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Symplectic Topology and Geometric Quantum Mechanics


Abstract The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system, with a projective Hilbert space regarded as the phase space. This thesis extends the theory by including some aspects of the symplectic topology of the quantum phase space. It is shown that the quantum mechanical uncertainty principle is a special case of an inequality from J-holomorphic map theory, that is, J-holomorphic curves minimize the difference between the quantum covariance matrix determinant and a symplectic area. An immediate consequence is that a minimal determinant is a topological invariant, within a fixed homology class of the curve. Various choices of quantum operators are studied with reference to the implications of the ... (more)
Created Date 2011
Contributor Sanborn, Barbara (Author) / Suslov, Sergei K (Advisor) / Suslov, Sergei (Committee member) / Spielberg, John (Committee member) / Quigg, John (Committee member) / Menendez, Jose (Committee member) / Jones, Donald (Committee member) / Arizona State University (Publisher)
Subject Mathematics / Quantum physics / Condensed Matter Physics / adiabatic theorem / geometric quantum mechanics / J-holomorphic curves / mean curvature / symplectic topology / uncertainty principle
Type Doctoral Dissertation
Extent 101 pages
Language English
Copyright
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Note Ph.D. Mathematics 2011
Collaborating Institutions Graduate College / ASU Library
Additional Formats MODS / OAI Dublin Core / RIS


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Description Dissertation/Thesis