Topology of parametrized systems

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Abstract
The discussion is limited to finite-dimensional parametrized systems with one constraint—the so-called super-Hamiltonian. The topology of the corresponding Hamiltonian vector field on the constraint hypersurface, in particular the question of the existence of cross sections, is studied. This has some bearing on the applicability of the reduction method, as well as on unitarity within the Dirac method, of canonical quantization [see the previous paper, Phys. Rev. D 34, 1040 (1986)]. The main theorem of the present paper states that a cross section will exist if and only if the following conditions are both satisfied: (i) no dynamical trajectory is closed or almost closed; and (ii) the quotient set topology of the set of all dynamical trajectories is Hausdorff. Examples of systems that separately violate each of these conditions are given.

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