Borel Quantization and Nonlinear Quantum Mechanics

J. Tolar

Slawomir Bugajski, Zbigniew Motyka

H. D. Doebner, H. J. Elmers, W. Heidenreich

Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold ℳ as its configuration space is based on a quantization method, which, in general, reflects global properties of ℳ, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on ℳ. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold ℳ, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on ℳ, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self-adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self-adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.

Gilbert Reinisch

S. Twareque Ali, H. D. Doebner

Starting from the idea that physical measurements may have residual imprecisions, the possibility of replacing the nonrelativistic, three-dimensional configuration space by a so-called fuzzy configuration space, having an isomorphic Borel structure, is discussed. A quantization procedure with respect to such a space is developed, and the invariance of nonrelativistic quantum mechanics under such Borel isomorphisms is exploited to prove the equivalence of this quantization procedure to the usual quantization procedure on a fuzzy-free configuration space. Further, any Galilean invariant dynamics is shown to be insensitive to such imprecisions in the measurements of position and momentum.

Pang Xiao Feng