We present a method, using the Bethe Ansatz, to produce exact dynamics for a collection of interacting particle systems. The Bethe Ansatz was introduced in 1931 by Hans Bethe to find eigenvectors and eigenvalues for the generator of a one-dimensional quantum spin system, which is also the setting for this talk. While the Bethe Ansatz proved to be an effective computational tool, providing deep insight into quantum spin chains, it lacked many rigorous theoretical guarantees for many years. There has been much elucidating mathematical activity over the decades to understand the hidden symmetries and analytical consequences of the Bethe Ansatz, and many theoretical questions have been answered. In this talk, we address how to obtain exact dynamics for the Heisenberg-Ising XXZ spin-1/2 periodic chain using the Bethe Ansatz. We provide precise formulas to take a linear combination of the eigenvectors from the Bethe Ansatz and obtain exact dynamics. The formulas presented have been rigorously confirmed for some special cases and the general case has only been confirmed numerically.
This talk is aimed at a general math audience with no special background needed beyond linear algebra and analysis.