Further studies in aesthetic field theory. II: Equations fore αi

In a previous paper (Muraskin, 1973), we obtained a bounded particle in aesthetic field theory. The field equations there are implied by a set of equations for a system of basis vector variables,e _i . In this paper, we propose a simpler set of field equations fore _i . We find that a bounded particle solution to the equations still appears (as determined by axes runs). The particle appears basically similar to the particle found previously.     

Links between microscopic and macroscopic fluid mechanics

The microscopic and macroscopic versions of fluid mechanics differ qualitatively. Microscopic particles obey time-reversible ordinary differential equations. The resulting particle trajectories {q(t)} may be time-averaged or ensemble-averaged so as to generate field quantities corresponding to macroscopic variables. On the other hand, the macroscopic continuum fields described by fluid mechanics follow irreversible partial differential equations. Smooth particle methods bridge the gap separating these two views of fluids by solving the macroscopic field equations with particle dynamics that resemble molecular dynamics. Recently, nonlinear dynamics have provided some useful tools for understanding the relationship between the microscopic and macroscopic points of view. Chaos and fractals play key roles in this new understanding. Non-equilibrium phase-space averages look very different from their equilibrium counterparts. Away from equilibrium the smooth phase-space distributions are replaced by fractional-dimensional singular distributions that exhibit time irreversibility.     

Relativistic dynamics of stochastic particles

Particle motion in stochastic space, i.e., space whose coordinates consist of small, regular stochastic parts, is considered. A free particle in this space resembles a Brownian particle the motion of which is characterized by a dispersionD dependent on the universal length l. It is shown that in the first approximation in the parameter l the particle motion in an external force field is described by equations coincident in form with equations of stochastic mechanics due to Nelson, Kershow, and de la Pena-Auerbach. A method is proposed for the relativization of the scheme used to describe the processes in the stochastic space; by using this method, the equations of particle motion can be written in a covariant form.     

Field theory onR×S 3 topology. V:SU 2 gauge theory

A gauge theory on R×S ³ topology is developed. It is a generalization to the previously obtained field theory on R×S ³ topology and in which equations of motion were obtained for a scalar particle, a spin one-half particle, the electromagnetic field of magnetic moments, and a Shrödinger-type equation, as compared to ordinary field equations defined on a Minkowskian manifold. The new gauge field equations are presented and compared to the ordinary Yang-Mills field equations, and the mathematical and physical differences between them are discussed.On leave from Center for Theoretical Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel.     

Schrödinger equations for su q (2)-invariant quantum systems

By deforming the Hamiltonian of a spinless particle in a central potential we set up su _q (2)-invariant Schrödinger equations within the usual framework of quantum mechanics. Different deformations correspond to a given Hamiltonian. We explicitly solve different stationary Schrödinger equations for the free particle and for the hydrogen atom, and compare the associated energy spectra.     

A simple model of the classicalZitterbewegung: Photon wave function

We propose a simple classical model of the zitterbewegung. In this model spin is proportional to the velocity of the particle, the component parallel top is constant and the orthogonal components are oscillating with2p frequency. The quantization of the system gives wave equations for spin,0, 1/2, 1, 3/2,…, etc. respectively. These equations are convenient for massless particles. The wave equation of the spin-1, massless free particle is equivalent to the Maxwell equations and the state functions have a probability interpretation and exhibit conserved current densities. The ground state has zero energy.     

On the Renormalization Group Method for the Problem of Random Walk without Self-Intersections

Random walk without self-intersections for a particle in the Euclidean n-dimensional space is investigated. Renormalization group equations for the probability density of the end-to-end distance of a particle trajectory are derived. The asymptotics for this probability density are defined for 4 - n ? 1.     

A classical Proca particle

An elementary particle is described as a spherically symmetric solution of the Proca equations and the Einstein general relativity equations. The mass is found to be of the order of the Planck mass. If the motion of its center of mass is determined by the Dirac equations, it has a spin 1/2.This work is parallel to an earlier one involving the Klein- Gordon equation.     

Three-body formalism for deuteron stripping reactions

A three-body formalism for deuteron stripping reactions has been developed. The equations of Altet al (1967) (AGS) for the three particle system (target A, n, p) are reduced to a set of coupled one-dimensional integral equations with the use of (i) angular momentum basis for representation and (ii) separable approximation for the two bodyt-matrices (which delineate the interactions between the particle pairs). The on-shell solutions of this set of integral equations are then related to the cross sections of the rearrangement processes. The inputs in this calculation, viz., the separable interactions between the particle pairs in the respective channels are simply constructed from the respective two body bound state in accordance with the bound state approximation (BSA) conforming to the ‘unitarity’ requirement. Using this formalism preliminary calculations for the (d, p) and (d, n) reaction cross sections on¹⁶O have been carried out and they seem to have considerable semblance with the observed cross sections.     

TheN/D method off the mass shell for the Klein-Gordon equation and the inversion problem

The relativistic analogon of a procedure demonstrating the link between theS-wave off-the-mass- shellN/D equations (variables: the momentum, energy and radial coordinate of a scattered particle) and the Marchenko equations of the inversion problem is presented in the static scattering. For the Klein-Gordon formalism the transition from the former type of the equations to the latter requires: a decomposition of theN/D equation quantities into the components without theE-branch points, a suitable deformation of the integration path in theN/D equations and an assumption on the regular behaviour of the off-the-mass-shellN function discontinuities.     

