Classical and quantum mechanics on the unit ball in C n

Abstract

In this paper we show that the complex unit ball B in C n is a strictly Hamiltonian SU(1, n) space and after reduction to the Heisenberg group (which is a subgroup of SU(1, n)) it consists of one parameter family of standard phase spaces (T ∗ R (n-1), − 4h μ dx k ∧ dp k) . We construct the quantum bundle over B and prove that the deformation of its holomorphic and metric structure gives the Hamiltonian on T ∗ R (n −1) . Moreover, we describe explicitly the coherent states and point out their connection with the path integrals.