Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fermionic integrable models and graded Borchers triples

Henning Bostelmann, Daniela Cadamuro

Abstract

We provide an operator-algebraic construction of integrable models of quantum field theory on 1+1 dimensional Minkowski space with fermionic scattering states. These are obtained by a grading of the wedge-local fields or, alternatively, of the underlying Borchers triple defining the theory. This leads to a net of graded-local field algebras, of which the even part can be considered observable, although it is lacking Haag duality. Importantly, the nuclearity condition implying nontriviality of the local field algebras is independent of the grading, so that existing results on this technical question can be utilized. Application of Haag-Ruelle scattering theory confirms that the asymptotic particles are indeed fermionic. We also discuss connections with the form factor programme.

Fermionic integrable models and graded Borchers triples

Abstract

We provide an operator-algebraic construction of integrable models of quantum field theory on 1+1 dimensional Minkowski space with fermionic scattering states. These are obtained by a grading of the wedge-local fields or, alternatively, of the underlying Borchers triple defining the theory. This leads to a net of graded-local field algebras, of which the even part can be considered observable, although it is lacking Haag duality. Importantly, the nuclearity condition implying nontriviality of the local field algebras is independent of the grading, so that existing results on this technical question can be utilized. Application of Haag-Ruelle scattering theory confirms that the asymptotic particles are indeed fermionic. We also discuss connections with the form factor programme.
Paper Structure (12 sections, 20 theorems, 78 equations)

This paper contains 12 sections, 20 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Grading
  4. Minkowski space
  5. Nets of algebras
  6. Graded Borchers triples and field algebras
  7. Observable algebras and disorder operators
  8. Scattering theory
  9. Graded integrable models
  10. Wedge-local fields and algebras
  11. Connection to the form factor programme
  12. Conclusions and Outlook

Key Result

Proposition 3.2

$\mathcal{F}$ is a twisted-local net of algebras, covariant under an extension of $U$ to a representation of $\mathcal{P}_+$ with $U(\mathsf{b}_\theta)=\Delta^{i\theta/2\pi}$ and $U(\mathsf{j})=ZJ$, where $(\Delta,J)$ are the Tomita-Takesaki modular data of $(\mathcal{M},\Omega)$.

Theorems & Definitions (42)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • ...and 32 more