Karel V. Kuchař

A Bubble‐Time Canonical Formalism for Geometrodynamics

A functional differential version of the ADM canonical formalism is proposed. The existence of a canonical transformation separating canonical variables into internal coordinates, energy‐momentum densities, and two pairs of true dynamical variables is assumed. The evolution of dynamical variables is governed by functional differential Hamiltons equations. They satisfy certain integrability conditions ensuring the internal path independence of dynamical evolution. The change of dynamical variables along any spacelike hypersurface is given by their Lie derivatives. This allows an elimination of 3∞3 components of the Hamilton equation, leading to a functional differential Hamilton equation based on a single bubble time. The Hamilton‐Jacobi theory is built along the same lines. The formalism is illustrated in the mini‐phase‐space of the cylindrical Einstein‐Rosen wave.

General relativity: Dynamics without symmetry

The concept of conditional symmetry is introduced for a parametrized relativistic particle model and generalized to geometrodynamics. Its role in maintaining a one‐system interpretation of the quantized theory is emphasized. It is shown that geometrodynamics does not have any conditional symmetry: Such a symmetry should be generated by a dynamical variable K[gab, pab] which is linear and homogeneous in the gravitational momentum pab and which has a weakly vanishing Poisson bracket with the super‐Hamiltonian and supermomentum. The generators K fall into equivalence classes modulo the supermomentum constraint. It is shown that each equivalence class can be represented by a member which is a spatial invariant. The remaining weak equations are turned into strong equations by the method of Lagrange multipliers. The local structure of the super‐Hamiltonian and supermomentum imposes locality restrictions on the multipliers. These restrictions imply that the generator must be weakly equivalent to a local generato...

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.

Gravitational constraints that generate a Lie algebra

The coupling of gravity to dust helps one discover simple quadratic combinations of the gravitational super-Hamiltonian and supermomentum whose Poisson brackets strongly vanish. This leads to a new form of vacuum constraints which generate a true Lie algebra. We show that the coupling of gravity to a massless scalar field leads to yet another set of constraints with the same property, albeit not as simple as that based on the coupling to dust.

Canonical geometrodynamics and general covariance

By extending geometrodynamical phase space by embeddings and their conjugate momenta, one can homomorphically map the Lie algebra of space-time diffeomorphisms into the Poisson algebra of dynamical variables on the extended phase space.

Measure for measure: Covariant skeletonizations of phase space path integrals for systems moving on Riemannian manifolds

We define phase space path integrals for systems moving on a Riemannian manifold and subject to a generalized potential by a skeletonization procedure which is manifestly covariant under point transformations. We achieve this goal by introducing a natural analog S(x″,t″‖x′,p′,t′) of the Hamilton principal function with phase space initial data. One class of such functions is based on the parallel transport of momentum, a second class is obtained by a modification of the first class, and a third class is based on the geodesic deviation transport of momentum. The third class of principal functions is geometrically privileged. We skeletonize the canonical action integral by replacing it by a manifestly covariant chain of phase space principal functions. Different functions lead to the same functional as we infinitely refine the skeletonization along a smooth path. Our phase space path integral is always taken with the natural Liouville measure, but the integration over momentum variables brings down a nontri...

Conditional symmetries in parametrized field theories

In parametrized field theories, spacelike hypersurfaces and fields which they carry are evolved by a Hamiltonian which is a linear combination of the super‐Hamiltonian and supermomentum constraints. We say that a dynamical variable K generates a conditional symmetry of the Hamiltonian when it is linear both in the hypersurface and the field momenta and its Poisson bracket with the Hamiltonian vanishes by virtue of the constraints. Generators are classified by their dependence on the momenta: P‐restricted generators depend only on the hypersurface momenta, π‐restricted generators depend only on the field momenta, while mixed generators depend on both kinds of momenta. Conditional symmetries in a parametrized Hamiltonian theory are then linked either with ordinary symmetries (isometries, conformal motions, or homothetic motions) of the spacetime background, or with internal symmetries of the fields. In particular, we prove that a generic field with nonderivative gravitational coupling and a quadratic energy...

Path integrals in parametrized theories: Newtonian systems

Constraints in dynamical systems typically arise either from gauge or from parametrization. We study Newtonian systems moving in curved configuration spaces and parametrize them by adjoining the absolute time and energy as conjugate canonical variables to the dynamical variables of the system. The extended canonical data are restricted by the Hamiltonian constraint. The action integral of the parametrized system is given in various extended spaces: Extended configuration space or phase space and with or without the lapse multiplier. The theory is written in a geometric form which is manifestly covariant under point transformations and reparametrizations. The quantum propagator of the system is represented by path integrals over different extended spaces. All path integrals are defined by a manifestly covariant skeletonization procedure. It is emphasized that path integrals for parametrized systems characteristically differ from those for gauge theories. Implications for the general theory of relativity ar...

On equivalence of parabolic and hyperbolic super‐Hamiltonians

Three types of super‐Hamiltonians occur in generally covariant field theories: linear in the momenta (hypersurface kinematics), parabolic in the momenta (parametrized field theories on a given Riemannian background), and hyperbolic in the momenta (geometrodynamics). Three simple models are discussed in which the linear or parabolic super‐Hamiltonian can be cast, essentially by a canonical transformation, into an equivalent hyperbolic form: (1) The scalar field propagating on a (1+1) ‐dimensional flat Minkowskian background, (2) hypersurface kinematics on a (1+n) ‐dimensional flat Minkowskian background, and (3) geometrodynamics of a (1+2) ‐dimensional vacuum spacetime. The implications for constraint quantization are mentioned.

Initial Value Problem Report of Workshop A3

The logical distinction between intensive and extensive definitions finds an amusing reflection in the rather different ways in which contributions are solicited for a scientific meeting and in which the outcome of the meeting is reported in conference proceedings. While the instructions to participants describe what they ought to talk about, the proceedings summarize what they actually did talk about. In this spirit, the present report is purely extensive. It is not intended to impose on the reader my ideas about what is the contemporary worldwide status of the initial value problem and Hamiltonian formulation of the general theory of relativity, and even less my ideas about what this status ought to be. It is simply a brief summary of what happened in the two afternoon sessions devoted to the workshop. Anyone interested in details should go to the original papers by the participants.