156 research outputs found
The Poisson bracket on free null initial data for gravity
Free initial data for general relativity on a pair of intersecting null
hypersurfaces are well known, but the lack of a Poisson bracket and concerns
about caustics have stymied the development of a constraint free canonical
theory. Here it is pointed out how caustics and generator crossings can be
neatly avoided and a Poisson bracket on free data is given. On sufficiently
regular functions of the solution spacetime geometry this bracket matches the
Poisson bracket defined on such functions by the Hilbert action via Peierls'
prescription. The symplectic form is also given in terms of free data.Comment: 4 pages,1 figure. Some changes to text to improve clarity of
presentation, this is the final published versio
so(4) Plebanski Action and Relativistic Spin Foam Model
In this note we study the correspondence between the ``relativistic spin
foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski
action. We argue that the Plebanski model is the continuum analog of
the relativistic spin foam model. We prove that the Plebanski action possess
four phases, one of which is gravity and outline the discrepancy between this
model and the model of Euclidean gravity. We also show that the Plebanski model
possess another natural dicretisation and can be associate with another, new,
spin foam model that appear to be the counterpart of the spin foam
model describing the self dual formulation of gravity.Comment: 12 pages, REVTeX using AMS fonts. Some minor corrections and
improvement
3+1 spinfoam model of quantum gravity with spacelike and timelike components
We present a spinfoam formulation of Lorentzian quantum General Relativity.
The theory is based on a simple generalization of an Euclidean model defined in
terms of a field theory over a group. The model is an extension of a recently
introduced Lorentzian model, in which both timelike and spacelike components
are included. The spinfoams in the model, corresponding to quantized
4-geometries, carry a natural non-perturbative local causal structure induced
by the geometry of the algebra of the internal gauge (sl(2,C)). Amplitudes can
be expressed as integrals over the spacelike unit-vectors hyperboloid in
Minkowski space, or the imaginary Lobachevskian space.Comment: 16 pages, 1 figur
Classical GR as a topological theory with linear constraints
We investigate a formulation of continuum 4d gravity in terms of a
constrained topological (BF) theory, in the spirit of the Plebanski
formulation, but involving only linear constraints, of the type used recently
in the spin foam approach to quantum gravity. We identify both the continuum
version of the linear simplicity constraints used in the quantum discrete
context and a linear version of the quadratic volume constraints that are
necessary to complete the reduction from the topological theory to gravity. We
illustrate and discuss also the discrete counterpart of the same continuum
linear constraints. Moreover, we show under which additional conditions the
discrete volume constraints follow from the simplicity constraints, thus
playing the role of secondary constraints. Our analysis clarifies how the
discrete constructions of spin foam models are related to a continuum theory
with an action principle that is equivalent to general relativity.Comment: 4 pages, based on a talk given at the Spanish Relativity Meeting 2010
(ERE2010, Granada, Spain
Classical GR as a topological theory with linear constraints
We investigate a formulation of continuum 4d gravity in terms of a
constrained topological (BF) theory, in the spirit of the Plebanski
formulation, but involving only linear constraints, of the type used recently
in the spin foam approach to quantum gravity. We identify both the continuum
version of the linear simplicity constraints used in the quantum discrete
context and a linear version of the quadratic volume constraints that are
necessary to complete the reduction from the topological theory to gravity. We
illustrate and discuss also the discrete counterpart of the same continuum
linear constraints. Moreover, we show under which additional conditions the
discrete volume constraints follow from the simplicity constraints, thus
playing the role of secondary constraints. Our analysis clarifies how the
discrete constructions of spin foam models are related to a continuum theory
with an action principle that is equivalent to general relativity.Comment: 4 pages, based on a talk given at the Spanish Relativity Meeting 2010
(ERE2010, Granada, Spain
Classical GR as a topological theory with linear constraints
We investigate a formulation of continuum 4d gravity in terms of a
constrained topological (BF) theory, in the spirit of the Plebanski
formulation, but involving only linear constraints, of the type used recently
in the spin foam approach to quantum gravity. We identify both the continuum
version of the linear simplicity constraints used in the quantum discrete
context and a linear version of the quadratic volume constraints that are
necessary to complete the reduction from the topological theory to gravity. We
illustrate and discuss also the discrete counterpart of the same continuum
linear constraints. Moreover, we show under which additional conditions the
discrete volume constraints follow from the simplicity constraints, thus
playing the role of secondary constraints. Our analysis clarifies how the
discrete constructions of spin foam models are related to a continuum theory
with an action principle that is equivalent to general relativity.Comment: 4 pages, based on a talk given at the Spanish Relativity Meeting 2010
(ERE2010, Granada, Spain
The volume operator in covariant quantum gravity
A covariant spin-foam formulation of quantum gravity has been recently
developed, characterized by a kinematics which appears to match well the one of
canonical loop quantum gravity. In particular, the geometrical observable
giving the area of a surface has been shown to be the same as the one in loop
quantum gravity. Here we discuss the volume observable. We derive the volume
operator in the covariant theory, and show that it matches the one of loop
quantum gravity, as does the area. We also reconsider the implementation of the
constraints that defines the model: we derive in a simple way the boundary
Hilbert space of the theory from a suitable form of the classical constraints,
and show directly that all constraints vanish weakly on this space.Comment: 10 pages. Version 2: proof extended to gamma > 1
Spin foam model for Lorentzian General Relativity
We present a spin foam formulation of Lorentzian quantum General Relativity.
The theory is based on a simple generalization of an Euclidean model defined in
terms of a field theory over a group. Its vertex amplitude turns out to be the
one recently introduced by Barrett and Crane. As in the case of its Euclidean
relatives, the model fully implements the desired sum over 2-complexes which
encodes the local degrees of freedom of the theory.Comment: 8 pages, 1 figur
The fermionic contribution to the spectrum of the area operator in nonperturbative quantum gravity
The role of fermionic matter in the spectrum of the area operator is analyzed
using the Baez--Krasnov framework for quantum fermions and gravity. The result
is that the fermionic contribution to the area of a surface is equivalent
to the contribution of purely gravitational spin network's edges tangent to
. Therefore, the spectrum of the area operator is the same as in the pure
gravity case.Comment: 10 pages, revtex file. Revised versio
Spacetime as a Feynman diagram: the connection formulation
Spin foam models are the path integral counterparts to loop quantized
canonical theories. In the last few years several spin foam models of gravity
have been proposed, most of which live on finite simplicial lattice spacetime.
The lattice truncates the presumably infinite set of gravitational degrees of
freedom down to a finite set. Models that can accomodate an infinite set of
degrees of freedom and that are independent of any background simplicial
structure, or indeed any a priori spacetime topology, can be obtained from the
lattice models by summing them over all lattice spacetimes. Here we show that
this sum can be realized as the sum over Feynman diagrams of a quantum field
theory living on a suitable group manifold, with each Feynman diagram defining
a particular lattice spacetime. We give an explicit formula for the action of
the field theory corresponding to any given spin foam model in a wide class
which includes several gravity models. Such a field theory was recently found
for a particular gravity model [De Pietri et al, hep-th/9907154]. Our work
generalizes this result as well as Boulatov's and Ooguri's models of three and
four dimensional topological field theories, and ultimately the old matrix
models of two dimensional systems with dynamical topology. A first version of
our result has appeared in a companion paper [gr-qc\0002083]: here we present a
new and more detailed derivation based on the connection formulation of the
spin foam models.Comment: 32 pages, 2 figure
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