954 research outputs found
On the geometry of loop quantum gravity on a graph
We discuss the meaning of geometrical constructions associated to loop
quantum gravity states on a graph. In particular, we discuss the "twisted
geometries" and derive a simple relation between these and Regge geometries.Comment: 6 pages, 1 figure. v2: some typos corrected, references update
A semiclassical tetrahedron
We construct a macroscopic semiclassical state state for a quantum
tetrahedron. The expectation values of the geometrical operators representing
the volume, areas and dihedral angles are peaked around assigned classical
values, with vanishing relative uncertainties.Comment: 10 pages; v2 revised versio
Twisted geometries: A geometric parametrisation of SU(2) phase space
A cornerstone of the loop quantum gravity program is the fact that the phase
space of general relativity on a fixed graph can be described by a product of
SU(2) cotangent bundles per edge. In this paper we show how to parametrize this
phase space in terms of quantities describing the intrinsic and extrinsic
geometry of the triangulation dual to the graph. These are defined by the
assignment to each triangle of its area, the two unit normals as seen from the
two polyhedra sharing it, and an additional angle related to the extrinsic
curvature. These quantities do not define a Regge geometry, since they include
extrinsic data, but a looser notion of discrete geometry which is twisted in
the sense that it is locally well-defined, but the local patches lack a
consistent gluing among each other. We give the Poisson brackets among the new
variables, and exhibit a symplectomorphism which maps them into the Poisson
brackets of loop gravity. The new parametrization has the advantage of a simple
description of the gauge-invariant reduced phase space, which is given by a
product of phase spaces associated to edges and vertices, and it also provides
an abelianisation of the SU(2) connection. The results are relevant for the
construction of coherent states, and as a byproduct, contribute to clarify the
connection between loop gravity and its subset corresponding to Regge
geometries.Comment: 28 pages. v2 and v3 minor change
Physical boundary state for the quantum tetrahedron
We consider stability under evolution as a criterion to select a physical
boundary state for the spinfoam formalism. As an example, we apply it to the
simplest spinfoam defined by a single quantum tetrahedron and solve the
associated eigenvalue problem at leading order in the large spin limit. We show
that this fixes uniquely the free parameters entering the boundary state.
Remarkably, the state obtained this way gives a correlation between edges which
runs at leading order with the inverse distance between the edges, in agreement
with the linearized continuum theory. Finally, we give an argument why this
correlator represents the propagation of a pure gauge, consistently with the
absence of physical degrees of freedom in 3d general relativity.Comment: 20 pages, 6 figure
Grasping rules and semiclassical limit of the geometry in the Ponzano-Regge model
We show how the expectation values of geometrical quantities in 3d quantum
gravity can be explicitly computed using grasping rules. We compute the volume
of a labelled tetrahedron using the triple grasping. We show that the large
spin expansion of this value is dominated by the classical expression, and we
study the next to leading order quantum corrections.Comment: 18 pages, 1 figur
Towards the graviton from spinfoams: higher order corrections in the 3d toy model
We consider the recent calculation gr-qc/0508124 of the graviton propagator
in the spinfoam formalism. Within the 3d toy model introduced in gr-qc/0512102,
we test how the spinfoam formalism can be used to construct the perturbative
expansion of graviton amplitudes. Although the 3d graviton is a pure gauge, one
can choose to work in a gauge where it is not zero and thus reproduce the
structure of the 4d perturbative calculations. We compute explicitly the next
to leading and next to next to leading orders, corresponding to one-loop and
two-loop corrections. We show that while the first arises entirely from the
expansion of the Regge action around the flat background, the latter receives
contributions from the microscopic, non Regge-like, quantum geometry.
Surprisingly, this new contribution reduces the magnitude of the next to next
to leading order. It thus appears that the spinfoam formalism is likely to
substantially modify the conventional perturbative expansion at higher orders.
This result supports the interest in this approach. We then address a number
of open issues in the rest of the paper. First, we discuss the boundary state
ansatz, which is a key ingredient in the whole construction. We propose a way
to enhance the ansatz in order to make the edge lengths and dihedral angles
conjugate variables in a mathematically well-defined way. Second, we show that
the leading order is stable against different choices of the face weights of
the spinfoam model; the next to leading order, on the other hand, is changed in
a simple way, and we show that the topological face weight minimizes it.
Finally, we extend the leading order result to the case of a regular, but not
equilateral, tetrahedron.Comment: 24 pages, many figure
Numerical indications on the semiclassical limit of the flipped vertex
We introduce a technique for testing the semiclassical limit of a quantum
gravity vertex amplitude. The technique is based on the propagation of a
semiclassical wave packet. We apply this technique to the newly introduced
"flipped" vertex in loop quantum gravity, in order to test the intertwiner
dependence of the vertex. Under some drastic simplifications, we find very
preliminary, but surprisingly good numerical evidence for the correct classical
limit.Comment: 4 pages, 8 figure
Coupling gauge theory to spinfoam 3d quantum gravity
We construct a spinfoam model for Yang-Mills theory coupled to quantum
gravity in three dimensional riemannian spacetime. We define the partition
function of the coupled system as a power series in g_0^2 G that can be
evaluated order by order using grasping rules and the recoupling theory. With
respect to previous attempts in the literature, this model assigns the
dynamical variables of gravity and Yang-Mills theory to the same simplices of
the spinfoam, and it thus provides transition amplitudes for the spin network
states of the canonical theory. For SU(2) Yang-Mills theory we show explicitly
that the partition function has a semiclassical limit given by the Regge
discretization of the classical Yang-Mills action.Comment: 18 page
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