886 research outputs found
Complexifier Coherent States for Quantum General Relativity
Recently, substantial amount of activity in Quantum General Relativity (QGR)
has focussed on the semiclassical analysis of the theory. In this paper we want
to comment on two such developments: 1) Polymer-like states for Maxwell theory
and linearized gravity constructed by Varadarajan which use much of the Hilbert
space machinery that has proved useful in QGR and 2) coherent states for QGR,
based on the general complexifier method, with built-in semiclassical
properties. We show the following: A) Varadarajan's states {\it are}
complexifier coherent states. This unifies all states constructed so far under
the general complexifier principle. B) Ashtekar and Lewandowski suggested a
non-Abelean generalization of Varadarajan's states to QGR which, however, are
no longer of the complexifier type. We construct a new class of non-Abelean
complexifiers which come close to the one underlying Varadarajan's
construction. C) Non-Abelean complexifiers close to Varadarajan's induce new
types of Hilbert spaces which do not support the operator algebra of QGR. The
analysis suggests that if one sticks to the present kinematical framework of
QGR and if kinematical coherent states are at all useful, then normalizable,
graph dependent states must be used which are produced by the complexifier
method as well. D) Present proposals for states with mildened graph dependence,
obtained by performing a graph average, do not approximate well coordinate
dependent observables. However, graph dependent states, whether averaged or
not, seem to be well suited for the semiclassical analysis of QGR with respect
to coordinate independent operators.Comment: Latex, 54 p., no 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
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
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
Free vacuum for loop quantum gravity
We linearize extended ADM-gravity around the flat torus, and use the
associated Fock vacuum to construct a state that could play the role of a free
vacuum in loop quantum gravity. The state we obtain is an element of the
gauge-invariant kinematic Hilbert space and restricted to a cutoff graph, as a
natural consequence of the momentum cutoff of the original Fock state. It has
the form of a Gaussian superposition of spin networks. We show that the peak of
the Gaussian lies at weave-like states and derive a relation between the
coloring of the weaves and the cutoff scale. Our analysis indicates that the
peak weaves become independent of the cutoff length when the latter is much
smaller than the Planck length. By the same method, we also construct
multiple-graviton states. We discuss the possible use of these states for
deriving a perturbation series in loop quantum gravity.Comment: 30 pages, 3 diagrams, treatment of phase factor adde
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
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
Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories
We extend the recently developed kinematical framework for diffeomorphism
invariant theories of connections for compact gauge groups to the case of a
diffeomorphism invariant quantum field theory which includes besides
connections also fermions and Higgs fields. This framework is appropriate for
coupling matter to quantum gravity. The presence of diffeomorphism invariance
forces us to choose a representation which is a rather non-Fock-like one : the
elementary excitations of the connection are along open or closed strings while
those of the fermions or Higgs fields are at the end points of the string.
Nevertheless we are able to promote the classical reality conditions to quantum
adjointness relations which in turn uniquely fixes the gauge and diffeomorphism
invariant probability measure that underlies the Hilbert space. Most of the
fermionic part of this work is independent of the recent preprint by Baez and
Krasnov and earlier work by Rovelli and Morales-Tec\'otl because we use new
canonical fermionic variables, so-called Grassman-valued half-densities, which
enable us to to solve the difficult fermionic adjointness relations.Comment: 26p, LATE
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
Disordered locality in loop quantum gravity states
We show that loop quantum gravity suffers from a potential problem with
non-locality, coming from a mismatch between micro-locality, as defined by the
combinatorial structures of their microscopic states, and macro-locality,
defined by the metric which emerges from the low energy limit. As a result, the
low energy limit may suffer from a disordered locality characterized by
identifications of far away points. We argue that if such defects in locality
are rare enough they will be difficult to detect.Comment: 11 pages, 4 figures, revision with extended discussion of result
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