Point Particles as 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.