Geometric Phases and Related Structures +

The phase of a single state is not an observable quantity. In particular, the phase commutes with the observables which define the system. Nevertheless the change of states is generally accompanied by a change of the phase that can be called phase transport. If a state w is changed in two different ways to become another state w’, the transport of the phases may yield different phases. Then their “difference”, the relative phase, may become observable by virtue of the superposition principle. A particular case is the cyclic change, where w comes back to itself, and the change of the phase will be compared with that of a “trivial” process, where w remains stationary. All this is obvious, both experimentally and theoretically, for pure states. But it should remain true, to a certain instant, for mixed states: At first, in deviating from the pure to the mixed states, i.e. in going from the extreme part into the inner parts of the state space, coherence and correlations will not be destroyed suddenly but gradually, continuously. Secondly, if embedded in a larger system, the mixed states may be seen as restrictions of pure states. Then some “parts” of the relative phase of a cyclic change in the larger system may become decodable already by observables of the smaller system in which the states appear as mixed ones. The phase transport and the relative phase consist (at least) of two parts, a dynamical and a geometrical one. The geometric part depends only on the shape of the curve in state space which describes the changes of the system, but not on the time needed for that changes. It is a feature that allows to distinct the geometric phase and its transport from the total phase change. This remarkable fact also opens a heuristic way to see why the geometric phase survives the adiabatic approximation in which the changes become “infinitely slow”.