30 research outputs found
LTB spacetimes in terms of Dirac observables
The construction of Dirac observables, that is gauge invariant objects, in
General Relativity is technically more complicated than in other gauge theories
such as the standard model due to its more complicated gauge group which is
closely related to the group of spacetime diffeomorphisms. However, the
explicit and usually cumbersome expression of Dirac observables in terms of
gauge non invariant quantities is irrelevant if their Poisson algebra is
sufficiently simple. Precisely that can be achieved by employing the relational
formalism and a specific type of matter proposed originally by Brown and
Kucha{\v r}, namely pressureless dust fields. Moreover one is able to derive a
compact expression for a physical Hamiltonian that drives their physical time
evolution. The resulting gauge invariant Hamiltonian system is obtained by
Higgs -- ing the dust scalar fields and has an infinite number of conserved
charges which force the Goldstone bosons to decouple from the evolution. In
previous publications we have shown that explicitly for cosmological
perturbations. In this article we analyse the spherically symmetric sector of
the theory and it turns out that the solutions are in one--to--one
correspondence with the class of Lemaitre--Tolman--Bondi metrics. Therefore the
theory is capable of properly describing the whole class of gravitational
experiments that rely on the assumption of spherical symmetry.Comment: 29 pages, no figure
Born--Oppenheimer decomposition for quantum fields on quantum spacetimes
Quantum Field Theory on Curved Spacetime (QFT on CS) is a well established theoretical framework which intuitively should be a an extremely effective description of the quantum nature of matter when propagating on a given background spacetime. If one wants to take care of backreaction effects, then a theory of quantum gravity is needed. It is now widely believed that such a theory should be formulated in a non-perturbative and therefore background independent fashion. Hence, it is a priori a puzzle how a background dependent QFT on CS should emerge as a semiclassical limit out of a background independent quantum gravity theory. In this article we point out that the Born-Oppenheimer decomposition (BOD) of the Hilbert space is ideally suited in order to establish such a link, provided that the Hilbert space representation of the gravitational field algebra satisfies an important condition. If the condition is satisfied, then the framework of QFT on CS can be, in a certain sense, embedded into a theory of quantum gravity. The unique representation of the holonomy-flux algebra underlying Loop Quantum Gravity (LQG) violates that condition. While it is conceivable that the condition on the representation can be relaxed, for convenience in this article we consider a new classical gravitational field algebra and a Hilbert space representation of its restriction to an algebraic graph for which the condition is satisfied. An important question that remains and for which we have only partial answers is how to construct eigenstates of the full gravity-matter Hamiltonian whose BOD is confined to a small neighbourhood of a physically interesting vacuum spacetime
Spinor Representation for Loop Quantum Gravity
We perform a quantization of the loop gravity phase space purely in terms of
spinorial variables, which have recently been shown to provide a direct link
between spin network states and simplicial geometries. The natural Hilbert
space to represent these spinors is the Bargmann space of holomorphic
square-integrable functions over complex numbers. We show the unitary
equivalence between the resulting generalized Bargmann space and the standard
loop quantum gravity Hilbert space by explicitly constructing the unitary map.
The latter maps SU(2)-holonomies, when written as a function of spinors, to
their holomorphic part. We analyze the properties of this map in detail. We
show that the subspace of gauge invariant states can be characterized
particularly easy in this representation of loop gravity. Furthermore, this map
provides a tool to efficiently calculate physical quantities since integrals
over the group are exchanged for straightforward integrals over the complex
plane.Comment: 36 pages, minor corrections and improvements, matches published
versio
Non-commutative flux representation for loop quantum gravity
The Hilbert space of loop quantum gravity is usually described in terms of
cylindrical functionals of the gauge connection, the electric fluxes acting as
non-commuting derivation operators. It has long been believed that this
non-commutativity prevents a dual flux (or triad) representation of loop
quantum gravity to exist. We show here, instead, that such a representation can
be explicitly defined, by means of a non-commutative Fourier transform defined
on the loop gravity state space. In this dual representation, flux operators
act by *-multiplication and holonomy operators act by translation. We describe
the gauge invariant dual states and discuss their geometrical meaning. Finally,
we apply the construction to the simpler case of a U(1) gauge group and compare
the resulting flux representation with the triad representation used in loop
quantum cosmology.Comment: 12 pages, matches published versio
Cosmological vector modes and quantum gravity effects
In contrast to scalar and tensor modes, vector modes of linear perturbations
around an expanding Friedmann--Robertson--Walker universe decay. This makes
them largely irrelevant for late time cosmology, assuming that all modes
started out at a similar magnitude at some early stage. By now, however,
bouncing models are frequently considered which exhibit a collapsing phase.
