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Foundations of Physics

The Equiareal Archimedean Synchronization Method of the Quantum Symplectic Phase Space: II. Circle-Valued Moment Map, Integrality, and Symplectic Abelian Shadows

Elias Zafiris

Published Apr 1, 2022• 2 citations
Publisher
DOI
10.1007/s10701-022-00559-9

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Phase space regions operators and ISp(2) maps

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Region operators (generalized observable) (RO), defined as operator valued probability measures over classical phase space ensuing from quantization of phase space characteristic functions are investigated. The problem of constructing of RO is treated by means of positive trace increasing maps (pti maps). Exemplary domains of RO include: sets of points, line segments and rotated line segments, circles and disks, straight and rotated lines, canonical polygons, squeezed line segments, and lines related to Radon transform in phase space. Phase plane RO constructing pti maps, in their operator sum representation, are generated by elements of the quantum symplectic in-homogeneous group ISp(2). The group ISp(2) and its double cover metaplectic group Mp(2) are the kinematical groups for RO construction on the plane.

Journal of Physics Conference Series•Apr 1, 2019•1 citation
Abstract. We argue that the classical description of a symplectic manifold endowed with a Hamil-tonian action of an abelian Lie group G and the corresponding quantum theory can be understood as different aspects of the unitary representation theory of G. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry (notably, Souriau’s moment map and the Mardsen-Weinstein symplectic reduction formalism) and group representation theory (notably Kirillov’s orbit method). The proposed argumentative line strongly relies on the conjecture pro-posed by Guillemin and Sternberg according to which “quantization commutes with (symplectic) reduction”. By using the generalization of this conjecture to non-zero coadjoint orbits, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric un-derpinning, namely the symplectic reduction formalism. This fact points towards a gauge-theoretic interpretation of Heisenberg indeterminacy principle. Key words: quantum mechanics; gauge theories; moment map; symplectic reduction. 1.
Region operators (generalized observable) (RO), defined as operator valued probability measures over classical phase space ensuing from quantization of phase space characteristic functions are investigated. The problem of constructing of RO is treated by means of positive trace increasing maps (pti maps). Exemplary domains of RO include: sets of points, line segments and rotated line segments, circles and disks, straight and rotated lines, canonical polygons, squeezed line segments, and lines related to Radon transform in phase space. Phase plane RO constructing pti maps, in their operator sum representation, are generated by elements of the quantum symplectic in-homogeneous group ISp(2). The group ISp(2) and its double cover metaplectic group Mp(2) are the kinematical groups for RO construction on the plane.