141 research outputs found
Quantization of generally covariant systems with extrinsic time
A generally covariant system can be deparametrized by means of an
``extrinsic'' time, provided that the metric has a conformal ``temporal''
Killing vector and the potential exhibits a suitable behavior with respect to
it. The quantization of the system is performed by giving the well ordered
constraint operators which satisfy the algebra. The searching of these
operators is enlightned by the methods of the BRST formalism.Comment: 10 pages. Definite published versio
Internal Time Formalism for Spacetimes with Two Killing Vectors
The Hamiltonian structure of spacetimes with two commuting Killing vector
fields is analyzed for the purpose of addressing the various problems of time
that arise in canonical gravity. Two specific models are considered: (i)
cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes,
which respectively involve open and closed universe boundary conditions. For
each model canonical variables which can be used to identify points of space
and instants of time, {\it i.e.}, internally defined spacetime coordinates, are
identified. To do this it is necessary to extend the usual ADM phase space by a
finite number of degrees of freedom. Canonical transformations are exhibited
that identify each of these models with harmonic maps in the parametrized field
theory formalism. The identifications made between the gravitational models and
harmonic map field theories are completely gauge invariant, that is, no
coordinate conditions are needed. The degree to which the problems of time are
resolved in these models is discussed.Comment: 36 pages, Te
Dust as a Standard of Space and Time in Canonical Quantum Gravity
The coupling of the metric to an incoherent dust introduces into spacetime a
privileged dynamical reference frame and time foliation. The comoving
coordinates of the dust particles and the proper time along the dust worldlines
become canonical coordinates in the phase space of the system. The Hamiltonian
constraint can be resolved with respect to the momentum that is canonically
conjugate to the dust time. Imposition of the resolved constraint as an
operator restriction on the quantum states yields a functional Schr\"{o}dinger
equation. The ensuing Hamiltonian density has an extraordinary feature: it
depends only on the geometric variables, not on the dust coordinates or time.
This has three important consequences. First, the functional Schr\"{o}dinger
equation can be solved by separating the dust time from the geometric
variables. Second, the Hamiltonian densities strongly commute and therefore can
be simultaneously defined by spectral analysis. Third, the standard constraint
system of vacuum gravity is cast into a form in which it generates a true Lie
algebra. The particles of dust introduce into space a privileged system of
coordinates that allows the supermomentum constraint to be solved explicitly.
The Schr\"{o}dinger equation yields a conserved inner product that can be
written in terms of either the instantaneous state functionals or the solutions
of constraints. Examples of gravitational observables are given, though neither
the intrinsic metric nor the extrinsic curvature are observables. Disregarding
factor--ordering difficulties, the introduction of dust provides a satisfactory
phenomenological approach to the problem of time in canonical quantum gravity.Comment: 56 pages (REVTEX file + 3 postscipt figure files
Dirac Quantization of Parametrized Field Theory
Parametrized field theory (PFT) is free field theory on flat spacetime in a
diffeomorphism invariant disguise. It describes field evolution on arbitrary
foliations of the flat spacetime instead of only the usual flat ones, by
treating the `embedding variables' which describe the foliation as dynamical
variables to be varied in the action in addition to the scalar field. A formal
Dirac quantization turns the constraints of PFT into functional Schrodinger
equations which describe evolution of quantum states from an arbitrary Cauchy
slice to an infinitesimally nearby one.This formal Schrodinger picture- based
quantization is unitarily equivalent to the standard Heisenberg picture based
Fock quantization of the free scalar field if scalar field evolution along
arbitrary foliations is unitarily implemented on the Fock space. Torre and
Varadarajan (TV) showed that for generic foliations emanating from a flat
initial slice in spacetimes of dimension greater than 2, evolution is not
unitarily implemented, thus implying an obstruction to Dirac quantization.
We construct a Dirac quantization of PFT,unitarily equivalent to the standard
Fock quantization, using techniques from Loop Quantum Gravity (LQG) which are
powerful enough to super-cede the no- go implications of the TV results. The
key features of our quantization include an LQG type representation for the
embedding variables, embedding dependent Fock spaces for the scalar field, an
anomaly free representation of (a generalization of) the finite transformations
generated by the constraints and group averaging techniques. The difference
between 2 and higher dimensions is that in the latter, only finite gauge
transformations are defined in the quantum theory, not the infinitesimal ones.Comment: 33 page
Classical and quantum geometrodynamics of 2d vacuum dilatonic black holes
We perform a canonical analysis of the system of 2d vacuum dilatonic black
holes. Our basic variables are closely tied to the spacetime geometry and we do
not make the field redefinitions which have been made by other authors. We
present a careful discssion of asymptotics in this canonical formalism.
