1,106 research outputs found
The Proca-field in Loop Quantum Gravity
In this paper we investigate the Proca-field in the framework of Loop Quantum
Gravity. It turns out that the methods developed there can be applied to the
symplectically embedded Proca-field, giving a rigorous, consistent,
non-perturbative quantization of the theory. This can be achieved by
introducing a scalar field, which has completely different properties than the
one used in spontaneous symmetry breaking. The analysis of the kernel of the
Hamiltonian suggests that the mass term in the quantum theory has a different
role than in the classical theory.Comment: 15 pages. v2: 19 pages, amended sections 2 and 6, references added
v3: 20 pages, amended section 6 and minor correction
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
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
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
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
Towards the QFT on Curved Spacetime Limit of QGR. I: A General Scheme
In this article and a companion paper we address the question of how one
might obtain the semiclassical limit of ordinary matter quantum fields (QFT)
propagating on curved spacetimes (CST) from full fledged Quantum General
Relativity (QGR), starting from first principles. We stress that we do not
claim to have a satisfactory answer to this question, rather our intention is
to ignite a discussion by displaying the problems that have to be solved when
carrying out such a program. In the present paper we propose a scheme that one
might follow in order to arrive at such a limit. We discuss the technical and
conceptual problems that arise in doing so and how they can be solved in
principle. As to be expected, completely new issues arise due to the fact that
QGR is a background independent theory. For instance, fundamentally the notion
of a photon involves not only the Maxwell quantum field but also the metric
operator - in a sense, there is no photon vacuum state but a "photon vacuum
operator"! While in this first paper we focus on conceptual and abstract
aspects, for instance the definition of (fundamental) n-particle states (e.g.
photons), in the second paper we perform detailed calculations including, among
other things, coherent state expectation values and propagation on random
lattices. These calculations serve as an illustration of how far one can get
with present mathematical techniques. Although they result in detailed
predictions for the size of first quantum corrections such as the gamma-ray
burst effect, these predictions should not be taken too seriously because a)
the calculations are carried out at the kinematical level only and b) while we
can classify the amount of freedom in our constructions, the analysis of the
physical significance of possible choices has just begun.Comment: LaTeX, 47 p., 3 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
Real and complex connections for canonical gravity
Both real and complex connections have been used for canonical gravity: the
complex connection has SL(2,C) as gauge group, while the real connection has
SU(2) as gauge group. We show that there is an arbitrary parameter
which enters in the definition of the real connection, in the Poisson brackets,
and therefore in the scale of the discrete spectra one finds for areas and
volumes in the corresponding quantum theory. A value for could be could
be singled out in the quantum theory by the Hamiltonian constraint, or by the
rotation to the complex Ashtekar connection.Comment: 8 pages, RevTeX, no figure
Thiemann transform for gravity with matter fields
The generalised Wick transform discovered by Thiemann provides a
well-established relation between the Euclidean and Lorentzian theories of
general relativity. We extend this Thiemann transform to the Ashtekar
formulation for gravity coupled with spin-1/2 fermions, a non-Abelian
Yang-Mills field, and a scalar field. It is proved that, on functions of the
gravitational and matter phase space variables, the Thiemann transform is
equivalent to the composition of an inverse Wick rotation and a constant
complex scale transformation of all fields. This result holds as well for
functions that depend on the shift vector, the lapse function, and the Lagrange
multipliers of the Yang-Mills and gravitational Gauss constraints, provided
that the Wick rotation is implemented by means of an analytic continuation of
the lapse. In this way, the Thiemann transform is furnished with a geometric
interpretation. Finally, we confirm the expectation that the generator of the
Thiemann transform can be determined just from the spin of the fields and give
a simple explanation for this fact.Comment: LaTeX 2.09, 14 pages, no figure
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