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arXiv · 1004.3450

Non-commutative flux representation for loop quantum gravity

Aristide Baratin·Bianca Dittrich·Daniele Oriti·Johannes Tambornino
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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. It has long been believed that this non-commutativity prevents a dual flux (or triad) representation of loop quantum gravity to exist. We show here, instead, that such a representation can be explicitly defined, by means of a non-commutative Fourier transform defined on the loop gravity state space. In this dual representation, flux operators act by *-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.

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Aristide Baratin, Bianca Dittrich, Daniele Oriti, Johannes Tambornino. 2012-02-06. Non-commutative flux representation for loop quantum gravity. https://doi.org/10.1088/0264-9381%2F28%2F17%2F175011

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