263 research outputs found
Algebraic Quantum Gravity (AQG) III. Semiclassical Perturbation Theory
In the two previous papers of this series we defined a new combinatorical
approach to quantum gravity, Algebraic Quantum Gravity (AQG). We showed that
AQG reproduces the correct infinitesimal dynamics in the semiclassical limit,
provided one incorrectly substitutes the non -- Abelean group SU(2) by the
Abelean group in the calculations. The mere reason why that
substitution was performed at all is that in the non -- Abelean case the volume
operator, pivotal for the definition of the dynamics, is not diagonisable by
analytical methods. This, in contrast to the Abelean case, so far prohibited
semiclassical computations. In this paper we show why this unjustified
substitution nevertheless reproduces the correct physical result: Namely, we
introduce for the first time semiclassical perturbation theory within AQG (and
LQG) which allows to compute expectation values of interesting operators such
as the master constraint as a power series in with error control. That
is, in particular matrix elements of fractional powers of the volume operator
can be computed with extremely high precision for sufficiently large power of
in the expansion. With this new tool, the non -- Abelean
calculation, although technically more involved, is then exactly analogous to
the Abelean calculation, thus justifying the Abelean analysis in retrospect.
The results of this paper turn AQG into a calculational discipline
On a partially reduced phase space quantisation of general relativity conformally coupled to a scalar field
The purpose of this paper is twofold: On the one hand, after a thorough
review of the matter free case, we supplement the derivations in our companion
paper on 'loop quantum gravity without the Hamiltonian constraint' with
calculational details and extend the results to standard model matter, a
cosmological constant, and non-compact spatial slices. On the other hand, we
provide a discussion on the role of observables, focussed on the situation of a
symmetry exchange, which is key to our derivation. Furthermore, we comment on
the relation of our model to reduced phase space quantisations based on
deparametrisation.Comment: 51 pages, 5 figures. v2: Gauge condition used shown to coincide with
CMC gauge. Minor clarifications and correction
Properties of the Volume Operator in Loop Quantum Gravity I: Results
We analyze the spectral properties of the volume operator of Ashtekar and
Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the
classical volume expression for regions in three dimensional Riemannian space.
Our analysis considers for the first time generic graph vertices of valence
greater than four. Here we find that the geometry of the underlying vertex
characterizes the spectral properties of the volume operator, in particular the
presence of a `volume gap' (a smallest non-zero eigenvalue in the spectrum) is
found to depend on the vertex embedding. We compute the set of all
non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of
valence 5--7, and argue that these sets can be used to label spatial
diffeomorphism invariant states. We observe how gauge invariance connects
vertex geometry and representation properties of the underlying gauge group in
a natural way. Analytical results on the spectrum on 4-valent vertices are
included, for which the presence of a volume gap is proved. This paper presents
our main results; details are provided by a companion paper arXiv:0706.0382v1.Comment: 36 pages, 7 figures, LaTeX. See also companion paper
arXiv:0706.0382v1. Version as published in CQG in 2008. See arXiv:1003.2348
for important remarks regarding the sigma configurations. Subsequent
computations have revealed some minor errors, which do not change the
qualitative results but modify some of the numbers presented her
Properties of the Volume Operator in Loop Quantum Gravity II: Detailed Presentation
The properties of the Volume operator in Loop Quantum Gravity, as constructed
by Ashtekar and Lewandowski, are analyzed for the first time at generic
vertices of valence greater than four. The present analysis benefits from the
general simplified formula for matrix elements of the Volume operator derived
in gr-qc/0405060, making it feasible to implement it on a computer as a matrix
which is then diagonalized numerically. The resulting eigenvalues serve as a
database to investigate the spectral properties of the volume operator.
Analytical results on the spectrum at 4-valent vertices are included. This is a
companion paper to arXiv:0706.0469, providing details of the analysis presented
there.Comment: Companion to arXiv:0706.0469. Version as published in CQG in 2008.
More compact presentation. Sign factor combinatorics now much better
understood in context of oriented matroids, see arXiv:1003.2348, where also
important remarks given regarding sigma configurations. Subsequent
computations revealed some minor errors, which do not change qualitative
results but modify some numbers presented her
The Holst Spin Foam Model via Cubulations
Spin foam models are an attempt for a covariant, or path integral formulation
of canonical loop quantum gravity. The construction of such models usually rely
on the Plebanski formulation of general relativity as a constrained BF theory
and is based on the discretization of the action on a simplicial triangulation,
which may be viewed as an ultraviolet regulator. The triangulation dependence
can be removed by means of group field theory techniques, which allows one to
sum over all triangulations. The main tasks for these models are the correct
quantum implementation of the Plebanski constraints, the existence of a
semiclassical sector implementing additional "Regge-like" constraints arising
from simplicial triangulations, and the definition of the physical inner
product of loop quantum gravity via group field theory. Here we propose a new
approach to tackle these issues stemming directly from the Holst action for
general relativity, which is also a proper starting point for canonical loop
quantum gravity. The discretization is performed by means of a "cubulation" of
the manifold rather than a triangulation. We give a direct interpretation of
the resulting spin foam model as a generating functional for the n-point
functions on the physical Hilbert space at finite regulator. This paper focuses
on ideas and tasks to be performed before the model can be taken seriously.
However, our analysis reveals some interesting features of this model: first,
the structure of its amplitudes differs from the standard spin foam models.
