Nonlocal conserved quantities,balance laws,and equations of motion

A formalism is developed whereby balance laws are directly obtained from nonlocal (integrodifferential) linear second-order equations of motion for systems described by several dependent variables. These laws augment the equations of motion as further useful information about the physical system and, under certain conditions, are shown to reduce to conservation laws. The formalism can be applied to physical systems whose equations of motion may be relativistic and either classical or quantum. It is shown to facilitate obtaining global conservation laws for quantities which include energy and momentum. Applications of the formalism are given for a nonlocal Schrödinger equation and for a system of local relativistic equations of motion describing particles of arbitrary integral spin.     

Quantum mechanics as relativistic statistics. I: The two-particle case

It is shown that non-relativistic quantum mechanics can be treated as a kind of relativistic statistical theory, which describes the indeterministic motion of classical particles. The theory is relativistic in the sense that the relativistic notion of the state and two-time equations of motion are used. The principles and relations of quantum mechanics are obtained from the principles of statistics and those of classical mechanics.     

Low-diffraction direct particle acceleration by a radially polarized laser beam

Two constants of the motion, which simplify the relativistic particle dynamics in a laser beam of radial polarization, are identified. Many-particle simulations based on the reduced set of equations of motion in a beam of relativistic intensity, demonstrate acceleration in vacuum to GeV energies of electrons, alpha particles and oxygen bare nuclei. The axially-injected particles are shown to be accelerated with negligible diffraction.     

Electromagnetic Field Theory Without Divergence Problems 1. The Born Legacy

Born's quest for the elusive divergence problem-free quantum theory of electromagnetism led to the important discovery of the nonlinear Maxwell–Born–Infeld equations for the classical electromagnetic fields, the sources of which are classical point charges in motion. The law of motion for these point charges has however been missing, because the Lorentz self-force in the relativistic Newtonian (formal) law of motion is ill-defined in magnitude and direction. In the present paper it is shown that a relativistic Hamilton–Jacobi type law of point charge motion can be consistently coupled with the nonlinear Maxwell–Born–Infeld field equations to obtain a well-defined relativistic classical electrodynamics with point charges. Curiously, while the point charges are spinless, the Pauli principle for bosons can be incorporated. Born's reasoning for calculating the value of his aether constant is re-assessed and found to be inconclusive.     

Dynamical derivation of momentum diffusion coefficients at collisions of relativistic charged particles

An expression has been obtained for the diffusion tensor of particles in the momentum space on the basis of the dynamics of particles motion. The general equations have been used to determine the rms momentum spread at collisions of relativistic charged particles at times shorter than the time of randomization of particles motion and at greater times when motion is completely random.     

f(R,L m ) gravity

We generalize the f(R) type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the matter Lagrangian L _m . We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the energy-momentum tensor. The equations of motion for test particles can also be derived from a variational principle in the particular case in which the Lagrangian density of the matter is an arbitrary function of the energy density of the matter only. Generally, the motion is non-geodesic, and it takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equation of motion is also considered, and a procedure for obtaining the energy-momentum tensor of the matter is presented. The gravitational field equations and the equations of motion for a particular model in which the action of the gravitational field has an exponential dependence on the standard general relativistic Hilbert–Einstein Lagrange density are also derived.     

On the pressure of gravitational waves

A system of coupled point masses under the influence of gravitational waves is considered. By means of the geodesic deviation equation as the equation of motion it is shown, taking into account the second order small terms, that there exist forces which cause the acceleration of the system in the longitudinal direction. The longitudinal force is due to the fact that simultaneously with energy momentum is also absorbed from waves. It is proved directly on the basis of the equations of motion of the point masses that the energy and momentum absorbed by the test system obey the special relativistic relationship of a zero rest mass particle. The case when the Weber oscillator moves at a relativistic speed with respect to the source of gravitational waves is also examined. In this case, the absorption of energy and momentum by the Weber oscillator is much larger or smaller compared to the stationary situation.     

Quantum-classical correspondence of the Dirac equation with a scalar-like potential

Quantum matrix elements of the coordinate, momentum and the velocity operator for a spin-1/2 particle moving in a scalar-like potential are calculated. In the large quantum number limit, these matrix elements give classical quantities for a relativistic system with a position-dependent mass. Meanwhile, the Klein-Gordon equation for the spin-0 particle is discussed too. Though the Heisenberg equations for both the spin-0 and spin-1/2 particles are unlike the classical equations of motion, they go to the classical equations in the classical limit.      

