14 research outputs found
Asymptotics of 4d spin foam models
We study the asymptotic properties of four-simplex amplitudes for various
four-dimensional spin foam models. We investigate the semi-classical limit of
the Ooguri, Euclidean and Lorentzian EPRL models using coherent states for the
boundary data. For some classes of geometrical boundary data, the asymptotic
formulae are given, in all three cases, by simple functions of the Regge action
for the four-simplex geometry.Comment: 10 pages, Proceedings for the 2nd Corfu summer school and workshop on
quantum gravity and quantum geometry, talk given by Winston J. Fairbair
Euclidean three-point function in loop and perturbative gravity
We compute the leading order of the three-point function in loop quantum
gravity, using the vertex expansion of the Euclidean version of the new spin
foam dynamics, in the region of gamma<1. We find results consistent with Regge
calculus in the limit gamma->0 and j->infinity. We also compute the tree-level
three-point function of perturbative quantum general relativity in position
space, and discuss the possibility of directly comparing the two results.Comment: 16 page
Spin foams with timelike surfaces
Spin foams of 4d gravity were recently extended from complexes with purely
spacelike surfaces to complexes that also contain timelike surfaces. In this
article, we express the associated partition function in terms of vertex
amplitudes and integrals over coherent states. The coherent states are
characterized by unit 3--vectors which represent normals to surfaces and lie
either in the 2--sphere or the 2d hyperboloids. In the case of timelike
surfaces, a new type of coherent state is used and the associated completeness
relation is derived. It is also shown that the quantum simplicity constraints
can be deduced by three different methods: by weak imposition of the
constraints, by restriction of coherent state bases and by the master
constraint.Comment: 22 pages, no figures; v2: remarks on operator formalism added in
discussion; correction: the spin 1/2 irrep of the discrete series does not
appear in the Plancherel decompositio
A new look at loop quantum gravity
I describe a possible perspective on the current state of loop quantum
gravity, at the light of the developments of the last years. I point out that a
theory is now available, having a well-defined background-independent
kinematics and a dynamics allowing transition amplitudes to be computed
explicitly in different regimes. I underline the fact that the dynamics can be
given in terms of a simple vertex function, largely determined by locality,
diffeomorphism invariance and local Lorentz invariance. I emphasize the
importance of approximations. I list open problems.Comment: 15 pages, 5 figure
Canonical path integral measures for Holst and Plebanski gravity. I. Reduced Phase Space Derivation
An important aspect in defining a path integral quantum theory is the
determination of the correct measure. For interacting theories and theories
with constraints, this is non-trivial, and is normally not the heuristic
"Lebesgue measure" usually used. There have been many determinations of a
measure for gravity in the literature, but none for the Palatini or Holst
formulations of gravity. Furthermore, the relations between different resulting
measures for different formulations of gravity are usually not discussed.
In this paper we use the reduced phase technique in order to derive the
path-integral measure for the Palatini and Holst formulation of gravity, which
is different from the Lebesgue measure up to local measure factors which depend
on the spacetime volume element and spatial volume element.
From this path integral for the Holst formulation of GR we can also give a
new derivation of the Plebanski path integral and discover a discrepancy with
the result due to Buffenoir, Henneaux, Noui and Roche (BHNR) whose origin we
resolve. This paper is the first in a series that aims at better understanding
the relation between canonical LQG and the spin foam approach.Comment: 27 pages, minor correction
Asymptotics of the Wigner 9j symbol
We present the asymptotic formula for the Wigner 9j-symbol, valid when all
quantum numbers are large, in the classically allowed region. As in the
Ponzano-Regge formula for the 6j-symbol, the action is expressed in terms of
lengths of edges and dihedral angles of a geometrical figure, but the angles
require care in definition. Rules are presented for converting spin networks
into the associated geometrical figures. The amplitude is expressed as the
determinant of a 2x2 matrix of Poisson brackets. The 9j-symbol possesses
caustics associated with the fold and elliptic and hyperbolic umbilic
catastrophes. The asymptotic formula obeys the exact symmetries of the
9j-symbol.Comment: 17 pages, 7 figure
Generating Functions for Coherent Intertwiners
We study generating functions for the scalar products of SU(2) coherent
intertwiners, which can be interpreted as coherent spin network evaluations on
a 2-vertex graph. We show that these generating functions are exactly summable
for different choices of combinatorial weights. Moreover, we identify one
choice of weight distinguished thanks to its geometric interpretation. As an
example of dynamics, we consider the simple case of SU(2) flatness and describe
the corresponding Hamiltonian constraint whose quantization on coherent
intertwiners leads to partial differential equations that we solve.
Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2)
flatness on coherent spin networks for arbitrary graphs.Comment: 31 page
Holomorphic Simplicity Constraints for 4d Spinfoam Models
Within the framework of spinfoam models, we revisit the simplicity
constraints reducing topological BF theory to 4d Riemannian gravity. We use the
reformulation of SU(2) intertwiners and spin networks in term of spinors, which
has come out from both the recently developed U(N) framework for SU(2)
intertwiners and the twisted geometry approach to spin networks and spinfoam
boundary states. Using these tools, we are able to perform a
holomorphic/anti-holomorphic splitting of the simplicity constraints and define
a new set of holomorphic simplicity constraints, which are equivalent to the
standard ones at the classical level and which can be imposed strongly on
intertwiners at the quantum level. We then show how to solve these new
holomorphic simplicity constraints using coherent intertwiner states. We
further define the corresponding coherent spin network functionals and
introduce a new spinfoam model for 4d Riemannian gravity based on these
holomorphic simplicity constraints and whose amplitudes are defined from the
evaluation of the new coherent spin networks.Comment: 27 page
Second-order amplitudes in loop quantum gravity
We explore some second-order amplitudes in loop quantum gravity. In
particular, we compute some second-order contributions to diagonal components
of the graviton propagator in the large distance limit, using the old version
of the Barrett-Crane vertex amplitude. We illustrate the geometry associated to
these terms. We find some peculiar phenomena in the large distance behavior of
these amplitudes, related with the geometry of the generalized triangulations
dual to the Feynman graphs of the corresponding group field theory. In
particular, we point out a possible further difficulty with the old
Barrett-Crane vertex: it appears to lead to flatness instead of Ricci-flatness,
at least in some situations. The observation raises the question whether this
difficulty remains with the new version of the vertex.Comment: 22 pages, 18 figure
(Broken) Gauge Symmetries and Constraints in Regge Calculus
We will examine the issue of diffeomorphism symmetry in simplicial models of
(quantum) gravity, in particular for Regge calculus. We find that for a
solution with curvature there do not exist exact gauge symmetries on the
discrete level. Furthermore we derive a canonical formulation that exactly
matches the dynamics and hence symmetries of the covariant picture. In this
canonical formulation broken symmetries lead to the replacements of constraints
by so--called pseudo constraints. These considerations should be taken into
account in attempts to connect spin foam models, based on the Regge action,
with canonical loop quantum gravity, which aims at implementing proper
constraints. We will argue that the long standing problem of finding a
consistent constraint algebra for discretized gravity theories is equivalent to
the problem of finding an action with exact diffeomorphism symmetries. Finally
we will analyze different limits in which the pseudo constraints might turn
into proper constraints. This could be helpful to infer alternative
discretization schemes in which the symmetries are not broken.Comment: 32 pages, 15 figure