ar X iv : 0 70 9 . 34 65 v 1 [ gr - q c ] 2 1 Se p 20 07 Dynamical Quantum Geometry ( DQG Programme )

In this brief note (written as a lengthy letter), we describe the construction of a representation for the Weyl-algebra underlying Loop Quantum Geometry constructed from a diffeomorphism variant state, which corresponds to a ”condensate” of Loop Quantum Geometry, resembling a static spatial geometry. We present the kinematical GNS-representation and the gaugeand diffeomorphism invariant Hilbert space representation and show that the expectation values of the geometric operators take essentialy classical values plus quantum corrections, which is similar to a ”local condensate” of quantum geometry. We describe the idea for the construction of a scale dependent asymptotic map into a family of scale dependent lattice gauge theories, where scale separates the essential geometry and a low energy effective theory, which is described as degrees of freedom in the lattice gauge theory. If this idea can be implemented then it is likely to turn out that this Hilbert space contains in addition to gravity also gauge coupled ”extra degrees of freedom”, which may not be dynamically irrelevant. The algebra that underlies Loop Quantum Gravity is generated by matrix elements he(A)IJ of SU(2)-holonomies along arbitrary piecewise analytical curves e in the Cauchy surface Σ and the fluxes E(S) of the conjugated electric fields through arbitrary piecewise analytical surfaces S. This algebra carries a canonical representation of the SU(2)-gauge transformations and the piecewise analytical diffeomorphisms on Σ. A C-algebra version of this algebra was introduced by Fleischhack [1]. This algebra is constructed as a subalgebra of B(H = L(A, dμAL)), the bounded operators on the Hilbert space of w.r.t. the Ashtekar-Lewandowski measure square integrable functions on the