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

Foundations of Physics 52 (2):1-32 (2022)
  Copy   BIBTEX

Abstract

The quantum transition probability assignment is an equiareal transformation from the annulus of symplectic spinorial amplitudes to the disk of complex state vectors, which makes it equivalent to the equiareal projection of Archimedes. The latter corresponds to a symplectic synchronization method, which applies to the quantum phase space in view of Weyl’s quantization approach involving an Abelian group of unitary ray rotations. We show that Archimedes’ method of synchronization, in terms of a measure-preserving transformation to an equiareal disk, imposes the integrality of the quantum of action, and requires the extension of the classical moment map from the real line to the circle. Additionally, the same synchronization method is encoded in the structure of the Heisenberg group, viewed as a principal bundle with a connection, whose curvature and anholonomy is expressed in terms of area bounding loops in relation to the underlying Abelian shadow on a symplectic plane. In this manner, we show that the geometric phase pertains to the minimal synchronized area \ of the 2-d symplectic Abelian shadow of the symplectic ball, modulo \. The integrality condition naturally leads to the consideration of modular commutative observables pertaining to the role of the discrete Heisenberg group. We prove that the structural transition from non-commutativity to modular commutativity in accordance to Weyl’s group-theoretic commutation relations takes place via universal factorization through the discrete Heisenberg group. In this way, we derive a homology-theoretic formulation of the synchronization method in terms of the area-bounding cells of the modular lattice \ in relation to any Abelian symplectic shadow. Thus, we finally obtain the physical interpretation of the analytic representation of quantum states as theta functions corresponding to the sections of a complex line bundle with an integral symplectic structure.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 76,346

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

On the Relation Between Gauge and Phase Symmetries.Gabriel Catren - 2014 - Foundations of Physics 44 (12):1317-1335.
Quantum Blobs.Maurice A. de Gosson - 2013 - Foundations of Physics 43 (4):440-457.
Geometry and Structure of Quantum Phase Space.Hoshang Heydari - 2015 - Foundations of Physics 45 (7):851-857.
A geometric approach to quantum mechanics.J. Anandan - 1991 - Foundations of Physics 21 (11):1265-1284.
Symplectic Reduction and the Problem of Time in Nonrelativistic Mechanics.Karim P. Y. Thébault - 2012 - British Journal for the Philosophy of Science 63 (4):789-824.
Coherent phase spaces. Semiclassical semantics.Sergey Slavnov - 2005 - Annals of Pure and Applied Logic 131 (1-3):177-225.

Analytics

Added to PP
2022-04-05

Downloads
3 (#1,315,069)

6 months
2 (#299,341)

Historical graph of downloads
How can I increase my downloads?