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Quantum energy inequalities in integrable models with several particle species and bound states

Henning Bostelmann, Daniela Cadamuro, Jan Mandrysch

Abstract

We investigate lower bounds to the time-smeared energy density, so-called quantum energy inequalities (QEI), in the class of integrable models of quantum field theory. Our main results are a state-independent QEI for models with constant scattering function and a QEI at one-particle level for generic models. In the latter case, we classify the possible form of the stress-energy tensor from first principles and establish a link between the existence of QEIs and the large-rapidity asymptotics of the two-particle form factor of the energy density. Concrete examples include the Bullough-Dodd, the Federbush, and the $O(n)$-nonlinear sigma models.

Quantum energy inequalities in integrable models with several particle species and bound states

Abstract

We investigate lower bounds to the time-smeared energy density, so-called quantum energy inequalities (QEI), in the class of integrable models of quantum field theory. Our main results are a state-independent QEI for models with constant scattering function and a QEI at one-particle level for generic models. In the latter case, we classify the possible form of the stress-energy tensor from first principles and establish a link between the existence of QEIs and the large-rapidity asymptotics of the two-particle form factor of the energy density. Concrete examples include the Bullough-Dodd, the Federbush, and the -nonlinear sigma models.
Paper Structure (22 sections, 25 theorems, 163 equations)

This paper contains 22 sections, 25 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. One-particle space and scattering function
  5. Integrable models, form factors, and the stress-energy tensor
  6. The stress-energy tensor at one-particle level
  7. A state-independent QEI for constant scattering functions
  8. QEI at one-particle level for general integrable models
  9. The connection between the S-function and the minimal solution
  10. QEIs in examples
  11. (Generalized) Bullough-Dodd model
  12. Federbush model
  13. O(n)-nonlinear sigma model
  14. Conclusion and outlook
  15. What is the nature of the freedom in the form of the stress-energy tensor?
  16. ...and 7 more sections

Key Result

Theorem 3.2

Given a little space $(\mathcal{K}, V, J, M)$, an S-function $S$, and a subset $\mathfrak{P} \subset \mathbb{S}(0,\pi)$, then $F_2$ is a stress-energy tensor at one-particle level (with poles $\mathfrak{P}$) iff it is of the form where $F:\mathbb{C}\to \mathcal{K}^{\otimes 2}$ is a meromorphic function which satisfies for all $\zeta \in \mathbb{C}$ that It is parity covariant iff, in addition,

Theorems & Definitions (69)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • proof : Proof of Theorem \ref{['thm:tformgen']}
  • Corollary 3.4
  • ...and 59 more