arXiv subjects

Anatoli Babin

Publications and source records attributed to Anatoli Babin.

9 recordsLinked to original sources

arXiv2015

Explicit mass renormalization and consistent derivation of radiative response of classical electron

The radiative response of the classical electron is commonly described by the Lorentz-Abraham-Dirac (LAD) equation. Dirac's derivation of this equation is based on energy and momentum conservation laws and on regularization of the field singularities and infinite energies of the point charge by subtraction of certain quantities: "We ... shall try to get over difficulties associated with the infinite energy of the process by a process of direct omission or subtraction of unwanted terms". To substantiate Dirac's approach and clarify the mass renormalization, we introduce the point charge as a limit of extended charges contracting to a point; the fulfillment of conservation laws follows from the relativistic covariant Lagrangian formulation of the problem. We derive the relativistic point charge dynamics described by the LAD equation from the extended charge dynamics in a localization limit by a method which can be viewed as a refinement of Dirac's approach in the spirit of Ehrenfest theorem. The model exhibits the mass renormalization as the cancellation of Coulomb energy with the Poincaré cohesive energy. The value of the renormalized mass is not postulated as an arbitrary constant, but is explicitly calculated. The analysis demonstrates that the local energy-momentum conservation laws yield dynamics of a point charge which involves three constants: mass, charge and radiative response coefficient θ. The value of θ depends on the composition of the adjacent potential which generates Poincaré forces. The classical value of the radiative response coefficient is singled out by the global requirement that the adjacent potential does not affect the radiated energy balance and affects only the local energy balance involved in the renormalization.

physics.class-ph↗

arXiv2014

Neoclassical Theory of Elementary Charges with Spin of 1/2

We advance here our neoclassical theory of elementary charges by integrating into it the concept of spin of 1/2. The developed spinorial version of our theory has many important features identical to those of the Dirac theory such as the gyromagnetic ratio, expressions for currents including the spin current, and antimatter states. In our theory the concepts of charge and anticharge relate naturally to their "spin" in its rest frame in two opposite directions. An important difference with the Dirac theory is that both the charge and anticharge energies are positive whereas their frequencies have opposite signs.

physics.gen-ph↗

arXiv2013

Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation

One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to nonlinear Klein-Gordon equation, we derived relativistic Newton's law and Einstein's formula for inertial mass. Here we apply the same approach to nonlinear Schrodinger equation and derive non-relativistic Newton's law for the trajectory of concentration.

math.AP↗

arXiv2012

Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation

Einstein's relation E=Mc^2 between the energy E and the mass M is the cornerstone of the relativity theory. This relation is often derived in a context of the relativistic theory for closed systems which do not accelerate. By contrast, Newtonian approach to the mass is based on an accelerated motion. We study here a particular neoclassical field model of a particle governed by a nonlinear Klein-Gordon (KG) field equation. We prove that if a solution to the nonlinear KG equation and its energy density concentrate at a trajectory, then this trajectory and the energy must satisfy the relativistic version of Newton's law with the mass satisfying Einstein's relation. Therefore the internal energy of a localized wave affects its acceleration in an external field as the inertial mass does in Newtonian mechanics. We demonstrate that the "concentration" assumptions hold for a wide class of rectilinear accelerating motions.

math-ph↗

arXiv2012

Relativistic dynamics of accelerating particles derived from field equations

In relativistic mechanics the energy-momentum of a free point mass moving without acceleration forms a four-vector. Einstein's celebrated energy-mass relation E=mc^2 is commonly derived from that fact. By contrast, in Newtonian mechanics the mass is introduced for an accelerated motion as a measure of inertia. In this paper we rigorously derive the relativistic point mechanics and Einstein's energy-mass relation using our recently introduced neoclassical field theory where a charge is not a point but a distribution. We show that both the approaches to the definition of mass are complementary within the framework of our field theory. This theory also predicts a small difference between the electron rest mass relevant to the Penning trap experiments and its mass relevant to spectroscopic measurements.

