504 research outputs found
The Canonical Approach to Quantum Gravity: General Ideas and Geometrodynamics
We give an introduction to the canonical formalism of Einstein's theory of
general relativity. This then serves as the starting point for one approach to
quantum gravity called quantum geometrodynamics. The main features and
applications of this approach are briefly summarized.Comment: 21 pages, 6 figures. Contribution to E. Seiler and I.-O. Stamatescu
(editors): `Approaches To Fundamental Physics -- An Assessment Of Current
Theoretical Ideas' (Springer Verlag, to appear
de Sitter symmetry of Neveu-Schwarz spinors
We study the relations between Dirac fields living on the 2-dimensional
Lorentzian cylinder and the ones living on the double-covering of the
2-dimensional de Sitter manifold, here identified as a certain coset space of
the group . We show that there is an extended notion of de Sitter
covariance only for Dirac fields having the Neveu-Schwarz anti-periodicity and
construct the relevant cocycle. Finally, we show that the de Sitter symmetry is
naturally inherited by the Neveu-Schwarz massless Dirac field on the cylinder.Comment: 24 page
On the "renormalization" transformations induced by cycles of expansion and contraction in causal set cosmology
We study the ``renormalization group action'' induced by cycles of cosmic
expansion and contraction, within the context of a family of stochastic
dynamical laws for causal sets derived earlier. We find a line of fixed points
corresponding to the dynamics of transitive percolation, and we prove that
there exist no other fixed points and no cycles of length two or more. We also
identify an extensive ``basin of attraction'' of the fixed points but find that
it does not exhaust the full parameter space. Nevertheless, we conjecture that
every trajectory is drawn toward the fixed point set in a suitably weakened
sense.Comment: 22 pages, 1 firgure, submitted to Phys. Rev.
A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
This paper is the first in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper we discuss two different types of language that can be
attached to a system, S. The first is a propositional language, PL(S); the
second is a higher-order, typed language L(S). Both languages provide deductive
systems with an intuitionistic logic. The reason for introducing PL(S) is that,
as shown in paper II of the series, it is the easiest way of understanding, and
expanding on, the earlier work on topos theory and quantum physics. However,
the main thrust of our programme utilises the more powerful language L(S) and
its representation in an appropriate topos.Comment: 36 pages, no figure
Quantum superposition principle and gravitational collapse: Scattering times for spherical shells
A quantum theory of spherically symmetric thin shells of null dust and their
gravitational field is studied. In Nucl. Phys. 603 (2001) 515 (hep-th/0007005),
it has been shown how superpositions of quantum states with different
geometries can lead to a solution of the singularity problem and black hole
information paradox: the shells bounce and re-expand and the evolution is
unitary. The corresponding scattering times will be defined in the present
paper. To this aim, a spherical mirror of radius R_m is introduced. The
classical formula for scattering times of the shell reflected from the mirror
is extended to quantum theory. The scattering times and their spreads are
calculated. They have a regular limit for R_m\to 0 and they reveal a resonance
at E_m = c^4R_m/2G. Except for the resonance, they are roughly of the order of
the time the light needs to cross the flat space distance between the observer
and the mirror. Some ideas are discussed of how the construction of the quantum
theory could be changed so that the scattering times become considerably
longer.Comment: 30 pages and 5 figures; the post-referee version: shortened and some
formulations improved; to be published in Physical Revie
Unification of the conditional probability and semiclassical interpretations for the problem of time in quantum theory
We show that the time-dependent Schr\"odinger equation (TDSE) is the
phenomenological dynamical law of evolution unraveled in the classical limit
from a timeless formulation in terms of probability amplitudes conditioned by
the values of suitably chosen internal clock variables, thereby unifying the
conditional probability interpretation (CPI) and the semiclassical approach for
the problem of time in quantum theory. Our formalism stems from an exact
factorization of the Hamiltonian eigenfunction of the clock plus system
composite, where the clock and system factors play the role of marginal and
conditional probability amplitudes, respectively. Application of the Variation
Principle leads to a pair of exact coupled pseudoeigenvalue equations for these
