EPRL/FK asymptotics and the flatness problem

Abstract

Spin foam models are an approach to quantum gravity based on the concept of sum over states, which aims to describe quantum spacetime dynamics in a way that its parent framework, loop quantum gravity, has not as of yet succeeded. Since these models' relation to classical Einstein gravity is not explicit, an important test of their viabilitiy is the study of asymptotics -the classical theory should be obtained in a limit where quantum effects are negligible, taken to be the limit of large triangle areas in a triangulated manifold with boundary. In this paper we will briefly introduce the EPRL/FK spin foam model and known results about its asymptotics, proceeding then to describe a practical computation of spin foam and semiclassical geometric data for a simple triangulation with only one interior triangle. The results are used to comment on the "flatness problem" -a hypothesis raised by Bonzom (2009) suggesting that EPRL/FK's classical limit only describes flat geometries in vacuum. *

Addendum: EPRL/FK Asymptotics and the Flatness Problem

“…Equation ( 1) is thus violated, so that the results of [6][7][8] imply that the amplitude should be suppressed. How can this be consistent with the conclusion of [3] that it is not suppressed? As we shall show below, the calculation in [3] makes an assumption, which, when removed, leads exactly to the constraint (1), thereby providing a new derivation of the constraint in this case.…”

“…At present, the spin-foam model which has passed the most tests of viability, and on which the most work has been done, is the EPRL/FK model, which comes in both Lorentzian and Euclidean versions [1,2]. The work [3] investigates the classical limit of the transition amplitude predicted by this model for the simplest triangulation allowing for space-time curvature. That is, it investigates the transition amplitude for the simplest triangulation with an internal triangle, in the limit of large boundary quantum numbers.…”

“…This is a version of the so-called 'flatness problem' [9], but more precise in that it considers the discrete nature of the sums over internal spins. The data for the curved triangulation considered by [3] is that of the three 4-simplices adjoining a common triangle in the boundary of a regular 5-simplex. Each of these 4-simplices is regular, whence they all have dihedral angle arccos(1/4), yielding the internal deficit angle…”

SU(2) graph invariants, Regge actions and polytopes

“…The numerical study of Barrett's asymptotic formula for the Lorentzian EPRL model will appear in a companion paper [21]. Future related work is to investigate the implications and extensions of our analytic results to the EPRL model with both simplicial and generalised vertices, and study the problem of asymptotics on extended 4d triangulations, for which the relation to solutions of the Regge equations is subject of a crucial debate [49,50,51,52,53,54,55]. It would also be interesting to explore if our techniques can be generalised to quantum deformations, see e.g.…”

“…A key open question is whether this difference is enough to make the EPRL model capture the correct Regge dynamics on the full triangulation, something that BF theory does not do. A difficulty in this respect is associated with properly treating the variation with respect to the bulk spins, leading to a possible flatness problem and an ongoing open debate [49,50,51,52,53,54,55]. A more detailed understanding of the dynamics on a full triangulation is needed, and this is one of our goals.…”