Second, the tetrad n-point functions admit a "Wick-like" structure. Third, the
restriction to simple representations does not automatically occur -- unless
one makes use of the time gauge, just as in the classical theory.Comment: 25 pages, 1 figure; v3: published version. arXiv admin note:
substantial text overlap with arXiv:0911.213
Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory
We study the large-j asymptotics of the Euclidean EPRL/FK spin foam amplitude
on a 4d simplicial complex with arbitrary number of simplices. We show that for
a critical configuration (j_f, g_{ve}, n_{ef}) in general, there exists a
partition of the simplicial complex into three regions: Non-degenerate region,
Type-A degenerate region and Type-B degenerate region. On both the
non-degenerate and Type-A degenerate regions, the critical configuration
implies a non-degenerate Euclidean geometry, while on the Type-B degenerate
region, the critical configuration implies a vector geometry. Furthermore we
can split the Non-degenerate and Type-A regions into sub-complexes according to
the sign of Euclidean oriented 4-simplex volume. On each sub-complex, the spin
foam amplitude at critical configuration gives a Regge action that contains a
sign factor sgn(V_4(v)) of the oriented 4-simplices volume. Therefore the Regge
action reproduced here can be viewed as a discretized Palatini action with
on-shell connection. The asymptotic formula of the spin foam amplitude is given
by a sum of the amplitudes evaluated at all possible critical configurations,
which are the products of the amplitudes associated to different type of
geometries.Comment: 27 pages, 5 figures, references adde
Loop quantum gravity without the Hamiltonian constraint
We show that under certain technical assumptions, including the existence of
a constant mean curvature (CMC) slice and strict positivity of the scalar
field, general relativity conformally coupled to a scalar field can be
quantised on a partially reduced phase space, meaning reduced only with respect
to the Hamiltonian constraint and a proper gauge fixing. More precisely, we
introduce, in close analogy to shape dynamics, the generator of a local
conformal transformation acting on both, the metric and the scalar field, which
coincides with the CMC gauge condition. A new metric, which is invariant under
this transformation, is constructed and used to define connection variables
which can be quantised by standard loop quantum gravity methods. While it is
hard to address dynamical problems in this framework (due to the complicated
'time' function), it seems, due to good accessibility properties of the CMC
gauge, to be well suited for problems such as the computation of black hole
entropy, where actual physical states can be counted and the dynamics is only
of indirect importance. The corresponding calculation yields the surprising
result that the usual prescription of fixing the Barbero-Immirzi parameter beta
to a constant value in order to obtain the well-known formula S = a(Phi) A/(4G)
does not work for the black holes under consideration, while a recently
proposed prescription involving an analytic continuation of beta to the case of
a self-dual space-time connection yields the correct result. Also, the
interpretation of the geometric operators gets an interesting twist, which
exemplifies the deep relationship between observables and the choice of a time
function and has consequences for loop quantum cosmology.Comment: 8 pages. v2: Journal version. Black hole state counting based on
physical states added. Applications to loop quantum cosmology discussed.
Gauge condition used shown to coincide with CMC gauge. Minor clarifications.
v3: Erroneous topology dependence of the entropy in journal version
corrected, conclusions fixed accordingly. Main results unaffecte
Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity
We perform a canonical, reduced phase space quantisation of General
Relativity by Loop Quantum Gravity (LQG) methods. The explicit construction of
the reduced phase space is made possible by the combination of 1. the Brown --
Kuchar mechanism in the presence of pressure free dust fields which allows to
deparametrise the theory and 2. Rovelli's relational formalism in the extended
version developed by Dittrich to construct the algebra of gauge invariant
observables. Since the resulting algebra of observables is very simple, one can
quantise it using the methods of LQG. Basically, the kinematical Hilbert space
of non reduced LQG now becomes a physical Hilbert space and the kinematical
results of LQG such as discreteness of spectra of geometrical operators now
have physical meaning. The constraints have disappeared, however, the dynamics
of the observables is driven by a physical Hamiltonian which is related to the
Hamiltonian of the standard model (without dust) and which we quantise in this
paper.Comment: 31 pages, no figure
Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
We analyze combinatorial structures which play a central role in determining
spectral properties of the volume operator in loop quantum gravity (LQG). These
structures encode geometrical information of the embedding of arbitrary valence
vertices of a graph in 3-dimensional Riemannian space, and can be represented
by sign strings containing relative orientations of embedded edges. We
demonstrate that these signature factors are a special representation of the
general mathematical concept of an oriented matroid. Moreover, we show that
oriented matroids can also be used to describe the topology (connectedness) of
directed graphs. Hence the mathematical methods developed for oriented matroids
can be applied to the difficult combinatorics of embedded graphs underlying the
construction of LQG. As a first application we revisit the analysis of [4-5],
and find that enumeration of all possible sign configurations used there is
equivalent to enumerating all realizable oriented matroids of rank 3, and thus
can be greatly simplified. We find that for 7-valent vertices having no
coplanar triples of edge tangents, the smallest non-zero eigenvalue of the
volume spectrum does not grow as one increases the maximum spin \jmax at the
vertex, for any orientation of the edge tangents. This indicates that, in
contrast to the area operator, considering large \jmax does not necessarily
imply large volume eigenvalues. In addition we give an outlook to possible
starting points for rewriting the combinatorics of LQG in terms of oriented
matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos
corrected, presentation slightly extende
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