258 research outputs found
Dynamics of Totally Constrained Systems II. Quantum Theory
In this paper a new formulation of quantum dynamics of totally constrained
systems is developed, in which physical quantities representing time are
included as observables. In this formulation the hamiltonian constraints are
imposed on a relative probability amplitude functional which determines
the relative probability for each state to be observed, instead of on the state
vectors as in the conventional Dirac quantization. This leads to a foliation of
the state space by linear manifolds on each of which is constant, and
dynamics is described as linear mappings among acausal subspaces which are
transversal to these linear manifolds. This is a quantum analogue of the
classical statistical dynamics of totally constrained systems developed in the
previous paper. It is shown that if the von Neumann algebra \C generated by
the constant of motion is of type I, can be consistently normalizable on
the acausal subspaces on which a factor subalgebra of \C is represented
irreducibly, and the mappings among these acausal subspaces are conformal. How
the formulation works is illustrated by simple totally constrained systems with
a single constraint such as the parametrized quantum mechanics, a relativistic
free particle in Minkowski and curved spacetimes, and a simple minisuperspace
model. It is pointed out that the inner product of the relative probability
amplitudes induced from the original Hilbert space picks up a special
decomposition of the wave functions to the positive and the negative frequency
modes.Comment: 57 pages, some unexpected control codes in the original file, which
may cause errors for some LaTeX compilers, were remove
Dynamics of Totally Constrained Systems I. Classical Theory
This is the first of a series of papers in which a new formulation of quantum
theory is developed for totally constrained systems, that is, canonical systems
in which the hamiltonian is written as a linear combination of constraints
with arbitrary coefficients. The main purpose of the present paper
is to make clear that classical dynamics of a totally constrained system is
nothing but the foliation of the constraint submanifold in phase space by the
involutive system of infinitesimal canonical transformations
generated by the constraint functions. From this point of view it is shown that
statistical dynamics for an ensemble of a totally constrained system can be
formulated in terms of a relative distribution function without gauge fixing or
reduction. There the key role is played by the fact that the canonical measure
in phase space and the vector fields induce natural conservative
measures on acausal submanifolds, which are submanifolds transversal to the
dynamical foliation. Further it is shown that the structure coefficients
defined by should weakly commute with ,
, in order that the
description in terms of the relative distribution function is consistent. The
overall picture on the classical dynamics given in this paper provides the
basic motivation for the quantum formulation developed in the subsequent
papers.Comment: 31 pages, LaTeX fil
Behavior of Cosmological Perturbations in the Brane-World Mode
In this paper we present a gauge-invariant formalism for perturbations of the
brane-world model developed by the author, A. Ishibashi and O. Seto recently,
and analyze the behavior of cosmological perturbations in a spatially flat
expanding universe realized as a boundary 3-brane in AdS in terms of this
formalism. For simplicity we restrict arguments to scalar perturbations. We
show that the behavior of cosmological perturbations on superhorizon scales in
the brane-world model is the same as that in the standard no-extradimension
model, irrespective of the initial condition for bulk perturbations, in the
late stage when the cosmic expansion rate is smaller than the inverse of
the bulk curvature scale . Further, we give rough estimates which
indicate that in the early universe when is much larger than ,
perturbations in these two models behave quite differently, and the
conservation of the Bardeen parameter does not hold for superhorizon
perturbations in the brane-world model.Comment: 4 pages in the revtex style. A talk in the conference CAPP2000 to be
published in the proceeding
Rigidity theorems in the braneworld model
In the present paper, we give some theorems representing ridigity of a vacuum
brane in static bulk spacetimes. As an application, we show that a static bulk
spacetime with dimension D>3 and spatial symmetry IO(D-2), O(D-1) or O_+(D-2,1)
does not allow a vacuum brane with a black hole on it. We also show that if a
static bulk spacetime with dimension D>4 satsifying the vacuum Einstein
equations can be foliated by a continuous family of vacuum branes with
asymptotically constant curvature, it is a black string solution.Comment: 7 pages in LaTeX with the PTP style. No figure. To be published in
the proceedings of the workshop "Braneworld -- Dynamics of spacetime with
boundary --
A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions
We show that in four or more spacetime dimensions, the Einstein equations for
gravitational perturbations of maximally symmetric vacuum black holes can be
reduced to a single 2nd-order wave equation in a two-dimensional static
spacetime for a gauge-invariant master variable, irrespective of the mode of
perturbations. Our formulation applies to the case of vanishing as well as
non-vanishing cosmological constant Lambda. The sign of the sectional curvature
K of each spatial section of equipotential surfaces is also kept general. In
the four-dimensional Schwarzschild background, this master equation for a
scalar perturbation is identical to the Zerilli equation for the polar mode and
the master equation for a vector perturbation is identical to the Regge-Wheeler
equation for the axial mode. Furthermore, in the four-dimensional
Schwarzschild-anti-de Sitter background with K=0,1, our equation coincides with
those derived by Cardoso and Lemos recently. As a simple application, we prove
the perturbative stability and uniqueness of four-dimensional non-extremal
spherically symmetric black holes for any Lambda. We also point out that there
exists no simple relation between scalar-type and vector-type perturbations in
higher dimensions, unlike in four dimensions. Although we only treat maximally
symmetric black holes in the present paper, the final master equations are
valid even when the hirozon geometry is described by a generic Einstein
manifold.Comment: 22 pages in the PTP TeX style, no figure. The published versio
- …