726 research outputs found
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
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 and energy density are related by . The
long time limit of the solutions are black holes whose horizon structures
depend on the value of . The solution is the
Banados-Teitelboim-Zanelli black hole metric in the long time static limit,
while the 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
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