Distributions in CFT. Part II. Minkowski space

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, $$ \overline{\rho} $$ ρ ¯ . We prove a key fact that |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ < 1 inside the forward tube, and set bounds on how fast |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ 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).

Notes on flat-space limit of AdS/CFT

“…Then we might be able to investigate the novel analytic region by directly studying analytic aspects of eq. (2.27), provided with axioms of CFT e.g., [43]. We leave this interesting question to future research.…”

Distributions in CFT I. Cross-Ratio Space

“…In our second paper [26], CFT Wightman four-point functions in Lorentzian space will be shown to be tempered distributions, thus establishing Wightman axioms. In the third paper [27], we will study analytic continuations of CFT correlation functions to the Lorentzian cylinder (also known as the boundary of the AdS space).…”

Spinors and conformal correlators

“…One aspect in which our approach differs from the standard literature is that it relies on the Lorentzian conformal group SO(d, 2). However, one might argue that the two are substantially equivalent thanks to the simple analytic continuation that is possible between Euclidean correlators and Wightman functions in Minkowski space-time [35].…”