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 ρ, overline{rho} . We prove a key fact that |ρ|, left|overline{rho}right| < 1 inside the forward tube, and set bounds on how fast |ρ|, left|overline{rho}right| 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).
Highlights
Quantum Field Theory (QFT) can be studied via constructive or axiomatic approaches
Our focus here will be on conformal field theories (CFTs) in dimension d 2, i.e. QFTs invariant under the action of conformal group, which are nowadays studied via the conformal bootstrap
In this paper we studied the relationship between the modern Euclidean CFT axioms and the more traditional Osterwalder-Schrader and Wightman axioms
Summary
Quantum Field Theory (QFT) can be studied via constructive or axiomatic approaches. The former use microscopic formulations, while the latter rely on axioms. Some additional technical details are given in appendices B)–(D
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