A Gauge Model for Quantum Mechanics on a Stratified Space

In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified Kähler space, and we make explicit the requisite singular holomorphic quantization procedure on this space....

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Bibliographic Details
Published in:Communications in mathematical physics 2009-03, Vol.286 (2), p.459-494
Main Authors: Huebschmann, J., Rudolph, G., Schmidt, M.
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Combinatorics. Ordered structures
Complex Systems
Exact sciences and technology
Functional analysis
Global analysis, analysis on manifolds
Mathematical analysis
Mathematical and Computational Physics
Mathematical methods in physics
Mathematical Physics
Mathematics
Order, lattices, ordered algebraic structures
Other topics in mathematical methods in physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Sciences and techniques of general use
Theoretical
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. 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 yields a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the 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, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-008-0693-0