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A Cosmological Sector in Loop Quantum Gravity

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A classical Hamiltonian system can be reduced to a subsystem of "relevant observables" using the pull-back under a Poisson embedding of the "relevant phase space" into the "full phase space". Since a quantum theory can be thought of a noncommutative phase space, one encounters the problem of the embedding of noncommutative spaces, when one tries to extend the reduction via a pull-back to a quantum theory. This problem can be solved for a class of physically interesting quantum systems and embeddings using an analogy to finding the base space of an embedded fibre bundle via the projection in the full fibre bundle. The resulting construction is then applied to Loop Quantum Gravity to extract a cosmological sector. This sector turns out to be similar but not equivalent to Loop Quantum Cosmology.