Abstract
We argue that every CFT contains light-ray operators labeled by a continuous spin J. When J is a positive integer, light-ray operators become integrals of local operators over a null line. However for non-integer J , light-ray operators are genuinely nonlocal and give the analytic continuation of CFT data in spin described by Caron-Huot. A key role in our construction is played by a novel set of intrinsically Lorentzian integral transforms that generalize the shadow transform. Matrix elements of light-ray operators can be computed via the integral of a double-commutator against a conformal block. This gives a simple derivation of Caron-Huot’s Lorentzian OPE inversion formula and lets us generalize it to arbitrary four-point functions. Furthermore, we show that light-ray operators enter the Regge limit of CFT correlators, and generalize conformal Regge theory to arbitrary four-point functions. The average null energy operator is an important example of a light-ray operator. Using our construction, we find a new proof of the average null energy condition (ANEC), and furthermore generalize the ANEC to continuous spin.
Highlights
Singularities of Euclidean correlators in conformal field theory (CFT) are described by the operator product expansion (OPE)
We show that light-ray operators enter the Regge limit of CFT correlators, and generalize conformal Regge theory to arbitrary fourpoint functions
By expressing E as the residue of an integral of a pair of real operators φ(x1)φ(x2), we find a new proof of the average null energy condition (ANEC) in section 6.7 E is part of a family of light-ray operators EJ labeled by continuous spin J, and our construction of light-ray operators applies to this entire family
Summary
Singularities of Euclidean correlators in conformal field theory (CFT) are described by the operator product expansion (OPE). The analytic continuation of OPE data in a scalar four-point function φ1φ2φ3φ4 can be computed by a “Lorentzian inversion formula,” given by the integral of a doublecommutator [φ4, φ1][φ2, φ3] times a conformal block GJ+d−1,∆−d+1 with unusual quantum numbers. We generalize conformal Regge theory to arbitrary operator representations as well, along the way showing that light-ray operators describe part of the Regge limit of four-point functions as conjectured in [6]. By expressing E as the residue of an integral of a pair of real operators φ(x1)φ(x2), we find a new proof of the ANEC in section 6.7 E is part of a family of light-ray operators EJ labeled by continuous spin J, and our construction of light-ray operators applies to this entire family. (1.22) and (1.23) do not include OPE coefficients
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