De Broglie waves and the nature of mass

In this paper an attempt is made to interpret inertial mass as a consequence of the invariant periodicity associated with physical de Broglie waves. In the case of a free particle, such waves, observed from an arbitrary reference frame, would exhibit the velocity-dependent wavelength given by de Broglie's relation; and it is conjectured that the inertial and additive properties of mass (or, more precisely, the conservation of momentum and energy) can be related to nonlinear interference effects occurring between the de Broglie waves for different particles. This picture could throw light on the physical meaning of quantization and suggests the possibility of reformulating classical and quantum mechanics in terms of a quasi-classical nonlinear field theory in which both inertial and quantization effects result essentially from the periodicity of de Broglie waves.     

The motion of wavelets—An interpretation of the Schrödinger equation

There are stable wavelets which satisfy the Schrödinger equation. The motion of a wavelet is determined by a set of ordinary differential equations. In a certain limit, a wavelet turns out to be the known representation of a classical material point. A de Broglie wave is constructed by superposing similar free wavelets. Conventional energy eigensolutions of the Schrödinger equation can be interpreted as ensembles of wavelets. If the dynamics of wavelets form the quantum mechanical counterpart of Newton's dynamics of particles, then conventional quantum mechanics is the counterpart of Gibbs's mechanics of ensembles. In this way, conventional quantum mechanics is reinterpreted on a deterministic basis. A difficulty of quantum field theory is predictable from this point of view.     

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.     

Gravitation and mass decrease

Consequences in physical theory of assuming the general relativistic time transformation for the de Broglie frequencies of matter, v = E/h = mc²/h, are investigated in this paper. Experimentally it is known that electromagnetic waves from a source in a gravitational field are decreased in frequency, in accordance with the Einstein general relativity time transformation. An extension to de Broglie frequencies implies mass decrease in a gravitational field. Such a decrease gives an otherwise missing energy conservation for some processes; also, a physical alteration is then associated with change in gravitational potential. Further, the general relativity time transformation that is the source of gravitational action in the weak field (Newtonian) approximation then has a physical correlate in the proposed gravitational mass loss. Rotational motion and the associated equivalent gravitational-field mass loss are considered; an essential formal difference between metric (gravitational) mass loss and special relativity mass increase is discussed. For a spherical, nonrotating mass collapsed to its Schwarzschild radius the postulated mass loss is found to give a 25% decrease in the mass acting as origin of an external gravitational field.     

Equivalent Forms of Dirac Equations in Curved Space-times and Generalized de Broglie Relations

One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved space-time. This canonical form is needed to apply Whitham’s Lagrangian method. The latter method, unlike the Wentzel–Kramers–Brillouin method, places no restriction on the magnitude of Planck’s constant to obtain wave packets and furthermore preserves the symmetries of the Dirac Lagrangian. We show by using canonical Dirac fields in a curved space-time that the probability current has a Gordon decomposition into a convection current and a spin current and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck’s constant. We further discuss the classical-quantum correspondence in a curved space-time based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved space-time are a direct consequence of Whitham’s Lagrangian method and not just a physical hypothesis as introduced by Einstein and de Broglie and by many quantum mechanics textbooks.     

The Bohm-Vigier subquantum fluctuations and nonlinear field theory

Utilising as starting point the double solution theory of Louis de Broglie and the Bohm-Vigier hypothesis of subquantum fluctuations, an attempt is made to give an example of the nonlinear field theory in which the guidance theorem of de Broglie can be realised. The simplest scalar model is considered.     

On the Interference of Fullerenes and Other Massive Particles

We report the results of an optical analogue of the fullerene molecule diffraction experiment. Our results, and an analysis of the fullerene experiment, suggest that the patterns observed in the latter can be explained using a localized particle model. There is no evidence that the grating period contributed to the published fullerene diffraction pattern. De Broglie waves, if they exist, are unlikely to have played a significant part in the fullerene diffraction experiment. The observed patterns are not consistent with those expected according to wave theory for the experimental geometry corresponding to the slit-detector system and the de Broglie wavelength. The measurements were performed in the near field, making the demonstration of wave properties difficult. We outline a new classical approach to the electron and neutron interference experiments. The magnetic moment is crucial to this model, which emphasizes a mechanism for generating narrow-band continuum X-radiation. Some experiments are proposed which can decide between the suggested model and quantum mechanics, and which can also rule out an alternative stochastic model.     

Wave-Corpuscle Mechanics for Electric Charges

It is well known that the concept of a point charge interacting with the electromagnetic (EM) field has a problem. To address that problem we introduce the concept of wave-corpuscle to describe spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over the 4-dimensional space time continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient—a nonlinear self-interaction term providing for a cohesive force assigned to every charge. An ideal wave-corpuscle is an exact solution to the Euler-Lagrange equations describing both free and accelerated motions. It carries explicitly features of a point charge and the de Broglie wave. Our analysis shows that a system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton equations of motion for point charges interacting with the Lorentz forces. In this regime the nonlinearities are “stealthy” and don’t show explicitly anywhere, but they provide for the binding forces that keep localized every individual charge. The theory can also be applied to closely interacting charges as in hydrogen atom where it produces discrete energy spectrum.     

Does the fine-structure constant have a dynamical origin

The electromagnetic interaction of a bagged complex scalar field is investigated. The de Broglie frequency relation, and quasi-quantization of charge emerge as dynamical consequences of a many-body stability condition. The estimated fine- structure constant is much too large.     

