Stability of Exponential Euler Method for Stochastic Systems under Poisson White Noise Excitations

The stability of stochastic systems under Poisson white noise excitations which based on the quantum theory is investigated in this paper. In general, the exact solution of the most of the stochastic systems with jumps is not easy to get. So it is very necessary to investigate the numerical solution of equations. On the one hand, exponential Euler method is applied to study stochastic delay differential equations, we can find the sufficient conditions for keeping mean square stability by investigating numerical method of systems. Through the comparison, we get the step-size of this method which is longer than the Euler-Maruyama method. On the other hand, mean square exponential stability of exponential Euler method for semi-linear stochastic delay differential equations under Poisson white noise excitations is confirmed.     

Realizations of Affine Weyl Group Symmetries on the Quantum Painlevé Equations by Fractional Calculus

We realize affine Weyl group symmetries on the Schr?dinger equations for the quantum Painlevé equations, by fractional calculus. This realization enables us to construct an infinite number of hypergeometric solutions to the Schr?dinger equations for the quantum Painlevé equations. In other words, since the Schr?dinger equations for the quantum Painlevé equations are equivalent to the Knizhnik?CZamolodchikov equations, we give one method of constructing hypergeometric solutions to the Knizhnik?CZamolodchikov equations.     

Parametric instabilities in single-walled carbon nanotubes

Parametric instabilities induced by the coupling excitation between the high frequency quantum Langmuir waves and the low frequency quantum ion-acoustic waves in single-walled carbon nanotubes are studied with a quantum Zakharov model. By linearizing the quantum hydrodynamic equations, we get the dispersion relations for the high frequency quantum Langmuir wave and the low frequency quantum ion-acoustic wave. Using two-time scale method, we obtain the quantum Zaharov model in the cylindrical coordinates. Decay instability and four-wave instability are discussed in detail. It is shown that the carbon nanotube＇s radius, the equilibrium discrete azimuthal quantum number, the perturbed discrete azimuthal quantum number, and the quantum parameter all play a crucial role in the instabilities.     

The time dependent Ginzburg-Landau equation in fractal space-time

We use the hydrodynamic formulation of Scale Relativity Theory to analyze the TDGL equation. As a result, London equations come naturally from the system, when equating to zero the real velocity, the imaginary one turns real, the superconducting fluid act as a subquantum medium energy accumulator, the vector potential, the real and the imaginary velocity are all written in terms of the elliptic function. When solving the resulted system by means of WKBJ method, we get tunneling and quantization. In other words, scale transformation laws produce, on the motion equation of particles governed by the TDGL equation, under some peculiar assumptions, effects which are analogous to those of a “macroscopic quantum mechanics”.     

Mathematisches zur Theorie der Linienbreite

In the present paper we consider the emission of light quanta by an atom using the methods of non-relativistic quantum theory. The differential equations occurring there are solved using a Laplace transformation. By means of an analytic continuation and a displacement of the integration path we get exact solutions whose approximations lead to the usual formulae for the intensity distribution and for the natural line width.     

Finslerian Multi-Dimensionality,Associated Gauge Fields,and Stochastic Space-Time

The theory is constructed which elucidates all the gauge fields associated with fibration of the tangent vectors and of the vectors of higher degrees of tangency. The approach synthesizes intrinsically the Einstein equations with the Yang-Mills equations and gives an adequate framework for extending the physical field equations. Then, by considering the statistical behaviour of internal variables, it is possible to get a deeper insight in understanding the origin of quantum laws. Actually, we construct what may be called the general relativity in vector fibrations over space-time manifold.     

Electromagnetic and Force Field Equations

In classical physics the electromagnetic equations are described by Maxwell's equations. Maxwell's equations proved to be invariant under gauge, or Lorentz transformations. Also, Einstein's equations of the special theory of relativity are invariant under Lorentz transformations. On the other hand classical mechanics and quantum mechanics laws are invariant under Galilean transformations. This means that, there are two different dynamical structures describing our universe. Einstein's unified field theory failled in putting our universe in one dynamical structure. New electromagnetic and force field equations are going to be derived. They have the same shape like Maxwell's equations, but with different dynamical structure. Those equations are invariant under Galilean transformations and in the density matrix formalism of quantum mechanics.     

A probabilistic analysis of the difficulties of unifying quantum mechanics with the theory of relativity

A procedure is given for the transformation of quantum mechanical operator equations into stochastic equations. The stochastic equations reveal a simple correlation between quantum mechanics and classical mechanics: Quantum mechanics operates with “optimal estimations,” classical mechanics is the limit of “complete information.” In this connection, Schrödinger's substitution relationsp _x → -i? ?/?x, etc, reveal themselves as exact mathematical transformation formulas. The stochastic version of quantum mechanical equations provides an explanation for the difficulties in correlating quantum mechanics and the theory of relativity: In physics “time” is always thought of as a numerical parameter; but in the present formalism of physics “time” is described by two formally totally different quantities. One of these two “times” is a numerical parameter and the other a random variable. This last concept of time shows all the properties required by the theory of relativity and is therefore to be considered as the relativistic time.     

Solutions of the factorization equations from Toda field theory

We apply the quantum inverse scattering method to a discretization of the generalized Toda field theory. In this way we get a family of solutions of the factorization equations. These solutions provide (i) a solvable many particle system with different types of particles interacting via non-relativistic complex potentials; (ii) a presumably solvable generalization of the six-vertex model in a planar lattice.     

