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

We show that the Siegel upper half space ${\mathrm{\Sigma }}_d$ is identified with the Marsden-Weinstein quotient obtained by symplectic reduction of the cotangent bundle $T^{\ast }{\mathbb{R}}^{2d^2}$ with $\mathsf{O}(2d)$-symmetry. The reduced symplectic form on ${\mathrm{\Sigma }}_d$ corresponding to the standard symplectic form on $T^{\ast }{\mathbb{R}}^{2d^2}$ turns out to be a constant multiple of the symplectic form on ${\mathrm{\Sigma }}_d$ 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.