  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).
