Aspects of affine Toda field theory on a half line

TL;DR

This paper investigates the integrability of affine Toda field theory on a half line, showing that boundary conditions preserving integrability are highly constrained and exploring their implications for classical and quantum scattering.

Contribution

It demonstrates that boundary conditions maintaining integrability in affine Toda field theory are strongly limited, with no free parameters except sign choices, and relates classical boundary states to quantum reflection factors.

Findings

Boundary conditions are highly constrained for integrability.

Classical scattering data aligns with quantum reflection bootstrap.

Special boundary conditions may relate classical states to quantum reflection factors.

Abstract

The question of the integrability of real-coupling affine toda field theory on a half line is discussed. It is shown, by examining low-spin conserved charges, that the boundary conditions preserving integrability are strongly constrained. In particular, among the cases treated so far, $e_6^{(1)}$, $d_n^{(1)}$ and $a_n^{(1)}, \ n\ge 2$, there can be no free parameters introduced by such boundary conditions; indeed the only remaining freedom (apart from choosing the simple condition $\partial_1\phi =0$), resides in a choice of signs. For a special case of the boundary condition, accessible only for $a_n^{(1)}$, it is pointed out that the classical boundary bound state spectrum may be related to a set of reflection factors in the quantum field theory. Some preliminary calculations are reported for other boundary conditions, demonstrating that the classical scattering data satisfies the weak coupling limit of the reflection bootstrap equation. (Invited talk at \lq Quantum field theory, integrable models and beyond', Yukawa Institute for Theoretical Physics, Kyoto University, 14-18 February 1994.)