The boundary sine-Gordon theory: classical and semi-classical analysis

TL;DR

This paper analyzes the boundary sine-Gordon model both classically and semi-classically, computing time delays and confirming quantum S-matrix predictions, thus bridging classical solutions with quantum integrability.

Contribution

It provides a detailed classical and semi-classical analysis of the boundary sine-Gordon model, including time delay calculations and validation of the quantum S-matrix conjecture.

Findings

Classical time delay computed for various parameters.

Method of images remains valid despite non-linearity.

Semi-classical results agree with the quantum S-matrix conjecture.

Abstract

We consider the sine-Gordon model on a half-line, with an additional potential term of the form $-M\cos{\beta\over 2}(\varphi-\varphi_0)$ at the boundary. We compute the classical time delay for general values of $M$, $\beta$ and $\varphi_0$ using $\tau$-function methods and show that in the classical limit, the method of images still works, despite the non-linearity of the problem. We also perform a semi-classical analysis, and find agreement with the exact quantum S-matrix conjectured by Ghoshal and Zamolodchikov.