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 $G$ on $\sim((n+rc)/(c+rc))^{r+1}$ vertices such that each ball of radius $r$ is properly $c$-colorable, we have $\chi(G)\leq n$

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 $\lfloor r/(2n)\rfloor^n$ vertices such that each ball of radius $r$ in it is $c$-colorable should have chromatic number at most $n(c-1)+1$. We show that this estimate is sharp in $r$. Namely, for every $n$, $r$, and $c$ we construct a graph $G$ containing $O((2rc)^{n-1}c)$ vertices such that $\chi(G)\geq n(c-1)+1$, although each ball of radius $r$ in $G$ is $c$-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 $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least $(p/q)(u_1+...+u_N)$. For fixed $d$ and $a$, this bound is sharp if $N$ 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