Dispersion Formulas in QFTs, CFTs, and Holography

Abstract

We study momentum space dispersion formulas in general QFTs and their applications for CFT correlation functions. We show, using two independent methods, that QFT dispersion formulas can be written in terms of causal commutators. The first derivation uses analyticity properties of retarded correlators in momentum space. The second derivation uses the largest time equation and the defining properties of the time-ordered product. At four points we show that the momentum space QFT dispersion formula depends on the same causal double-commutators as the CFT dispersion formula. At $n$-points, the QFT dispersion formula depends on a sum of nested advanced commutators. For CFT four-point functions, we show that the momentum space dispersion formula is equivalent to the CFT dispersion formula, up to possible semi-local terms. We also show that the Polyakov-Regge expansions associated to the momentum space and CFT dispersion formulas are related by a Fourier transform. In the process, we prove that the momentum space conformal blocks of the causal double-commutator are equal to cut Witten diagrams. Finally, by combining the momentum space dispersion formulas with the AdS Cutkosky rules, we find a complete, bulk unitarity method for AdS/CFT correlators in momentum space.

arXiv comment63 pages plus appendices, 3 figures

References in corpus (96)

• Non-Gaussian features of primordial fluctuations in single field inflationary models
• A bound on chaos
• Causality, Analyticity and an IR Obstruction to UV Completion
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Holography from Conformal Field Theory
• Conformal collider physics: Energy and charge correlations
• Writing CFT correlation functions as AdS scattering amplitudes
• Causality Constraints on Corrections to the Graviton Three-Point Coupling
• The Analytic Bootstrap and AdS Superhorizon Locality
• Analyticity in Spin in Conformal Theories
• Convexity and Liberation at Large Spin
• Conformal Regge theory
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Implications of conformal invariance in momentum space
• Light-ray operators in conformal field theory
• Cosmological Collider Physics
• Large Spin Perturbation Theory
• Causality Constraints in Conformal Field Theory
• Unitarity and the Holographic S-Matrix
• Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
• Positive Moments for Scattering Amplitudes
• Loops in AdS from Conformal Field Theory
• Bootstrapping Inflationary Correlators in Mellin Space
• Positivity Bounds for Scalar Theories
• Sharp Boundaries for the Swampland
• A spacetime derivation of the Lorentzian OPE inversion formula
• On Four-point Functions in the CFT/AdS Correspondence
• Conformal Field Theory Correlators from Classical Scalar Field Theory on $Ad{S}_{d+1}$
• The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds
• New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators
• Local bulk S-matrix elements and CFT singularities
• Softness and Amplitudes' Positivity for Spinning Particles
• Holography in the Flat Space Limit
• A Mellin Space Approach to Cosmological Correlators
• The Cosmological Optical Theorem
• Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
• UV complete me: Positivity Bounds for Particles with Spin
• The Duality Between Color and Kinematics and its Applications
• Crossing Symmetric Dispersion Relations in QFTs
• Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT
• Einstein gravity 3-point functions from conformal field theory
• S-Matrices from AdS Spacetime
• TASI Lectures on the Conformal Bootstrap
• Scattering Amplitudes
• Deep Inelastic Scattering in Conformal QCD
• Double copy structure of CFT correlators
• Dispersive CFT Sum Rules
• Bounds for OPE coefficients on the Regge trajectory
• Shocks, Superconvergence, and a Stringy Equivalence Principle
• On duality of color and kinematics in (A)dS momentum space
• Recursion Relations for AdS/CFT Correlators
• Bulk Phase Shift, CFT Regge Limit and Einstein Gravity
• Color/Kinematics Duality in AdS${}_4$
• BCFW for Witten Diagrams
• Cosmological Polytopes and the Wavefunction of the Universe
• Landau diagrams in AdS and S-matrices from conformal correlators
• Four Point Functions of the Stress Tensor and Conserved Currents in AdS_4/CFT_3
• Beyond $a=c$: Gravitational Couplings to Matter and the Stress Tensor OPE
• Momentum space approach to crossing symmetric CFT correlators
• Unitarity Methods in AdS/CFT
• CFT Unitarity and the AdS Cutkosky Rules
• Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications
• Double copy structure and the flat space limit of conformal correlators in even dimensions
• Crossing Symmetric Dispersion Relations for Mellin Amplitudes
• An étude of momentum space scalar amplitudes in AdS
• A Conformal Collider for Holographic CFTs
• New relation for AdS amplitudes
• Einstein gravity from ANEC correlators
• The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT
• Momentum space approach to crossing symmetric CFT correlators II: General spacetime dimension
• Anomalous Dimensions from Crossing Kernels
• Classification of out-of-time-order correlators
• Scattering Equations in AdS: Scalar Correlators in Arbitrary Dimensions
• The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization
• Towards the higher point holographic momentum space amplitudes II: Gravitons
• Dynamical Constraints on RG Flows and Cosmology
• On the Positive Geometry of Conformal Field Theory
• On spinning loop amplitudes in Anti-de Sitter space
• From bulk loops to boundary large-N expansion
• Scale Anomalies, States, and Rates in Conformal Field Theory
• Helicity basis for three-dimensional conformal field theory
• Higher-Point Positivity
• A Basis of Analytic Functionals for CFTs in General Dimension
• Graviton Scattering and a Sum Rule for the c Anomaly in 4D CFT
• Bounds on Crossing Symmetry
• AdS/CFT Unitarity at Higher Loops: High-Energy String Scattering
• The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop
• Wavefunctions in dS/CFT revisited: principal series and double-trace deformations
• Conformal partial waves in momentum space
• Symbol Recursion for the dS Wave Function
• An Effective Theory for Holographic RG Flows
• The Unique Polyakov Blocks
• A brief Introduction to Dispersion Relations and Analyticity
• The Inversion Formula and 6j Symbol for 3d Fermions
• Ambitwistor Strings and the Scattering Equations on AdS${}_3\times$S${}^3$
• Dimensional renormalization in AdS/CFT

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