Mode decomposition and unitarity in quantum cosmology, Talk given at the Second Meeting on Constrained Dynamics and Quantum gravity, Santa Margherita Ligure, September 17-21, 1996

Contrary to common belief, there are perspectives for generalizing the notion of positive and negative frequency in minisuperspace quantum cosmology, even when the wave equation does not admit symmetries. We outline a strategy in doing so when the potential is positive. Also, an underlying unitarity structure shows up. Starting in the framework of the Klein-Gordon type quantization, I am led to a result that relies on global features on the model, and that is possibly related to structures encountered in the refined algebraic quantization scheme.Comment: 5 pages, LaTeX (no figures

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

Operator ordering for generally covariant systems

The constraint operators belonging to a generally covariant system are found out within the framework of the BRST formalism. The result embraces quadratic Hamiltonian constraints whose potential can be factorized as a never null function times a gauge invariant function. The building of the inner product between physical states is analyzed for systems featuring either intrinsic or extrinsic time.Comment: 4 pages. Talk given at the Third Conference on "Constrained Dynamics and Quantum Gravity" held in Sardinia (Italy), September 1999. Journal reference:Nucl. Phys. B (Proc. Suppl.) 88 (2000) 322-32

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

What simplified models say about unitarity and gravitational collapse

This paper is an extended version of a talk at the conference Constrained Dynamics and Quantum Gravity QG99. It reviews some work on the quantum collapse of the spherically symmetric gravitating thin shell of zero rest mass. Recent results on Kucha\v{r} decomposition are applied. The constructed version of quantum mechanics is unitary, although the shell falls under its Schwarzschild radius if its energy is high enough. Rather that a permanent black hole, something like a transient black and white hole pair seems to be created in such a case.Comment: 17 pages, uses amstex, no figure

Ehrenfest's Principle and the Problem of Time in Quantum Gravity

We elaborate on a proposal made by Greensite and others to solve the problem of time in quantum gravity. The proposal states that a viable concept of time and a sensible inner product can be found from the demand for the Ehrenfest equations to hold in quantum gravity. We derive and discuss in detail exact consistency conditions from both Ehrenfest equations as well as from the semiclassical approximation. We also discuss consistency conditions arising from the full field theory. We find that only a very restricted class of solutions to the Wheeler-DeWitt equation fulfills all consistency conditions. We conclude that therefore this proposal must either be abandoned as a means to solve the problem of time or, alternatively, be used as an additional boundary condition to select physical solutions from the Wheeler-DeWitt equation.Comment: 20 pages, LATE

Generalized Stochastic Gauge Fixing

We propose a generalization of the stochastic gauge fixing procedure for the stochastic quantization of gauge theories where not only the drift term of the stochastic process is changed but also the Wiener process itself. All gauge invariant expectation values remain unchanged. As an explicit example we study the case of an abelian gauge field coupled with three bosonic matter fields in 0+1 dimensions. We nonperturbatively prove quivalence with the path integral formalism.Comment: 6 pages, latex, no figure

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