Before this phase reaches a minimum size and re-expands, vector modes grow.
Such modes are thus relevant for the bounce and may even signal the breakdown
of perturbation theory if the growth is too strong. Here, a gauge invariant
formulation of vector mode perturbations in Hamiltonian cosmology is presented.
This lays out a framework for studying possible canonical quantum gravity
effects, such as those of loop quantum gravity, at an effective level. As an
explicit example, typical quantum corrections, namely those coming from inverse
densitized triad components and holonomies, are shown to increase the growth
rate of vector perturbations in the contracting phase, but only slightly.
Effects at the bounce of the background geometry can, however, be much
stronger.Comment: 20 page
Gauge invariant perturbations around symmetry reduced sectors of general relativity: applications to cosmology
We develop a gauge invariant canonical perturbation scheme for perturbations
around symmetry reduced sectors in generally covariant theories, such as
general relativity. The central objects of investigation are gauge invariant
observables which encode the dynamics of the system. We apply this scheme to
perturbations around a homogeneous and isotropic sector (cosmology) of general
relativity. The background variables of this homogeneous and isotropic sector
are treated fully dynamically which allows us to approximate the observables to
arbitrary high order in a self--consistent and fully gauge invariant manner.
Methods to compute these observables are given. The question of backreaction
effects of inhomogeneities onto a homogeneous and isotropic background can be
addressed in this framework. We illustrate the latter by considering
homogeneous but anisotropic Bianchi--I cosmologies as perturbations around a
homogeneous and isotropic sector.Comment: 39 pages, 1 figur
Effective relational dynamics
We provide a synopsis of an effective approach to the problem of time in the
semiclassical regime. The essential features of this new approach to evaluating
relational quantum dynamics in constrained systems are illustrated by means of
a simple toy model.Comment: 4 pages, based on a talk given at Loops '11 in Madrid, to appear in
Journal of Physics: Conference Series (JPCS
Effective Hamiltonian Constraint from Group Field Theory
Spinfoam models provide a covariant formulation of the dynamics of loop
quantum gravity. They are non-perturbatively defined in the group field theory
(GFT) framework: the GFT partition function defines the sum of spinfoam
transition amplitudes over all possible (discretized) geometries and
topologies. The issue remains, however, of explicitly relating the specific
form of the group field theory action and the canonical Hamiltonian constraint.
Here, we suggest an avenue for addressing this issue. Our strategy is to expand
group field theories around non-trivial classical solutions and to interpret
the induced quadratic kinematical term as defining a Hamiltonian constraint on
the group field and thus on spin network wave functions. We apply our procedure
to Boulatov group field theory for 3d Riemannian gravity. Finally, we discuss
the relevance of understanding the spectrum of this Hamiltonian operator for
the renormalization of group field theories.Comment: 14 page
Entropy in the Classical and Quantum Polymer Black Hole Models
We investigate the entropy counting for black hole horizons in loop quantum
gravity (LQG). We argue that the space of 3d closed polyhedra is the classical
counterpart of the space of SU(2) intertwiners at the quantum level. Then
computing the entropy for the boundary horizon amounts to calculating the
density of polyhedra or the number of intertwiners at fixed total area.
Following the previous work arXiv:1011.5628, we dub these the classical and
quantum polymer models for isolated horizons in LQG. We provide exact
micro-canonical calculations for both models and we show that the classical
counting of polyhedra accounts for most of the features of the intertwiner
counting (leading order entropy and log-correction), thus providing us with a
simpler model to further investigate correlations and dynamics. To illustrate
this, we also produce an exact formula for the dimension of the intertwiner
space as a density of "almost-closed polyhedra".Comment: 24 page
Many-nodes/many-links spinfoam: the homogeneous and isotropic case
I compute the Lorentzian EPRL/FK/KKL spinfoam vertex amplitude for regular
graphs, with an arbitrary number of links and nodes, and coherent states peaked
on a homogeneous and isotropic geometry. This form of the amplitude can be
applied for example to a dipole with an arbitrary number of links or to the
4-simplex given by the compete graph on 5 nodes. All the resulting amplitudes
have the same support, independently of the graph used, in the large j (large
volume) limit. This implies that they all yield the Friedmann equation: I show
this in the presence of the cosmological constant. This result indicates that
in the semiclassical limit quantum corrections in spinfoam cosmology do not
come from just refining the graph, but rather from relaxing the large j limit.Comment: 8 pages, 4 figure