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Loop quantum gravity (LQG) is a quantization program for gravity based on theprinciples of QFT and general covariance of general relativity. Quantum statesof LQG describe gravitational excitations based on graphs embedded in a spatialslice of spacetime. We show that, under certain assumptions on the class ofdiffeomorphisms, the space of diffeomorphism invariant states carries a Fockspace structure. The role of one-particle excitations for this structure isplayed by the diffeomorphism invariant states based on graphs with a single(linked) component. This means, however, that a lot of the structure of thediffeomorphism invariant Hilbert space remains unresolved by this structure. Weshow how the Fock structure allows to write at least some condensate states ofgroup field theory as diffeomorphism invariant coherent states of LQG in aprecise sense. We also show how to construct other interesting states usingthis Fock structure. We finally explore the quantum geometry of single- andmulti-particle states and tentatively observe some resemblance to geometrieswith a single or multiple components, respectively.

A Fock space structure for the diffeomorphism invariant Hilbert space of loop quantum gravity and its applications