New strings for old Veneziano amplitudes IV.Connections with spin chains and other stochastic systems

TL;DR

This paper reveals a deep connection between Veneziano amplitudes, spin chains, and stochastic systems, showing how these mathematical structures relate to various physical and non-physical processes through exact mappings and determinantal conditions.

Contribution

It demonstrates an exact mapping of Veneziano amplitude partition functions to spin chain models and links them to solutions of K-Z equations, broadening their applicability beyond high energy physics.

Findings

Partition function maps to Polychronakos-Frahm spin chain

Determinantal conditions relate to Veneziano amplitudes

Applications extend to stochastic processes in diverse fields

Abstract

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)