Completing the Lorentzian OPE inversion formula: higher-point correlators, low-spin data, canonical subtractions, and perturbative spin bounds

The Lorentzian OPE inversion formula expresses the operator product expansion (OPE) data of a conformal field theory (CFT) as a weighted integral of a four-point function over a Lorentzian region of cross-ratio space, with the recovered coefficient function analytic in the exchanged spin $J$. Simmons-Duffin, Stanford and Witten (JHEP 2018) re-derived the formula directly in position space via Wick rotation, expressing it through integrals of double commutators over causal domains, and explicitly acknowledged the derivation's limitations and hoped-for generalizations: external operators with spin; higher-point functions and analyticity in other $SO(d)$ Dynkin labels; the missing spin-$J\le1$ contributions; case-by-case subtraction prescriptions for non-normalizable external dimensions; and the behavior of the spin bound in $1/N$ expansions. The external-spin case has since been settled by the light-ray-operator generalization of Kravchuk and Simmons-Duffin. The remaining components are open and constitute the problem stated here: construct a higher-point Lorentzian inversion formula with analyticity in the exchanged representation labels; characterize, by construction or impossibility, the spin-0 and spin-1 data accessible to Lorentzian double-commutator observables; formulate a general canonical prescription for subtractions and analytic continuation in the external dimensions; and determine the spin bound on the formula's validity at each order of a $1/N$ expansion.

Background

Conformal field theories and their OPE data. A conformal field theory (CFT) is a quantum field theory invariant under the conformal group $SO(d,2)$ of $d$-dimensional Minkowski space. Its local operators carry a scaling dimension $\Delta$ and, for spinning operators, representation data under the spatial rotation group $SO(d)$ — most commonly symmetric-traceless tensors of integer spin $J$. The operator product expansion (OPE) expresses the product of two local operators as a sum over local operators weighted by OPE coefficients, so that correlation functions are determined, channel by channel, by this 'OPE data' (the spectrum of dimensions and spins together with the OPE coefficients). Four-point functions of scalar primaries decompose into theory-independent conformal blocks $G_{\Delta,J}(z,\bar z)$ — eigenfunctions of the conformal Casimir depending only on the two cross-ratios $z,\bar z$ (the two independent invariant combinations of the four insertion points) — with the OPE data as expansion coefficients. The conformal bootstrap program constrains or solves CFTs by imposing consistency (crossing symmetry) of these decompositions.

The Lorentzian inversion formula. Caron-Huot (2017) showed that the OPE data in a crossed channel can be recovered from a weighted integral of the four-point function over a Lorentzian region of cross-ratio space. The integrand involves the double discontinuity (dDisc), i.e. the failure of the correlator to return to itself after analytic continuation around the branch point $\bar z=1$ in either of the two possible ways. The output is a coefficient function $c(\Delta,J)$ whose poles on the real $\Delta$ axis sit at physical operators with residues proportional to OPE coefficients, and — the formula's key structural property — $c(\Delta,J)$ is analytic in the spin $J$ ('analyticity in spin'). Convergence relies on boundedness of the correlator in the Regge limit (the high-boost, high-energy limit of Lorentzian kinematics, in which an individual conformal block of spin $J$ grows like $e^{(J-1)t}$ with boost parameter $t$), and the formula is proven only for exchanged spin $J>1$.

The spacetime derivation and its limitations. Simmons-Duffin, Stanford and Witten (2018) re-derived the formula by Wick rotation in position space rather than in cross-ratio space: the coefficient function is written as an integral of double commutators $[\mathcal{O}_1,\mathcal{O}_4][\mathcal{O}_2,\mathcal{O}_3]$ (products of commutators evaluated in causally ordered Lorentzian configurations, which vanish unless the operator pairs are causally connected) over intersections of light cones. This makes causality and positivity manifest, yields a one-dimensional inversion formula relevant to the chaos regime of the SYK model (a solvable model of fermions with all-to-all random interactions, in which out-of-time-ordered correlators can grow exponentially, universally bounded by the 'chaos bound'), and clarifies why $J>1$ is needed: partial waves of spin $J=0,1$ can be modified without spoiling Regge boundedness, and their double commutators vanish identically, so they are invisible to the formula. The derivation also requires subtractions when the external dimensions are 'non-normalizable': the harmonic-analysis inner product on partial waves converges for external dimensions on the principal series $\Delta=d/2+is$, but degenerate external dimensions, the identity operator, and low-dimension scalar exchanges produce divergences that must be subtracted case by case, and extra discrete states can then appear in the partial-wave expansion.

