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Abelian gerbes as a gauge theory of quantum mechanics on phase space

AutorIsidro, José M.; Gosson, Maurice A. de
Palabras clavePath-integral Duality
Schrödinger equation
[PACS] General structures of groups
[PACS] Algebraic topology
[PACS] Algebraic methods
Fecha de publicación30-mar-2007
EditorInstitute of Physics Publishing
CitaciónJournal of Physics A - Mathematical and Theoretical 40(13): 3549-3567 (2007)
ResumenWe construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space P. The connection is given by a triple of forms A, B, H: a potential 1-form A, a Neveu-Schwarz potential 2-form B, and a field-strength 3-form H = dB. All three of them are defined exclusively in terms of elements already present in P, the only external input being Planck's constant h. U(1) gauge transformations acting on the triple A, B, H are also defined, parametrized either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness versus classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H]/2πi is an integral class in de Rham cohomology is related to the discretization of symplectic area on P. This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction to generalized complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to P as an Abelian gauge theory.
Descripción19 pages, 1 figure.-- PACS nrs.: 02.20.Bb, 02.40.Re, 03.65.Fd.-- ISI Article Identifier: 000245037900017.-- ArXiv pre-print available at: http://arxiv.org/abs/hep-th/0608087
Versión del editorhttp://dx.doi.org/10.1088/1751-8113/40/13/016
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