Symmetry reduction of Brownian motion and Quantum Calogero-Moser systems

TL;DR

This paper explores how symmetry reduction techniques applied to Brownian motion on Riemannian manifolds can lead to quantum Calogero-Moser systems, revealing connections between stochastic processes and quantum integrable systems.

Contribution

It introduces a novel approach to derive quantum Calogero-Moser systems via symmetry reduction of Brownian motion, linking stochastic analysis with quantum integrable models.

Findings

Derived quantum Calogero-Moser systems from stochastic symmetry reduction.

Established parallels between stochastic Hamilton-Jacobi equations and Schrödinger equations.

Provided examples illustrating the reduction scheme in a quantum context.

Abstract

Let $Q$ be a Riemannian $G$-manifold. This paper is concerned with the symmetry reduction of Brownian motion in $Q$ and ramifications thereof in a Hamiltonian context. Specializing to the case of polar actions we discuss various versions of the stochastic Hamilton-Jacobi equation associated to the symmetry reduction of Brownian motion and observe some similarities to the Schr\"odinger equation of the quantum free particle reduction as described by Feher and Pusztai. As an application we use this reduction scheme to derive examples of quantum Calogero-Moser systems from a stochastic setting.