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Free keywords:
High Energy Physics - Theory, hep-th,General Relativity and Quantum Cosmology, gr-qc,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
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.
