_i=1+i\lambda_i, \qquad \lambda_i\in\mathbb R$$

for the standard massless normalizable basis. This proves global two-dimensional conformal covariance of the transformed scattering states Pasterski, Shao, and Strominger 2017, §§2–4.

It does not prove a discrete spectrum bounded below, a Euclidean vacuum state–operator correspondence, or reflection positivity. Momentum conservation makes low-point celestial objects distributions; for four massless particles, physical kinematics restricts the complex cross-ratio to a real locus in the appropriate signature and channel. Soft and collinear singularities, loop-level infrared divergences, and the choice of dressed or inclusive states can also alter the function space. Ordinary block or inversion formulas may be used only after these features have been incorporated into their domain.

Celestial “crossing” begins with scattering crossing and permutation of incoming and outgoing legs. It can induce relations among celestial channel representations, but the analytic continuation in energies and the Mellin contours are part of that operation. It is not automatically the equality of two absolutely convergent Euclidean OPEs with positive squared coefficients.

Cosmological boundary data depend on state and observable

Approximate de Sitter isometries act as Euclidean conformal transformations on a late-time spatial slice. A bulk field of mass $m$ has late-time weights

$$\Delta_\pm=\frac d2\pm\sqrt{\frac{d^2}{4}-\frac{m^2}{H^2}},$$

or $\Delta_\pm=d/2\pm i\mu$ above the principal-series threshold. Complex weights and oscillatory squeezed-limit signals are compatible with unitary time evolution in the bulk; they do not furnish reflection-positive Euclidean boundary data.

Three commonly conflated objects are:

1. a coefficient $\psi_n$ in the late-time wavefunction $\Psi[\varphi]$;
2. an in-in expectation value $\langle\varphi_{\mathbf k_1}\cdots\varphi_{\mathbf k_n}\rangle$ in a specified initial state;
3. a proposed boundary correlator in a dS/CFT dictionary.

Even at tree level, the in-in correlator is obtained from $\lvert\Psi\rvert^2$ and includes inverse quadratic kernels and contractions of lower wavefunction coefficients. Therefore a singularity or sign of $\psi_n$ need not transfer unchanged to an observable. The Bunch–Davies $i\epsilon$ prescription, initial-state choice, gauge-invariant variable, boundary counterterms, and treatment of late-time logarithms must be declared Maldacena 2003, §§2–4.

The cosmological bootstrap can constrain wavefunction coefficients from spatial conformal symmetry, singularities, and factorization. Those are powerful inputs, but their positivity and locality structures are cosmological ones; see Arkani-Hamed et al. 2018, §§2–4. An ordinary CFT unitarity bound cannot be applied to a complex late-time weight without a separate dictionary and inner-product theorem.

Compatibility table

Ordinary CFT input	Ordinary Euclidean CFT	Celestial transform	Cosmological late-time object	Safe conclusion
Global conformal covariance	Generated by $SO(d+1,1)$ on local operators	Follows from the four-dimensional Lorentz group for boost eigenstates	Follows from exact or approximate de Sitter isometries on a spatial slice	Conformal tensor structures and Ward identities appropriate to the object
Spectrum	Local dimensions, usually discrete on the cylinder and bounded by unitarity	Principal-series boost weights are continuous; soft sectors may add distributions	Complementary or principal-series weights; time dependence and mixing may remain	Use the actual measure; do not assume a discrete positive OPE
State–operator correspondence	Radial quantization identifies local insertions with Hilbert-space states	Scattering states live at null infinity; the Mellin basis is not a proved Euclidean radial Hilbert space	Boundary data encode a wavefunction or in-in observable, not generally cylinder states	No imported norm bound without a new inner-product construction
Reflection positivity	Gives nonnegative identical-scalar OPE squares in a Euclidean ordering	Not supplied by Lorentz covariance; the scattering inner product is different	Not supplied by late-time conformal covariance; complex weights can occur	Positivity methods require an object-specific proof
Local OPE and convergence	Convergent inside radial domains away from singularities	Celestial OPEs can encode collinear factorization and soft currents, often distributionally	Boundary expansions encode bulk factorization and late-time limits	Match factorization residues before claiming CFT OPE completeness
Crossing	Associativity of local OPEs in overlapping domains	Scattering crossing plus energy and Mellin continuation	Exchange symmetry, factorization, and wavefunction/in-in relations	Write the exact channel equation before applying a bootstrap functional
Distributional support	Correlators are distributions but Euclidean separated-point functions are ordinary functions	Momentum conservation can restrict cross-ratios to lower-dimensional support	Momentum conservation gives a spatial delta function; trimmed correlators remain state dependent	Keep delta functions and support conditions through transforms
Observable map	Correlator itself is the specified observable	Inverse Mellin transform returns a scattering amplitude with an infrared prescription	$\lvert\Psi\rvert^2$ and in-in rules convert wavefunction data to expectation values	Check the round trip to the physical observable
Lorentzian inversion or dispersion	Requires a declared sheet, growth bound, and appropriate discontinuity	Requires a celestial analytic domain compatible with scattering support and Mellin contours	Requires the wavefunction or in-in analytic structure and its boundary terms	Transfer only the theorem whose hypotheses have been re-established