Gauge theory of gravitation: (4+N)-dimensional theory

(4+N)-dimensional theory is studied using the method of differential geometry. The invariant line element is uniquely determined by the connection one-form which is invariant under the local gauge transformations. Generalized Lorentz equations are derived as the geodesic equations. One of these equations is that for a spinning point particle in gravitation which violates the strong equivalence principle.     

A comprehensive approach to an equation of state for hard spheres and Lennard-Jones fluids

We present a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way.We will guess equations of state by using suitable axiomatic functional forms (n=1,2,3,4,5) for surface tension S n m (r),r ≥ d/2 with intermolecular separation r as a variable,where m is an arbitrary real number (pole).Among the equations of state obtained in this way are Percus-Yevick,scaled particle theory and Carnahan-Starling equations of state.In addition,we have found a simple equation of state for the hard sphere fluid in the region that represents the simulation data accurately.It is found that for both hard sphere fluids as well as Lennard-Jones fluids,with m=3/4 the derived equation of state (EOS) gives results which are in good agreement with computer simulation results.Furthermore,this equation of state gives the Percus-Yevick (pressure) EOS for the m=0,the Carnahan-Starling EOS for m=4/5,while for the value of m=1 it corresponds to a scaled particle theory EOS.     

New Discrete Model Boltzmann Equations for Arbitrary Partitions of the Velocity Space

Modified discrete Boltzmann equations for arbitrary partitions of the velocity space are established. The new equations can be derived from the continuous Boltzmann equation and are a generalization of previous discrete-velocity models. They preserve mass, momentum, and energy, and an H-theorem holds. The new model equations are tested by comparing their solutions with the analytical ones of the continuous Boltzmann equation for the Krook–Wu and the very hard particle models.     

Lattice Boltzmann simulation of solid particles suspended in fluid

The lattice Boltzmann method, an alternative approach to solving a fluid flow system, is used to analyze the dynamics of particles suspended in fluid. The interaction rule between the fluid and the suspended particles is developed for real suspensions where the particle boundaries are treated as no-slip impermeable surfaces. This method correctly and accurately determines the dynamics of single particles and multi-particles suspended in the fluid. With this method, computational time scales linearly with the number of suspensions,N, a significant advantage over other computational techniques which solve the continuum mechanics equations, where the computational time scales asN ³. Also, this method solves the full momentum equations, including the inertia terms, and therefore is not limited to low particle Reynolds number.     

Green’s function for an electron model with a plane wave

The Green function for a Dirac particle subject to a plane wave field is constructed according to the path integral approach and the Barut’s electron model. Then it is exactly determined after having fixed a matrix U chosen so that the equations of motion are those of a free particle, and by using the properties of the plane wave and also with some shifts.      

Bhabha Equations for Unique Mass and Spin

Linear relativistic wave equations of Bhabha's type are discussed. Conditions for unique mass and unique spin are analysed in detail. It is shown that if one relaxes the diagonalisability of β_o by a weaker block form one gets a class of equations describing a spin-half particle. Several algebras which emerge and their representations are studied. The existence or otherwise of the hermitianising matrix in such cases is explicitly discussed. The properties of the new equations in the presence of minimally coupled electro-magnetic field are discussed.     

坐标算符本征矢的表示与不对称投影算符的积分

借助于粒子数算符的本征态和坐标算符函数的本征方程,把坐标算符的本征矢〈f(x)|表示为一个算符对坐标本征矢〈x|的作用.由此,把不对称的坐标投影算符转换为对称的坐标投影算符,再利用坐标本征矢的完备性,给出不对称坐标投影算符的积分. 关键词： 本征矢 算符的积分 本征方程 完备性     

Heisenberg Equations of Motion in a Nonabelian Gauge Theory

Heisenberg type equations of motion are established in a nonabelian gauge theory with minimal and nonminimal couplings and various relativistic particle equations of motion are derived from them. These equations for pointlike particles possessing a nonabelian gauge interaction (chosen for definiteness to be of SO(4,1) type) ore obtained in classical limit, ħ → 0, or in a semiclassical limit in which contributions of first order in ħ are retained. As a byproduct of the formalism, which can be applied to an arbitrary gauge group, a simple derivation of the Lorentz equation and the Bargmann-Michel-Telegdi equation from spinor electrodynamics with anomalous (i.e. nonminimal) coupling is given starting from the associated quantum mechanical Heisenberg equations of motion and specializing the gauge group to the electromagnetic U(1) group.     

Stability of the solution of a special type of self-consistent problem

In preceding studies by the present author, it has been shown that in quantum mechanics formulated in the framework of theK-field formalism, Lyapunov-stable solutions of the equations ofK-motions describing the dynamics of a test point particle which models the behavior of a microscopic particle from the standpoint of classical mechanics may be selected as the quantization criterion. The present article is concerned with methods of obtaining a stability condition in explicit form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 42–45, January, 1996.     

su q (2)-invariant Schrödinger equations

We set upsu _q (2)-invariant Schrödinger equations within the usual framework of quantum mechanics. We show that the stationary equations reduce to radial ordinary differential equations by using the q-Spherical Harmonics. We apply these results to deformations of the free particle and of the hydrogen atom, for which we explicitly solve the equations and compare the deformed and undeformed theories.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.