50 research outputs found
Dirac versus Reduced Quantization of the Poincar\'{e} Symmetry in Scalar Electrodynamics
The generators of the Poincar\'{e} symmetry of scalar electrodynamics are
quantized in the functional Schr\"{o}dinger representation. We show that the
factor ordering which corresponds to (minimal) Dirac quantization preserves the
Poincar\'{e} algebra, but (minimal) reduced quantization does not. In the
latter, there is a van Hove anomaly in the boost-boost commutator, which we
evaluate explicitly to lowest order in a heat kernel expansion using zeta
function regularization. We illuminate the crucial role played by the gauge
orbit volume element in the analysis. Our results demonstrate that preservation
of extra symmetries at the quantum level is sometimes a useful criterion to
select between inequivalent, but nevertheless self-consistent, quantization
schemes.Comment: 24 page
Nonlinear quantum gravity on the constant mean curvature foliation
A new approach to quantum gravity is presented based on a nonlinear
quantization scheme for canonical field theories with an implicitly defined
Hamiltonian. The constant mean curvature foliation is employed to eliminate the
momentum constraints in canonical general relativity. It is, however, argued
that the Hamiltonian constraint may be advantageously retained in the reduced
classical system to be quantized. This permits the Hamiltonian constraint
equation to be consistently turned into an expectation value equation on
quantization that describes the scale factor on each spatial hypersurface
characterized by a constant mean exterior curvature. This expectation value
equation augments the dynamical quantum evolution of the unconstrained
conformal three-geometry with a transverse traceless momentum tensor density.
The resulting quantum theory is inherently nonlinear. Nonetheless, it is
unitary and free from a nonlocal and implicit description of the Hamiltonian
operator. Finally, by imposing additional homogeneity symmetries, a broad class
of Bianchi cosmological models are analyzed as nonlinear quantum
minisuperspaces in the context of the proposed theory.Comment: 14 pages. Classical and Quantum Gravity (To appear
Semiclassical States in Quantum Cosmology: Bianchi I Coherent States
We study coherent states for Bianchi type I cosmological models, as examples
of semiclassical states for time-reparametrization invariant systems. This
simple model allows us to study explicitly the relationship between exact
semiclassical states in the kinematical Hilbert space and corresponding ones in
the physical Hilbert space, which we construct here using the group averaging
technique. We find that it is possible to construct good semiclassical physical
states by such a procedure in this model; we also discuss the sense in which
the original kinematical states may be a good approximation to the physical
ones, and the situations in which this is the case. In addition, these models
can be deparametrized in a natural way, and we study the effect of time
evolution on an "intrinsic" coherent state in the reduced phase space, in order
to estimate the time for this state to spread significantly.Comment: 21 pages, 1 figure; Version to be published in CQG; The discussion
has been slightly reorganized, two references added, and some typos correcte
An action principle for the quantization of parametric theories and nonlinear quantum cosmology
By parametrizing the action integral for the standard Schrodinger equation we
present a derivation of the recently proposed method for quantizing a
parametrized theory. The reformulation suggests a natural extension from
conventional to nonlinear quantum mechanics. This generalization enables a
unitary description of the quantum evolution for a broad class of constrained
Hamiltonian systems with a nonlinear kinematic structure. In particular, the
new theory is applicable to the quantization of cosmological models where a
chosen gravitational degree of freedom acts as geometric time. This is
demonstrated explicitly using three cosmological models: the Friedmann universe
with a massless scalar field and Bianchi type I and IX models. Based on these
investigations, the prospect of further developing the proposed quantization
scheme in the context of quantum gravity is discussed.Comment: 14 page
Unification of the conditional probability and semiclassical interpretations for the problem of time in quantum theory
We show that the time-dependent Schr\"odinger equation (TDSE) is the
phenomenological dynamical law of evolution unraveled in the classical limit
from a timeless formulation in terms of probability amplitudes conditioned by
the values of suitably chosen internal clock variables, thereby unifying the
conditional probability interpretation (CPI) and the semiclassical approach for
the problem of time in quantum theory. Our formalism stems from an exact
factorization of the Hamiltonian eigenfunction of the clock plus system
composite, where the clock and system factors play the role of marginal and
conditional probability amplitudes, respectively. Application of the Variation
Principle leads to a pair of exact coupled pseudoeigenvalue equations for these
amplitudes, whose solution requires an iterative self-consistent procedure. The
