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Group quantization of parametrized systems. II. Pasting Hilbert spaces

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JournalJournal of Mathematical Physics
DatePublished - 1995
Issue number9
Volume36
Number of pages27
Pages (from-to)4639-4665
Original languageEnglish

Abstract

The method of group quantization described in the preceding paper [J. Math. Phys. 36, 4612 (1995)] is extended so that it becomes applicable to some parametrized systems that do not admit a global transversal surface. A simple completely solvable toy model is studied that admits a pair of maximal transversal surfaces intersecting all orbits. The corresponding two quantum mechanics are constructed. The similarity of the canonical group actions in the classical phase spaces on the one hand and in the quantum Hilbert spaces on the other hand suggests how the two Hilbert spaces are to be pasted together. The resulting quantum theory is checked to be equivalent to that constructed directly by means of Dirac's operator constraint method. The complete system of partial Hamiltonians for any of the two transversal surfaces is chosen and the quantum Schrödinger or Heisenberg pictures of time evolution are constructed.

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