On the structure of the quantum-mechanical probability models

In this paper the role of the mathematical probability models in the classical and quantum physics is shortly analyzed. In particular the formal structure of the quantum probability spaces (QPS) is contrasted with the usual Kolmogorovian models of probability by putting in evidence the connections between this structure and the fundamental principles of the quantum mechanics. The fact that there is no unique Kolmogorovian model reproducing a QPS is recognized as one of the main reasons of the paradoxical behaviors pointed out in the quantum theory from its early days.Paper written in honor of L. de Broglie.     

Classical tunneling

A classical representation of an extended body over barriers of height greater than the energy of the incident body is shown to have many features in common with quantum tunneling as the center-of-mass literally goes through the barrier. It is even classically possible to penetrate any finite barrier with a body of arbitrarily low energy if the body is sufficiently long. A distribution of body lengths around the de Broglie wavelength leads to reasonable agreement with the quantum transmission coefficient.     

High-order Talbot fringes for atomic matter waves

We observe the interference of de Broglie waves in the diffraction near field of a microfabricated grating. The reduction of the grating period by self-imaging of second to seventh order is spatially resolved. We investigate the dependence of this effect on the de Broglie wavelength by a time-to-flight technique.     

Nonlinear Theory of Quantum Brownian Motion

A nonlinear theory of quantum Brownian motion in classical environment is developed based on a thermodynamically enhanced nonlinear Schrödinger equation. The latter is transformed via the Madelung transformation into a nonlinear quantum Smoluchowski-like equation, which is proven to reproduce key results from the quantum and classical physics. The application of the theory to a free quantum Brownian particle results in a nonlinear dependence of the position dispersion on time, being quantum generalization of the Einstein law of Brownian motion. It is shown that the time of decoherence from quantum to classical diffusion is proportional to the square of the thermal de Broglie wavelength divided by the classical Einstein diffusion constant.     

Playing with Planets,by Gerard 't Hooft

Open channels exist between the rows or planes of atoms in a crystal. In experiments with carefully defined beams of energetic ions it is possible to inject particles into these channels. There they travel with relative ease, being reflected back and forth between the channel walls by a correlated series of atomic collisions. The main types of experiment on this phenomenon are summarized and the classical theory is developed. Classical treatment is not adequate for light particles, such as electrons or positrons because their de Broglie wavelength is greater and a wave mechanical version is outlined in a form which is also suitable for describing protons. This confirms that for heavy particles classical theory can explain any experiments performed so far, but it is just possible that the interpretation of more precise proton experiments may require wave mechanics.     

无限深势阱中粒子的态密度

对于自由粒子在有限容器中的能态密度,热力学统计教材一般根据半经典量子图像,由驻波条件和德布罗意关系,以动量分立值为基础出发得到;然而根据量子理论,无限深势阱中的粒子存在能量本征态,而非动量本征态.本文以能量本征态为统计对象推导了有限体积中的自由粒子的能态密度,结果与教材一致.但是我们的处理方式显得更为自然.     

DERIVATION OF THE LEHNERT FIELD EQUATIONS FROM GAUGE THEORY IN VACUUM: SPACE CHARGE AND CURRENT

It is shown that the Lehnert field equations in vacuum, with concomitant space charge and current, can be derived straightforwardly from standard gauge theory applied in vacuum, using the concept of covariant derivative and Feynman's universal influence. The Lehnert and Proca field equations are shown to be inter-related through the well-known de Broglie theorem, in which the photon mass can be interpreted as finite. These ideas go some way towards addressing the inconsistency inherent in Maxwell's famous displacement current, which has no concomitant vacuum space charge.     

Quantumlike dynamics of classical particles in ponderomotive potentials

The average dynamics of a classical particle under the action of a high-frequency radiation resembles quantum particle motion in a conservative field with an effective de Broglie wavelength lambda equal to the particle average displacement on the oscillation period. In a quasiclassical field, with a spatial scale large compared to lambda, the guiding-center motion is adiabatic. Otherwise, a particle exhibits quantized eigenstates in ponderomotive potential wells, tunnels through "classically forbidden" regions, and experiences stochastic reflection from attractive potentials.     

The density of energy flow of quasiparticles with arbitrary energy–momentum relation

In this paper, we consider the energy conservation law in a continuous medium with arbitrary energy–momentum relation. We use a new theoretical approach in which both the long wavelength and short wavelength thermal excitations are described in a unified way. The theory is based on the fact that in a quantum fluid, the thermal de Broglie wavelengths of the atoms overlap each other. In this case, the atoms are delocalized in space and we can treat a quantum fluid as a continuous medium without any restriction on length scale. So, in quantum liquids, we can determine the probabilistic values of the parameters of the continuous medium in every mathematical point of space. From the Hamiltonian of this system, we derive a system of linear equations for the general case of an ideal liquid, which has a nonlocal relationship between pressure and density. In the frame of this model from the energy conservation law, a general expression for the energy density flow is obtained. It is shown that for the wave packet, it is not affected by the freedom in its definition. A clear relation for the energy density flow of a wave packet is derived that generalizes the ordinary form of it to the case of arbitrary dispersion.     

Noether's theorem on a quasistochastic half-group of relativistic endomorphisms (QHGRE)

The quantum theory of free fields is used with Noether's theorem to discuss the polarization features of vector and spinor fields in the QHGRE (generalized de Broglie group) from the viewpoint of local isomorphism of the QHGRE for the product of two ordinary groups (the groups of x , t deformations and rotations). Estimates are made of the contributions to the polarization tensors for the corresponding fields. In both cases the spin tensors are induced by space and time generators of the generalized de Broglie group, and the rotational component of the QHGRE is responsible for the spin properties.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, Vol. 12, No. 1, pp. 32–36, January, 1969.We are indebted to D. D. Ivanenko for discussion of these topics.