Nonlinear dynamics of the infinite classical Heisenberg model: Existence proof and classical limit of the corresponding quantum time evolution

For any initial spin configuration we prove the existence, unicity and regularity properties of the solution of Hamilton's equations for the infinite classical Heisenberg model with stable interactions, along with the essential selfadjointness of the associated Liouville operator. We also prove new properties of SU (2)-coherent states which, together with the concept of Trotter approximations for one-parameter (semi-) groups, are used to show that this dynamics is nothing but the classical limit of the time evolution generated by the infinite quantum (suitably normalized) Heisenberg Hamiltonian. The classical limit emerges when the spin magnitude S goes to infinity while Plank's constant goes to zero, their product S remaining fixed. The main results are stated in the form of intertwining relations between the classical and the quantum unitary group.Work supported in part by the Swiss National Science Foundation under Grant 820-436-76 and in part by the U.S. Department of Energy under contract EG-77-C-03-1538.