Distributions in CFT II. Minkowski Space

TL;DR

This paper rigorously derives Lorentzian properties of 4-point functions in conformal field theories from Euclidean axioms, establishing key axioms and bounds in Minkowski space without extra assumptions.

Contribution

It provides a constructive derivation of Lorentzian Wightman axioms and distributional OPE convergence in CFTs directly from Euclidean axioms, clarifying assumptions needed.

Findings

Proves all Wightman axioms for Lorentzian CFTs from Euclidean principles.

Establishes Lorentzian conformal invariance and distributional OPE convergence.

Sets bounds on the approach of cross-ratios to the Minkowski space limit.

Abstract

CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios $\rho, \bar{\rho}$. We prove a key fact that $|\rho|, |\bar{\rho}| < 1$ inside the forward tube, and set bounds on how fast $|\rho|, |\bar{\rho}|$ may tend to 1 when approaching the Minkowski space. We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).