(v) Identification of the (local) operator content

The central concepts of relativistic QFT, like Einstein causality and Poincaré covariance, are captured in local field equations and commutation relations. As a matter of fact local quantum physics (algebraic quantum field theory) [85 ] takes the collection of all operators localized in a particular region, which generate a von Neumann algebra, as its very starting point. On the other hand, as we outlined above in the formulation of a QFT one may alternatively start from a particle picture and investigate the corresponding scattering theories. Ignoring subtleties of non-asymptotic states, it is essentially possible to obtain the latter picture from the former by means of the LSZ-reduction formalism [99 ]. However, the question of how to reconstruct at least part of the field content from the scattering theory is in general still an outstanding issue. In the context of 1+1 dimensional IQFTs the identification of the operators is based on the assumption, dating back to the initial papers [ 117 ], that each solution to the form factor consistency equations corresponds to a particular local operator. Based on this, numerous authors have used various ways to identify and constrain the specific nature of the operator, e.g. by looking at asymptotic behaviours, performing perturbation theory, taking symmetries into account, formulating quantum equations of motion, etc. We carried out an analysis [20 ] for the SU(3)$_{2}$ -HSG model identifying each of our many solutions with a local operator and matching it with a counterpart in the ultraviolet conformal field theory, hence organizing and identifying the operator content. An additional organizing principle comes from the cluster properties. In space they are quite familiar and intuitive, corresponding to the observation that far separated operators do not interact. In 1+1 dimensions a similar property has also been noted in momentum space. It states that whenever the first, say $\kappa$ , rapidities of an $n$ -particle form factor are shifted to infinity, the $n$ -particle form factor factorizes into a $%% \kappa$ and an ($n-\kappa$ )-particle form factor which are possibly related to different types of operators. This means the cluster property can be used as an organizing principle amongst different types of operators and also to construct new solutions. These properties have been analyzed explicitly for several specific models [20 ]. In general, the problem of how to reconstruct the field content from the scattering theory is unresolved. As not many models exist for which the above mentioned analysis has been carried out, one certainly needs first of all more examples. At present the identification and naming of the operators in this context is carried out with regard to the corresponding conformal dimension in the UV-limit. Possibly one needs also to find different types of identification criteria or quantities as it is not guaranteed that there is a one-to-one correspondence between the operators in the massive model and those in the corresponding conformal field theory. Conceptually this is not clear yet and identical to the question whether so-called `` shadow operators" [113 ] exist or not.