Gravitational Wilson loop and large scale curvature

Abstract

In a quantum theory of gravity the gravitational Wilson loop, defined as a suitable quantum average of a parallel transport operator around a large near-planar loop, provides important information about the large-scale curvature properties of the geometry. Here we shows that such properties can be systematically computed in the strong coupling limit of lattice regularized quantum gravity, by performing local averages over loop bivectors, and over lattice rotations, using an assumed near-uniform measure in group space. We then relate the resulting quantum averages to an expected semiclassical form valid for macroscopic observers, which leads to an identification of the gravitational correlation length appearing in the Wilson loop with an observed large-scale curvature. Our results suggest that strongly coupled gravity leads to a positively curved (de Sitter-like) quantum ground state, implying a positive effective cosmological constant at large distances.

Wilson loops in nonperturbative quantum gravity

“…We will investigate how this influences the measured invariants t i . More specifically, we will extract from them the distribution of the angles θ i and compare it to the distribution one 6 The fact that two angles are necessary to characterize holonomy matrices up to conjugation, as soon as one considers nonminimal loops in a four-dimensional simplicial manifold T , seems to have been overlooked by the authors of [16]. This also holds when T is almost flat and holonomies do not deviate much from the identity matrix.…”

“…In order to do this, we need to derive the theoretical distribution of the θ i on SOð4Þ. Recall that these angles were introduced in the context of the maximal torus (16). They are two of a total of six parameters needed to label points of SOð4Þ.…”

“…After a brief description 3 We note in passing that the analogous averaging problem in classical general relativity has not been resolved (see, for example, [15]). 4 Theoretical arguments were put forward in a different formulation of lattice gravity based on Regge calculus, promoting large Wilson loops as carriers of nonperturbative information [16]. Although sympathetic to the aim, we are unable to follow the technical claims in [16] or to understand how the construction can be implemented meaningfully in a nonperturbative context.…”

Ultraviolet Divergences and Scale-Dependent Gravitational Couplings

“…(16). An equivalent way of phrasing the last result is the suggestion that 1/ξ 2 , where ξ is the renormalization group invariant gravitational correlation length of Eq.…”

“…In the gravity case the analogs of the gauge variables of Yang-Mills theories are given by the connections, so it is natural when computing the gravitational Wilson loop [15] to look for a first order formulation of Regge gravity [16]. For each neighboring pair of simplices s, s + 1 one can associate a Lorentz transformation R µ ν (s, s + 1), and one then might want to consider a near-planar closed loop C, such as the one shown schematically in Fig.2.…”