ON SOLITON QUANTUM S-MATRICES IN SIMPLY LACED AFFINE TODA FIELD-THEORIES

Exact solutions to the quantum S-matrices for solitons in simply laced affine Toda field theories are obtained, except for certain factors o f simple type which remain undetermined in some cases. These are found by postulating solutions which are consistent with the semi-classical limit, h --> 0, and the known time delays for a classical two-soliton interaction. This is done by a 'q-deformation' procedure, to move fro m the classical time delay to the exact S-matrix, by inserting a speci al function called the 'regularised' quantum dilogarithm, which only h olds when \q\ = 1. It is then checked that the solutions satisfy the c rossing, unitarity and bootstrap constraints of S-matrix theory. These properties essentially follow from analogous properties satisfied by the classical time delay. Furthermore, the lowest mass breather S-matr ices are computed by the bootstrap, and it is shown, module the undete rmined factors, that these agree with the particle S-matrices known al ready in the affine Toda field theories, in all simply laced cases. (C ) 1997 Elsevier Science B.V.