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We present an analytic computation of an explicit renormalisation group flowfor cosmological states in loop quantum gravity. A key ingredient in ouranalysis are Perelomov coherent states for the Lie group SU(1,1) whoserepresentation spaces are embedded into the standard loop quantum cosmology(LQC) Hilbert space. The SU(1,1) group structure enters our analysis byconsidering a classical set of phase space functions that generates the Liealgebra su(1,1). We implement this Poisson algebra as operators on the LQCHilbert space in a non-anomalous way. This task requires a rather involvedordering choice, whose existence is one of the main results of the paper. As aconsequence, we can transfer recently established results on coarse grainingcosmological states from direct quantisations of the above Poisson algebra tothe standard LQC Hilbert space and full theory embeddings thereof. Weexplicitly discuss how the su(1,1) representation spaces used in this latterapproach are embedded into the LQC Hilbert space and how the su(1,1)representation label sets a lower cut-off for the loop quantum gravity spins (=U(1) representation labels in LQC). Our results provide an explicit example ofa non-trivial renormalisation group flow with a scale set by the su(1,1)representation label and interpreted as the minimally resolved geometric scale.

Renormalisation with SU(1, 1) coherent states on the LQC Hilbert space