Abstract
We present an analytic computation of an explicit renormalisation group flow for cosmological states in loop quantum gravity. A key ingredient in our analysis are Perelomov coherent states for the Lie group SU(1, 1) whose representation spaces are embedded into the standard loop quantum cosmology (LQC) Hilbert space. The SU(1, 1) group structure enters our analysis by considering a classical set of phase space functions that generates the Lie algebra . We implement this Poisson algebra as operators on the LQC Hilbert space in a non-anomalous way. This task requires a rather involved ordering choice, whose existence is one of the main results of the paper. As a consequence, we can transfer recently established results on coarse graining cosmological states from direct quantisations of the above Poisson algebra to the standard LQC Hilbert space and full theory embeddings thereof. We explicitly discuss how the representation spaces used in this latter approach are embedded into the LQC Hilbert space and how the 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 of a non-trivial renormalisation group flow with a scale set by the representation label and interpreted as the minimally resolved geometric scale.
Highlights
Loop quantum gravity [1, 2] is a non-perturbative approach to a quantum theory of the gravitational field
To avoid unnecessary technicalities and references to full loop quantum gravity, we introduce loop quantum cosmology (LQC) as the synthesis of a spatially flat, homogeneous and isotropic quantum cosmological model in the presence of a spatial volume quantised in integer multiples of a fundamental scale λ > 0
We should study the properties of the representation we found, i.e. find the representation label j and study how the representation space is embedded in the LQC Hilbert space
Summary
Loop quantum gravity [1, 2] is a non-perturbative approach to a quantum theory of the gravitational field. If one is far away from the Planck scale, such states may be represented well by many small spins on fine lattices, or few large ones on coarse lattices, as far as coarse observables are concerned These descriptions should be connected via a renormalisation group flow that renormalises the operators involved in the description. The su(1, 1) structure will immediately yield an explicit and non-trivial renormalisation group flow under a change of scale, i.e. the transition from many small to few large spins In this process, the su(1, 1) representation label j functions as a lower cutoff for the involved U(1)-analogues of the SU(2) spins jSU(2) in loop quantum cosmology.
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