109 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
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
Continuum spin foam model for 3d gravity
An example illustrating a continuum spin foam framework is presented. This
covariant framework induces the kinematics of canonical loop quantization, and
its dynamics is generated by a {\em renormalized} sum over colored polyhedra.
Physically the example corresponds to 3d gravity with cosmological constant.
Starting from a kinematical structure that accommodates local degrees of
freedom and does not involve the choice of any background structure (e. g.
triangulation), the dynamics reduces the field theory to have only global
degrees of freedom. The result is {\em projectively} equivalent to the
Turaev-Viro model.Comment: 12 pages, 3 figure
Relating Covariant and Canonical Approaches to Triangulated Models of Quantum Gravity
In this paper explore the relation between covariant and canonical approaches
to quantum gravity and theory. We will focus on the dynamical
triangulation and spin-foam models, which have in common that they can be
defined in terms of sums over space-time triangulations. Our aim is to show how
we can recover these covariant models from a canonical framework by providing
two regularisations of the projector onto the kernel of the Hamiltonian
constraint. This link is important for the understanding of the dynamics of
quantum gravity. In particular, we will see how in the simplest dynamical
triangulations model we can recover the Hamiltonian constraint via our
definition of the projector. Our discussion of spin-foam models will show how
the elementary spin-network moves in loop quantum gravity, which were
originally assumed to describe the Hamiltonian constraint action, are in fact
related to the time-evolution generated by the constraint. We also show that
the Immirzi parameter is important for the understanding of a continuum limit
of the theory.Comment: 28 pages, 10 figure
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
Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory
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 this paper we reconsider the implementation
of the constraints that defines the model. We define in a simple way the
boundary Hilbert space of the theory, introducing a slight modification of the
embedding of the SU(2) representations into the SL(2,C) ones. We then show
directly that all constraints vanish on this space in a weak sense. The
vanishing is exact (and not just in the large quantum number limit.) We also
generalize the definition of the volume operator in the spinfoam model to the
Lorentzian signature, and show that it matches the one of loop quantum gravity,
as does in the Euclidean case.Comment: 11 page
A left-handed simplicial action for euclidean general relativity
An action for simplicial euclidean general relativity involving only
left-handed fields is presented. The simplicial theory is shown to converge to
continuum general relativity in the Plebanski formulation as the simplicial
complex is refined. This contrasts with the Regge model for which Miller and
Brewin have shown that the full field equations are much more restrictive than
Einstein's in the continuum limit. The action and field equations of the
proposed model are also significantly simpler then those of the Regge model
when written directly in terms of their fundamental variables.
An entirely analogous hypercubic lattice theory, which approximates
Plebanski's form of general relativity is also presented.Comment: Version 3. Adds current home address + slight corrections to
references of version 2. Version 2 = substantially clarified form of version
1. 29 pages, 4 figures, Latex, uses psfig.sty to insert postscript figures.
psfig.sty included in mailing, also available from this archiv
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