physics.class-ph↗

arXiv2010

Electrodynamics of balanced charges

In this work we modify the wave-corpuscle mechanics for elementary charges introduced by us recently. This modification is designed to better describe electromagnetic (EM) phenomena at atomic scales. It includes a modification of the concept of the classical EM field and a new model for the elementary charge which we call a balanced charge (b-charge). A b-charge does not interact with itself electromagnetically, and every b-charge possesses its own elementary EM field. The EM energy is naturally partitioned as the interaction energy between pairs of different b-charges. We construct EM theory of b-charges (BEM) based on a relativistic Lagrangian with the following properties: (i) b-charges interact only through their elementary EM potentials and fields; (ii) the field equations for the elementary EM fields are exactly the Maxwell equations with proper currents; (iii) a free charge moves uniformly preserving up to the Lorentz contraction its shape; (iv) the Newton equations with the Lorentz forces hold approximately when charges are well separated and move with non-relativistic velocities. The BEM theory can be characterized as neoclassical one which covers the macroscopic as well as the atomic spatial scales, it describes EM phenomena at atomic scale differently than the classical EM theory. It yields in macroscopic regimes the Newton equations with Lorentz forces for centers of well separated charges moving with nonrelativistic velocities. Applied to atomic scales it yields a hydrogen atom model with a frequency spectrum matching the same for the Schrodinger model with any desired accuracy.

physics.class-ph↗

arXiv2010

Some mathematical problems in a neoclassical theory of electric charges

We study here a number of mathematical problems related to our recently introduced neoclassical theory for the electromagnetic phenomena in which charges are represented by complex valued wave functions as in the Schrodinger wave mechanics. Dynamics of charges in the non-relativistic case is governed by a system of nonlinear Schrodinger equations coupled with the electromagnetic fields, and we prove for it that the centers of wave functions converge in macroscopic regimes to trajectories of points governed by the Newton's equations with the Lorentz forces provided the wave functions remain localized. Exact solutions in the form of localized accelerating solitons are found. Our studies of a class of time multiharmonic solutions of the same field equations show that they satisfy Planck-Einstein relation and that the energy levels of the nonlinear eigenvalue problem for the hydrogen atom converge to the well-known energy levels of the linear Schrodinger operator when the free charge size is much larger than the Bohr radius.

math-ph↗

arXiv2010

Wave-corpuscle mechanics for elementary charges

It is well known that a concept of point charge interacting with electromagnetic (EM) field has a problem. To address that problem we introduce a 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 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 or accelerated motion. It carries explicitly features of a point charge and the de Broglie wave. A system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton motion equations 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 binding forces that keep localized every individual charge.

physics.class-ph↗

arXiv2005

Nonlinear Photonic Crystals: IV. Nonlinear Schrodinger Equation Regime

We study here the nonlinear Schrodinger Equation (NLS) as the first term in a sequence of approximations for an electromagnetic (EM) wave propagating according to the nonlinear Maxwell equations (NLM). The dielectric medium is assumed to be periodic, with a cubic nonlinearity, and with its linear background possessing inversion symmetric dispersion relations. The medium is excited by a current $\mathbf{J}$ producing an EM wave. The wave nonlinear evolution is analyzed based on the modal decomposition and an expansion of the exact solution to the NLM into an asymptotic series with respect to some three small parameters $α$, $β$ and $\varrho $. These parameters are introduced through the excitation current $\mathbf{J}$ to scale respectively (i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the range of wavevectors involved in its modal composition, with $β^{-1}$ scaling its spatial extension; (iii) its frequency bandwidth, with $\varrho ^{-1}$ scaling its time extension. We develop a consistent theory of approximations of increasing accuracy for the NLM with its first term governed by the NLS. We show that such NLS regime is the medium response to an almost monochromatic excitation current $\mathbf{J} $. The developed approach not only provides rigorous estimates of the approximation accuracy of the NLM with the NLS in terms of powers of $α$, $β$ and $\varrho $, but it also produces new extended NLS (ENLS) equations providing better approximations. Remarkably, quantitative estimates show that properly tailored ENLS can significantly improve the approximation accuracy of the NLM compare with the classical NLS.

math-ph↗