A new spin foam model for 4D gravity

Abstract

Starting from Plebanski formulation of gravity as a constrained BF theory we propose a new spin foam model for 4d Riemannian quantum gravity that generalises the well-known Barrett-Crane model and resolves the inherent to it ultra-locality problem. The BF formulation of 4d gravity possesses two sectors: gravitational and topological ones. The model presented here is shown to give a quantization of the gravitational sector, and is dual to the recently proposed spin foam model of Engle et al. which, we show, corresponds to the topological sector. Our methods allow us to introduce the Immirzi parameter into the framework of spin foam quantisation. We generalize some of our considerations to the Lorentzian setting and obtain a new spin foam model in that context as well.

Semiclassical regime of Regge calculus and spin foams

“…There are strong indications that this is likely to be true. On one side, recent numerical and analytical work on the so-called "flipped" vertex [72,73] indicates that it has a wellbehaved semiclassical regime with a geometrical interpretation; on the other, a stationary phase analysis of the Freidel-Krasnov (FK) model [33] after writing it in path integral form [74,75] finds not only that the semiclassical limit of the vertex is the cosine of the Regge action, but also that the mismatch between area and length variables is sidestepped since area configurations that do not correspond to simplicial geometries drop out from the semiclassical limit. This last development is especially encouraging since the variable mismatch [76] and the presence of discontinuous boundary geometries [23,64] is another general problem the spin foam formalism seems to face when trying to obtain a semiclassical limit with a clear geometrical interpretation.…”

“…Comparing (33) with (32), we see that the two-point correlation function to lowest order is given by the inverse of an M × M matrix which is the sum of two terms: the Hessian matrix of the Hamilton function and the correlation matrix of the boundary state…”

“…The graviton propagator can be extracted from the connected part of (1). This expression has been thoroughly analyzed for the Barrett-Crane [24] spin foam model [9,25,26,27,28,29] and has motivated as well the recent introduction of new models [30,31,32,33,34]. However, the analysis has been so far restricted to the case where the boundary state has support only on a single graph corresponds to the boundary of a single simplex, with the internal spin foam consisting only in the vertex amplitude that is dual to this simplex.…”

Asymptotic analysis of the Engle–Pereira–Rovelli–Livine four-simplex amplitude

“…To study the semi-classical approximation of the theory, we are interested in the limit of expression (14) or (19) when λ → ∞. Our strategy is to use extended stationary phase methods, that is, stationary phase generalized to (non purely imaginary) complex functions.…”

“…Meanwhile Livine and Speziale introduced coherent states to the analysis and definition of spin foam models [12], and suggested a way to construct new models in [13]. In parallel, Freidel and Krasnov defined and developed a model along these lines in [14]. A refined version of the original EPR model was published in [15] together with Livine.…”