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Light-ray operators in conformal field theory

Petr Kravchuk, David Simmons-Duffin

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.

Light-ray operators in conformal field theory

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.

Paper Structure

This paper contains 70 sections, 2 theorems, 211 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. The light transform
  3. Review: Lorentzian cylinder
  4. Symmetry between different Poincare patches
  5. Causal structure
  6. Review: Representation theory of the conformal group
  7. Weyl reflections and integral transforms
  8. Transforms for $\mathrm{S}_\Delta,\mathrm{S}_J,\mathrm{S}$
  9. Transform for $\mathrm{L}$
  10. Transforms for $\mathrm{F}, \mathrm{R}, \mathrm{\overline{R}}$
  11. Some properties of the light transform
  12. Light transform of a Wightman function
  13. Light transform of a time-ordered correlator
  14. Algebra of integral transforms
  15. Light-ray operators
  16. ...and 55 more sections

Key Result

Lemma 2.1

The light transform of a local primary operator, when it exists (i.e. $\Delta+J>1$), annihilates the vacuum,For general spin representations $J$ must be replaced by the sum of all Dynkin labels with spinor labels taken with weight $\frac{1}{2}$.

Figures (17)

  • Figure 1: The Regge limit of a four-point function: the points $x_1,\dots,x_4$ approach null infinity, with the pairs $x_1,x_2$ and $x_3,x_4$ becoming nearly lightlike separated.
  • Figure 2: Poincare patch $\mathcal{M}_d$ (blue, shaded) inside the Lorentzian cylinder $\widetilde{\mathcal{M}}_d$ in the case of 2 dimensions. The spacelike infinity of $\mathcal{M}_d$ is marked by $\infty$. The dashed lines should be identified.
  • Figure 3: $1$ is spacelike from $2$ ($1\approx 2$) if and only if $1$ is in the future of $2^-$ and the past of $2^+$ ($2^-<1<2^+$). The figure shows the Lorentzian cylinder in 2-dimensions. The dashed lines should be identified.
  • Figure 4: The contour prescription for the light-transform. The contour starts at $x\in \mathcal{M}_d$ and moves along the $z$ direction to the point $x^+=\mathcal{T} x$ in the next Poincare patch $\mathcal{T} \mathcal{M}_d$.
  • Figure 5: Relationships between the imaginary parts $\zeta_k$. A deformation of $v$ in the positive imaginary direction is shown in blue.
  • ...and 12 more figures

Theorems & Definitions (3)

  • Lemma 2.1
  • proof
  • Lemma 2.2