Commun. Math. Phys. 127, 637-651 (1990) Mathematical
© Springer-Verlag 1990
On the Classical Limit
of Berry's Phase Integrable Systems
Technische Universitat Berlin, Fachbereich Mathematik, Ma7-1,
StraBe des 17. Juni 136, D-1000 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].
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