Distributions in CFT. Part II. Minkowski space

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, (rho) over bar. We prove a key fact that vertical bar rho vertical bar, vertical bar(rho) over bar vertical bar < 1 inside the forward tube, and set bounds on how fast vertical bar rho vertical bar, vertical bar<(rho)over bar>vertical bar 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).

Automated summary

This paper investigates scalar primary Wightman 4-point functions in Lorentzian signature, deriving their properties solely from Euclidean unitary CFT axioms, including Wightman axioms and Lorentzian conformal invariance. The study constructsively establishes key facts about the s-channel Lorentzian OPE and provides a guide to axiomatic QFT literature for modern CFT audience.