221 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
Kruskal coordinates as canonical variables for Schwarzschild black holes
We derive a transformation from the usual ADM metric-extrinsic curvature
variables on the phase space of Schwarzschild black holes, to new canonical
variables which have the interpretation of Kruskal coordinates. We explicitly
show that this transformation is non-singular, even at the horizon. The
constraints of the theory simplify in terms of the new canonical variables and
are equivalent to the vanishing of the canonical momenta. Our work is based on
earlier seminal work by Kuchar in which he reconstructed curvature coordinates
and a mass function from spherically symmetric canonical data. The key feature
in our construction of a nonsingular canonical transformation to Kruskal
variables, is the scaling of the curvature coordinate variables by the mass
function rather than by the mass at left spatial infinity.Comment: 18 pages, no figure
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
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
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
On Group Averaging for SO(n,1)
The technique known as group averaging provides powerful machinery for the
study of constrained systems. However, it is likely to be well defined only in
a limited set of cases. Here, we investigate the possibility of using a
`renormalized' group averaging in certain models. The results of our study may
indicate a general connection between superselection sectors and the rate of
divergence of the group averaging integral.Comment: Minor corrections, 17 pages,RevTe
Loop constraints: A habitat and their algebra
This work introduces a new space \T'_* of `vertex-smooth' states for use in
the loop approach to quantum gravity. Such states provide a natural domain for
Euclidean Hamiltonian constraint operators of the type introduced by Thiemann
(and using certain ideas of Rovelli and Smolin). In particular, such operators
map \T'_* into itself, and so are actual operators in this space. Their
commutator can be computed on \T'_* and compared with the classical
hypersurface deformation algebra. Although the classical Poisson bracket of
Hamiltonian constraints yields an inverse metric times an infinitesimal
diffeomorphism generator, and despite the fact that the diffeomorphism
generator has a well-defined non-trivial action on \T'_*, the commutator of
quantum constraints vanishes identically for a large class of proposals.Comment: 30 pages RevTex, 2 figures include
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
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
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