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

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##### Fulltext (public)

0608087v1.pdf

(Preprint), 194KB

a7_13_016.pdf

(Publisher version), 286KB

##### Supplementary Material (public)

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##### Citation

Isidro, J. M., & de Gosson, M. A. (2007). Abelian gerbes as a gauge theory of quantum
mechanics on phase space.* Journal of Physics A,* *40*(13),
3549-3567. doi:10.1088/1751-8113/40/13/016.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-4865-7

##### Abstract

We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space {\bb 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 {\bb P} , the only external input being Planck's constant planck. 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 {\bb 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 {\bb P} as an Abelian gauge theory.