(ii) Form factors

Wightman's reconstruction theorem [118 ] states essentially that a QFT is regarded as solved once all n-point functions are known. The n-point functions on the other hand can be expanded in terms of form factors, which are matrix elements of local operators between the vacuum and an $n$ -particle state.

$$F_{n}^{{\cal O\mid \mu }_{1}\ldots \mu _{n}}(\theta _{1},\d... ...\dagger }\left( \theta _{n}\right) \right\rangle _{\rm {in}}$$

Thus the determination of form factors is an intermediate step towards a construction of correlation functions, and therefore the entire QFT. This idea of a form factor approach to 1+1 dimensional integrable quantum field theories was initiated in the late seventies by Karowski and Weisz [ 117 ] and then carried forward for a long period mainly by Smirnov [ 110 ] until it received some wider attention about fourteen years ago [ 121 ]. The main strategy is similar to the above mentioned procedure of computing the scattering matrices, that is setting up an axiomatic system (for proofs see e.g. [5 ]) for the n-particle form factors, in the hope that this system of equations is so restrictive that it allows for their explicit computation. Remarkably, it has turned out that this method indeed works and that the equations are knitted in such a way that they lead to meaningful solutions. The ambiguities in the solutions may be related to different types of operators. Based on my contributions to the subject, I will now provide a brief account of the state of the art on the technical status. To solve the set of consistency equations at least for the lowest n-particle form factors is a relatively well established general procedure, which consists of the following principle steps: Taking the scattering matrix as an input one may anticipate the pole structure of the n-particle form factors, independently of the nature of the operator ${\cal O}$ , and extract it explicitly in form of a factorizing ansatz. This might turn out to be a relatively involved matter due to the occurrence of higher order poles in the scattering matrix of some IQFT, but nonetheless it is always possible. Thereafter the task of finding solutions may be reduced to the evaluation of the so-called minimal form factors and to the problem of solving a (usually two if bound states may be formed in the theory) recursive equation for a polynomial which results from substituting the mentioned ansatz into the `` form factor residue equations" which are part of the axioms. The first task of finding the minimal form factors can be carried out relatively easy as it only involves to solve simultaneously two functional equations, the so-called Watson's equations. The second task is rather more complicated and the heart of the whole problem in this approach. Having a seed for the recursive equations, that is the lowest non-vanishing form factor one can in general compute from them several form factors which involve more particles. This seed could be either a known form factor when the model reduces to some solved case or possibly the vacuum expectation value of the operator ${\cal O}$ , which is not known in most cases. Unfortunately, the equations become relatively involved after several steps. Several examples [ 77 ,78 ,61 ,17 ,19 ,20 ,18 ] have shown that often the general solution may be cast into a form of determinants whose entries are elementary symmetric polynomials. Presuming such a structure which, at present, may be obtained by extrapolating from lower particle solutions to higher ones or by some inspired guess, one can thereafter rigorously formulate proofs for such solutions. These determinant expressions allow directly to write down equivalent integral representations, see e.g. [17 ]. There exist also different types of universal expressions like for instance the integral representations with complicated pathes in the complex rapidity plane presented in [110 ,5 ]. However, these type of expressions are sometimes only of a very formal nature since to evaluate them concretely for higher $n$ -particle form factors requires still a considerable amount of computational effort. Besides these structures for the functional relations, for theories which allow for backscattering, one can use the method of the nested Bethe ansatz [5 ] to capture the non-diagonal nature of some models. For the HSG models generic expressions have been developed [17 ,19 ,20 ,18 ] already up to the general case of $SU(N)_{2}$ , albeit not for the entire operator content. Instead of solving the consistency equations one may alternatively try to find representations for particle creation operators [62 ] and the local operators, using them thereafter to compute directly the corresponding matrix elements. For the example of the Federbush model [ 49 ] we succeeded in completing this [23 ] idea and compared with the outcome from the axiomatic system. We clarified at the same time how exotic statistics (it is well known that in low dimensions besides bosonic and fermionic statistics it is also possible to have more generic types of statistical interactions, such as Haldane [ 86 ] or Gentile [83 ]) can be implemented into the axiomatic system [ 28 ,64 ] and demonstrated that these equations select out solutions which correspond to local operators in the true sense of QFT, that is they (anti)-commute for space-like separations. In general, it remains a challenge for most theories to find closed analytic solutions for all $n$ -particle form factors of a particular operator ${\cal O}$ and more concrete examples are needed to be able to aim at generic structures. Possibly they consist of building blocks made of determinants as in [77 ,78 ,61 ,17 ,19 ,20 ,18 ].