Gravitational Constraint Combinations Generate a Lie Algebra

We find a first--order partial differential equation whose solutions are all ultralocal scalar combinations of gravitational constraints with Abelian Poisson brackets between themselves. This is a generalisation of the Kucha\v{r} idea of finding alternative constraints for canonical gravity. The new scalars may be used in place of the hamiltonian constraint of general relativity and, together with the usual momentum constraints, replace the Dirac algebra for pure gravity with a true Lie algebra: the semidirect product of the Abelian algebra of the new constraint combinations with the algebra of spatial diffeomorphisms.Comment: 10 pages, latex, submitted to Classical and Quantum Gravity. Section 3 is expanded and an additional solution provided, minor errors correcte

Action functionals of single scalar fields and arbitrary--weight gravitational constraints that generate a genuine Lie algebra

We discuss the issue initiated by Kucha\v{r} {\it et al}, of replacing the usual Hamiltonian constraint by alternative combinations of the gravitational constraints (scalar densities of arbitrary weight), whose Poisson brackets strongly vanish and cast the standard constraint-system for vacuum gravity into a form that generates a true Lie algebra. It is shown that any such combination---that satisfies certain reality conditions---may be derived from an action principle involving a single scalar field and a single Lagrange multiplier with a non--derivative coupling to gravity.Comment: 26 pages, plain TE

Mass Superselection, Canonical Gauge Transformations, and Asymptotically Flat Variational Principles

The phase space reduction of Schwarzschild black holes by Thiemann and Kastrup and by Kucha\v{r} is reexamined from a different perspective on gauge freedom. This perspective introduces additional gauge transformations which correspond to asymptotically nontrivial diffeomorphisms. Various subtleties concerning variational principles for asymptotically flat systems are addressed which allow us to avoid the usual conclusion that treating such transformations as gauge implies the vanishing of corresponding total charges. Instead, superselection rules are found for the (nonvanishing) ADM mass at the asymptotic boundaries. The addition of phenomenological clocks at each asymptotic boundary is also studied and compared with the `parametrization clocks' of Kucha\v{r}.Comment: 15 pages, ReVTeX, Minor changes made in response to referee's commment

Consistency of Semiclassical Gravity

We discuss some subtleties which arise in the semiclassical approximation to quantum gravity. We show that integrability conditions prevent the existence of Tomonaga-Schwinger time functions on the space of three-metrics but admit them on superspace. The concept of semiclassical time is carefully examined. We point out that central charges in the matter sector spoil the consistency of the semiclassical approximation unless the full quantum theory of gravity and matter is anomaly-free. We finally discuss consequences of these considerations for quantum field theory in flat spacetime, but with arbitrary foliations.Comment: 12 pages, LATEX, Report Freiburg THEP-94/2

The Dispersion of Newton's Constant: A Transfer Matrix Formulation of Quantum Gravity

A transfer matrix formalism applicable to certain reparametrization invariant theories, including quantum gravity, is proposed. In this formulation it is found that every stationary state in quantum gravity satisfies a Wheeler-DeWitt equation, but each with a different value of the Planck mass; the value $m_{Planck}^4$ turns out to be proportional to the eigenvalue of the evolution operator. As a consequence, the fact that the Universe is non-stationary implies that it is not in an eigenstate of Newton's constant.Comment: 24 pages, plain LaTeX, NBI-HE-93-5

Functional Evolution of Free Quantum Fields

We consider the problem of evolving a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation is computed; it does not correspond to a unitary transformation on the Fock space. This means that functional evolution of the quantum state as originally envisioned by Tomonaga, Schwinger, and Dirac is not a viable concept. Nevertheless, we demonstrate that functional evolution of the quantum state can be satisfactorily described using the formalism of algebraic quantum field theory. We discuss possible implications of our results for canonical quantum gravity.Comment: 21 pages, RevTeX, minor improvements in exposition, to appear in Classical and Quantum Gravit

Black hole solutions in 2+1 dimensions

We give circularly symmetric solutions for null fluid collapse in 2+1-dimensional Einstein gravity with a cosmological constant. The fluid pressure $P$ and energy density $\rho$ are related by $P=k\rho$ $(k\le 1)$. The long time limit of the solutions are black holes whose horizon structures depend on the value of $k$. The $k=1$ solution is the Banados-Teitelboim-Zanelli black hole metric in the long time static limit, while the $k<1$ solutions give other, `hairy' black hole metrics in this limit.Comment: 8 pages, RevTeX (to appear in Phys. Rev. D) References to Mann and Ross, and Mann, Chan and Chan adde

Canonical Gravity, Diffeomorphisms and Objective Histories

This paper discusses the implementation of diffeomorphism invariance in purely Hamiltonian formulations of General Relativity. We observe that, if a constrained Hamiltonian formulation derives from a manifestly covariant Lagrangian, the diffeomorphism invariance of the Lagrangian results in the following properties of the constrained Hamiltonian theory: the diffeomorphisms are generated by constraints on the phase space so that a) The algebra of the generators reflects the algebra of the diffeomorphism group. b) The Poisson brackets of the basic fields with the generators reflects the space-time transformation properties of these basic fields. This suggests that in a purely Hamiltonian approach the requirement of diffeomorphism invariance should be interpreted to include b) and not just a) as one might naively suppose. Giving up b) amounts to giving up objective histories, even at the classical level. This observation has implications for Loop Quantum Gravity which are spelled out in a companion paper. We also describe an analogy between canonical gravity and Relativistic particle dynamics to illustrate our main point.Comment: Latex 16 Pages, no figures, revised in the light of referees' comments, accepted for publication in Classical and Quantum Gravit

A nonlinear quantum model of the Friedmann universe

A discussion is given of the quantisation of a physical system with finite degrees of freedom subject to a Hamiltonian constraint by treating time as a constrained classical variable interacting with an unconstrained quantum state. This leads to a quantisation scheme that yields a Schrodinger-type equation which is in general nonlinear in evolution. Nevertheless it is compatible with a probabilistic interpretation of quantum mechanics and in particular the construction of a Hilbert space with a Euclidean norm is possible. The new scheme is applied to the quantisation of a Friedmann Universe with a massive scalar field whose dynamical behaviour is investigated numerically.Comment: 11 pages of text + 4 pages for 8 figure