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Andrey Akhmeteli

Publications and source records attributed to Andrey Akhmeteli.

15 recordsLinked to original sources
arXiv2026

On the possibility of differential-algebraic elimination of the spinor field from the Maxwell--Dirac electrodynamics

We investigate whether the spinor field can be differential-algebraically eliminated from the Maxwell--Dirac equations in a particular gauge. To this end, we construct a generic truncated power-series solution and linearize the prolonged system of the Maxwell--Dirac equations about this solution. We then analyze the ranks of the coefficient matrices associated with the linearized system. Our results indicate that, generically, the spinor components are uniquely determined by the electromagnetic field and its derivatives. Furthermore, the fourth-order time derivatives of the components of the electromagnetic four-potential are uniquely determined by derivatives of the lower order with respect to time. These findings strongly suggest that the spinor field can be differential-algebraically eliminated, and the resulting equations describe independent evolution of the electromagnetic field, i.e., a Cauchy problem can be formulated in terms of the electromagnetic variables alone.

quant-ph↗
arXiv2024

The Dirac equation as a linear tensor equation for one component

The Dirac equation is one of the most fundamental equations of modern physics. It is a spinor equation, but some tensor equivalents of the equation were proposed previously. Those equivalents were either nonlinear or involved several components of the Dirac field. On the other hand, the author showed previously that the Dirac equation in electromagnetic field is equivalent to a fourth-order equation for one component of the Dirac spinor. The equivalency is used in this work to derive a linear tensor equivalent of the Dirac equation for just one component. This surprising result can be used in applications of the Dirac equation, for example, in general relativity or for lattice approximation of the Dirac field and can improve our understanding of the Dirac equation.

quant-ph↗
arXiv2022

Some Classical Models of Particles and Quantum Gauge Theories

The article contains a review and new results of some mathematical models relevant to the interpretation of quantum mechanics and emulating well-known quantum gauge theories, such as scalar electrodynamics (Klein-Gordon-Maxwell electrodynamics), spinor electrodynamics (Dirac-Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations. In the case of scalar electrodynamics, the scalar complex wave function can be made real by a gauge transformation, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe the independent evolution of the electromagnetic field. Similar results were obtained for spinor electrodynamics. Three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. A similar result was obtained for the Dirac equation in the Yang-Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general. One-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. This seems to enable the simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative.

physics.gen-ph↗
arXiv2022

The Dirac equation as one fourth-order equation for one function -- a general, manifestly covariant form

Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function. This was done for a specific (chiral) representation of gamma-matrices and for a specific component. In the current work, the result is generalized for a general representation of gamma-matrices and a general component (satisfying some conditions). The resulting equivalent of the Dirac equation is also manifestly relativistically covariant and should be useful in applications of the Dirac equation.

quant-ph↗
arXiv2022

Plasma-like Description for Elementary and Composite Quantum Particles

Schr{ö}dinger noticed in 1952 that a scalar complex wave function can be made real by a gauge transformation. The author showed recently that one real function is also enough to describe matter in the Dirac equation in an arbitrary electromagnetic or Yang--Mills field. This suggests some "symmetry" between positive and negative frequencies and, therefore, particles and antiparticles, so the author previously considered a description of one-particle wave functions as plasma-like collections of a large number of particles and antiparticles. The description has some similarities with Bohmian mechanics. This work offers a criterion for approximation of continuous charge density distributions by discrete ones with quantized charge based on the equality of partial Fourier sums, and an example of such approximation is computed using the homotopy continuation method. An example mathematical model of the description is proposed. The description is also extended to composite particles, such as nucleons or large molecules, regarded as collections including a composite particle and a large number of pairs of elementary particles and antiparticles. While it is not clear if this is a correct description of the reality, it can become a basis of an interesting model or useful picture of quantum mechanics.

physics.gen-ph↗
arXiv2021

Vacuum balloon -- a 350-year-old dream

The centuries-old idea of a lighter-than-air vacuum balloon has not materialized yet as such structure needs to be both light enough to float in the air and strong enough to withstand atmospheric pressure. We propose a design of a rigid spherical sandwich shell and demonstrate that it can satisfy these stringent conditions with commercially available materials, such as boron carbide ceramic and aluminum alloy honeycomb. A finite element analysis was employed to demonstrate that buckling can be prevented in the proposed structure. Also discussed are other modes of failure and approach to manufacturing.

physics.app-ph↗
arXiv2019

Towards differential elimination of spinor field from spinor electrodynamics

A system of PDEs for the electromagnetic field and one real component of the spinor field is generally equivalent to spinor electrodynamics. There are reasons to believe that the component can be differentially eliminated from the system. A Lagrangian depending on the electromagnetic field and one real component of the spinor field generally describes the same physics as spinor electrodynamics.

physics.gen-ph↗
arXiv2019

No Drama Quantum Electrodynamics?

