 
Summary: Communicationsin
Commun. Math. Phys. 127, 637651 (1990) Mathematical
Physics
© SpringerVerlag 1990
On the Classical Limit
of Berry's Phase Integrable Systems
J. Asch
Technische Universitat Berlin, Fachbereich Mathematik, Ma71,
StraBe des 17. Juni 136, D1000 Berlin12
Abstract. Berry's Phase is given by integration of acharacteristic two form. We
consider integrable systems defined by Weyl quantized classical Hamiltonians.
It is shown that the limit of h/i times this two form is the curvature of the classical
connection whose holonomy isthe Hannay angles. A result ofthis type was
derived byBerry [B2].
Introduction
Consider aquantum system whose dynamics is determined by afamily of self adjoint
Hamiltonians on a Hubert space depending smoothly on several parameters.
Consider furthermore a region of energies such that the corresponding spectral
subspace  defined bya projection P  also varies ina smooth manner.
Thus a vector bundle over themanifold of parameters is defined. Asitis
