Berry’s phase and the symplectic character of quantum time evolution

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Abstract
General time-dependent, bilinear classical Lagrangian mechanics can be interpreted as a charged particle interacting with a gauge scalar and vector potential. For an arbitrary discrete number of degrees of freedom, there are enough symplectic symmetries in classical mechanics to make this system canonically identical to the most general discrete spectrum quantum system. It follows that the general invariance group of the Schrödinger system of equations is the symplectic group Sp(2N), not U(N). Several examples show the physical naturalness of the classical coordinates. In the adiabatic limit Berry’s phase is identified as the cumulative change of the angle of action-angle coordinates of a normal mode. A generalization of Berry’s phase, the time component of the adiabatic connection and its associated electric curvature, is shown to be on an equal footing with the usual Berry phase.