On the Bound State Problem in 1 + 1 Dimensional Field Theories

In theoretical elementary particle physics quantum field theory has gained renewed interest in the last years. Non-Abelian gauge theories unify weak and electromagnetic interactions and QCD seems to be a good candidate for the description of strong interactions. Since these theories in four dimensions are very complicated it is useful to study simpler models in two space-time dimensions with similar properties e.g., “asymptotic freedom”, “ confinement”, non-trivial topological structure, e-vacua, etc. There are models possessing some of these properties which have a chance to be explicitly solvable. This class of 2-dimensional field theories, the so-called soliton field theories, is characterized by an infinite set of conservation laws which imply the factorization of the S-matrix. It is amazing that the procedure used to solve these models is just the old analytic S-matrix program. First, by constraints due to unitarity, crossing, internal symmetries, and the special property of factorization, the S-matrix can be determined [la]*, then matrix elements of local operators [2], and finally the correlation functions. The whole program has been carried out until now only in a very simple soliton field theory, the Ising model in the scaling limit [3]. The procedure is at several stages non-unique but minimality assumptions are necessary. Under the constraints mentioned above, the S-matrix is unique up to CDD-like singularities. It is the purpose of this paper to give more restrictions in order to select allowed S-matrices. We shall give a necessary condition for a CDDlike pole in a two-particle S-matrix to be connected with bound states. Otherwise the pole has to be redundant [4]. This condition is based on the positivity of the