14 research outputs found
Relational Observables in Gravity: a Review
We present an overview on relational observables in gravity mainly from a
loop quantum gravity perspective. The gauge group of general relativity is the
diffeomorphism group of the underlying manifold. Consequently, general
relativity is a totally constrained theory with vanishing canonical
Hamiltonian. This fact, often referred to as the problem of time, provides the
main conceptual difficulty towards the construction of gauge-invariant local
observables. Nevertheless, within the framework of complete observables, that
encode relations between dynamical fields, progress has been made during the
last 20 years. Although analytic control over observables for full gravity is
still lacking, perturbative calculations have been performed and within
de-parameterizable toy models it was possible for the first time to construct a
full set of gauge invariant observables for a background independent field
theory. We review these developments and comment on their implications for
quantum gravity
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.Comment: 38 pages, 2 figure
Spinors and Twistors in Loop Gravity and Spin Foams
Spinorial tools have recently come back to fashion in loop gravity and spin
foams. They provide an elegant tool relating the standard holonomy-flux algebra
to the twisted geometry picture of the classical phase space on a fixed graph,
and to twistors. In these lectures we provide a brief and technical
introduction to the formalism and some of its applications.Comment: 16 pages; to appear in the Proceedings of the 3rd Quantum Gravity and
Quantum Geometry School, February 28 - March 13, 2011 Zakopane, Poland. v2:
minor amendment
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
Twistor Networks and Covariant Twisted Geometries
We study the symplectic reduction of the phase space of two twistors to the
cotangent bundle of the Lorentz group. We provide expressions for the Lorentz
generators and group elements in terms of the spinors defining the twistors. We
use this to define twistor networks as a graph carrying the phase space of two
twistors on each edge. We also introduce simple twistor networks, which provide
a classical version of the simple projected spin networks living on the
boundary Hilbert space of EPRL/FK spin foam models. Finally, we give an
expression for the Haar measure in terms of spinors.Comment: 18 pages. v2: minor amendments and some typos correcte
A perturbative approach to Dirac observables and their space-time algebra
We introduce a general approximation scheme in order to calculate gauge
invariant observables in the canonical formulation of general relativity. Using
this scheme we will show how the observables and the dynamics of field theories
on a fixed background or equivalently the observables of the linearized theory
can be understood as an approximation to the observables in full general
relativity. Gauge invariant corrections can be calculated up to an arbitrary
high order and we will explicitly calculate the first non--trivial correction.
Furthermore we will make a first investigation into the Poisson algebra between
observables corresponding to fields at different space--time points and
consider the locality properties of the observables.Comment: 23 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