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Wightman axiomatics of the bootstrap construction of the 1+1 dimensional Sinh-Gordon model

Karol K. Kozlowski, Alex Simon

TL;DR

This work tackles the problem of placing the integrable bootstrap construction of the 1+1D Sinh-Gordon quantum field theory on rigorous footing by aiming to satisfy the Wightman axioms for its multi-point correlation functions. The authors assemble the correlation functions from the S-matrix and form factors via a K-transform, defining $k$-point distributions $\mathcal{W}_{\boldsymbol{\alpha}_k}$ as sums of $r$-truncated terms, and they prove that these distributions satisfy covariance, spectral, hermiticity, locality, positivity, and clustering under a convergence conjecture for the bootstrap series. The key methodological steps rely on the Lorentz-boost behavior of form factors, the pole/analytic structure of the S-matrix, and contour-deformation techniques that preserve the correlators while exchanging operator order. The results establish that, once the convergence of the bootstrap series is proven, the bootstrap-based correlation functions indeed define a bona fide quantum field theory for the Sinh-Gordon model, with potential universality to other integrable QFTs.

Abstract

The integrable bootstrap program allows one to express the tempered distributions associated with the multipoint functions of the integrable 1+1 dimensional Sinh-Gordon quantum field theory by means of explicit series. The convergence of the latter is an open problem that was only solved for the two-point case. In this work, by taking for granted the convergence of these series, we show that these expressions satisfy all of the Wightman axioms. This thus shows that, upon a yet to be proven convergence property, the integrable bootstrap based construction of correlation functions does lead to a quantum field theory.

Wightman axiomatics of the bootstrap construction of the 1+1 dimensional Sinh-Gordon model

TL;DR

This work tackles the problem of placing the integrable bootstrap construction of the 1+1D Sinh-Gordon quantum field theory on rigorous footing by aiming to satisfy the Wightman axioms for its multi-point correlation functions. The authors assemble the correlation functions from the S-matrix and form factors via a K-transform, defining -point distributions as sums of -truncated terms, and they prove that these distributions satisfy covariance, spectral, hermiticity, locality, positivity, and clustering under a convergence conjecture for the bootstrap series. The key methodological steps rely on the Lorentz-boost behavior of form factors, the pole/analytic structure of the S-matrix, and contour-deformation techniques that preserve the correlators while exchanging operator order. The results establish that, once the convergence of the bootstrap series is proven, the bootstrap-based correlation functions indeed define a bona fide quantum field theory for the Sinh-Gordon model, with potential universality to other integrable QFTs.

Abstract

The integrable bootstrap program allows one to express the tempered distributions associated with the multipoint functions of the integrable 1+1 dimensional Sinh-Gordon quantum field theory by means of explicit series. The convergence of the latter is an open problem that was only solved for the two-point case. In this work, by taking for granted the convergence of these series, we show that these expressions satisfy all of the Wightman axioms. This thus shows that, upon a yet to be proven convergence property, the integrable bootstrap based construction of correlation functions does lead to a quantum field theory.

Paper Structure

This paper contains 14 sections, 13 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Preliminary setup
  3. Wightman Axioms
  4. The correlation function built from the bootstrap program
  5. The $\boldsymbol{ \texttt{S} }$-matrix and form factors
  6. The multi-point functions
  7. Proof of the Wightman Axioms
  8. The covariance axiom
  9. Spectral condition
  10. Hermiticity
  11. Local Commutativity
  12. Positivity
  13. Cluster Decomposition
  14. Support of tempered distributions

Key Result

Theorem 2.2

Let $\{ \boldsymbol{ \texttt{W} } ^{\, (k)} \}_{k \geq 1}$ be a sequence of tempered distributions, $\boldsymbol{ \texttt{W} } ^{\, (k)}$ acting on $\mathcal{S}( (\mathbb{R}^{1,1})^k)$, that satisfy the Wightman axioms. Then, there exist a separable Hilbert space $\mathfrak{h}$, a continuous unitar The vacuum correlation functions are exactly the Wightman functions. Moreover, this vacuum is cycli

Theorems & Definitions (16)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Definition 2.4
  • Conjecture 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 6 more