The Conformal Group SU(2,2) and Integrable Systems on a Lorentzian Hyperboloid

Abstract

Eleven different types of "maximally superintegrable" Hamiltonian systems on the real hyperboloid $(s^0{)}^2-(s^1{)}^2+(s^2{)}^2-(s^3{)}^2=1$ are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space $SU(2,2)/U(2,1)$, but to reductions by different maximal abelian subgroups of $SU(2,2)$. Each of the obtained systems allows 5 functionally independent integrals of motion, from which it is possible to form two or more triplets in involution (each of them includes the hamiltonian). The corresponding classical and quantum equations of motion can be solved by separation of variables on the $O(2,2)$ space.

37 pages (CRM 2194)

References in corpus (1)

• Path Integral Discussion for Smorodinsky-Winternitz Potentials: I.\ Two- and Three Dimensional Euclidean Space

Cited by in corpus (10)

• Intertwining symmetry algebras of quantum superintegrable systems on the hyperboloid
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: I. Darboux Spaces DI and DII
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV
• On the oscillator realization of conformal U(2,2) quantum particles and their particle-hole coherent states
• Groups, Jacobi functions and rigged Hilbert spaces
• Intertwining Symmetry Algebras of Quantum Superintegrable Systems
• Path Integration on Darboux Spaces
• Harmonic Oscillator on the $\mathrm{SO}(2,2)$ Hyperboloid
• Path Integration on Hermitian Hyperbolic Space
• Non-Hermitian superintegrable systems