Dynamical Quantum Geometry (DQG Programme)

Abstract

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 gauge- and 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.

Figures (3)

• Figure 1: This figure illustrates the relation between the "used diagonal cross sections" and the "used faces" in the moved diagonals. The numbers indicate the pair $(i,j)$ in the labeling of the face. The coordinate chart $U$ is assumed to be right-handed cartesian and the 1-direction is assumed to be going from left to right.
• Figure 2: An example of a solution to the Gauss- and difeomrophism constraint: The graph is embedded up to isometries of the background, contains additional couplings to the background (black dots) and disconnected regions of the graph are not physically disconnected, due to the occurrence of the background geometry.
• Figure 3: Some of the "extra vertex degrees of freedom" arising in our construction. Notice that these "extra degrees of freedom" do not occur in a lattice version of Ashtekar gravity. The relevance of the degrees of freedom is decided by the dynamics. The usual argument from perturbatively renormizable QFT that the new degrees of freedom on a finer lattice effectively decouple from degrees of freedom on the coarser lattice is not obvious in our case due to the occurrence of couplings to the background, that may occur between extra degrees of freedom on the finer lattice in the $E_o$-representation.