Riemann surfaces, separation of variables and classical and quantum integrability

TL;DR

This paper demonstrates how Riemann surfaces and separated variables can be used to construct classical and quantum integrable systems without relying on the Yang-Baxter equation.

Contribution

It introduces a simple, general method to derive classical and quantum commuting Hamiltonians from Riemann surfaces and Baxter's equations, bypassing traditional Yang-Baxter reliance.

Findings

Classical Poisson commuting Hamiltonians derived from Riemann surfaces.

Quantum commuting Hamiltonians obtained from Baxter's equations.

Method is simple, general, and Yang-Baxter independent.

Abstract

We show that Riemann surfaces, and separated variables immediately provide classical Poisson commuting Hamiltonians. We show that Baxter's equations for separated variables immediately provide quantum commuting Hamiltonians. The construction is simple, general, and does not rely on the Yang--Baxter equation.