Reduction formulae for Euclidean lattice theories
This work is devoted to the rigorous derivations of the LSZ reduction formulas in the context of Euclidean lattice field theories with a Euclidean dynamics defined by the transfer matrix formalism. The existence of a well-defined scattering theory for theories of this sort exhibiting massive one-particle states has been established in a previous work with \textit{K. Fredenhagen} [Commun. Math. Phys. 138, No. 3, 507-519 (1991; Zbl 0741.47017)]. There, the methods of the Haag-Ruelle scattering theory, successfully employed in relativistic quantum field theories, have been used. The main problem to be overcome was the lack of locality (Einstein causality) present in Euclidean theories. As a by-product a clustering theorem for lattice Wightman functions of fields with an energy cut-off has been proven in the massive case. To derive the reduction formulas in the Euclidean context we employed the same techniques developed by Hepp in the sixties in the context of Wightman theories. However some adaptations were necessary, mainly due to two sorts of problems. First, Hepp made use of locality in a strong sense, a property which has to be replaced by clustering in our case. Second, the clustering property we have proven for our Wightman functions is not uniform in the whole region of space-like separated points. At the end the results obtained are natural lattice generalizations of those of Lehmann, Symanzik and Zimmermann, but for smooth time-ordered functions of fields with an energy cut-off. As in the case of Hepp's work a restriction to wave functions with non-overlapping velocities is necessary. These results can be interesting for theoretical or numerical computations of \(S\)-matrix elements in Euclidean lattice theories.
scientific article; zbMATH DE number 54726
scientific article