Light-ray operators in conformal field theory (1805.00098v3)

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.

• The paper introduces light-ray operators as a continuous spin generalization of local operator data in conformal field theory.
• It presents new Lorentzian integral transforms that extend the Euclidean shadow transform for analyzing nonlocal operator behavior.
• The work extends conformal Regge theory and provides new proofs for the average null energy condition (ANEC) for operators with continuous spin.

Light-ray Operators in Conformal Field Theory: An Overview

The paper "Light-ray operators in conformal field theory," authored by Petr Kravchuk and David Simmons-Duffin, explores an intriguing aspect of conformal field theories (CFTs): the existence and properties of light-ray operators. These operators provide a continuous spin generalization of local operators' data in a conformal field theory, offering a new perspective on the analytic continuation in spin discussed by Simon Caron-Huot.

Key Contributions

1. Introduction of Light-ray Operators: The authors propose that every conformal field theory contains light-ray operators, which are labeled by a continuous spin $J$. For integral $J$, these operators reduce to integrals of local operators over a null line. However, for non-integer $J$, they genuinely become nonlocal and represent the analytic continuation of conformal data in spin.
2. Lorentzian Integral Transforms: A novel set of intrinsically Lorentzian integral transforms is introduced, generalizing the Euclidean shadow transform. These transforms facilitate the paper of light-ray operators.
3. Conformal Regge Theory and ANEC: The paper extends conformal Regge theory to arbitrary four-point functions in CFT and provides new proofs for the average null energy condition (ANEC). The ANEC is further generalized to include operators with continuous spin.

Theoretical Background

Conformal field theories exhibit elegant symmetry properties governed by the conformal group. Traditional analyses often involve the operator product expansion (OPE), particularly useful in the Euclidean regime. However, in Lorentzian signature, the OPE faces limitations, especially in describing certain singularities, such as those arising in the Regge limit.

The Regge limit corresponds to high-energy scattering processes, focusing on the behavior of correlators as two operators approach their lightlike separation. This regime is particularly relevant in holographic theories where it relates to the bulk scattering processes.

Light-ray Operators and Lorentzian Signature

The light-ray operators extend the notion of local operators to nonlocal entities along null lines and allow for an analytic continuation in the spin $J$. This continuation becomes particularly insightful in understanding CFT data in the Regge regime. They also enter the discussion of non-vacuum state OPEs, which are crucial for understanding a CFT's dynamics in more general scenarios than traditional vacuum states.

Implications and Prospects

1. Generalizing Operator Product Expansion: Light-ray operators provide a natural framework for discussing the OPE beyond the vacuum, potentially leading to a more comprehensive understanding of non-vacuum CFT dynamics.
2. Positivity in CFTs: The proof of ANEC and its generalization to continuous spins using light-ray operators offers new insights into the positivity conditions in CFTs, suggesting stronger connections with information-theoretic aspects of quantum field theory.
3. Future of Conformal Theory: The paper opens up new avenues for exploration in conformal Regge theory across different operator representations and dimensions, potentially impacting our understanding of high-energy behaviors in various physical systems.

Conclusion

This paper enriches the landscape of conformal field theory by offering a deeper understanding of light-ray operators and their profound implications for both theoretical constructs and potential phenomenological applications. The work of Kravchuk and Simmons-Duffin marks a significant step forward in relating CFT correlators' behavior in different regimes and extending foundational concepts like the OPE to more general contexts. The analysis promises fruitful directions for future research in analytic continuations, positivity conditions, and high-energy limits within the field of theoretical physics.