Bulg. J. Phys. vol.35 no.s1 (2008), pp. 190-210

Quantum Mechanical Tunneling on a Space with Singularities

Johannes Huebschmann1, Gerd Rudolph2, Matthias Schmidt2
1USTL, UFR de Mathématiques, CNRS-UMR 8524, 59655 Villeneuve d'Ascq Cédex, France
2Institute for Theoretical Physics, University of Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany
Abstract. We review the construction of a quantum lattice gauge theory on a single spatial plaquette carried out in detail elsewhere. This approach incorporates the classical phase space singularities. Here we concentrate on a special case. The reduced phase space is a stratified Kähler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure furnishes a costratified Hilbert space, that is, a Hilbert space together with a system which consists of subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is SU(2), we discuss the tunneling probabilities between the strata, the energy eigenstates, and the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata.

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