What is already settled. For external operators with spin, the source paper notes that weight-shifting operators (conformally covariant differential operators that shift the dimension and spin of an external operator) reduce the inversion to the scalar case by integration by parts, and expresses the hope for a more direct formula. Kravchuk and Simmons-Duffin (2018) supplied it via light-ray operators: nonlocal operators obtained by integrating a local operator along a null ray, which exist for continuous (non-integer) spin $J$, and whose matrix elements are computed by double-commutator integrals. This work explicitly generalizes the Lorentzian inversion formula to four-point correlators of operators in arbitrary Lorentz representations, so the external-spin direction is no longer open. In higher-point functions ($n\ge5$), exchanged operators can carry more general $SO(d)$ representation labels (Dynkin indices beyond the symmetric-traceless spin), and composites of three or more primaries ('multi-twist' operators, generalizing the 'double-twist' families that dominate large-spin expansions) contribute; this is the setting the remaining open components target.

Problem Statement

The problem is to complete the Lorentzian OPE inversion formula — established for four-point functions of scalar external operators, with output analytic in the exchanged spin $J$ and validity proven for $J>1$ — along the directions explicitly left open by the spacetime derivation of Simmons-Duffin, Stanford and Witten. These directions share one objective: extending the double-commutator/analyticity-in-representation-labels framework to the general correlators and regimes that its applications require. (The external-spin case, also listed as open in the source, is excluded because it was settled by the light-ray inversion formula of Kravchuk and Simmons-Duffin.) The components are:

1. Higher-point inversion. Construct and prove a Lorentzian inversion formula for conformal $n$-point functions with $n\ge5$: an explicit integral transform of the correlator, or of causally ordered commutators built from it, over a Lorentzian region, whose output is the coefficient function of a conformal partial wave decomposition as an analytic function of the representation labels of the exchanged operators, together with its convergence conditions (Regge-type boundedness hypotheses on the correlator). This includes the sharply posed subquestion from the source: whether CFT data can be extended to analytic functions of $SO(d)$ Dynkin indices other than the symmetric-traceless spin label $J$.

2. Low-spin data. Characterize what the Lorentzian framework can access about exchanged operators of spin $J=0$ and $J=1$: either construct an explicit Lorentzian observable (an extension of the double-commutator integrals) from which the spin-0 and spin-1 CFT data of a general four-point function can be recovered, or prove a rigorous impossibility result delimiting the class of double-commutator-type observables that cannot access them — the obstruction being that the double commutator of an individual spin-$J\le1$ partial wave vanishes identically.

3. Canonical subtractions. Replace the case-by-case subtraction and contour prescriptions with a general, canonical prescription for defining the inversion integral when the external operator dimensions are non-normalizable — degenerate external dimensions, identity and low-dimension scalar exchanges, and the associated discrete states — including a statement of existence and uniqueness (up to stated conventions) of the analytic continuation in the external dimensions.

4. Perturbative validity. Determine how the spin bound on the formula's validity behaves order by order in a $1/N$ expansion (the expansion governing CFTs with weakly curved holographic duals, in which individual orders can grow in the Regge limit even though the exact correlator is bounded): the chaos bound implies validity only for $J>2$ at the leading nontrivial order for the growing terms, and the source expects further restrictions at higher orders; prove, disprove, or sharply delimit the general bound $J_k$ at order $1/N^{2k}$.

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