Quantum Corrections to the Classical Dynamics: Application to the SU(2) Lipkin Model

We show how nonrelativistic many body techniques can be used to study quantum corrections to the classical limit, in particular of the $SU(2)$ Lipkin Model. We show that the quantum corrections are essentially of two types: unitary and nonunitary. In this work we perform a detailed study of the unitary corrections. They can be cast in Hamiltonian form and are shown to double the number of degrees of freedom. As a consequence chaotic behavior emerges. We show that this semiquantal chaos is the mechanism trough which tunneling is effected. We also show that these corrections systematically improve the classical results and propose some quantitative measure of this improvement.