14 Marsden, Montgomery, and Ratiu
theory of Thurston. Here one considers a linkage of rods, say four rods linked by pivot joints as in
Figure 1D-2. The system is free to rotate without external forces or torques, but there are
assumed to be torques at the joints. When one turns the small "crank" the whole assemblage turns
even though the angular momentum, as in the previous example, stays zero.
overall phase rotation
of the assemblage
Figure 1D-2
§1E Quantum mechanics
The original motivation for geometic phases came from quantum mechanics. Here the
important contributions historically were by Kato in 1950 (for the quantum adiabatic theorem),
Longuet-Higgins in 1958 for anomalous spectra in rotating molecules, Berry [1984] who first saw
the geometry of the phenomena for a variety of systems, and Simon [1983] who explicitly realized
the phases as the holonomy of the Chern-Bott connection. For more information on quantum
mechanical phases, and for the references quoted, see the collection of papers in Shapere and
Wilczek [1988].
For the purposes of the present work, the paper of Aharonov and Anandan [1987] plays an
important role. They got rid of the adiabaticity and showed that the phase for a closed loop in
projectivized complex Hilbert space is the exponential of the symplectic area of a two-dimensional
manifold whose boundary is the given loop. The symplectic form in question is naturally induced
on the projective space from the canonical symplectic form of complex Hilbert space (minus the
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