Symmetry, Reduction, and Phases in Mechanics 15
imaginary part of the inner product) via reduction, as in Abraham and Marsden [1978]. We shall
show in §4 that this formula is the holonomy of the closed loop relative to a principal
connection on complex Hilbert space and is a particular case of the holonomy formula in principal
bundles with abelian structure group.
Littlejohn [1988] has shown that the Bohr-Sommerfeld and Maslov phases of semi-
classical mechanics can be viewed as incarnations of Berry's phase. To do this he notes that
Gaussian wave-packets define an embedding of classical phase space into Hilbert space, then uses
the Aharonov-Anandan point of view on phases, together with the variational formulation of
quantum mechanics. The quantum-classical relation between the phases is also considered in
Hannay [1985], Anandan [1988], and Weinstein [1989a,b].
§1F Integrable systems
Consider an integrable system with action-angle variables (I
I2, ..., In, Qv 82, » ®n)
and with a Hamiltonian H(Ij, I2, ... In, Bv 82,... 9n; m) that depends on a parameter m e M.
Let c be a loop based at a point m0 in M. We want to compare the angular variables in the torus
over m0, once the system is slowly changed as the parameters undergo the circuit c. Since the
dynamics in the fiber varies as we move along c, even if the actions vary by a negligible amount,
there will be a shift in the angle variables due to the frequencies
of the integrable
system; correspondingly, one defines
dynamic phase = co^I, c(t))) dt .
Here we assume that the loop is contained in a neighborhood whose standard action coordinates
are defined. In completing the circuit c , we return to the same torus, so a comparison between the
angles makes sense. The actual shift in the angular variables during the circuit is the dynamic
phase plus a correction term called the geometric phase. One of the key results is that this
geometric phase is the holonomy of an appropriately constructed connection called the Hannay-
Berry connection on the torus bundle over M which is constructed from the action-angle
variables. The corresponding angular shift, computed by Hannay [1985], is called Hannay's
angles, so the actual phase shift is given by
I A8 = dynamic phases + Hannay's angles .
The geometric construction of the Hannay-Berry connection for classical systems is given in terms
of momentum maps and averaging in Golin, Knauf, and Marmi [1989] and Montgomery [1988].
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