Embedding of Particle Waves in a Schwarzschild Metric Background

The special and general relativity theories are used to demonstrate that the velocity of an unradiative particle in a Schwarzschild metric background, and in an electrostatic field, is the group velocity of a wave that we call a particle wave, which is a monochromatic solution of a standard equation of wave motion and possesses the following properties. It generalizes the de Broglie wave. The rays of a particle wave are the possible particle trajectories, and the motion equation of a particle can be obtained from the ray equation. The standing particle wave equation generalizes the Schrödinger equation of wave amplitudes. The particle wave motion equation generalizes the Klein–Gordon equation; this result enables us to analyze the essence of the particle wave frequency. The equation of the eikonal of a particle wave generalizes the Hamilton–Jacobi equation; this result enables us to deduce the general expression for the linear momentum. The Heisenberg uncertainty relation expresses the diffraction of the particle wave, and the uncertainty relation connecting the particle instant of presence and energy results from the fact that the group velocity of the particle wave is the particle velocity. A single classical particle may be considered as constituted of geometrical particle wave; reciprocally, a geometrical particle wave may be considered as constituted of classical particles. The expression for a particle wave and the motion equation of the particle wave remain valid when the particle mass is zero. In that case, the particle is a photon, the particle wave is a component a classical electromagnetic wave that is embedded in a Schwarzschild metric background, and the motion equation of the wave particle is the motion equation of an electromagnetic wave in a Schwarzschild metric background. It follows that a particle wave possesses the same physical reality as a classical electromagnetic wave. This last result and the fact that the particle velocity is the group velocity of its wave are in accordance with the opinions of de Broglie and of Schrödinger. We extend these results to the particle subjected to any static field of forces in any gravitational metric background. Therefore we have achieved a synthesis of undulatory mechanics, classical electromagnetism, and gravitation for the case where the field of forces and the gravitational metric background are static, and this synthesis is based only on special and general relativity.     

Complex Vector Formalism of Harmonic Oscillator in Geometric Algebra: Particle Mass,Spin and Dynamics in Complex Vector Space

Elementary particles are considered as local oscillators under the influence of zeropoint fields. Such oscillatory behavior of the particles leads to the deviations in their path of motion. The oscillations of the particle in general may be considered as complex rotations in complex vector space. The local particle harmonic oscillator is analyzed in the complex vector formalism considering the algebra of complex vectors. The particle spin is viewed as zeropoint angular momentum represented by a bivector. It has been shown that the particle spin plays an important role in the kinematical intrinsic or local motion of the particle. From the complex vector formalism of harmonic oscillator, for the first time, a relation between mass $m$ and bivector spin $S$ has been derived in the form $\varvec{\sigma }_3 mc^2{\mathcal {J}}_{\pm } =\lambda \Omega _{\mathbf{s}} \cdot \mathrm{{S}} {\mathcal {J}}_{\pm }$ . Where, $\Omega _{s}$ is the angular velocity bivector of complex rotations, $c$ is the velocity of light. The unit vector $\varvec{\sigma }_3$ acts as an operator on the idempotents ${\mathcal {J}}_{+}$ and ${\mathcal {J}}_{-}$ to give the eigen values $\lambda =\pm 1.$ The constant $\lambda $ represents two fold nature of the equation corresponding to particle and antiparticle states. Further the above relation shows that the mass of the particle may be interpreted as a local spatial complex rotation in the rest frame. This gives an insight into the nature of fundamental particles. When a particle is observed from an arbitrary frame of reference, it has been shown that the spatial complex rotation dictates the relativistic particle motion. The mathematical structure of complex vectors in space and spacetime is developed.     

Motion of test particles in a torsion field

The solution of the Mathisson-Papapetrou equations generalized to the case of the Einstein-Cartan theory, when they describe the motion of a test particle in an external torsion field, is considered. It is shown that a particle of nonzero rest mass moves inertially in a constant polarized torsion field, but its spin precesses around the direction of polarization of the spin of the torsion source. It is also found that the motion of a massless particle in a variable torsion field leads to a torsion frequency displacement effect of a photon under the assumption that the photon spin interacts with the torsion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 84–87, August, 1980.     

Visualization techniques in plasma numerical simulations

Numerical simulations of plasma processes usually yield a huge amount of raw numerical data. Information about electric and magnetic fields and particle positions and velocities can be typically obtained. There are two major ways of elaborating these data. First of them is calledplasma diagnostics. We can calculate average values, variances, correlations of variables, etc. These results may be directly comparable with experiments and serve as the typical quantitative output of plasma simulations. The second possibility is theplasma visualization. The results are qualitative only, but serve as vivid display of phenomena in the plasma followed-up. An experience with visualizing electric and magnetic fields via Line Integral Convolution method is described in the first part of the paper. The LIC method serves for visualization of vector fields in two dimensional section of the three dimensional plasma. The field values can be known only in grid points of three-dimensional grid. The second part of the paper is devoted to the visualization techniques of the charged particle motion. The colour tint can be used for particle’s temperature representation. The motion can be visualized by a trace fading away with the distance from the particle. In this manner the impressive animations of the particle motion can be achieved.     

