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Distributions in CFT II. Minkowski Space

Petr Kravchuk, Jiaxin Qiao, Slava Rychkov

TL;DR

The paper shows that Euclidean CFT axioms imply OS reflection positivity and Wightman positivity for scalar 2–4 point functions, constructing Minkowski correlators as tempered distributions via analytic continuation in the forward tube. Central to the construction are precise bounds on the radial cross‑ratios ρ and ρ, and the use of Vladimirov’s theorem to guarantee distributional limits and conformal invariance in Minkowski space. It provides a detailed analysis of the scalar 4‑point function, establishing conformal invariance, positivity, clustering, and local commutativity, and proves distributional OPE convergence consistent with Mack’s earlier work. The results solidify the link between Euclidean CFT data and Lorentzian QFT structure, offering a rigorous Lorentzian footing for the conformal bootstrap and guiding extension to spinning operators and higher-point functions.

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 $ρ, \barρ$. We prove a key fact that $|ρ|, |\barρ| < 1$ inside the forward tube, and set bounds on how fast $|ρ|, |\barρ|$ 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).

Distributions in CFT II. Minkowski Space

TL;DR

The paper shows that Euclidean CFT axioms imply OS reflection positivity and Wightman positivity for scalar 2–4 point functions, constructing Minkowski correlators as tempered distributions via analytic continuation in the forward tube. Central to the construction are precise bounds on the radial cross‑ratios ρ and ρ, and the use of Vladimirov’s theorem to guarantee distributional limits and conformal invariance in Minkowski space. It provides a detailed analysis of the scalar 4‑point function, establishing conformal invariance, positivity, clustering, and local commutativity, and proves distributional OPE convergence consistent with Mack’s earlier work. The results solidify the link between Euclidean CFT data and Lorentzian QFT structure, offering a rigorous Lorentzian footing for the conformal bootstrap and guiding extension to spinning operators and higher-point functions.

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 . We prove a key fact that inside the forward tube, and set bounds on how fast 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).

Paper Structure

This paper contains 58 sections, 20 theorems, 428 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Executive summary of results for CFT experts
  3. Axioms
  4. Wightman axioms
  5. Osterwalder-Schrader axioms
  6. Euclidean CFT axioms
  7. Euclidean CFT $\Rightarrow$ Osterwalder-Schrader
  8. OS reflection positivity
  9. Denseness and Hilbert space implications
  10. OS clustering
  11. Euclidean CFT $\Rightarrow$ Wightman: Basic strategy
  12. Recovering Minkowski averages from Euclidean averages
  13. Two- and 3-point functions
  14. Comparison with the $i \varepsilon$-prescription
  15. Scalar 4-point function
  16. ...and 43 more sections

Key Result

Theorem 4.1

Suppose that the function $G_n$ is translation- and rotation-invariant, satisfies Gass and in addition satisfies everywhere on $\mathcal{D}_n$ the following 'powerlaw bound' with some positive constants $C_n, A_n, B_n$: Then:

Figures (15)

  • Figure 4.1: Inequalities \ref{['ftineq']} should be satisfied along the analytic continuation contour.
  • Figure 6.1: Illustration of the discussion in Sec. \ref{['sec:informal']}.
  • Figure 6.2: Illustration of a potential difficulty if the set $\omega (\mathcal{T}_4)$ were not simply connected (see Sec. \ref{['sec:informal']}).
  • Figure 6.3: In the 2d case, the special conformal transformation \ref{['fb']} is singular on the blue light cone $x^0 = \pm | x^1 + \beta^{- 1} |$. Suppose $\varphi$ is supported as shown on the right of the light cone. As $\beta \rightarrow 0$, the light cones moves towards the left infinity and does not touch $\operatorname{supp} (\varphi)$. Therefore, such a $\varphi$ satisfies the condition for the invariance under a finite special conformal transformation \ref{['fb']}.
  • Figure 6.4: Location of supports of $\varphi_{\lambda}$ and $\chi$ with respect to the singularity light cone of $f$.
  • ...and 10 more figures

Theorems & Definitions (42)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 4.1: Vladimirov's theorem
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.2
  • ...and 32 more