A convenient coordinatization of Siegel-Jacobi domains

Abstract

We determine the homogeneous Kähler diffeomorphism $FC$ which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler two-form on $\C^n$ and the one on the Siegel ball $\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$ determined by a hermitian linear Hamiltonian in the generators of the Jacobi group $G^J_n=H_n\rtimes\text{Sp}(n,\R)_{\C}$ are described by a matrix Riccati equation on $\mc{D}_n$ and a linear first order differential equation in $z\in\C^n$, with coefficients depending also on $W\in\mc{D}_n$. $H_n$ denotes the $(2n+1)$-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on $\mc{D}_n$. When the transform $FC:(\eta ,W)\to (z,W)$ is applied, the first order differential equation in the variable $η=(\un-W\bar{W})^{-1}(z+W\bar{z})\in\C^n$ becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel-Jacobi upper half plane $\mc{X}^J_n=\C^n\times\mc{X}_n$, where $\mc{X}_n$ denotes the Siegel upper half plane.

32 pages, corrected typos, Latex, amsart, AMS fonts

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Cited by in corpus (9)

• Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball
• Bergman representative coordinates on the Siegel-Jacobi disk
• Invariant metric on the extended Siegel-Jacobi upper half space
• The Real Jacobi Group Revisited
• Geodesics associated to the balanced metric on the Siegel-Jacobi ball
• Remarks on the geometry of the extended Siegel--Jacobi upper half-plane
• Wei-Norman and Berezin's equations of motion on the Siegel-Jacobi disk
• Geodesics on the extended Siegel-Jacobi upper half-plane
• Mean-field theory based on the \mathfrak{Jacobi~hsp} := semi-direct sum \mathfrak{h}_N \rtimes \mathfrak{sp}(2N,\mathbb{R})_\mathbb{C} algebra of boson operators