The book !

I just pre-published a very first draft on the usual site for independent thinkers. Ines Urdaneta has kindly agreed to go through it – and she will probably take over from here. That is good – because I need to get back to my day job. 🙂 She’s probably going to make a proper book of it. We’ll keep you updated on progress ! It’s sure not going to be a book like the others. Its working title is still the same: The Emperor has No Clothes: A Classical Explanation of Quantum Physics.

The whole idea is that a good electron and photon model can take all of the weirdness out of the QED sector of the Standard Model. What we’ve written so far, looks promising. Let’s see where it all goes ! 🙂

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Introductions…

As mentioned in my previous post, I thought it might be a good idea to release bits and pieces of my book on the go. It gives me the feeling I get something done, at least. 🙂

Introduction

If you are reading this book, then you are like me. You want to understand. Something inside of you tells you that the idea that we will never be able to understand quantum mechanics “the way we would like to understand it” does not make any sense. The quote is from Richard Feynman – probably the most eminent of all post-World War II physicists – and, yes, this is an aggressive opening. But you are right: our mind is flexible. We can imagine weird shapes and hybrid models. Hence, we should be able to understand quantum physics in some kind of intuitive way.

But what is intuitive? A lot of the formulas in this book feel intuitive to me, but that is only because I have been working with them for quite a while now. They may not feel very intuitive to you. However, I have confidence you will also sort of understand what they represent – intuitively, that is – because all of the formulas I use represent something we can imagine in space and in time – and I mean 3D space here: our Universe. Not the Universe of strings and hidden dimensions. An intuitive understanding of things means an understanding in terms of their geometry and their physicality.

You bought the right book. No hocus-pocus here. All physicists – and popular writers on physics – will tell you it is not possible. You see, the wavefunction of a particle – say, an electron – has this weird 720-degree symmetry, which we cannot really imagine. Of course, we have these professors doing the Dirac belt trick on YouTube – and wonderful animations (Jason Hise – whom I’ve been in touch with – makes the best ones) but, still, these visualizations all assume some weird relation between the object and the subject. In short, we cannot really imagine an object with a 720-degree symmetry.

The good news is: you don’t have to. The early theorists made a small mistake: they did not fully exploit the power of Euler’s ubiquitous a·eiθ function. The mistake is illustrated below – but don’t worry if this looks like you won’t understand: we’ll come back to it. It is a very subtle thing. Quantum physicists will tell you they don’t really think of the elementary wavefunction as representing anything real but, in fact, they do. And they will tell you, rather reluctantly because they are not so sure about what is what, that it might represent some theoretical spin-zero particle. Now, we all know spin-zero particles do not exist. All real particles – electrons, photons, anything – have spin, and spin (a shorthand for angular momentum) is always in one direction or the other: it is just the magnitude of the spin that differs. So it’s completely odd that the plus (+) or the minus (-) sign of the imaginary unit (i) in the a·e±iθ function is not being used to include the spin direction in the mathematical description.

clock

Figure 1: The meaning of +i and –i

Indeed, most introductory courses in quantum mechanics will show that both a·ei·θ = a·ei·(wtkx) and a·e+i·θ = a·e+i·(wtkx) are acceptable waveforms for a particle that is propagating in a given direction (as opposed to, say, some real-valued sinusoid). We would think physicists would then proceed to provide some argument showing why one would be better than the other, or some discussion on why they might be different, but that is not the case. The professors usually conclude that “the choice is a matter of convention” and, that “happily, most physicists use the same convention.” In case you wonder, this is a quote from the MIT’s edX course on quantum mechanics (8.01.1x).

That leads to the false argument that the wavefunction of spin-½ particles have a 720-degree symmetry. Again, you should not worry if you don’t get anything of what I write here – because I will come back to it – but the gist of the matter is the following: because they think the elementary wavefunction describes some theoretical zero-spin particle, physicists treat -1 as a common phase factor: they think we can just multiply a set of amplitudes – let’s say two amplitudes, to focus our mind (think of a beam splitter or alternative paths here) – with -1 and we’re going to get the same states. We find it rather obvious that that is not necessarily the case: -1 is not necessarily a common phase factor. We should think of -1 as a complex number itself: the phase factor may be +π or, alternatively, -π. To put it simply, when going from +1 to -1, it matters how you get there – and vice versa – as illustrated below.

rotation

Figure 2: e+iπ ¹ eiπ

I know this sounds like a bad start for a book that promises to be easy – but I just thought it would be good to be upfront about why this book is very different than anything you’ve ever read about quantum physics. If we exploit the full descriptive power of Euler’s function, then all weird symmetries disappear – and we just talk standard 360-degree symmetries in space. Also, weird mathematical conditions – such as the Hermiticity of quantum-mechanical operators – can easily be explained as embodying some common-sense physical law. In this particular case (Hermitian operators), we are talking physical reversibility: when we see something happening at the elementary particle level, then we need to be able to play the movie backwards. Physicists refer to it as CPT-symmetry, but that’s what it is really: physical reversibility.

