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this is not based on detailed work but just an idea which arised comparing causal dynamical triangulations and loop quantum gravity.

In CDT it seems reasonable to treat spacetime as a fractal. That means there is no limit or minimum length in the triangulations, but the triangulations are "nested" like a fractal. This corresponds to coarse graining like the block-spin transformations in the renormalization group approach for the Ising model.

Wouldn't it be possible to apply a similar method in LQG? That would mean nesting of spin networks. In a sense a vertex would be an effective description of a volume which (via coarse graining) resolves into finer volumina with new vertices. In one paper studying LQG black holes I found a similar idea. The isolated horizon in LQG translates into the idea that one can replace all vertices (intertwiners) forming the spacetime inside the horizon by one single huge (!) interwiner representing the whole black hole.

If one applies this idea to LQG as a whole one must answer the question what happens with a minimum length. This could be achieved via renormalization of the Immirzi parameter. So coarse graining of spin networks goes hand in hand with a running Immirzi parameter. Btw.: the length, area and volume operators are not necessarily Dirac observables, therefore their quantized spectrum does not automatically carry over to physical observables.

Does this idea make sense?

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# Fractal LQG spacetime and renormalization of the Immirzi parameter

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