Wilson loops in nonperturbative quantum gravity

Abstract

By explicit construction, we show that one can in a simple way introduce and measure gravitational holonomies and Wilson loops in lattice formulations of nonperturbative quantum gravity based on (causal) dynamical triangulations. We use this setup to investigate a class of Wilson line observables associated with the world line of a point particle coupled to quantum gravity, and deduce from their expectation values that the underlying holonomies cover the group manifold of SO(4) uniformly.

Quantum gravity from causal dynamical triangulations: a review

“…Few such observables are currently known. On the conceptual side, as illustrated by studies of CDT coupled to a single massive particle [42,43], there are subtleties in relating Euclidean and Lorentzian results, which still need to be understood better.…”

“…There are different ways of constructing observables from Wilson loops, for example by "marking" the location of the loop in terms of matter degrees of freedom or by performing averages over loops or subsets of loops that share certain invariant geometric features regarding their length and shape. These considerations are relevant for CDT, where parallel transport and Wilson loops can be defined in a straightforward way, as described in [43]. Because of the piecewise flat nature of the geometry this turns out to be far easier than on a smooth curved manifold.…”

“…30 However, it is a priori conceivable that in nonperturbative quantum gravity like CDT, Wilson loops which probe the quantum geometry on scales sufficiently large compared to the Planck scale but are still infinitesimal from a classical point of view display a semiclassical behaviour, where quantum fluctuations mostly "average out", leading to expectation values for the Wilson loops close to (4 times) the identity, with small deviations that characterize some effective curvature scale. This hypothesis was investigated in simulations of four-dimensional CDT with spherical slices at (κ 0 , ∆) = (2.2, 0.6), for t tot = 80 and N 4 = 20k and loops of winding number 1 with respect to the compactified time direction [43]. To obtain a genuine Wilson loop observable, the underlying loop γ was identified with the spacetime trajectory of a massive particle moving forward in time along one of the piecewise straight paths mentioned earlier.…”

“…Histogram of the invariant angles (θ 1 , θ 2 ), extracted from Monte Carlo measurements of Wilson loops in CDT[43], matching the functional form(37) up to an overall normalization.…”

“…Barycentric coordinates are a convenient choice[43] 30. A gravitational version of the nonabelian Stokes' theorem is unlikely to be useful because curvature appears only inside a complicated, nonlocal expression involving area ordering[99].…”