Nonlocal correlations for semiclassical states in loop quantum gravity

TL;DR

The paper investigates semiclassical states in loop quantum gravity by computing two-point area correlations on a cubic lattice in the large-spin limit. It shows that Livine-Speziale intrinsic and heat-kernel extrinsic coherent states yield exponentially decaying, short-range correlations, consistent with a localized semiclassical geometry. The authors then construct perturbed LS coherent states by adjoining correlated two-loop excitations, producing long-range correlations that scale as $1/d^2$, akin to massless graviton fluctuations on a background geometry while preserving peakedness and gauge invariance. This perturbative framework provides a bridge between semiclassical LQG states and perturbative quantum gravity, suggesting a route to semiclassical solutions with graviton-like fluctuations. The work also clarifies differences from squeezed-vacuum approaches and outlines future work on higher-order perturbations and the Hamiltonian constraint.

Abstract

We compute the two-point correlation function of the area operator for semiclassical states of loop quantum gravity in the limit of large spins. The cases of intrinsic and extrinsic coherent states are considered, along with a new class of semiclassical states constructed as perturbations of Livine-Speziale coherent states. For the usual coherent states, the correlations are shown to be short-ranged, decaying exponentially with the distance. Introducing perturbations given by correlated elementary excitations and decays of the gravitational field along pairs of loops, we obtain new states that, while preserving the peakedness properties of the unperturbed states, can also display long-ranged correlations. The perturbed coherent states include examples reproducing the typical decay of correlations for quantum fluctuations of the geometry associated with free gravitons on a background metric. Such a behavior is a natural requirement for the compatibility of semiclassical states in quantum gravity with the physical regime pictured by perturbative quantum gravity.