We compute metric correlations in loop quantum gravity with the dynamics
defined by the new spin foam models. The analysis is done at the lowest order
in a vertex expansion and at the leading order in a large spin expansion. The
result is compared to the graviton propagator of perturbative quantum gravity.Comment: 28 page
In this paper we discuss a proposal of coherent states for Loop Quantum
Gravity. These states are labeled by a point in the phase space of General
Relativity as captured by a spin-network graph. They are defined as the gauge
invariant projection of a product over links of Hall's heat-kernels for the
cotangent bundle of SU(2). The labels of the state are written in terms of two
unit-vectors, a spin and an angle for each link of the graph. The heat-kernel
time is chosen to be a function of the spin. These labels are the ones used in
the Spin Foam setting and admit a clear geometric interpretation. Moreover, the
set of labels per link can be written as an element of SL(2,C). Therefore,
these states coincide with Thiemann's coherent states with the area operator as
complexifier. We study the properties of semiclassicality of these states and
show that, for large spins, they reproduce a superposition over spins of
spin-networks with nodes labeled by Livine-Speziale coherent intertwiners.
Moreover, the weight associated to spins on links turns out to be given by a
Gaussian times a phase as originally proposed by Rovelli.Comment: 15 page
We study a holomorphic representation for spinfoams. The representation is
obtained via the Ashtekar-Lewandowski-Marolf-Mour\~ao-Thiemann coherent state
transform. We derive the expression of the 4d spinfoam vertex for Euclidean and
for Lorentzian gravity in the holomorphic representation. The advantage of this
representation rests on the fact that the variables used have a clear
interpretation in terms of a classical intrinsic and extrinsic geometry of
space. We show how the peakedness on the extrinsic geometry selects a single
exponential of the Regge action in the semiclassical large-scale asymptotics of
the spinfoam vertex.Comment: 10 pages, 1 figure, published versio
We give the definition of a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity covariant dynamics. The coupling takes a surprisingly simple form. Here we only define the dynamics; physical implications are considered in a subsequent paper.
PACS 04.60.Pp -Loop quantum gravity, quantum geometry, spin foams PACS 04.60.Gw -Covariant and sum-over-histories quantization PACS 04.60.Nc -Lattice and discrete methods Abstract -We find a nontrivial regime of spinfoam quantum gravity that reproduces classical Einstein equations. This is the double scaling limit of small Immirzi parameter (gamma), large spins (j) with physical area (gamma times j) constant. In addition to quantum corrections in the Planck constant, we find new corrections in the Immirzi parameter due to the quantum discreteness of spacetime. The result is a strong evidence that the spinfoam covariant quantization of general relativity possesses the correct classical limit.
We consider spinfoam quantum gravity for general triangulations in the regime l 2 P a a/γ, namely in the combined classical limit of large areas a and flipped limit of small Barbero-Immirzi parameter γ, where lP is the Planck length. Under few working hypotheses we find that the flipped limit enforces the constraints that turn the spinfoam theory into an effective Regge-like quantum theory with lengths as variables, while the classical limit selects among the possible geometries the ones satisfying the Einstein equations. Two kinds of quantum corrections appear in terms of powers of l 2 P /a and γl 2 P /a. The result also suggests that the Barbero-Immirzi parameter may run to smaller values under coarse-graining of the triangulation.
We consider spinfoam quantum gravity. We show in a simple case that the amplitude projects over a nontrivial (curved) classical geometry. This suggests that, at least for spinfoams without bubbles and for large values of the boundary spins, the amplitude takes the form of a path integral over Regge metrics, thus enforcing discrete Einstein equations in the classical limit. The result relies crucially on a new interpretation of the semiclassical limit for the amplitudes truncated to a fixed 2-complex.
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