Consequences of the fundamental conjecture for the motion on the Siegel-Jacobi disk

2011/10/25 by Stefan Berceanu, Berceanu, Stefan
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math-ph #math.DG #math.MP #msc:32Q15 #msc:34A05 #msc:81R30 #msc:81S10 #msc:81V80

paper · pdf · doi:10.48550/arxiv.1110.5469

16 pages, corrected typos, improved notation, Latex, amsart, AMS fonts, to appeaar in Int. J. Geom. Methods Mod. Phys. v10 n1 (January 2013)

arxiv created 2012/04/23 · arxiv updated 2012/04/24

Abstract

We find the homogenous Kähler isomorphism FC which expresses the Kähler two-form on the Siegel-Jacobi domain D^J₁=ℂ\timesD₁ as the sum of the Kähler two-form on ℂ and the one on the Siegel ball D₁. The classical motion and quantum evolution on D^J₁ determined by a linear Hamiltonian in the generators of the Jacobi group G^J₁=H₁\rtimesSU(1,1) is described by a Riccati equation on D₁ and a linear first order differential equation in z∈ℂ, where H₁ denotes the real 3-dimensional Heisenberg group. When the transformation FC is applied, the first order differential equation for the variable z∈ ℂ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel-Jacobi space X^J₁=ℂ\timesX₁, where X₁ denotes the Siegel upper half plane.

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