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Construction of Quantum Field Theories with Factorizing S-Matrices

Gandalf Lechner

TL;DR

This work completes a two-step program for constructing interacting quantum field theories in two-dimensional Minkowski space from a prescribed factorizing S-matrix: first, build wedge-local fields whose commutation relations realize the Zamolodchikov-Faddeev algebra, then extract local observables by establishing the modular nuclearity condition. For a large class of regular scattering functions $S_2$, the authors prove the existence of a local net with the Reeh-Schlieder property, construct explicit collision states, and solve the inverse scattering problem by showing the resulting S-matrix matches the prescribed $S_2$-dependent scattering form. They also prove asymptotic completeness by explicit computation of scattering states, and identify a tight link between the analytic properties of wedge-local form factors and the nuclearity condition. The approach provides a robust algebraic framework that yields nontrivial interacting QFTs (including Sinh-Gordon) without relying on a Lagrangian, and clarifies the role of form factors in establishing locality and scattering within integrable models.

Abstract

A new approach to the construction of interacting quantum field theories on two-dimensional Minkowski space is discussed. In this program, models are obtained from a prescribed factorizing S-matrix in two steps. At first, quantum fields which are localized in infinitely extended, wedge-shaped regions of Minkowski space are constructed explicitly. In the second step, local observables are analyzed with operator-algebraic techniques, in particular by using the modular nuclearity condition of Buchholz, d'Antoni and Longo. Besides a model-independent result regarding the Reeh-Schlieder property of the vacuum in this framework, an infinite class of quantum field theoretic models with non-trivial interaction is constructed. This construction completes a program initiated by Schroer in a large family of theories, a particular example being the Sinh-Gordon model. The crucial problem of establishing the existence of local observables in these models is solved by verifying the modular nuclearity condition, which here amounts to a condition on analytic properties of form factors of observables localized in wedge regions. It is shown that the constructed models solve the inverse scattering problem for the considered class of S-matrices. Moreover, a proof of asymptotic completeness is obtained by explicitly computing total sets of scattering states. The structure of these collision states is found to be in agreement with the heuristic formulae underlying the Zamolodchikov-Faddeev algebra.

Construction of Quantum Field Theories with Factorizing S-Matrices

TL;DR

This work completes a two-step program for constructing interacting quantum field theories in two-dimensional Minkowski space from a prescribed factorizing S-matrix: first, build wedge-local fields whose commutation relations realize the Zamolodchikov-Faddeev algebra, then extract local observables by establishing the modular nuclearity condition. For a large class of regular scattering functions , the authors prove the existence of a local net with the Reeh-Schlieder property, construct explicit collision states, and solve the inverse scattering problem by showing the resulting S-matrix matches the prescribed -dependent scattering form. They also prove asymptotic completeness by explicit computation of scattering states, and identify a tight link between the analytic properties of wedge-local form factors and the nuclearity condition. The approach provides a robust algebraic framework that yields nontrivial interacting QFTs (including Sinh-Gordon) without relying on a Lagrangian, and clarifies the role of form factors in establishing locality and scattering within integrable models.

Abstract

A new approach to the construction of interacting quantum field theories on two-dimensional Minkowski space is discussed. In this program, models are obtained from a prescribed factorizing S-matrix in two steps. At first, quantum fields which are localized in infinitely extended, wedge-shaped regions of Minkowski space are constructed explicitly. In the second step, local observables are analyzed with operator-algebraic techniques, in particular by using the modular nuclearity condition of Buchholz, d'Antoni and Longo. Besides a model-independent result regarding the Reeh-Schlieder property of the vacuum in this framework, an infinite class of quantum field theoretic models with non-trivial interaction is constructed. This construction completes a program initiated by Schroer in a large family of theories, a particular example being the Sinh-Gordon model. The crucial problem of establishing the existence of local observables in these models is solved by verifying the modular nuclearity condition, which here amounts to a condition on analytic properties of form factors of observables localized in wedge regions. It is shown that the constructed models solve the inverse scattering problem for the considered class of S-matrices. Moreover, a proof of asymptotic completeness is obtained by explicitly computing total sets of scattering states. The structure of these collision states is found to be in agreement with the heuristic formulae underlying the Zamolodchikov-Faddeev algebra.

Paper Structure

This paper contains 8 sections, 19 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Construction of Local Nets from a Wedge Algebra
  3. A Class of Models with Factorizing S-Matrices
  4. Analytic Properties of Wedge-Local Form Factors
  5. Proof of the Nuclearity Condition
  6. Collision States and Reconstruction of the S-Matrix
  7. Conclusions
  8. Proof of Lemma \ref{['lemma:tech']}

Key Result

Lemma 2.1

Consider a triple $(\mathcal{M},U,\mathcal{H})$ satisfying the assumptions A1)-A3). Then $\mathcal{M}$ is a type III$_1$ factor according to the classification of Connes connes.

Figures (2)

  • Figure 1: The double cone $O_{x,y}$\ref{['def:dc']} and its causal complement ${O_{x,y}}'=(W_L+x)\cup(W_R+y)$.
  • Figure 2: The two-dimensional bases $-\Lambda_2$ (triangle on the left), $-\mathcal{B}_2(\kappa)$ (pentagon on the right) and ${\hbox{\boldmath{$\lambda$}}}_0+\mathcal{C}_2(\kappa)$ (square on the right), inscribed in the square $[-\pi,0]\times[-\pi,0]$.

Theorems & Definitions (23)

  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4: Müger
  • Theorem 2.5
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Lemma 4.1
  • Lemma 4.2
  • ...and 13 more