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Register a new userThe Siegel Upper Half Space is a Marsden-Weinstein Quotient: Symplectic Reduction and Gaussian Wave Packets
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Authors
Tomoki Ohsawa
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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.
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We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostants coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.
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