The Cryptanalysis of Yuan et al.’s Multiparty Quantum Secret Sharing Protocol

Recently, Yuan et al. summarized some previous analysis of quantum secret sharing task and designed a multiparty quantum secret sharing protocol based on the continuous variable operations (CVO) with the ideas of dense coding and ping-pong technique (YMQSS). However, our research shows that the YMQSS protocol is unable to complete the quantum secret sharing task securely if the particular participant, Bob₀, is dishonest. In order to show that, we describe the following two attack strategies: one is that Bob₀ can get the accurate shared secrets himself and leave the other sharers get his forged ones; the other one is that Bob₀ can conspire with another sharer Bob _i (i≠0) to get Alice’s secrets without anyone’s help. Finally, our discussions and conclusions are proposed.     

有限温度下一维Gaudin-Yang模型的热力学性质

本文通过数值求解有限温度下一维均匀费米Gaudin-Yang模型的热力学Bethe-ansatz方程, 研究了此模型的基本性质,得到了在给定的温度或给定的相互作用下, 化学势、相互作用、粒子密度和熵的相互变化图像. 对结果分析发现, 在给定温度和相互作用下, 熵随着化学势的变化有一个量子临界区域.     

Landau problem in noncommutative quantum mechanics

The Landau problem in non-commutative quantum mechanics (NCQM) is studied. First by solving the Schrodinger equations on noncommutative (NC) space we obtain the Landau energy levels and the energy correction that is caused by space-space noncommutativity. Then we discuss the noncommutative phase space case, namely, space-space and momentum-momentum non-commutative case, and we get the explicit expression of the Hamiltonian as well as the corresponding eigenfunctions and eigenvalues.     

Backreaction in acoustic black holes

The backreaction equations for the linearized quantum fluctuations in an acoustic black hole are given. The solution near the horizon, obtained within a dimensional reduction, indicates that acoustic black holes, unlike Schwarzschild ones, get cooler as they radiate phonons. They show remarkable analogies with near-extremal Reissner-Nordstr?m black holes.     

A unified approach to quantum and classical TTW systems based on factorizations

A unifying method based on factorization properties is introduced for finding symmetries of quantum and classical superintegrable systems using the example of the Tremblay–Turbiner–Winternitz (TTW) model. It is shown that the symmetries of the quantum system can be implemented in a natural way to its classical version. Besides, by this procedure we get also other type of constants of motion depending explicitly on time that allow to find directly the motion of the system whose corresponding trajectories coincide with those obtained previously by using its symmetries.     

Distillation protocols for mixed states of multilevel qubits and the quantum renormalization group

We study several properties of distillation protocols to purify multilevel qubit states (qudits) when applied to a certain family of initial mixed bipartite states. We find that it is possible to use qudits states to increase the stability region obtained with the flow equations to distill qubits. In particular, for qutrits we get the phase diagram of the distillation process with a rich structure of fixed points. We investigate the large-D limit of qudits protocols and find an analytical solution in the continuum limit. The general solution of the distillation recursion relations is presented in an appendix. We stress the notion of weight amplification for distillation protocols as opposed to the quantum amplitude amplification that appears in the Grover algorithm. Likewise, we investigate the relations between quantum distillation and quantum renormalization processes.Received: 23 April 2003, Published online: 12 August 2003PACS: 03.67.-a Quantum information - 03.67.Lx Quantum computation     

Quantum Protocol for Millionaire Problem

A quantum protocol for millionaire problem based on commutative encryption is proposed. In our protocol, Alice and Bob don’t have to use the entangled character, joint measurement of quantum states. They encrypt their private information and privately get the result of their private information with the help of a third party (TP). Correctness analysis shows that the proposed protocol can be used to get the result of their private information correctly. The proposed protocol can also resist various attacks and overcomes the problem of information leakage with acceptable efficiency. In theory, our protocol can be used to build complex secure protocols for other multiparty computation problems and also have lots of other important applications in distributed networks.

     

Landau problem in noncommutative quantum mechanics

The Landau problem in non-commutative quantum mechanics (NCQM) is studied.First by solving the Schr(o)dinger equations on noncommutative (NC) space we obtain the Landau energy levels and the energy correction that is caused by space-space noncommutativity.Then we discuss the noncommutative phase space case,namely,space-space and momentum-momentum non-commutative case,and we get the explicit expression of the Hamfltonian as well as the corresponding eigenfunctions and eigenvalues.     

Quantized Knizhnik-Zamolodchikov equations,quantum Yang-Baxter equation,and difference equations forq-hypergeometric functions

Thes₂ quantized Knizhnik-Zamolodchikov equations are solved inq-hypergeometric functions. New difference equations are derived for generalq-hypergeometric functions. The equations are given in terms of quantum Yang-Baxter matrices and have the form similar to quantum Knizhnik-Zamolodchikov equations for quantum affine algebras introduced by Frenkel and Reshetikhin.This work was supported by NSF grant DMS-9203929.     

Chiral invariant Gross-Neveu model: Classical versus quantum

We present a unification of different and independently investigated aspects of the chiral invariant Gross-Neveu model. Special emphasis is placed on the relevance of classical (c-number, non Grassmann) spinor solutions of the G-N field equations for the construction, and thus understanding of the respective quantized Fermi model. To get an insight into the “quantum meaning of classical field theory” if specialized to the G-N case, we perform the path integral quantization procedure which first leads to the Fermi oscillator problem, and then, after appropriate generalizations, to the quantum Fermi G-N model. Path integrals are carried out with respect to c-number spinor paths only, and in fact no reference is necessary to the Grassmann algebra methods, which are conventionally used to integrate out fermions.