Point Particles as Spin Chains

TL;DR

This paper explores a geometric approach to modeling free point particles on Riemannian manifolds, linking classical dynamics with quantum spin chains through the Kirillov orbit method and geometric quantization.

Contribution

It introduces a novel framework connecting point particle dynamics on manifolds with spin chain models via geometric quantization and Lagrangian submanifolds.

Findings

Establishes a spectral equivalence between Laplace-Beltrami operators and spin Hamiltonians.

Provides explicit examples on complex plane, sphere, flag manifolds, and hyperbolic plane.

Demonstrates the geometric quantization linking classical particle dynamics to quantum spin chains.

Abstract

This work surveys a recently developed approach to the study of free point particles on Riemannian manifolds, based on the Kirillov orbit method, geometric quantization, and the geometry of Lagrangian submanifolds. We discuss that given a Lagrangian submanifold $\mathcal{M}$ embedded in a product of coadjoint orbits and a Hamiltonian $H$ attaining its minimum on this submanifold, such a configuration naturally induces free point particle dynamics on $\mathcal{M}$. The metric governing this dynamics is precisely defined by the quadratic expansion of $H$ around its minimum. Upon quantization, this correspondence establishes a relation between the $L^2(\mathcal{M})$ and a corresponding spin chain Hilbert space as well as a spectral equivalence between Laplace-Beltrami operator on $L^2(\mathcal{M})$ and a spin Hamiltonian. Explicit examples of this construction are presented for particles moving on the complex plane, two-dimensional sphere, flag manifolds, and the hyperbolic plane.