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Journal Article

Non-commutative flux representation for loop quantum gravity

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons20702

Baratin,  Aristide
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons20705

Dittrich,  Bianca
Canonical and Covariant Dynamics of Quantum Gravity, AEI Golm, MPI for Gravitational Physics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons20698

Oriti,  Daniele
Microscopic Quantum Structure & Dynamics of Spacetime, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons2681

Tambornino,  Johannes
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Fulltext (public)

1004.3450
(Preprint), 307KB

CQG_28_17_175011.pdf
(Any fulltext), 305KB

Supplementary Material (public)
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Citation

Baratin, A., Dittrich, B., Oriti, D., & Tambornino, J. (2011). Non-commutative flux representation for loop quantum gravity. Classical and quantum gravity, 28(17): 175011. doi:10.1088/0264-9381/28/17/175011.


Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-03F3-5
Abstract
The Hilbert space of loop quantum gravity is usually described in terms of cylindrical functionals of the gauge connection, the electric fluxes acting as non-commuting derivation operators. Here we introduce a dual description of this space, by means of a Fourier transform mapping the usual loop gravity states to non-commutative functions on Lie algebras. We show that the Fourier transform defines a unitary equivalence of representations for loop quantum gravity. In the dual representation, flux operators act by star-multiplication and holonomy operators act by translation. We describe the gauge invariant dual states and discuss their geometrical meaning. Finally, we apply the construction to the simpler case of a U(1) gauge group and compare the resulting flux representation with the triad representation used in loop quantum cosmology.