4,533 research outputs found
Hyper-relativistic mechanics and superluminal particles
Recent experiments by OPERA with high energy neutrinos, as well as
astrophysics observation data, may possibly prove violations of underlying
principles of special relativity theory. This paper attempts to present an
elementary modification of relativistic mechanics that is consistent both with
the principles of mechanics and with Dirac's approach to derivation of
relativistic quantum equations. Our proposed hyper-relativistic model is based
on modified dispersion relations between energy and momentum of a particle.
Predictions of the new theory significantly differ from the standard model, as
the former implies large Lorentz gamma-factors (ratio of particle energy to its
mass). First of all, we study model relationships that describe hypothetical
motion of superluminal neutrinos. Next, we analyze characteristics of Cherenkov
radiation of photons and non-zero mass particles in vacuum. Afterwards, we
derive generalized Lorentz transformations for a hyper-relativistic case,
resulting in a radical change in the law of composition of velocities and
particle kinematics. Finally, we study a hyper-relativistic version of Dirac
equation and some of its properties. In present paper we attempted to use plain
language to make it accessible not only to scientists but to undergraduate
students as well.Comment: 30 pages, 7 figure
Quantum Mechanical View of Mathematical Statistics
Multiparametric statistical model providing stable reconstruction of
parameters by observations is considered. The only general method of this kind
is the root model based on the representation of the probability density as a
squared absolute value of a certain function, which is referred to as a psi
function in analogy with quantum mechanics. The psi function is represented by
an expansion in terms of an orthonormal set of functions. It is shown that the
introduction of the psi function allows one to represent the Fisher information
matrix as well as statistical properties of the estimator of the state vector
(state estimator) in simple analytical forms.Comment: OAO "Angstrem", Moscow, Russia 26 pages, 2 figur
Number of vertices in graphs with locally small chromatic number and large chromatic number
We discuss the minimal number of vertices in a graph with a large chromatic
number such that each ball of a fixed radius in it has a small chromatic
number. It is shown that for every graph on
vertices such that each ball of radius is properly -colorable, we have
Statistical Inverse Problem: Root Approach
Multiparametric statistical model providing stable reconstruction of
parameters by observations is considered. The only general method of this kind
is the root model based on the representation of the probability density as a
squared absolute value of a certain function, which is referred to as a
psi-function in analogy with quantum mechanics. The psi-function is represented
by an expansion in terms of an orthonormal set of functions. It is shown that
the introduction of the psi-function allows one to represent the Fisher
information matrix as well as statistical properties of the estimator of the
state vector (state estimator) in simple analytical forms. The chi-square test
is considered to test the hypotheses that the estimated vector converges to the
state vector of a general population. The method proposed may be applied to its
full extent to solve the statistical inverse problem of quantum mechanics (root
estimator of quantum states). In order to provide statistical completeness of
the analysis, it is necessary to perform measurements in mutually complementing
experiments (according to the Bohr terminology). The maximum likelihood
technique and likelihood equation are generalized in order to analyze quantum
mechanical experiments. It is shown that the requirement for the expansion to
be of a root kind can be considered as a quantization condition making it
possible to choose systems described by quantum mechanics from all statistical
models consistent, on average, with the laws of classical mechanics.Comment: 17 pages, 3 figures, 2nd Asia-Pacific Workshop on Quantum Information
Science, Singapore, National University of Singapore, 15-19 December 200
Examples of topologically highly chromatic graphs with locally small chromatic number
Kierstead, Szemer\'edi, and Trotter showed that a graph with at most vertices such that each ball of radius in it is
-colorable should have chromatic number at most . We show that
this estimate is sharp in . Namely, for every , , and we construct
a graph containing vertices such that , although each ball of radius in is -colorable. The core
idea is the construction of a graph whose neighborhood complex is homotopy
equivalent to the join of neighborhood complexes of two given graphs.Comment: 2 figure
Schmidt information and entanglement in quantum systems
The purpose of this paper is to study entanglement of quantum states by means
of Schmidt decomposition. The notion of Schmidt information which characterizes
the non-randomness of correlations between two observers that conduct
measurements of EPR-states is proposed. In two important particular cases - a
finite number of Schmidt modes with equal probabilities and Gaussian
correlations- Schmidt information is equal to Shannon information. A universal
measure of a dependence of two variables is proposed. It is based on Schmidt
number and it generalizes the classical Pearson correlation coefficient. It is
demonstrated that the analytical model obtained can be applied to testing the
numerical algorithm of Schmidt modes extraction. A thermodynamic interpretation
of Schmidt information is given. It describes the level of entanglement and
correlations of micro-system with its environmentComment: 9 pages, 1 figur
Analysis of localized Schmidt decomposition modes and of entanglement in atomic and optical quantum systems with continuous variables
We investigate the procedure of Schmidt modes extraction in systems with
continuous variables. An algorithm based on singular value matrix decomposition
is applied to the study of entanglement in an "atom-photon" system with
spontaneous radiation. Also, this algorithm is applied to the study of a
bi-photon system with spontaneous parametric down conversion with type-II phase
matching for broadband pump. We demonstrate that dynamic properties of
entangled states in an atom-photon system with spontaneous radiation are
defined by a parameter equal to the product of the fine structure constant and
the atom-electron mass ratio. We then consider the evolution of the system
during radiation and show that the atomic and photonic degrees of freedom are
entangling for the times of the same order of magnitude as the excited state
life-time. Then the degrees of freedom are de-entangling and asymptotically
approach to the level of small residual entanglement that is caused by momentum
dispersion of the initial atomic packet.Finally, we investigate the process of
coherence loss between modes in type-II parametric down conversion that is
caused by non-linear crystal properties.Comment: 20 pages, 6 figure
Finding a subset of nonnegative vectors with a coordinatewise large sum
Given a rational and nonnegative -dimensional real vectors
, ..., , we show that it is always possible to choose of them such that their sum is (componentwise) at least
. For fixed and , this bound is sharp if is
large enough. The method of the proof uses Carath\'eodory's theorem from linear
programming
Quantum Informatics View of Statistical Data Processing
Application of root density estimator to problems of statistical data
analysis is demonstrated. Four sets of basis functions based on
Chebyshev-Hermite, Laguerre, Kravchuk and Charlier polynomials are considered.
The sets may be used for numerical analysis in problems of reconstructing
statistical distributions by experimental data. Examples of numerical modeling
are given
Simple Modules of Exceptional Groups with Normal Closures of Maximal Torus Orbits
Let G be an exceptional simple algebraic group, and let T be a maximal torus
in G. In this paper, for every such G, we find all simple rational G-modules V
with the following property: for every vector v in V, the closure of its
T-orbit is a normal affine variety. For all G-modules without this property we
present a T-orbit with the non-normal closure. To solve this problem, we use a
combinatorial criterion of normality formulated in the terms of weights of a
simple G-module. This paper continues two papers of the second author, where
the same problem was solved for classical linear groups
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