Dispersion Formulas in QFTs, CFTs, and Holography

TL;DR

This paper develops and compares dispersion formulas in quantum field theories and conformal field theories, establishing their equivalence and connections to holography, and introduces a bulk unitarity method for AdS/CFT correlators in momentum space.

Contribution

It derives momentum space dispersion formulas in QFTs and CFTs using two methods, relates them to holography, and introduces a bulk unitarity approach for AdS/CFT correlators.

Findings

QFT dispersion formulas depend on causal double-commutators.

CFT four-point dispersion formulas are equivalent to QFT formulas up to semi-local terms.

Momentum space conformal blocks are equal to cut Witten diagrams.

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.