884 research outputs found
Exploring the diffeomorphism invariant Hilbert space of a scalar field
As a toy model for the implementation of the diffeomorphism constraint, the
interpretation of the resulting states, and the treatment of ordering
ambiguities in loop quantum gravity, we consider the Hilbert space of spatially
diffeomorphism invariant states for a scalar field. We give a very explicit
formula for the scalar product on this space, and discuss its structure.
Then we turn to the quantization of a certain class of diffeomorphism
invariant quantities on that space, and discuss in detail the ordering issues
involved. On a technical level these issues bear some similarity to those
encountered in full loop quantum gravity.Comment: 20 pages, no figures; v3: corrected typos, added reference, some
clarifications added; version as published in CQ
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
Dynamics of Scalar Field in Polymer-like Representation
In recent twenty years, loop quantum gravity, a background independent
approach to unify general relativity and quantum mechanics, has been widely
investigated. We consider the quantum dynamics of a real massless scalar field
coupled to gravity in this framework. A Hamiltonian operator for the scalar
field can be well defined in the coupled diffeomorphism invariant Hilbert
space, which is both self-adjoint and positive. On the other hand, the
Hamiltonian constraint operator for the scalar field coupled to gravity can be
well defined in the coupled kinematical Hilbert space. There are 1-parameter
ambiguities due to scalar field in the construction of both operators. The
results heighten our confidence that there is no divergence within this
background independent and diffeomorphism invariant quantization approach of
matter coupled to gravity. Moreover, to avoid possible quantum anomaly, the
master constraint programme can be carried out in this coupled system by
employing a self-adjoint master constraint operator on the diffeomorphism
invariant Hilbert space.Comment: 24 pages, accepted for pubilcation in Class. Quant. Gra
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
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
On (Cosmological) Singularity Avoidance in Loop Quantum Gravity
Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological
sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of
the phase space of classical General Relativity to spatially homogeneous
situations which is then quantized by the methods of LQG. Thus, LQC is a
quantum mechanical toy model (finite number of degrees of freedom) for LQG(a
genuine QFT with an infinite number of degrees of freedom) which provides
important consistency checks. However, it is a non trivial question whether the
predictions of LQC are robust after switching on the inhomogeneous fluctuations
present in full LQG. Two of the most spectacular findings of LQC are that 1.
the inverse scale factor is bounded from above on zero volume eigenstates which
hints at the avoidance of the local curvature singularity and 2. that the
Quantum Einstein Equations are non -- singular which hints at the avoidance of
the global initial singularity. We display the result of a calculation for LQG
which proves that the (analogon of the) inverse scale factor, while densely
defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in
full LQG, if curvature singularity avoidance is realized, then not in this
simple way. In fact, it turns out that the boundedness of the inverse scale
factor is neither necessary nor sufficient for curvature singularity avoidance
and that non -- singular evolution equations are neither necessary nor
sufficient for initial singularity avoidance because none of these criteria are
formulated in terms of observable quantities.After outlining what would be
required, we present the results of a calculation for LQG which could be a
first indication that our criteria at least for curvature singularity avoidance
are satisfied in LQG.Comment: 34 pages, 16 figure
Electric fields and double layers in plasmas
Various mechanisms for driving double layers in plasmas are briefly described, including applied potential drops, currents, contact potentials, and plasma expansions. Some dynamical features of the double layers are discussed. These features, as seen in simulations, laboratory experiments, and theory, indicate that double layers and the currents through them undergo slow oscillations which are determined by the ion transit time across an effective length of the system in which double layers form. It is shown that a localized potential dip forms at the low potential end of a double layer, which interrupts the electron current through it according to the Langmuir criterion, whenever the ion flux into the double is disrupted. The generation of electric fields perpendicular to the ambient magnetic field by contact potentials is also discussed. Two different situations were considered; in one, a low-density hot plasma is sandwiched between high-density cold plasmas, while in the other a high-density current sheet permeates a low-density background plasma. Perpendicular electric fields develop near the contact surfaces. In the case of the current sheet, the creation of parallel electric fields and the formation of double layers are also discussed when the current sheet thickness is varied. Finally, the generation of electric fields and double layers in an expanding plasma is discussed
Regularized Hamiltonians and Spinfoams
We review a recent proposal for the regularization of the scalar constraint
of General Relativity in the context of LQG. The resulting constraint presents
strengths and weaknesses compared to Thiemann's prescription. The main
improvement is that it can generate the 1-4 Pachner moves and its matrix
elements contain 15j Wigner symbols, it is therefore compatible with the
spinfoam formalism: the drawback is that Thiemann anomaly free proof is spoiled
because the nodes that the constraint creates have volume.Comment: 4 pages, based on a talk given at Loops '11 in Madrid, to appear in
Journal of Physics: Conference Series (JPCS
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
- …