Topological Quantum Numbers

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9528345 Thouless This is a proposal to continue work on two fundamental problems in materials theory, quantum Hall effect and superconductivity. The specific problems here are all derived from the topological symmetry derived conservation laws and the corresponding quantum numbers. The questions that need to be addressed, include the location where the current is generated in a Hall device and in a superfluid or a superconductor, the question of the mass density of a vortex and whether the Magnus force on a vortex is finite or zero. In addition there will be exploration of some aspects of quantum chaos, for example the notion of a distance between two states in quantum mechanics. %%% Much of the metrological work (measumenent of fundamental constants to high precision, or high precision measurements of voltage and resistance) involve use of topological quantum numbers. These quantum numbers are found in a variety of diverse situations and yet there are deep underlying similarities. The fundamental properties associated with the topological quantum numbers are the subject of this study. The applications include the stability of superconductor or a superfluid and more specifically the detailed dynamics of the vortices. A comprehensive review, in the form of lecture notes and reprints of classical papers on the subject of topological quantum numbers will be prepared.