Abstract
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 constantħ goes to zero, their product Sħ remaining fixed. The main results are stated in the form of intertwining relations between the classical and the quantum unitary group.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lanford, O.E., Lebowitz, J. L., Lieb, E. H.: J. Stat. Phys.16, 453–461 (1977)
Dobrushin, R. L., Fritz, J.: Commun. Math. Phys.55, 275–292 (1977)
Dobrushin, R. L., Fritz, J.: Commun. Math. Phys.57, 67–81 (1977)
Vuillermot, P. A.: Lett. Nuovo Cimento24, 333–338 (1979)
Lang, S.: Analysis II, London, Amsterdam, Paris: Addison Wesley (1968)
Ruelle, D.: Statistical mechanics, rigorous results. New York: Benjamin, 1969
Arnold, V.: Ordinary differential equations. Cambridge, Massachusetts: MIT Press, 1973
Schwartz, L.: Cours d'analyse. Paris: Hermann, 1967
Trotter, A. F.: Pac. J. Math.8, 887–919 (1958)
Kurtz, T. G.: Funct. Analy.3, 354–375 (1969)
Hepp, K., Lieb, E. H.: Ann. Phys.76, 360–404 (1973)
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. II New York: Academic Press, 1977
Riesz, F., Nagy, B.: Leçons d'analyse fonctionelle. Paris: Gauthiers-Villars 1968
Nelson, E.: Math.70, 572–615 (1959)
Robinson, D. W.: Commun. Math. Phys.7, 337–348 (1968)
Streater, R. F.: Commun. Math. Phys.6, 233–247 (1967)
Sakai, S.: C*-algebras and W*-algebras, Berlin, Heidelberg, New York: Springer 1971
Lieb, E. H.: Commun. Math. Phys.31, 327–340 (1973)
Yosida, K.: Functional analysis, Berlin, Heidelberg, New York: Springer 1965
Schwartz, L.: Theorie des distributions. Paris: Herman, 1973
Simon, B.: Commun. Math. Phys.71, 247–276 (1980)
Author information
Authors and Affiliations
Additional information
Communicated by E. Lieb
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.
Rights and permissions
About this article
Cite this article
Vuillermot, P.A. Nonlinear dynamics of the infinite classical Heisenberg model: Existence proof and classical limit of the corresponding quantum time evolution. Commun.Math. Phys. 76, 1–26 (1980). https://doi.org/10.1007/BF01197107
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01197107