8.3 Topologic Hole in Solutions of Maxwell's Equations

8.3  Topologic Hole in Solutions of Maxwell's Equations


We have inadvertently made a mistake. [In]

rψ = f(tr) (20.35)

We have not solved the free wave equation

(Ñ22/t2)ψ(r,t) = 0 (20.33)

everywhere; we have solved Eq. (20.33) with zero on the right everywhere, except at the origin. Our mistake crept in because some of the steps in our derivation are not «legal» when r=0. [28,Ch.20]

     So, although we thought we were solving [Laplacian's equation Ñ2ψ=0] for the potential in free space, our solution [ψ=a+b/r] also gives the field for a point source at the origin. [28,Ch.20]

Richard Feynman

These «confessions» are taken from:

The Feynman Lectures on PhysicsFLP
6  Electrodynamics [28]
Chapter 20  Solutions of Maxwell's Equations in Free Space
§4. Spherical waves

I believe it will not be a big stretch to attribute these «confessions» to Richard Feynman. It is necessary to specially emphasize the very fact of the well considered mentioning in FLP this feature of the solution of the wave equation in the form of spherical waves. Taking an opportunity, I express my deepest gratitude to the generous Genius of Richard Feynman for such caution signs placed by him in many node points of the choice of methods of the description of the physical reality. Feynman, while helping both, those only starting studying the difficult technique of walking the routes of his favourite Physics, and those who already saved up considerable experience in this field, specifies providently the major moments of turn of the thought, its choice of routes, and mathematical means used at it.

The constituent of Feynman's printed work, which to this day regularly captivates and inspires more and more new admirers and creators of Physics, is unique for us in its importance. Today as well it still forms successfully those so competent and devoted to it, so enthusiastically occupied with discovering and finishing the fragments of primeval Beauty and Simplicity of the Nature. But, – this component, as if by virtue of its nature, as an invisible Cinderella, escapes a due estimation and appropriate attention of the managers of balls. Especially that side of it which is faced with a set of the bright captivating images, and the bewitching panoramas, which in their turn are opening from the discovered by Feynman and carefully surveyed by him points of consideration, located far away from fashionable research routes.

Many items from this thesaurus were found and tested by him in the creative union with his seniour colleague and Teacher – John Wheeler during their joint activities at exhausting ascensions to the sacred truths. Much was discovered thanks to the ability of Feynman's Genius to perform a Deed, that received duly additional charging from such a powerful source his Teacher was. Both together and separately, again and again they were «overwinding», as it seemed, indisputable truths, and found necessary for them toolkit among the ideas left by others a long time ago. A lot of invaluable scetches from their individual «cabinets of curiosities» of physicists-theorists are still wandering around the world lonely in search of their named princes. Even to such a titled prince of Physics whom by general recognition became Richard Feynman, it was necessary to wait almost twenty years for the Nobel crowning in order to, in a dignified manner, present at the ball in his honour the doubtless hero of the occasion, – the Cinderella-assistant – the mistress of his youth, whereas now it's become an old lady.

We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing. Since winning the prize is a personal thing, I thought I could be excused in this particular situation, if I were to talk personally about my relationship to quantum electrodynamics, rather than to discuss the subject itself in a refined and finished fashion… So, what I would like to tell you about today are the sequence of events, really the sequence of ideas, which occurred, and by which I finally came out the other end with an unsolved problem for which I ultimately received a prize.[30][link]

The greatest merit of Feynman's Genius in Physics is that of the precise formulation and the subsequent true service to his basic THESIS, which managed all his researches, the one Feynman left to us in the form of the creative WILL:

The mechanisms responsible for reception
of the infinite charge and mass of electron in QED,
are inherited and transferred by it directly
from Classical Electrodynamics of Maxwell–Lorentz

The roots of this «feature» or «difficulty» are based in historically prevalent procedures of the choice of decisions of Maxwell's equations and in their physical interpretations, which we trust fully and which in many ways determine, in their turn, these very procedures of the choice and physical decisions. The main, entirely determining the final result, true bases of our fundamental problems have deeply sprouted into the evolutionally accumulated layers of our «natural» intuition. It appeared to be naturally laid out during the evolution (phylogenesis) of our collective cognizing «I» from a number of historically arising and consecutively laid on each other layers (accumulated paradigmatic rings). Each such next and newest layer-outgrowth on the tree of our intuition lay down directly on the previous one and entirely depended on its extensive ramified network of former semantic «world» connections and communications, subjected only to some rearrangement of the domination and degrees of necessity of the binding strings (ways) in early underlying layers.

