Introduction to classical and quantum integrability

TL;DR

This paper provides a comprehensive pedagogical overview of classical and quantum integrability, covering key concepts, models, and solution techniques like Lax pairs, r-matrices, and Bethe ansatz, with practical examples.

Contribution

It offers a systematic introduction to integrability concepts, including classical models and quantum spin chains, with detailed construction methods and interpretations, suitable for educational purposes.

Findings

Construction of conserved charges for classical models using Lax pairs.

Application of algebraic Bethe ansatz to quantum integrable systems.

Interpretation of the R-matrix as an S-matrix in scattering processes.

Abstract

In these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability. Starting from Liouville integrability and passing through Lax pair and r-matrix we discuss the construction of the conserved charges for classical integrable models taking as example the harmonic oscillator. The construction of these charges for 2D integrable field theories is also discussed using a Lax connection and the Sine-Gordon model as example. On the quantum side, the XXZ spin chain is used to explain the systematic construction of the conserved charges starting from a quantum R-matrix, solution of the quantum Yang-Baxter equation. The diagonalization of these charges is performed using the algebraic Bethe ansatz. At the end, the interpretation of the R-matrix as an S-matrix in a scattering process is also presented. These notes were written for the lectures delivered at the school "Integrability, Dualities and Deformations", that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.