New strings for old Veneziano amplitudes IV.Connections with spin chains and other stochastic systems
In a series of published papers we reanalyzed treatments of the Veneziano amplitudes and the models associated with them. In this work we demonstrate that the already obtained partition function for these amplitudes can be exactly mapped into that for the Polychronakos-Frahm spin chain which, in turn, is obtainable from the Richardson-Gaudin XXX model. Reshetikhin and Varchenko demonstrated that such a model is recoverable from their WKB-type analysis of solutions of the Knizhnik-Zamolodchikov (K-Z) equations. The linear independence of solutions of these equations is controlled by determinants whose form (up to a constant) coincides with the Veneziano (or Veneziano-like) amplitudes.In the simplest case, when the K-Z equations are reducible to the Gauss hypergeometric equation, the determinantal conditions coincide with those which were discovered by Kummer in 19th century. Kummer's results admit physical interpretation crucial for providing needed justification associating determinanatal formula(s) with Veneziano-like amplitudes. General results are illustrated by many examples. These include but are not limited to only high energy physics since all high energy physics scattering processes can be looked upon from much broader sctochastic theory of random fragmentation and coagulation processes recently undergoing active development in view of its applications in disciplines ranging from ordering in spin glasses and population genetics to computer science, linguistics and economics,etc. In this theory Veneziano amplitudes play the universal role since they are the Poisson-Dirichlet-type distributions for these processes (analogous to the more familiar Maxwell distribution for gases)