Path integral renormalization in loop quantum cosmology

A coarse graining technique akin to block spin transformations that groups together fiducial cells in a homogeneous and isotropic universe has been recently developed in the context of loop quantum cosmology. The key technical ingredient was an SU(1, 1) group and Lie algebra structure of the physical observables as well as the use of Perelomov coherent states for SU(1, 1). It was shown that the coarse graining operation is completely captured by changing group representations. Based on this result, it was subsequently shown that one can extract an explicit renormalisation group flow of the loop quantum cosmology Hamiltonian operator in a simple model with dust-clock. In this paper, we continue this line of investigation and derive a coherent state path integral formulation of this quantum theory and extract an explicit expression for the renormalisation-scale dependent classical Hamiltonian entering the path integral for a coarse grained description at that scale. We find corrections to the non-renormalised Hamiltonian that are qualitatively similar to those previously investigated via canonical quantisation. In particular, they are again most sensitive to small quantum numbers, showing that the large quantum number (spin) description captured by so called "effective equations" in loop quantum cosmology does not reproduce the physics of many small quantum numbers (spins). Our results have direct impact on path integral quantisation in loop quantum gravity, showing that the usually taken large spin limit should be expected not to capture (without renormalisation, as mostly done) the physics of many small spins that is usually assumed to be present in physically reasonable quantum states.