Mathematics – Differential Geometry
Scientific paper
2011-10-25
Mathematics
Differential Geometry
16 pages, corrected typos, improved notation, Latex, amsart, AMS fonts, to appeaar in Int. J. Geom. Methods Mod. Phys. v10 n1
Scientific paper
We find the homogenous K\"ahler isomorphism $FC$ which expresses the K\"ahler two-form on the Siegel-Jacobi domain $\mathcal{D}^J_1=\mathbb{C}\times\mathcal{D}_1$ as the sum of the K\"ahler two-form on $\mathbb{C}$ and the one on the Siegel ball $\mathcal{D}_1$. The classical motion and quantum evolution on $\mathcal{D}^J_1$ determined by a linear Hamiltonian in the generators of the Jacobi group $G^J_1=H_1\rtimes\text{SU}(1,1)$ is described by a Riccati equation on $\mathcal{D}_1$ and a linear first order differential equation in $z\in\mathbb{C}$, where $H_1$ denotes the real 3-dimensional Heisenberg group. When the transformation $FC$ is applied, the first order differential equation for the variable $z\in \mathbb{C}$ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel-Jacobi space $\mathcal{X}^J_1=\mathbb{C}\times\mathcal{X}_1$, where $\mathcal{X}_1$ denotes the Siegel upper half plane.
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