A convenient coordinatization of Siegel-Jacobi domains

TL;DR

This paper provides a new coordinate system for the Siegel-Jacobi domains that simplifies the description of classical and quantum dynamics, leading to decoupled differential equations and clearer geometric understanding.

Contribution

It introduces a specific homogeneous K"ahler diffeomorphism that simplifies the K"ahler structure on Siegel-Jacobi domains and clarifies the dynamics of Hamiltonian systems on these spaces.

Findings

The K"ahler two-form on $\\mathcal{D}_n^J$ can be expressed as a sum of forms on $\\mathcal{C}^n$ and $\\mathcal{D}_n$.

The dynamics reduce to a matrix Riccati equation and a linear differential equation, which can be decoupled via the new coordinate system.

The approach applies similarly to the Siegel-Jacobi upper half plane, providing a unified geometric framework.

Abstract

We determine the homogeneous K\"ahler diffeomorphism $FC$ which expresses the K\"ahler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the K\"ahler 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 $FC:(\eta,W)\rightarrow (z,W)$ is applied, the first order differential equation in the variable $\eta=(\un-W\bar{W})^{-1}(z+W\bar{z})\in\C^n$ 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.