Convergent Momentum-Space OPE and Bootstrap Equations in Conformal Field Theory

TL;DR

This paper establishes the convergence of the operator product expansion in Minkowski momentum space for conformal field theories, constructs conformal blocks explicitly in 2D, and formulates a bootstrap equation using momentum space correlators.

Contribution

It demonstrates the convergence of the momentum-space OPE in Minkowski space and constructs simple conformal blocks in 2D CFT, linking microcausality with bootstrap equations.

Findings

OPE converges in Minkowski momentum space as a distribution.

Conformal blocks are products of 3-point functions in 2D CFT.

Bootstrap equations can be formulated directly in momentum space.

Abstract

General principles of quantum field theory imply that there exists an operator product expansion (OPE) for Wightman functions in Minkowski momentum space that converges for arbitrary kinematics. This convergence is guaranteed to hold in the sense of a distribution, meaning that it holds for correlation functions smeared by smooth test functions. The conformal blocks for this OPE are conceptually extremely simple: they are products of 3-point functions. We construct the conformal blocks in 2-dimensional conformal field theory and show that the OPE in fact converges pointwise to an ordinary function in a specific kinematic region. Using microcausality, we also formulate a bootstrap equation directly in terms of momentum space Wightman functions.