Each new external layer of the «natural» intuition was accumulated from some, badly suited to scientific inheritance, set of individual auxiliary scaffolding and props, various sensually painted adaptations and motivations, binding elements of «faith» into the basic practicability of program expectations etc., claimed during the construction, further furnishing or transfer to the successors (heirs) of the newest theoretical scheme (the program, the paradigm, the picture of the world, …). These auxiliary individual (sensual, analog, right-brain) props to the newest theory, as a rule, are not inherited directly from its author complete with a determining mathematical formalism. They, generally, are selected independently by its supporters or are independently developed by the representatives of new generation during studying and developing this theory – its interweaving with always very intimate and unique, multilayered and multiscrappied a coverlet-network, thrown by everyone in their own way on the «general» World during their own formation-development (ontogenesis) in this very World.

At times, without good reasons, it is declared that Special Relativity (SR) and Quantum Theory have cardinally reconstructed the base of «Philosophiae Naturalis» laid in the works of Isaac Newton and James Maxwell. In reality, – «Principia Mathematica» of Newton have immortalized «Philosophiae Naturalis» laying behind them so thoroughly, that both the electromagnetic field theory of Faraday–Maxwell, and the physical revolutions of the 20th century, by and large, could not escape its sweet captivity. We still think in terms of corpuscles cooperating through field «in themselves», moving in some space and time. Yes, – we have corrected the concepts of both corpuscles, having given them in addition to mass another quantum numbers, and new kinematics, as well as containing them space, having supplied it with richer structure; have named the modernized corpuscles the quanta of the appropriate fields and have transformed laws of their movement and interaction…

Already from time when Faraday and Maxwell painfully searched for the means of describing the phenomena of electricity and magnetism, as an obstacle in a way of progress in fundamental physics «the sentry sphinx» of the initial concept of an unstructured CORPUSCLE acted. The classical electromagnetic field theory has given corpuscles in addition to mass also the electric charge, that was responsible for the creation of force field in space surrounding a charged corpuscle. As sources of the field the charged corpuscles began to act, being completely alien element in the field theory in which they are impossible by virtue of existing, according to this theory, electric forces of pushing away of the likely charged parts. The progress, achieved by the electronic theory of Lorentz, constituted for many physicists temporary justification for use of the theory constructed on so contradictory bases. SR, – only partly having corrected the classical field theory, at the same time, – has deepened the gap between a field and its charged sources-corpuscles.

It is even claimed that SR has finished the construction of classical electrodynamics (the field theory) of Maxwell–Lorentz. This is an error tragic for Physics. Having seized only a part of the truth correctly, SR, not in absolutely lawful though effective enough a way, absolutized one of the group properties of Maxwell–Lorentz equations (ML-equations). Wrongly believing observable electrons to be the initial sources of the field, described by the ML-equations, SR has extended kinematics of the charged sources of a field to all without exception corpuscles and fields, which can ever appear to be claimed in the future. Thus, for the whole century, the searches of true sources of a field which are distinct from corpuscles-electrons appeared to be frozen. Relativistic kinematics is certainly fair and necessary, but not in the least for electrons-corpuscles. Kinematics of Minkowski (M-kinematics) is unambiguously demanded by the ML-equations for their sources of a field which are included in the right parts of these equations. At the same time, by virtue of their very nature, electrons-corpuscles do not suit for the role of such initial sources of a field. Accordingly, – electrons-corpuscles follow M-kinematics only asymptotically, when it is possible to neglect their interactions with the radiation loss of energy.

Electron-corpuscles, as well as other long-living elementary particles, are still waiting for the hour to appear to our look as law-abiding products of the full field theory. Electron is the most symmetric and steady electrodynamic field structure, continuously self-reproducing in process of interaction between its own extensive spatial system of the field charged subcurrents of two charge signs – the sources of a field – and the collectively produced by them working field of electron.