This article builds on recent work (A. Akhmeteli, Int'l Journ. of Quantum Information, vol. 9, Suppl. (2011) p. 17, and A. Akhmeteli, Journ. Math. Phys., vol. 52 (2011) p. 082303), providing a theory that is based on spinor electrodynamics, is described by a system of partial differential equations in 3+1 dimensions, but reproduces unitary evolution of a quantum field theory in the Fock space. To this end, after introduction of a complex four-potential of electromagnetic field, which generates the same electromagnetic fields as the initial real four-potential, spinor field is algebraically eliminated from the equations of spinor electrodynamics. It is proven that the resulting equations for electromagnetic field describe independent evolution of the latter and can be embedded into a quantum field theory using a generalized Carleman linearization procedure. The theory provides a simple and at least reasonably realistic model, valuable for interpretation of quantum theory. The issues related to the Bell theorem are discussed.

math-ph↗
arXiv2018

The Dirac equation in a Yang-Mills field as an equation for just one real function

Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function, and the remaining component can be made real by a gauge transformation. This work extends the result to the case of the Dirac equation in the Yang-Mills field.

physics.gen-ph↗
arXiv2011

One real function instead of the Dirac spinor function

Three out of four complex components of the Dirac spinor can be algebraically eliminated from the Dirac equation (if some linear combination of electromagnetic fields does not vanish), yielding a partial differential equation of the fourth order for the remaining complex component. This equation is generally equivalent to the Dirac equation. Furthermore, following Schrödinger (Nature, \textbf{169}, 538 (1952)), the remaining component can be made real by a gauge transform, thus extending to the Dirac field the Schrödinger's conclusion that charged fields do not necessarily require complex representation. One of the two resulting real equations for the real function describes current conservation and can be obtained from the Maxwell equations in spinor electrodynamics (the Dirac-Maxwell electrodynamics). As the Dirac equation is one of the most fundamental, these results both belong in textbooks and can be used for development of new efficient methods and algorithms of quantum chemistry.

quant-ph↗
arXiv2010

No Drama Quantum Theory?

This work builds on the following result of a previous article (quant-ph/0509044): the matter field can be naturally eliminated from the equations of the scalar electrodynamics (the Klein-Gordon-Maxwell electrodynamics) in the unitary gauge. The resulting equations describe independent dynamics of the electromagnetic field (they form a closed system of partial differential equations). An improved derivation of this surprising result is offered in the current work. It is also shown that for this system of equations, a generalized Carleman linearization (Carleman embedding) procedure generates a system of linear equations in the Hilbert space, which looks like a second-quantized theory and is equivalent to the original nonlinear system on the set of solutions of the latter. Thus, the relevant local realistic model can be embedded into a quantum field theory. This model is equivalent to a well-established model - the scalar electrodynamics, so it correctly describes a large body of experimental data. Although it does not describe the electronic spin and possibly some other experimental facts, it may be of great interest as a "no drama quantum theory", as simple (in principle) as classical electrodynamics. Possible issues with the Bell theorem are discussed.

quant-ph↗
arXiv2008

Real-Valued Charged Fields and Interpretation of Quantum Mechanics II

In the first part of this work (http://www.arxiv.org/abs/quant-ph/0509044), it was shown that the Klein-Gordon-Maxwell electrodynamics in the unitary gauge allows natural elimination of the particle wave function and describes independent evolution of the electromagnetic field. Therefore, the electromagnetic field can be regarded as the guiding field in the Bohm interpretation of quantum mechanics. An extension of those results to the Dirac-Maxwell electrodynamics was less general, but represented at least an interesting toy model of quantum theory. Another model based on the Dirac-Maxwell electrodynamics is considered in this work. The model also typically allows elimination of the wave function and describes independent evolution of the electromagnetic field.

quant-ph↗
arXiv2006

Efficient Heating of Thin Cylindrical Targets by Broad Electromagnetic Beams II

In Part I (physics/0405091), it was shown that it is possible to achieve efficient heating of cylindrical targets by electromagnetic beams with transverse dimensions that are several orders of magnitude greater than those of the cylinder. Part II contains derivation of the detailed conditions of efficient heating in the longitudinal geometry and establishes a broader domain of parameters providing efficient heating in the transverse geometry. One possible implementation using currently available technology is considered: a nanotube (length 1 cm, diameter 15 nm) may be heated in the transverse geometry by a 30 femtosecond pulse of a 50 GW Ti:Sapphire laser (wavelength 800 nm) to a temperature of 1 keV with heating efficiency of 16 per cent. The resulting energy density exceeds 100 MJ/cm3. An exact formula in the form of a one-dimensional integral is obtained for the amount of energy absorbed in the cylinder heated by a gaussian electromagnetic beam in the longitudinal geometry.

physics.plasm-ph↗
arXiv2004

Efficient Heating of Thin Cylindrical Targets by Broad Electromagnetic Beams I

In many high-profile applications, such as nuclear fusion and pumping of active media of short-wavelength lasers, it is necessary to achieve high specific input of power of an electromagnetic beam in a target. Diffraction sets the lower limit to the transverse dimensions of electromagnetic beams and represents a fundamental obstacle for electromagnetic heating of small or inaccessible regions. It was found, however, that it is possible to achieve efficient heating of cylindrical targets by electromagnetic beams with transverse dimensions that are several orders of magnitude greater than those of the cylinder. These counter-intuitive results have the following physical mechanism: the absorption in the cylinder causes a deep fall in the field distribution, and this fall causes diffractive diffusion of the field towards the axis from a large volume of the beam. The heating efficiency was rigorously calculated using the exact solution of the problem of diffraction on an infinite homogeneous cylinder (J.R. Wait, 1955). Several non-resonant domains of parameters were found that provide efficient absorption of the energy of a broad beam in a thin conducting cylinder. The typical asymptotic efficiency for very thin cylinders is (2-5)/L, where L=ln(w/d), where d is the diameter of the cylinder and w is the width of the beam waist in the longitudinal geometry or the wavelength in the transverse geometry. This efficiency is high even if the beam is several orders of magnitude broader than the cylinder. The relevant conditions are not too rigid and may be used to heat cylindrical targets to high temperatures in a number of important applications. External magnetic field may further relax the conditions of efficient heating.

physics.plasm-ph↗