On the Wick Rotation of the Four-point Function in Conformal Field Theories

TL;DR

This paper investigates the relationship between Euclidean and Lorentzian conformal field theories, demonstrating that Euclidean axioms for four-point functions imply key quantum field theory axioms in both signatures.

Contribution

It clarifies that Euclidean CFT axioms for four-point functions imply the standard quantum field theory axioms, bridging the gap between Euclidean and Lorentzian formulations.

Findings

Euclidean CFT axioms imply Osterwalder-Schrader axioms in Euclidean signature.

Euclidean axioms imply Wightman axioms in Lorentzian signature.

The work clarifies assumptions needed for Lorentzian properties in CFTs.

Abstract

Conformal field theories (CFTs) in Euclidean signature satisfy well-accepted rules, such as conformal invariance and the convergent Euclidean operator product expansion (OPE). Nowadays, it is common to assume that CFT correlators exist and have various properties in the 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. In this thesis, we clarify that at the level of four-point functions, the Euclidean CFT axioms imply the standard quantum field theory axioms such as Osterwalder-Schrader axioms (in Euclidean) and Wightman axioms (in Lorentzian).