Coherent states for FLRW space-times in loop quantum gravity

We construct a class of coherent spin-network states that capture properties of curved space-times of the Friedmann-Lamaître-Robertson-Walker type on which they are peaked. The data coded by a coherent state are associated to a cellular decomposition of a spatial ($t=\mathrm{const}$) section with a dual graph given by the complete five-vertex graph, though the construction can be easily generalized to other graphs. The labels of coherent states are complex $SL(2,\mathbb{C})$ variables, one for each link of the graph, and are computed through a smearing process starting from a continuum extrinsic and intrinsic geometry of the canonical surface. The construction covers both Euclidean and Lorentzian signatures; in the Euclidean case and in the limit of flat space we reproduce the simplicial 4-simplex semiclassical states used in spin foams.

Figure 1

The complete graph with 5 nodes, 1-skeleton of the cellular decomposition $\Delta \simeq {\Delta }^{*}$.