Another obstacle in a way of evolution of the models applying for the «true» description of physical reality, is the sort of frank and very much militant, on its origin, protective «ethic», which spread widely approximately from the middle of XX century among a dominant part of new generation of physicist of quantum formation. At the cradle of this «ethic», and backdating at that, – there appeared to be put (in deformed, somewhat comical version) titanic efforts of Nils Bohr and his team, the efforts to protect their child from the attacks of eminent otherwise-minded contemporaries. It is well-known, that Copenhagen interpretation of the quantum mechanics was exposed to numerous and very sharp attacks of Albert Einstein, Max Planck, Arnold Sommerfeld, Max fon Laue, Lui de Broil', Erwin Schrodinger and others physicist, who didn't give up hope of successful development of the own programs essentially differing from the Copenhagen one. It is significant, that the largest contemporaries of Bohr, the physicists, who opened with their works the quantum epoch, who presented the world with the most part of fragments of quantum mosaic and mathematical formalisms of quantum physics, appeared then to be otherwise-minded.

This extremely unethical for scientists and gnoseologically absurd «ethic» was quickly multiplied and, not having met due repulse, spread to other areas of knowledge. It got the use in the form, unreasonably exaggerated to the epistemological bases that as if have to be considered in any realization of theoretical knowledge, practiced on the given historical moment and capable of calling itself scientific. This «new growth» not only allows, but imperatively orders in a circle of the selected to ask only «correct» questions and encourages to ignore «inconvenient» ones for the orthodox theory, attaching to them arbitrarily a label of «incorrectly put». And all these both shameful, and vulgar substitutions are presented to scientific community as a remarkable achievement, that is as if sanctified by the newest fundamental results, obtained by physicist as a result of a significant break-through in the ways of understanding secrets of the universe.

The vivid sketch from such an «ethic» afoot is given in the work

Emission–Absorption–Scattering (EAS) Particle Physics, Robert S. Fritzius,

In which the author acquaints us with the results of his own researches of interaction mechanisms of electrically charged corpuscles in Quantum Electrodynamics – QED – Dirac–Feynman. This work is the frank and colourfully performed report on experience of «walking on sorrows» in the process of making the individual «Philosophiae Naturalis».

The topological hole in decisions of the ML-equations «is done» by the efforts of our «natural» intuition or, more particularly, – by the concept of a relativistic dot corpuscle-electron together with a principle of relativistic causality. This principle leads us still further from the observance of symmetry of the ML-equations concerning the changing of a sign of time or speed of movement of a field source. The situation is aggravated with the next circumstance: it is very difficult for us to escape from under «the hypnosis» on the part of our extensive former positive experience in theoretical interpretation of many and many experimental situations in which the use of concept of corpuscle-electron was quite justified. All this historically caused MASS of the last successes of the theory, disarms us literally at the slightest attempts to weaken the chains of our «natural» intuition regarding the «valid» nature of sources of field.

Really, – teaches us our own «natural» intuition – can't you see that the charged corpuscle-electron as a source of a field can not move simultaneously in two opposite directions at once, since the electron-corpuscles observed in experiments are not capable of it. This «Gordian knot» can be split by one, but rather strong will effort. It is enough for this purpose to cut off all superfluous from the concept of charged corpuscles – sources of the field, the concept we got in the process of for a long time accumulating and assimilating our macroscopic experience. In the concept of a field source it is necessary to leave in inviolability only that is natural and is in agreement, without any violence on the part of our obliging «natural» intuition, with mathematical nature of the ML-equations and their symmetry. It is necessary to rely entirely, and without unnecessary looking back at rich macroscopic experience, on imperatives of mathematical structure of the ML-equations which tell us about the physical reality ciphered in them more authentically. And our obliging «natural» intuition, by the very virtue of its nature, is always in a captivity of its former achievements and is extremely inertial and dogmatic.

As «the proof» of possibility of putting the question as that, taken from the history of theoretical idea, I shall quote again the surprising remark of Hendrik Lorentz in his «Theory of Electrons», the remark having a shade of prophesy or even a will.

We might even go further and imagine any number of charges with density ρ1,ρ2,…, which have the property of mutual permeability, taking the same part of space, and every of them, at the same time, moves with its own speed. It takes us to the change of the terms ρ and ρv in [Maxwell equations] through ρ1+ρ2+… and ρ1v1+ρ2v2+…, where vectors v1,v2,… are the speed of individual charges. Such an assumption, whatever artificial it might seem, turns out to be useful in one of the problems we have to analyze.

 The translation from Russian was made by Masha and Natasha Zazerska
Last modifications: April 27 2003
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The Literature Quoted:
28. Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics, Vol.2 – Addison-Wesley Publishing Co., London, 1964
30. Feynman R. The Development of the Space-Time View of Quantum Electrodynamics, Physics Today, 19, 31, 1966
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