21 research outputs found
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.Comment: Version 8: slight simplificatio
Plasma-like Description for Elementary and Composite Quantum Particles
Schr{\"o}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.Comment: Formatting differs from that in the published versio
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.Comment: 30 pages, 1 figure (the number of pages differs from that of the
journal version due to different formatting). arXiv admin note: text overlap
with arXiv:2204.1029
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.Comment: 18 pages, 5 figures, 20 formula
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.Comment: 9 pages, no figures. Published in European Physical Journal C. A
clarification is added at the end of Section III. The journal version is at
http://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-013-2371-4.pdf
(open access