The Siegel Upper Half Space is a Marsden-Weinstein Quotient: Symplectic Reduction and Gaussian Wave Packets

TL;DR

This paper demonstrates that the Siegel upper half space can be derived as a Marsden-Weinstein quotient through symplectic reduction, linking geometric structures to Gaussian wave packet dynamics in semiclassical mechanics.

Contribution

It reveals the geometric relationship between two formulations of Gaussian wave packet dynamics via symplectic reduction, connecting Siegel's symplectic form to the standard form on the quotient.

Findings

Siegel upper half space identified with a Marsden-Weinstein quotient

Reduced symplectic form is a constant multiple of Siegel's form

Two formulations of Gaussian wave packet dynamics are related through symplectic reduction

Abstract

We show that the Siegel upper half space $\Sigma_{d}$ is identified with the Marsden-Weinstein quotient obtained by symplectic reduction of the cotangent bundle $T^{*}\mathbb{R}^{2d^{2}}$ with $\mathsf{O}(2d)$-symmetry. The reduced symplectic form on $\Sigma_{d}$ corresponding to the standard symplectic form on $T^{*}\mathbb{R}^{2d^{2}}$ turns out to be a constant multiple of the symplectic form on $\Sigma_{d}$ obtained by Siegel. Our motivation is to understand the geometry behind two different formulations of the Gaussian wave packet dynamics commonly used in semiclassical mechanics. Specifically, we show that the two formulations are related via the symplectic reduction.