Finite Hamiltonian systems: Linear transformations and aberrations

Finite Hamiltonian systems contain operators of position, momentum, and energy, having a finite number N of equally-spaced eigenvalues. Such systems are under the æis of the algebra su (2), and their phase space is a sphere. Rigid motions of this phase space form the group SU (2); overall phases complete this to U (2). But since N -point states can be subject to U ( N ) ⊃ U (2) transformations, the rest of the generators will provide all N 2 unitary transformations of the states, which appear as nonlinear transformations—aberrations—of the system phase space. They are built through the “finite quantization” of a classical optical system.