Where the analytic machinery can connect

The diagram places recoupling and multipoint methods before the cross-domain interface. Inspect the final arrows: they are conditional on the compatibility table rather than automatic consequences of conformal covariance.

Schematic method interface. Crossing kernels and multilightcone limits transfer only after the target object’s spectrum, ordering, distributional support, factorization, and observable map satisfy the hypotheses of the selected theorem.

A celestial partial-wave expansion can legitimately use $SL(2,\mathbb C)$ harmonic analysis on its continuous spectrum. That does not make its coefficients positive OPE squares. A cosmological wavefunction coefficient can legitimately obey conformal Ward identities and factorization on partial-energy poles. That does not make it a Euclidean Schwinger function. In both cases, dispersion reasoning may survive in a modified form because analyticity and discontinuities remain meaningful, but its contour, growth, and subtraction data must be derived for the physical object.

Two minimal checks

Celestial check. Apply a Lorentz transformation to $q^\mu(z,\bar z)$. The Mellin transform converts the energy rescaling into the primary factor $(cz+d)^{-2h}(\bar c\bar z+\bar d)^{-2\bar h}$ with $h=(\Delta+J)/2$ and $\bar h=(\Delta-J)/2$. This verifies covariance. The surviving momentum delta function shows immediately why covariance alone does not prove an ordinary separated-point correlator.

Cosmological check. A conformally covariant scalar two-point kernel has trimmed momentum dependence

$$\psi_2(k)\propto k^{2\Delta-d}.$$

For $\Delta=d/2+i\mu$, this is $k^{2i\mu}$, an oscillatory phase rather than a positive Euclidean spectral density. The transformation law is correct, but an ordinary reflection-positivity argument is unavailable without additional structure.

Exact handoffs

• The energy Mellin transform, incoming/outgoing prescription, infrared-safe scattering observable, and inverse transform remain with celestial amplitudes.
• Cosmological backgrounds, initial states, in-in contours, gauge-invariant perturbations, and observables continue in quantum fields in cosmology, especially vacuum choice and initial-state effects.
• A proposed dS/CFT dictionary or bulk interpretation continues with late-time wavefunctions and boundary data. Celestial holographic completeness questions continue with the celestial CFT and scattering interface.

This page’s conclusion is deliberately narrower: conformal kinematics may be transferred object by object, while positivity, locality, spectral discreteness, and inversion domains must be proved again.

Exercises

A celestial four-point transform is $SL(2,\mathbb C)$ covariant and supported on $z=\bar z$. Which ordinary scalar-CFT inputs have been established, and which remain missing before applying a positive double-discontinuity functional?

Solution

Global conformal covariance and the distributional support are established. One still needs a suitable local OPE with convergence on the support or its regulated extension, an inner product giving nonnegative coefficients, a precise channel continuation, and growth conditions that permit swapping and discard the arc. Scattering unitarity does not supply these ingredients in the ordinary Euclidean-CFT form without a derivation.

References

• Arkani-Hamed, Nima, Daniel Baumann, Hayden Lee, and Guilherme L. Pimentel. “The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities.” arXiv:1811.00024 [hep-th] (2018). arXiv:1811.00024.
• Maldacena, Juan. “Non-Gaussian Features of Primordial Fluctuations in Single Field Inflationary Models.” Journal of High Energy Physics 2003, no. 5 (2003): 013. doi:10.1088/1126-6708/2003/05/013.
• Pasterski, Sabrina, Shu-Heng Shao, and Andrew Strominger. “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere.” Physical Review D 96, no. 6 (2017): 065026. doi:10.1103/PhysRevD.96.065026.