amplitudes, whose solution requires an iterative self-consistent procedure. The
equation for the conditional amplitude constitutes an effective "equation of
motion" for the quantum state of the system with respect to the clock
variables. These coupled equations also provide a convenient framework for
treating the back-reaction of the system on the clock at various levels of
approximation. At the lowest level, when the WKB approximation for the marginal
amplitude is appropriate, in the classical limit of the clock variables the
TDSE for the system emerges as a matter of course from the conditional
equation. In this connection, we provide a discussion of the characteristics
required by physical systems to serve as good clocks. This development is seen
to be advantageous over the original CPI and semiclassical approach since it
maintains the essence of the conventional formalism of quantum mechanics,
admits a transparent interpretation, avoids the use of the Born-Oppenheimer
approximation, and resolves various objections raised about them.Comment: 10 pages. Typographical errors correcte
Interacting classical and quantum ensembles
A consistent description of interactions between classical and quantum
systems is relevant to quantum measurement theory, and to calculations in
quantum chemistry and quantum gravity. A solution is offered here to this
longstanding problem, based on a universally-applicable formalism for ensembles
on configuration space. This approach overcomes difficulties arising in
previous attempts, and in particular allows for backreaction on the classical
ensemble, conservation of probability and energy, and the correct classical
equations of motion in the limit of no interaction. Applications include
automatic decoherence for quantum ensembles interacting with classical
measurement apparatuses; a generalisation of coherent states to hybrid harmonic
oscillators; and an equation for describing the interaction of quantum matter
fields with classical gravity, that implies the radius of a Robertson-Walker
universe with a quantum massive scalar field can be sharply defined only for
particular `quantized' values.Comment: 31 pages, minor clarifications and one Ref. added, to appear in PR
Embedding variables in finite dimensional models
Global problems associated with the transformation from the Arnowitt, Deser
and Misner (ADM) to the Kucha\v{r} variables are studied. Two models are
considered: The Friedmann cosmology with scalar matter and the torus sector of
the 2+1 gravity. For the Friedmann model, the transformations to the Kucha\v{r}
description corresponding to three different popular time coordinates are shown
to exist on the whole ADM phase space, which becomes a proper subset of the
Kucha\v{r} phase spaces. The 2+1 gravity model is shown to admit a description
by embedding variables everywhere, even at the points with additional symmetry.
The transformation from the Kucha\v{r} to the ADM description is, however,
many-to-one there, and so the two descriptions are inequivalent for this model,
too. The most interesting result is that the new constraint surface is free
from the conical singularity and the new dynamical equations are linearization
stable. However, some residual pathology persists in the Kucha\v{r}
description.Comment: Latex 2e, 29 pages, no figure
Bohmian trajectories and Klein's paradox
We compute the Bohmian trajectories of the incoming scattering plane waves
for Klein's potential step in explicit form. For finite norm incoming
scattering solutions we derive their asymptotic space-time localization and we
compute some Bohmian trajectories numerically. The paradox, which appears in
the traditional treatments of the problem based on the outgoing scattering
asymptotics, is absent.Comment: 14 pages, 3 figures; minor format change
The issue of time in generally covariant theories and the Komar-Bergmann approach to observables in general relativity
Diffeomorphism-induced symmetry transformations and time evolution are
distinct operations in generally covariant theories formulated in phase space.
Time is not frozen. Diffeomorphism invariants are consequently not necessarily
constants of the motion. Time-dependent invariants arise through the choice of
an intrinsic time, or equivalently through the imposition of time-dependent
gauge fixation conditions. One example of such a time-dependent gauge fixing is
the Komar-Bergmann use of Weyl curvature scalars in general relativity. An
analogous gauge fixing is also imposed for the relativistic free particle and
the resulting complete set time-dependent invariants for this exactly solvable
model are displayed. In contrast with the free particle case, we show that
gauge invariants that are simultaneously constants of motion cannot exist in
general relativity. They vary with intrinsic time
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