Canonical transformations are made to variables which (on shell) have a clear
spacetime significance. We are able to deduce the location of the horizon on
the spatial slice (on shell) from the vanishing of a combination of canonical
data. The constraints dramatically simplify in terms of the new canonical
variables and quantization is easy. The physical interpretation of the variable
conjugate to the ADM mass is clarified. This work closely parallels that done
by Kucha{\v r} for the vacuum Schwarzschild black holes and is a starting point
for a similar analysis, now in progress, for the case of a massless scalar
field conformally coupled to a 2d dilatonic black hole.Comment: 21 pages, latex fil
Covariant gauge fixing and Kuchar decomposition
The symplectic geometry of a broad class of generally covariant models is
studied. The class is restricted so that the gauge group of the models
coincides with the Bergmann-Komar group and the analysis can focus on the
general covariance. A geometrical definition of gauge fixing at the constraint
manifold is given; it is equivalent to a definition of a background (spacetime)
manifold for each topological sector of a model. Every gauge fixing defines a
decomposition of the constraint manifold into the physical phase space and the
space of embeddings of the Cauchy manifold into the background manifold (Kuchar
decomposition). Extensions of every gauge fixing and the associated Kuchar
decomposition to a neighbourhood of the constraint manifold are shown to exist.Comment: Revtex, 35 pages, no figure
A Kucha\v{r} Hypertime Formalism For Cylindrically Symmetric Spacetimes With Interacting Scalar Fields
The Kucha\v{r} canonical transformation for vacuum geometrodynamics in the
presence of cylindrical symmetry is applied to a general non-vacuum case. The
resulting constraints are highly non-linear and non-local in the momenta
conjugate to the Kucha\v{r} embedding variables. However, it is demonstrated
that the constraints can be solved for these momenta and thus the dynamics of
cylindrically symmetric models can be cast in a form suitable for the
construction of a hypertime functional Schr\"odinger equation.Comment: 5 pages, LaTeX, UBCTP-93-02
Embedding variables in finite dimensional models
Global problems associated with the transformation from the Arnowitt, Deser
and Misner (ADM) to the Kucha\v{r} variables are studied. Two models are
considered: The Friedmann cosmology with scalar matter and the torus sector of
the 2+1 gravity. For the Friedmann model, the transformations to the Kucha\v{r}
description corresponding to three different popular time coordinates are shown
to exist on the whole ADM phase space, which becomes a proper subset of the
Kucha\v{r} phase spaces. The 2+1 gravity model is shown to admit a description
by embedding variables everywhere, even at the points with additional symmetry.
The transformation from the Kucha\v{r} to the ADM description is, however,
many-to-one there, and so the two descriptions are inequivalent for this model,
too. The most interesting result is that the new constraint surface is free
from the conical singularity and the new dynamical equations are linearization
stable. However, some residual pathology persists in the Kucha\v{r}
description.Comment: Latex 2e, 29 pages, no figure
Dirac Constraint Quantization of a Dilatonic Model of Gravitational Collapse
We present an anomaly-free Dirac constraint quantization of the
string-inspired dilatonic gravity (the CGHS model) in an open 2-dimensional
spacetime. We show that the quantum theory has the same degrees of freedom as
the classical theory; namely, all the modes of the scalar field on an auxiliary
flat background, supplemented by a single additional variable corresponding to
the primordial component of the black hole mass. The functional Heisenberg
equations of motion for these dynamical variables and their canonical
conjugates are linear, and they have exactly the same form as the corresponding
classical equations. A canonical transformation brings us back to the physical
geometry and induces its quantization.Comment: 37 pages, LATEX, no figures, submitted to Physical Review
Covariant perturbations of domain walls in curved spacetime
A manifestly covariant equation is derived to describe the perturbations in a
domain wall on a given background spacetime. This generalizes recent work on
domain walls in Minkowski space and introduces a framework for examining the
stability of relativistic bubbles in curved spacetimes.Comment: 15 pages,ICN-UNAM-93-0
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