1,314 research outputs found
Testing the Master Constraint Programme for Loop Quantum Gravity II. Finite Dimensional Systems
This is the second paper in our series of five in which we test the Master
Constraint Programme for solving the Hamiltonian constraint in Loop Quantum
Gravity. In this work we begin with the simplest examples: Finite dimensional
models with a finite number of first or second class constraints, Abelean or
non -- Abelean, with or without structure functions.Comment: 23 pages, no figure
Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models
This is the third paper in our series of five in which we test the Master
Constraint Programme for solving the Hamiltonian constraint in Loop Quantum
Gravity. In this work we analyze models which, despite the fact that the phase
space is finite dimensional, are much more complicated than in the second
paper: These are systems with an SL(2,\Rl) gauge symmetry and the
complications arise because non -- compact semisimple Lie groups are not
amenable (have no finite translation invariant measure). This leads to severe
obstacles in the refined algebraic quantization programme (group averaging) and
we see a trace of that in the fact that the spectrum of the Master Constraint
does not contain the point zero. However, the minimum of the spectrum is of
order which can be interpreted as a normal ordering constant arising
from first class constraints (while second class systems lead to normal
ordering constants). The physical Hilbert space can then be be obtained after
subtracting this normal ordering correction.Comment: 33 pages, no figure
Gauge Field Theory Coherent States (GCS) : II. Peakedness Properties
In this article we apply the methods outlined in the previous paper of this
series to the particular set of states obtained by choosing the complexifier to
be a Laplace operator for each edge of a graph. The corresponding coherent
state transform was introduced by Hall for one edge and generalized by
Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann to arbitrary, finite,
piecewise analytic graphs. However, both of these works were incomplete with
respect to the following two issues : (a) The focus was on the unitarity of the
transform and left the properties of the corresponding coherent states
themselves untouched. (b) While these states depend in some sense on
complexified connections, it remained unclear what the complexification was in
terms of the coordinates of the underlying real phase space. In this paper we
resolve these issues, in particular, we prove that this family of states
satisfies all the usual properties : i) Peakedness in the configuration,
momentum and phase space (or Bargmann-Segal) representation, ii) Saturation of
the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These
states therefore comprise a candidate family for the semi-classical analysis of
canonical quantum gravity and quantum gauge theory coupled to quantum gravity,
enable error-controlled approximations and set a new starting point for {\it
numerical canonical quantum general relativity and gauge theory}. The text is
supplemented by an appendix which contains extensive graphics in order to give
a feeling for the so far unknown peakedness properties of the states
constructed.Comment: 70 pages, LATEX, 29 figure
Gauge Field Theory Coherent States (GCS) : I. General Properties
In this article we outline a rather general construction of diffeomorphism
covariant coherent states for quantum gauge theories.
By this we mean states , labelled by a point (A,E) in the
classical phase space, consisting of canonically conjugate pairs of connections
A and electric fields E respectively, such that (a) they are eigenstates of a
corresponding annihilation operator which is a generalization of A-iE smeared
in a suitable way, (b) normal ordered polynomials of generalized annihilation
and creation operators have the correct expectation value, (c) they saturate
the Heisenberg uncertainty bound for the fluctuations of and
(d) they do not use any background structure for their definition, that is,
they are diffeomorphism covariant.
