Symplectic Manifolds, Coherent States and Semiclassical Approximation

TL;DR

This paper explores the symplectic geometry and Hamiltonian dynamics of Grassmannian manifolds, illustrating path integral evaluations via semiclassical methods with examples like the sphere and disc.

Contribution

It introduces a local coordinate framework for Grassmannians and demonstrates exact path integral evaluations using semiclassical techniques.

Findings

Exact path integral evaluations for $S^2$ and $D^2$

Localization formulas derived from semiclassical methods

Local coordinate descriptions for Grassmannians

Abstract

We describe the symplectic structure and Hamiltonian dynamics for a class of Grassmannian manifolds. Using the two dimensional sphere ($S^2$) and disc ($D^2$) as illustrative cases, we write their path integral representations using coherent state techniques. These path integrals can be evaluated exactly by semiclassical methods, thus providing examples of localisation formula. Along the way, we also give a local coordinate description for a class of Grassmannians.