Classically and Quantum Integrable Systems with Boundary

TL;DR

This paper explores integrable systems with boundaries, deriving conditions for classical and quantum cases, including generating functions, boundary conditions, and reflection equations, advancing understanding of boundary effects in integrable models.

Contribution

It introduces new generating functions for integrals of motion in boundary integrable systems and establishes quantum boundary reflection equations based on unitarity.

Findings

Three generating functions for classical integrals of motion.

Boundary conditions derived from boundary $K_{ ext{±}}$ equations.

Quantum reflection equations obtained from unitarity of the $R$-matrix.

Abstract

We study two-dimensional classically integrable field theory with independent boundary condition on each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically integrable boundary condition can be found in solving boundary $K_{\pm}$ equations. In quantum case, we also find that unitarity condition of quantum $R$- matrix is sufficient to construct commutative quantities with boundary, and its reflection equations are obtained.