equation for the conditional amplitude constitutes an effective "equation of
motion" for the quantum state of the system with respect to the clock
variables. These coupled equations also provide a convenient framework for
treating the back-reaction of the system on the clock at various levels of
approximation. At the lowest level, when the WKB approximation for the marginal
amplitude is appropriate, in the classical limit of the clock variables the
TDSE for the system emerges as a matter of course from the conditional
equation. In this connection, we provide a discussion of the characteristics
required by physical systems to serve as good clocks. This development is seen
to be advantageous over the original CPI and semiclassical approach since it
maintains the essence of the conventional formalism of quantum mechanics,
admits a transparent interpretation, avoids the use of the Born-Oppenheimer
approximation, and resolves various objections raised about them.Comment: 10 pages. Typographical errors correcte
Essential Constants for Spatially Homogeneous Ricci-flat manifolds of dimension 4+1
The present work considers (4+1)-dimensional spatially homogeneous vacuum
cosmological models. Exact solutions -- some already existing in the
literature, and others believed to be new -- are exhibited. Some of them are
the most general for the corresponding Lie group with which each homogeneous
slice is endowed, and some others are quite general. The characterization
``general'' is given based on the counting of the essential constants, the
line-element of each model must contain; indeed, this is the basic contribution
of the work. We give two different ways of calculating the number of essential
constants for the simply transitive spatially homogeneous (4+1)-dimensional
models. The first uses the initial value theorem; the second uses, through
Peano's theorem, the so-called time-dependent automorphism inducing
diffeomorphismsComment: 26 Pages, 2 Tables, latex2
Nucleation of Brane Universes
The creation of brane universes induced by a totally antisymmetric tensor
living in a fixed background spacetime is presented, where a term involving the
intrinsic curvature of the brane is considered. A canonical quantum mechanical
approach employing Wheeler-DeWitt equation is done. The probability nucleation
for the brane is calculated taking into account both an instanton method and a
WKB approximation. Some cosmological implications arose from the model are
presented.Comment: 19 pages, 2 figure
The Bohm Interpretation of Quantum Cosmology
I make a review on the aplications of the Bohm-De Broglie interpretation of
quantum mechanics to quantum cosmology. In the framework of minisuperspaces
models, I show how quantum cosmological effects in Bohm's view can avoid the
initial singularity, isotropize the Universe, and even be a cause for the
present observed acceleration of the Universe. In the general case, we
enumerate the possible structures of quantum space and time.Comment: 28 pages, 1 figure, contribution to the James Cushing festschrift to
appear in Foundations of Physic
Nontrivial Dynamics in the Early Stages of Inflation
Inflationary cosmologies, regarded as dynamical systems, have rather simple
asymptotic behavior, insofar as the cosmic baldness principle holds.
Nevertheless, in the early stages of an inflationary process, the dynamical
behavior may be very complex. In this paper, we show how even a simple
inflationary scenario, based on Linde's ``chaotic inflation'' proposal,
manifests nontrivial dynamical effects such as the breakup of invariant tori,
formation of cantori and Arnol'd's diffusion. The relevance of such effects is
highlighted by the fact that even the occurrence or not of inflation in a given
Universe is dependent upon them.Comment: 26 pages, Latex, 9 Figures available on request, GTCRG-94-1
Background Independent Quantum Gravity: A Status Report
The goal of this article is to present an introduction to loop quantum
gravity -a background independent, non-perturbative approach to the problem of
unification of general relativity and quantum physics, based on a quantum
theory of geometry. Our presentation is pedagogical. Thus, in addition to
providing a bird's eye view of the present status of the subject, the article
should also serve as a vehicle to enter the field and explore it in detail. To
aid non-experts, very little is assumed beyond elements of general relativity,
gauge theories and quantum field theory. While the article is essentially
self-contained, the emphasis is on communicating the underlying ideas and the
significance of results rather than on presenting systematic derivations and
detailed proofs. (These can be found in the listed references.) The subject can
be approached in different ways. We have chosen one which is deeply rooted in
well established physics and also has sufficient mathematical precision to
ensure that there are no hidden infinities. In order to keep the article to a
reasonable size, and to avoid overwhelming non-experts, we have had to leave
out several interesting topics, results and viewpoints; this is meant to be an
introduction to the subject rather than an exhaustive review of it.Comment: 125 pages, 5 figures (eps format), the final version published in CQ