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      Non-local, non-commutative picture in quantum mechanics and distinguished continuous canonical maps

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      Author(s)
      Hakioglu, T.
      Date
      2002
      Source Title
      Physica Scripta
      Print ISSN
      0031-8949
      Publisher
      IOP Science
      Volume
      66
      Issue
      5
      Pages
      345 - 353
      Language
      English
      Type
      Article
      Item Usage Stats
      184
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      Abstract
      It is shown that continuous classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate h-independent quantum nonlinear canonical maps. It is also shown that such maps act in the non-commutative phase space under the classical covariance. A crucial result of the work is that under the action of Poisson maps a local quantum mechanical picture is converted onto a non-local picture which is then represented in a non-local Hilbert space. On the other hand, it is known that a non-local picture is equivalent by the Weyl map to a non-commutative picture which, in the context of this work, corresponds to a phase space formulation of the theory. As a result of this equivalence, a phase space Schrödinger picture can be formulated. In particular, we obtain the *-genvalue equation of Fairlie [Proc. Camb. Phil. Soc., 60, 581 (1964)] and Curtright, Fairlie and Zachos [Phys. Rev., D 58, 025002 (1998)]. In a non-local picture entanglement becomes a crucial concept. The connection between the entanglement and non-locality is explored in the context of Poisson maps and specific examples of the generation of entanglement from a local wavefunction are provided by using the concept of generalized Bell states. The results obtained are also relevant for the non-commutative soliton picture in the non-commutative field theories. We elaborate on this in the context of the scalar non-commutative field theory.
      Keywords
      Eigenvalues and eigenfunctions
      Hamiltonians
      Mathematical transformations
      Phase space methods
      Polynomials
      Non-commutative pictures
      Quantum theory
      Permalink
      http://hdl.handle.net/11693/24630
      Published Version (Please cite this version)
      http://dx.doi.org/10.1238/Physica.Regular.066a00345
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