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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. 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.