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