均匀磁场中带电粒子运动的双波描述

用双波量子理论描述带电粒子在均匀磁场中的运动,得到对单个粒子运动状况的完全描述。在任何时刻都能说出任一力学量的确切数值。通常量子力学中的概率性和平均值公式来源于双波描述对某类系综的平均结果。量子力学中的规范变换特性也能由系综平均看出。 关键词：     

The Classical Coulomb Problem in Pre-Maxwell Electrodynamics

We explore certain difficulties in the covariant classical mechanics associated with off-shell electrodynamics, through an examination of the classical Coulomb problem. We present a straightforward solution of the classical equations of motion for a test event traversing the field induced by a fixed event (an event moving uniformly along the time axis at a fixed point in space). This solution reveals the essential difficulties in the formalism at the classical level. We then offer a new model of the particle, as a certain distribution of events on the worldline, which eliminates these difficulties and permits comparison of classical off-shell electrodynamics with the standard Maxwell theory In this model, the fixed event induces a Yukawa-type potential, permitting a semiclassical identification of the pre-Maxwell time scale with the inverse mass of the intervening photon. Numerical solutions to the equations of motion are compared with the standard Maxwell solutions—they are seen to coincide when > 10^–6 sec, providing an initial estimate of this parameter.     

Acceleration of charged particles and radiation reaction in strong plane and spherical waves. II

Particle motion in a superstrong wave field (Laser or Pulsar field) is considered in the presence of an additional longitudinal magnetic field. It is shown that the particle motion is distinctly different from that in a previously considered pure wave field and that the maximum obtainable energy is substantially reduced. We also reconsider the effects of radiation reaction on a particle moving in a plane linearily polarized wave.     

Resonance acceleration of ions in a variable magnetic field

The motion of a particle is considered in a variable magnetic field — the superposition of a driving field H(t) and a weak field oscillating at the frequency =eH(t)/mc. The law of motion is found by the averaging method, and it is shown that the particle energy increases exponentially.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 75–78, October, 1978.In conclusion, thanks are due to the participants in Prof. Sokolov's seminar for discussion of the work.     

正电子面沟道辐射的非线性特征

经典物理学指出,在电磁场中作加速运动的带电粒子将不断向外辐射能量.在晶体沟道中运动的带电粒子也不例外,晶格场可以使带电粒子的辐射能量达到很高.对于10MeV的正电子,辐射能量可达keV量级.粒子在沟道中的运动行为决定于粒子晶体的相互作用势,常用的相互作用势有Lindhard势、Moliere势和正弦平方势.由于粒子在沟道中的运动行为十分类似于震荡器中运动的自由电子,可望把沟道辐射改造为Χ射线激光或γ射线激光.从Lindhard势出发,将其展开到四次项,在经典力学框架内,粒子的运动方程可以化为含立方项的二阶非线性微分方程,并利用Jacobian椭圆函数和第一类全椭圆积分解析地表示了系统的解和粒子运动周期,导出了正电子面沟道辐射的瞬时辐射强度、平均辐射强度和最大辐射频率,指出了利用沟道辐射作为γ激光的可能性.     

Behavior of a charged particle in a schwarzschild field

The influence of the De Witt self-action force on the motion of and electromagnetic emission from a charged particle in a Schwarzschild field is considered. It is shown that a charged particle in a Schwarzschild field is equivalent to a neutral particle of the same mass in a certain Reissner-Nordstrom field. A relationship is found between the power of the electromagnetic emission from an accelerated charge and the power of the thermal emission generated in a reference frame with the same acceleration at the event horizon. The quantum-mechanical problem of the motion of and emission from a charge in the field of a minihole is considered. Wave functions, the energy spectrum, and the widths of quasi-stationary levels are found with allowance for the De Witt self-action force. It is shown that the latter is important for large charges, when the solution becomes oscillatory. "Brainstorm" Little Science and Technology Enterprise. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 75–82, July, 1998.     

Relativistic motion of a free particle in a uniform gravitational field

The problem of the motion of a free particle in a uniform gravitational field is considered. A relativistic solution based on the assumption that the motion is a consequence of the curvature of spacetime is obtained. The results are compared with various results based on the assumption that spacetime is flat in a region in which the gravitational field is uniform. In the curved spacetime approach, if a particle is projected from a point in a uniform gravitational field, the vertical distance covered by the particle in infinite coordinate time is infinite, but the horizontal distance covered and the elapsed proper time of the particle are finite. If spacetime is assumed to be flat and the gravitational motion of a particle a consequence of a relativistic force proportional to the relative mass of the particle, then the results obtained for the motion of a particle in a uniform gravitational field are close to the curved spacetime results. All other assumptions, including the assumption that the motion of a particle in a uniform gravitational field is equivalent to the motion of a particle in a uniformly accelerating frame of reference, lead to results in serious disagreement with the curved spacetime results.     

