(ix) Lattice statistical models

 There has always been a strong mutual influence of ideas between QFT and lattice statistical mechanics, in particular in 1+1 dimensions. The aforementioned TBA is one concrete example for a technique which has been developed in the context of lattice models and then adopted to the QFT context. The non-diagonal nature of the scattering matrices can be captured with the nested Bethe ansatz [5 ], which is another example for their mutual influence. For various other quantities there exist well known analogies, such as generating function vs partition function, action vs Hamiltonians, fields vs degrees of freedom, Euclidean Green functions vs correlation functions, quantum Hamiltonians vs transfer matrices, fields vs degrees of freedom, mass vs inverse correlation length, cut-off vs lattice spacing, bare parameters vs coupling constants, etc. But one should always keep some fundamental differences in mind: renormalized fields and coupling constants have no counterpart in statistical mechanics. With regard to the above mentioned program one can capitalize on the close relation between R-matrices occurring in the context of lattice statistical models and the S-matrices of QFT. It is well known that the Yang-Baxter equations can be solved by means of quantum group representations, which then is essentially an R-matrix corresponding to a lattice statistical model. In general these R-matrices are not yet unitary and also not crossing invariant, as this are QFT concepts. However, these properties can be implemented by means of some scalar functions and the resulting object is then a candidate for an S-matrix of a QFT. The remaining sector can be constructed by means of the bootstrap procedure. A similar picture holds when boundaries or defects are included. There are still many R-matrices for which the corresponding S-matrices have not been constructed yet.