  An adapted representation of quantum mechanics sheds new light on the
relationship between quantum states and classical states. In this approach the
space of quantum states splits into a product of the state space of classical
mechanics and a Hilbert space, and expectation values of observables decompose
into their classical value plus a quantum correction. The splitting is
preserved under time evolution of the Schr\"odinger equation under certain
assumptions, and the time evolution of the classical part of a quantum state is
governed by Hamilton's equation. The new representation is obtained from the
usual Hilbert space representation of quantum mechanics by introducing a gauge
degree of freedom in a time-dependent unitary transformation, followed by a
non-conventional gauge fixing condition.
