On the universal representation of the scattering matrix of affine Toda field theory

By exploiting the properties of q-deformed Coxeter elements, the scattering matrices of affine Toda field theories with real coupling constant related to any dual pair of simple Lie algebras may be expressed in a completely g eneric way. We discuss the governing equations for the existence of bound s tates, i.e. the fusing rules, in terms of q-deformed Coxeter elements, twis ted q-deformed Coxeter elements and undeformed Coxeter elements. We establi sh the precise relation between these different formulations and study thei r solutions. The generalized S-matrix bootstrap equations are shown to be e quivalent to the fusing rules. The relation between different versions of f using rules and quantum conserved quantities, which result as nullvectors o f a doubly q-deformed Cartan like matrix, is presented. The properties of t his matrix together with the so-called combined bootstrap equations are uti lised in order to derive generic integral representations for the scatterin g matrix in terms of quantities of either of the two dual algebras. We pres ent extensive case-by-case data, in particular on the orbits generated by t he various Coxeter elements. (C) 2000 Elsevier Science B.V. All rights rese rved.