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Journal Article

#### Non-commutative flux representation for loop quantum gravity

##### MPS-Authors

##### Locator

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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.