38,782 research outputs found
General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space
In our previous work, we proposed a mathematical framework for PT-symmetric
quantum theory, and in particular constructed a Krein space in which
PT-symmetric operators would naturally act. In this work, we explore and
discuss various general consequences and aspects of the theory defined in the
Krein space, not only spectral property and PT symmetry breaking but also
several issues, crucial for the theory to be physically acceptable, such as
time evolution of state vectors, probability interpretation, uncertainty
relation, classical-quantum correspondence, completeness, existence of a basis,
and so on. In particular, we show that for a given real classical system we can
always construct the corresponding PT-symmetric quantum system, which indicates
that PT-symmetric theory in the Krein space is another quantization scheme
rather than a generalization of the traditional Hermitian one in the Hilbert
space. We propose a postulate for an operator to be a physical observable in
the framework.Comment: 32 pages, no figures; explanation, discussion and references adde
Cosmic Censorship for Some Spatially Homogeneous Cosmological Models
The global properties of spatially homogeneous cosmological models with
collisionless matter are studied. It is shown that as long as the mean
curvature of the hypersurfaces of homogeneity remains finite no singularity can
occur in finite proper time as measured by observers whose worldlines are
orthogonal to these hypersurfaces. Strong cosmic censorship is then proved for
the Bianchi I, Bianchi IX and Kantowski-Sachs symmetry classes.Comment: 14 pages, Plain TeX, MPA-AR-93-
Post-Newtonian Freely Specifiable Initial Data for Binary Black Holes in Numerical Relativity
Construction of astrophysically realistic initial data remains a central
problem when modelling the merger and eventual coalescence of binary black
holes in numerical relativity. The objective of this paper is to provide
astrophysically realistic freely specifiable initial data for binary black hole
systems in numerical relativity, which are in agreement with post-Newtonian
results. Following the approach taken by Blanchet, we propose a particular
solution to the time-asymmetric constraint equations, which represent a system
of two moving black holes, in the form of the standard conformal decomposition
of the spatial metric and the extrinsic curvature. The solution for the spatial
metric is given in symmetric tracefree form, as well as in Dirac coordinates.
We show that the solution differs from the usual post-Newtonian metric up to
the 2PN order by a coordinate transformation. In addition, the solutions,
defined at every point of space, differ at second post-Newtonian order from the
exact, conformally flat, Bowen-York solution of the constraints.Comment: 41 pages, no figures, accepted for publication in Phys. Rev. D,
significant revision in presentation (including added references and
corrected typos
The Einstein-Vlasov system/Kinetic theory
The main purpose of this article is to provide a guide to theorems on global
properties of solutions to the Einstein-Vlasov system. This system couples
Einstein's equations to a kinetic matter model. Kinetic theory has been an
important field of research during several decades in which the main focus has
been on nonrelativistic and special relativistic physics, {\it i.e.} to model
the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems.
In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov
system. Since then many theorems on global properties of solutions to this
system have been established. The Vlasov equation describes matter
phenomenologically and it should be stressed that most of the theorems
presented in this article are not presently known for other such matter models
({\it i.e.} fluid models). This paper gives introductions to kinetic theory in
non-curved spacetimes and then the Einstein-Vlasov system is introduced. We
believe that a good understanding of kinetic theory in non-curved spacetimes is
fundamental to good comprehension of kinetic theory in general relativity.Comment: 40 pages, updated version, to appear in Living Reviews in Relativit
A note on self-adjoint extensions of the Laplacian on weighted graphs
We study the uniqueness of self-adjoint and Markovian extensions of the
Laplacian on weighted graphs. We first show that, for locally finite graphs and
a certain family of metrics, completeness of the graph implies uniqueness of
these extensions. Moreover, in the case when the graph is not metrically
complete and the Cauchy boundary has finite capacity, we characterize the
uniqueness of the Markovian extensions.Comment: 17 pages. The assumption of "finite jump size" found in Theorems 1
and 2 in the previous version has been replaced by a weaker condition
concerning the newly introduced notion of a "combinatorial neighborhood" in
Theorem 1 and has been removed altogether from Theorem 2. Some references
added. Final version to appear in J. Funct. Ana
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