Poincaré-invariant gravitational field and equations of motion of two pointlike objects: The postlinear approximation of general relativity

Using a fast-motion approximation method we obtain the second-order gravitational field and equations of motion for two pointlike objects in algebraically closed form. A regularization procedure is used which is shown to guarantee the consistency of the approximation scheme. The equations of motion are then transformed within the framework of relativistic predictive mechanics into a system of ordinary differential equations.     

Conservation of the Energy-Momentum

In Relativity the sum of 4?vectors in different points does not generally represent a 4?vector. By using this result, it is shown by simple methods that the total energy-momentum of a system of point particles represents a well-defined 4?vector if the particles do not interact. It is proved that this is equivalent to the no-interaction theorem in Classical Physics. This theorem difficulties the study of a system of interacting particles since it is not even possible to define the total energy-momentum nor the reference frame where the system is at rest. This impediment is avoided by adding to the energy-momentum tensor the stress tensor describing the interaction. As an example, this is applied to a system of charged particles. In the process, the equation of motion for a charged particle including the self-force is formally obtained. However, when a thermodynamic system is analyzed from two different reference frames with a relativistic relative velocity, the interaction between the particles and the walls of the volume cannot be described by means of a covariant stress tensor and consequently the proposed technique is not feasible. Despite the above mentioned drawbacks, a covariant theory of the relativistic transformation laws of the thermodynamic quantities is developed.     

Relativistic Two-Boson System in Presence of Electromagnetic Plane Wave

The relativistic two-body problem is considered for spinless particles subject to an external electromagnetic field. When this field is made of the monochromatic superposition of two counter-propagating plane waves (and provided the mutual interaction between particles is known), it is possible to write down explicitly a pair of coupled wave equations (corresponding to a pair of mass-shell constraints) which takes into account also the field contribution. These equations are manifestly covariant; constants of the motion are exhibited, so one ends up with a reduced problem involving five degrees of freedom.     

质点的相对论式的汉密尔敦运动式与相对论式的哈生堡方程式

在本文中,作者推得一组相对论式的汉密尔敦运动式;并根据此运动式,详细地讨论了一质点之运动;由此还可以很自然地看出,在量子力学中,狄拉克电子方程式似乎是一个必然的波动方程式。在狄拉克理论中的取平方根步骤在这里找到它在古典物理学中的对照。同时根据了上面的理论,作者还推得到一个相对论式的哈生堡方程式,最后则讨论了此方程式在狄拉克电子理论中的一些应用;同时并指出,在相对论的观点上,此方程式可以引导出一些比较对称的及有比较普遍形式的物理的量。     

Relativistic kinetic theory of quantum systems: III. Transport equation for a neutrino system

On the basis of the quantum-mechanical equations of motion and the statistical assumption that dynamical correlations present in the initial state may be ignored, a kinetic equation for a dilute neutrino system is derived. If the system is sufficiently uniform in space this equation reduces to the relativistic Boltzmann equation for massless particles with a quantum-mechanical transition probability.     

Fractal behavior in quantum statistical physics

The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.     

New electrostatic cylindrical energy analyzer

The focusing properties of an ideal cylindrical field in full relativistic field have been investigated for charged particles originating from an extended source and an analytical solution for the equations of motion has been provided. This focusing system can be used for production of electrostatic energy analyzer provided surface analysis from an extended area.     

Particles with curvature and torsion in three-dimensional pseudo-Riemannian space forms

We consider the motion of relativistic particles described by an action which is a function of the curvature and torsion of the particle path. The Euler–Lagrange equations and the dynamical constants of the motion are expressed in a simple way in terms of a suitable coordinate system. The moduli spaces of solutions in a three-dimensional pseudo-Riemannian space form are completely exhibited.     

Calculation of the Relativistic Correction in Terms of the Oscillator Representation

A procedure is proposed for the calculation of the corrections associated with the relativistic motion and spin interaction of particles in a system of a few bodies. The relativistic corrections for the Coulomb and Cornell potentials are determined from the relativistic generalization of the Schrödinger equation. The slope of the Regge trajectory and the masses of mesons are calculated taking the relativistic correction into account.     

On the theory of the relativistic motion of a charged particle in the field of intense electromagnetic radiation

Averaged relativistic equations of motion of a charged particle in the field of intense electromagnetic radiation have been obtained in the geometrical optics approximation using the Bogoliubov method. Constraints are determined under which these equations are valid. Oscillating additions to the smoothed dynamical variables of the particle have been found; they are reduced to known expressions in the case of the circularly and linearly polarized plane waves. It has been shown that the expressions for the averaged relativistic force in both cases contain new additional small terms weakening its action. The known difference between the expressions for the ponderomotive force in the cases of circularly and linearly polarized waves has been confirmed.