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 $\Psi$ 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 $\Psi$ 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, $\Psi$ 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 $h_\alpha$ 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 $Y_\alpha$ 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 $Y_\alpha$ induce natural conservative measures on acausal submanifolds, which are submanifolds transversal to the dynamical foliation. Further it is shown that the structure coefficients $c^\gamma_{\alpha\beta}$ defined by $\{h_\alpha,h_\beta\}=\sum_\gamma c^\gamma_{\alpha\beta}h_\gamma$ should weakly commute with $h_\alpha$, $\sum_\gamma\{h_\gamma,c^\gamma_{\alpha\beta}\}\approx0$, 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$^5$ 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 $H$ is smaller than the inverse of the bulk curvature scale $\ell$. Further, we give rough estimates which indicate that in the early universe when $H$ is much larger than $1/\ell$, 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