Experimental observation of optical rotation generated in vacuum by a magnetic field

We report the experimental observation of a light polarization rotation in vacuum in the presence of a transverse magnetic field. Assuming that data distribution is Gaussian, the average measured rotation is (3.9 +/- 0.5) x 10(-12) rad/pass, at 5 T with 44 000 passes through a 1 m long magnet, with lambda = 1064 nm. The relevance of this result in terms of the existence of a light, neutral, spin-zero particle is discussed.     

Author's reply to "Good news: you can patch active plasma and collisionless sheath"

Godyak and Sternberg (2003) reassert their contention that one can obtain a satisfactory physical solution to the active plasma-collisionless sheath by patching together plasma and sheath. They choose to do it at an arbitrary point where the sheath electric field is kT/sub e/ /e/spl lambda//sub D/. If one tacks their sheath solution to the full plasma solution, then the field is infinity on the plasma side and finite on the sheath side. Alternatively, if one terminates the plasma solution where the plasma field is kT/sub e//e/spl lambda//sub D/, then one has continuity of electric field, but not of its gradient, since on the sheath side it is zero and on the plasma side of order L//spl lambda//sub D/, where L is the size of the plasma. Furthermore, in achieving continuity of the field, one has introduced discontinuities in the ion speed and in the particle densities. Thus, in no sense is a joining which denies the existence of a transition layer, smooth. J. Ockendon and H. Ockendon, my colleagues in the production of our paper describing the transition layer (Franklin et al., 1970), privately expressed disappointment in not finding a proof of the existence and uniqueness of our solution. Such a formal mathematical proof has been given recently by Slemrod (2002). Smooth joining of active plasma and collisionless sheath within the context of a fluid model or free fall model of the ion motion, does require a transition layer and of length scale intermediate between L and /spl lambda//sub D/.     

Drift Lagrangian for a relativistic particle in an intense laser field

The Lagrangian and Hamiltonian functions describing the average motion of a relativistic particle under the action of a slightly inhomogeneous intense laser field are obtained. In weak low-frequency background fields, such a particle on average drifts with an effective relativistically invariant mass, which depends on the laser intensity. The essence of the proposed ponderomotive formulation is presented in a physically intuitive and mathematically simple form yet represents a powerful tool for studying various nonlinear phenomena caused by the interaction of currently available smooth ultraintense laser pulses with plasmas.     

Shock Fluctuations for the Hammersley Process

We consider the Hammersley interacting particle system starting from a shock initial profile with densities \(\rho ,\lambda \in {\mathbb R}\) (\( \rho < \lambda \)). The microscopic shock is taken as the position of a second-class particle initially at the origin, and the main results are: (i) a central limit theorem for the shock; (ii) the variance of the shock equals \(2[\lambda \rho (\lambda - \rho )]^{-1}t + O(t^{2/3})\). By using the same method of proof, we also prove similar results for first-class particles.     

An extension of the nonsymmetric unified field theory

The field equations, in the new formulation of Einstein's unified field theory, are extended from the present vacuum form to the general case in which sources are present. In this generalization the contracted torsion tensor corresponds to the electromagnetic four-potential. By this correspondence, Einsteins-gauge transformation becomes identical to the ordinary electromagnetic gauge symmetry. The generalized Bianchi identities are found and used to discuss deviations from the Einstein-Lorentz equations of motion.     

二维晶化束的平均场概念和单粒子模型(Ⅱ)

利用平均场概念和单粒子模型讨论了储存环中二维晶化束的大尺度运动行为,结果再现了粒子轨道的螺旋运动特征,并用天体物理学方法揭示了螺旋轨道不断进动的新运动形态     

Spin gauge fields: From Berry phase to topological spin transport and Hall effects

The paper examines the emergence of gauge fields during the evolution of a particle with a spin that is described by a matrix Hamiltonian with n different eigenvalues. It is shown that by introducing a spin gauge field a particle with a spin can be described as a spin multiplet of scalar particles situated in a non-Abelian pure gauge (forceless) field U (n). As the result, one can create a theory of particle evolution that is gauge-invariant with regards to the group Uⁿ (1). Due to this, in the adiabatic (Abelian) approximation the spin gauge field is an analogue of n electromagnetic fields U (1) on the extended phase space of the particle. These fields are force ones, and the forces of their action enter the particle motion equations that are derived in the paper in the general form. The motion equations describe the topological spin transport, pumping, and splitting. The Berry phase is represented in this theory analogously to the Dirac phase of a particle in an electromagnetic field. Due to the analogy with the electromagnetic field, the theory becomes natural in the four-dimensional form. Besides the general theory, the article considers a number of important particular examples, both known and new.     

Average Field Idea and Single Particle Model for 2 Dimension Crystallization Beams(Ⅱ)

The average field idea and the single particle model have been introduced, and the motion behaviours in large scale of beam particles have been disscussed for 2 Dimension crystallization beams in storage rings. It is shown that a particle orbit is the spiral line moved along z axis, and new configuration has been disclosed in the plane vertical to z axis.