This is the first paper in a series of articles entitled ``Gauge Field Theory
Coherent States (GCS)'' which aim at connecting non-perturbative quantum
general relativity with the low energy physics of the standard model. In
particular, coherent states enable us for the first time to take into account
quantum metrics which are excited {\it everywhere} in an asymptotically flat
spacetime manifold. The formalism introduced in this paper is immediately
applicable also to lattice gauge theory in the presence of a (Minkowski)
background structure on a possibly {\it infinite lattice}.Comment: 40 pages, LATEX, no figure
The large cosmological constant approximation to classical and quantum gravity: model examples
We have recently introduced an approach for studying perturbatively classical
and quantum canonical general relativity. The perturbative technique appears to
preserve many of the attractive features of the non-perturbative quantization
approach based on Ashtekar's new variables and spin networks. With this
approach one can find perturbatively classical observables (quantities that
have vanishing Poisson brackets with the constraints) and quantum states
(states that are annihilated by the quantum constraints). The relative ease
with which the technique appears to deal with these traditionally hard problems
opens several questions about how relevant the results produced can possibly
be. Among the questions is the issue of how useful are results for large values
of the cosmological constant and how the approach can deal with several
pathologies that are expected to be present in the canonical approach to
quantum gravity. With the aim of clarifying these points, and to make our
construction as explicit as possible, we study its application in several
simple models. We consider Bianchi cosmologies, the asymmetric top, the coupled
harmonic oscillators with constant energy density and a simple quantum
mechanical system with two Hamiltonian constraints. We find that the technique
satisfactorily deals with the pathologies of these models and offers promise
for finding (at least some) results even for small values of the cosmological
constant. Finally, we briefly sketch how the method would operate in the full
four dimensional quantum general relativity case.Comment: 21 pages, RevTex, 2 figures with epsfi
QSD VI : Quantum Poincar\'e Algebra and a Quantum Positivity of Energy Theorem for Canonical Quantum Gravity
We quantize the generators of the little subgroup of the asymptotic
Poincar\'e group of Lorentzian four-dimensional canonical quantum gravity in
the continuum. In particular, the resulting ADM energy operator is densely
defined on an appropriate Hilbert space, symmetric and essentially
self-adjoint. Moreover, we prove a quantum analogue of the classical positivity
of energy theorem due to Schoen and Yau. The proof uses a certain technical
restriction on the space of states at spatial infinity which is suggested to us
given the asymptotically flat structure available. The theorem demonstrates
that several of the speculations regarding the stability of the theory,
recently spelled out by Smolin, are false once a quantum version of the
pre-assumptions underlying the classical positivity of energy theorem is
imposed in the quantum theory as well. The quantum symmetry algebra
corresponding to the generators of the little group faithfully represents the
classical algebra.Comment: 24p, LATE
Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework
Recently the Master Constraint Programme for Loop Quantum Gravity (LQG) was
proposed as a classically equivalent way to impose the infinite number of
Wheeler -- DeWitt constraint equations in terms of a single Master Equation.
While the proposal has some promising abstract features, it was until now
barely tested in known models. In this series of five papers we fill this gap,
thereby adding confidence to the proposal. We consider a wide range of models
with increasingly more complicated constraint algebras, beginning with a finite
dimensional, Abelean algebra of constraint operators which are linear in the
momenta and ending with an infinite dimensional, non-Abelean algebra of
constraint operators which closes with structure functions only and which are
not even polynomial in the momenta. In all these models we apply the Master
Constraint Programme successfully, however, the full flexibility of the method
must be exploited in order to complete our task. This shows that the Master
Constraint Programme has a wide range of applicability but that there are many,
physically interesting subtleties that must be taken care of in doing so. In
this first paper we prepare the analysis of our test models by outlining the
general framework of the Master Constraint Programme. The models themselves
will be studied in the remaining four papers. As a side result we develop the
Direct Integral Decomposition (DID) for solving quantum constraints as an
alternative to Refined Algebraic Quantization (RAQ).Comment: 42 pages, no figure
Loop Quantum Cosmology III: Wheeler-DeWitt Operators
In the framework of loop quantum cosmology anomaly free quantizations of the
Hamiltonian constraint for Bianchi class A, locally rotationally symmetric and
isotropic models are given. Basic ideas of the construction in (non-symmetric)
loop quantum gravity can be used, but there are also further inputs because the
special structure of symmetric models has to be respected by operators. In
particular, the basic building blocks of the homogeneous models are point
holonomies rather than holonomies necessitating a new regularization procedure.
In this respect, our construction is applicable also for other
(non-homogeneous) symmetric models, e.g. the spherically symmetric one.Comment: 19 page
A Path-integral for the Master Constraint of Loop Quantum Gravity
In the present paper, we start from the canonical theory of loop quantum
gravity and the master constraint programme. The physical inner product is
expressed by using the group averaging technique for a single self-adjoint
master constraint operator. By the standard technique of skeletonization and
the coherent state path-integral, we derive a path-integral formula from the
group averaging for the master constraint operator. Our derivation in the
present paper suggests there exists a direct link connecting the canonical Loop
quantum gravity with a path-integral quantization or a spin-foam model of
General Relativity.Comment: 19 page
Latticing quantum gravity
I discuss some aspects of a lattice approach to canonical quantum gravity in
a connection formulation, discuss how it differs from the continuum
construction, and compare the spectra of geometric operators - encoding
information about components of the spatial metric - for some simple lattice
quantum states.Comment: 7 pages, TeX, 1 figure (epsf); contribution to Santa Margherita
Conference on Constrained Dynamics and Quantum Gravit
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