163,745 research outputs found
Conserved- and zero-mean quadratic quantities in oscillatory systems
We study quadratic functionals of the variables of a linear oscillatory system and their derivatives. We show that such functionals are partitioned in conserved quantities and in trivially- and intrinsic zero-mean quantities. We also state an equipartition of energy principle for oscillatory systems
Hyperbolic formulations of General Relativity with Hamiltonian structure
With the aim of deriving symmetric hyperbolic free-evolution systems for GR
that possess Hamiltonian structure and allow for the popular puncture gauge
condition we analyze the hyperbolicity of Hamiltonian systems. We develop
helpful tools which are applicable to either the first order in time, second
order in space or the fully second order form of the equations of motion. For
toy models we find that the Hamiltonian structure can simplify the proof of
symmetric hyperbolicity. In GR we use a special structure of the principal part
to prove symmetric hyperbolicity of a formulation that includes gauge
conditions which are very similar to the puncture gauge.Comment: Our mathematica scripts are available at
http://na.mathematik.uni-tuebingen.de/~richter
Completely Mixing Quantum Open Systems and Quantum Fractals
Departing from classical concepts of ergodic theory, formulated in terms of
probability densities, measures describing the chaotic behavior and the loss of
information in quantum open systems are proposed. As application we discuss the
chaotic outcomes of continuous measurement processes in the EEQT framework.
Simultaneous measurement of four noncommuting spin components is shown to lead
to a chaotic jump on quantum spin sphere and to generate specific fractal
images - nonlinear ifs (iterated function system). The model is purely
theoretical at this stage, and experimental confirmation of the chaotic
behavior of measuring instruments during simultaneous continuous measurement of
several noncommuting quantum observables would constitute a quantitative
verification of Event Enhanced Quantum Theory.Comment: Latex format, 20 pages, 6 figures in jpg format. New replacement has
two more references (including one to a paper by G. Casati et al on quantum
fractal eigenstates), adds example and comments concerning mixing properties
of of a two-level atom driven by a laser field, and also adds a number of
other remarks which should make it easier to follow mathematical argument
Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
A parameterization of the density operator, a coherence vector
representation, which uses a basis of orthogonal, traceless, Hermitian matrices
is discussed. Using this parameterization we find the region of permissible
vectors which represent a density operator. The inequalities which specify the
region are shown to involve the Casimir invariants of the group. In particular
cases, this allows the determination of degeneracies in the spectrum of the
operator. The identification of the Casimir invariants also provides a method
of constructing quantities which are invariant under {\it local} unitary
operations. Several examples are given which illustrate the constraints
provided by the positivity requirements and the utility of the coherence vector
parameterization.Comment: significantly rewritten and submitted for publicatio
Quantum Mechanics as a Simple Generalization of Classical Mechanics
A motivation is given for expressing classical mechanics in terms of diagonal
projection matrices and diagonal density matrices. Then quantum mechanics is
seen to be a simple generalization in which one replaces the diagonal real
matrices with suitable Hermitian matrices.Comment: 9 pages, LaTe
Localization and absence of Breit-Wigner form for Cauchy random band matrices
We analytically calculate the local density of states for Cauchy random band
matrices with strongly fluctuating diagonal elements. The Breit-Wigner form for
ordinary band matrices is replaced by a Levy distribution of index
and the characteristic energy scale is strongly enhanced as compared
to the Breit-Wigner width. The unperturbed eigenstates decay according to the
non-exponential law . We analytically determine
the localization length by a new method to derive the supersymmetric non-linear
model for this type of band matrices.Comment: 4 pages, 1 figur
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