Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity

Abstract

This paper focuses on the semiclassical behavior of the spinfoam quantum gravity in 4 dimensions. There has been long-standing confusion, known as the flatness problem, about whether the curved geometry exists in the semiclassical regime of the spinfoam amplitude. The confusion is resolved by the present work. By numerical computations, we explicitly find curved Regge geometries from the large-j Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam amplitudes on triangulations. These curved geometries are with small deficit angles and relate to the complex critical points of the amplitude. The dominant contribution from the curved geometry to the spinfoam amplitude is proportional to e iI , where I is the Regge action of the geometry plus corrections of higher order in curvature. As a result, the spinfoam amplitude reduces to an integral over Regge geometries weighted by e iI in the semiclassical regime. As a byproduct, our result also provides a mechanism to relax the cosine problem in the spinfoam model. Our results provide important evidence supporting the semiclassical consistency of the spinfoam quantum gravity. * hanm(AT)fau.edu † hzc881126(AT)hotmail.com ‡ hongguang.liu(AT)gravity.fau.de § dqu2017(AT)fau.edu

From spin foams to area metric dynamics to gravitons

“…These misunderstandings have also contributed to the so-called flatness problem for spin foams [2,10,14], which states the expectation that the semi-classical limit of spin foams suppresses curvature. There is more and more evidence [2,10,11,15,16] that this issue can be circumvented. The arguments and supporting examples in these works [2,10,15,16] are however based on triangulations with a small number of building blocks.…”

“…There is more and more evidence [2,10,11,15,16] that this issue can be circumvented. The arguments and supporting examples in these works [2,10,15,16] are however based on triangulations with a small number of building blocks. They require furthermore the implementation of area-length constraints, which force the areas to arise from a consistent length assignment.…”

“…Whereas the Barrett-Crane spin foam model [18] does not implement these constraints, the more recent spin foam models [2,10,19] do impose the constraints weakly, with the Barbero-Immirzi parameter controlling how much the constraints are allowed to fluctuate [2]. The works [2,10,15,16] therefore also suggest a bound on the Barbero-Immirzi parameter.…”

The accidental flatness constraint does not mean a wrong classical limit

“…Here, building on a number of recent results, in particular the analytical and numerical investigations in [13][14][15][16], as well as original ideas proposed in [11,17], we illustrate why the tension is only apparent, and that accidental constraints appear commonly from exchanging the order of limits.…”

Numerical evaluation of spin foam amplitudes beyond simplices

“…These algorithms furthermore lead to first studies of 2-complexes with several 4-simplices [29], e.g. to explore the so-called "flatness problem" in spin foams [30][31][32][33][34][35], yet they remain challenging due to rapidly growing numerical costs as the representation labels are increased. Additionally, these promising results are complemented by developments that will help to unlock larger 2-complexes: effective spin foam models [34,36,37] bypass this numerical challenge by directly starting from the asymptotic formula to investigate under which conditions reasonable semi-classical physics emerge, while Lefshetz thimbles enable the use of Monte Carlo methods to compute expectation values of observables on larger 2-complexes [38].…”