Unitarity and the Holographic S-Matrix (1112.4845v2)

Abstract: The bulk S-Matrix can be given a non-perturbative definition in terms of the flat space limit of AdS/CFT. We show that the unitarity of the S-Matrix, ie the optical theorem, can be derived by studying the behavior of the OPE and the conformal block decomposition in the flat space limit. When applied to perturbation theory in AdS, this gives a holographic derivation of the cutting rules for Feynman diagrams. To demonstrate these facts we introduce some new techniques for the analysis of conformal field theories. Chief among these is a method for conglomerating local primary operators to extract the contribution of an individual primary in their OPE. This provides a method for isolating the contribution of specific conformal blocks which we use to prove an important relation between certain conformal block coefficients and anomalous dimensions. These techniques make essential use of the simplifications that occur when CFT correlators are expressed in terms of a Mellin amplitude.

• The paper derives a non-perturbative holographic S-Matrix from the flat space limit of AdS/CFT by leveraging conformal field theory techniques.
• It uses the operator product expansion and conformal block decomposition to reproduce the optical theorem and establish cutting rules for Feynman diagrams.
• The work bridges CFT correlators and bulk scattering processes, offering new insights into quantum gravity and holographic methods.

Overview of "Unitarity and the Holographic S-Matrix"

The paper "Unitarity and the Holographic S-Matrix" by A. Liam Fitzpatrick and Jared Kaplan focuses on providing a non-perturbative definition of the S-Matrix in terms of the AdS/CFT correspondence. This work aims to show how the unitarity of the S-Matrix—the property that leads to the optical theorem—can be derived using the operator product expansion (OPE) and conformal block decomposition within the context of a flat space limit of AdS/CFT.

Key Contributions

1. Holographic Interpretation of the S-Matrix: The paper provides a framework for deriving the S-Matrix from the conformal field theory (CFT) by taking a flat space limit of AdS/CFT, which serves as a holographic dictionary between the boundary and bulk theories.
2. Derivation of the Optical Theorem: The research demonstrates that the optical theorem—which relates the imaginary part of the forward scattering amplitude to the sum over intermediate states—can be derived non-perturbatively from the CFT in the flat space limit.
3. Cutting Rules and Feynman Diagrams: Through new techniques developed for the analysis of conformal field theories, the authors derive holographic expressions for the cutting rules in the context of Feynman diagrams, which are essential for computing scattering processes in perturbation theory.
4. Conglomeration of Operators: A methodology is introduced for combining local primary operators to isolate specific contributions in their OPE, which is crucial for analyzing conformal blocks. The relation between conformal block coefficients and anomalous dimensions is proven, giving new insights into the structure of perturbative expansions.
5. Implications of the Mellin Amplitude: Through the use of the Mellin amplitude for CFT correlators, the authors provide a new avenue for computing analytic properties of scattering amplitudes. This approach highlights the analytical structure necessary for deriving S-Matrix elements holographically.

Numerical and Analytical Results

• Derivation of S-Matrix Unitarity: By using the flat space limit, the authors show that the conformal block decomposition naturally leads to expressions consistent with the unitarity of the S-Matrix, bridging the CFT framework to traditional scattering theory.
• Reduction to Phase Space Integrals: The paper explains how sums over $k$-trace operators in the conformal block decomposition translate into integrals over $k$-particle state phase spaces in the bulk, highlighting their correspondences in flat space.

Implications and Future Directions

This work extends our understanding of quantum gravity by suggesting that holographic principles are sufficient to describe bulk scattering processes non-perturbatively. It underscores the potential of the AdS/CFT correspondence as a tool for exploring unitarity and analytic properties of the S-Matrix. Future research directions include extending these concepts to higher-spin theories and exploring the role of supersymmetry in holographic scattering.

The techniques outlined offer a promising starting point for formulating holographic approaches to quantum gravity beyond the field of perturbative string theory, with the potential to better understand phenomena like black hole evaporation and the ultimate nature of spacetime singularities.

Continued exploration into conglomeration techniques and their connection to other theoretical frameworks could lead to broader applications, potentially advancing our understanding of non-locality and its manifestations in quantum gravity scenarios.