The argument above involved geometry, and this brings me to a second mistake of the early quantum physicists: a total neglect of what I refer to as the form factor in physics. Why would an electron be some perfect sphere, or some perfect disk? In fact, we will argue it is not. It is a regular geometric shape – the Dirac-Kerr-Newman model suggests it’s an oblate spheroid – but so that’s not a perfect sphere. Once you acknowledge that, the so-called anomalous magnetic moment is not-so-anomalous anymore.

The mistake is actually more general than what I wrote above. We are thinking of the key constants in Nature as some number. Most notably, we think of Planck’s quantum of action (h ≈ 6.626×10−34 N·m·s) as some (scalar) number. Why would it be? It is – obviously – some vector quantity or – let me be precise – some matrix quantity: h is the product of a force (some vector in three-dimensional space), a distance (another three-dimensional concept) and time (one direction only). Somehow, those dimensions disappeared in the analysis. Vector equations became flat: vector quantities became magnitudes. Schrödinger’s equation should be rewritten as a vector or matrix equation. We do think of Planck’s quantum of action as some vector. We, therefore, think that the uncertainty – or the probabilistic nature of Nature, so to speak[1] – is not in its magnitude: it’s in its direction. But we are getting ahead of ourselves here – as usual. We should go step by step. Let us first acknowledge where we came from.

Before doing so, I would like to make yet another remark – one that is actually not so relevant for what we are going to try to do this in this book – and that is to understand the QED sector of the Standard Model geometrically – or physically, I should say. The innate nature of man to generalize did not contribute to greater clarity – in my humble opinion, that is. Feynman’s weird Lecture (Volume III, Chapter 4) on the key difference between bosons and fermions does not have any practical value: it just confuses the picture.

Likewise, it makes perfect sense to me to think that each sector of the Standard Model requires its own mathematical approach. I will briefly summarize this idea in totally non-scientific language. We may say that mass comes in one ‘color’ only: it is just some scalar number. Hence, Einstein’s geometric approach to gravity makes total sense. In contrast, the electromagnetic force is based on the idea of an electric charge, which can come in two ‘colors’ (+ or -), so to speak. Maxwell’s equation seemed to cover it all until it was discovered the nature of Nature – sorry for the wordplay – might be discrete and probabilistic. However, that’s fine. We should be able to modify the classical theory to take that into account. There is no need to invent an entirely new mathematical framework (I am talking quantum field and gauge theories here). Now, the strong force comes in three colors, and the rules for mixing them, so to speak, are very particular. It is, therefore, only natural that its analysis requires a wholly different approach. Hence, I would think the new mathematical framework should be reserved for that sector. I don’t like the reference of Aitchison and Hey to gauge theories as ‘the electron-figure’. The electron figure is a pretty classical idea to me. Hence, I do hope one day some alien will show us that the application of the Dyson-Feynman-Schwinger-Tomonaga ‘electron-figure’ to what goes on inside of the nucleus of an atom was, perhaps, not all that useful. A simple exponential series should not be explained by calculating a zillion integrals over 891 Feynman diagrams. It should be explained by a simple set of equations. If I have not lost you by now, please follow me to the acknowledgments section.

[1] A fair amount of so-called thought experiments in quantum mechanics – and we are not (only) talking the more popular accounts on what quantum mechanics is supposed to be all about – do not model the uncertainty in Nature, but on our uncertainty on what might actually be going on. Einstein was not worried about the conclusion that Nature was probabilistic (he fully agreed we cannot know everything): a quick analysis of the full transcriptions of his oft-quoted remarks reveal that he just wanted to see a theory that explains the probabilities. A theory that just describes them didn’t satisfy him.

PS: The book might take a while – as I am probably going to co-publish with another author (it’s perhaps a bit too big for me alone). In the meanwhile, you can just read the articles that are going to make up its contents.

Chapter I…

As mentioned in my previous post, I am going to publish a book. The Emperor has No Clothes. This is the introduction. I am probably going to release the various chapters one by one for my readers here. Thanks for being there ! The working title of the book is still the same:

The Emperor has No Clothes

A classical interpretation of quantum mechanics

I. Introduction, history and acknowledgments

This book is the result of a long search for understanding. The journey started about thirty-five years ago when – I was a teenager then – I started reading popular physics books. Gribbin’s In Search of Schrödinger’s Cat is just one of the many that left me unsatisfied in my quest for knowledge.

However, my dad never pushed me and so I went the easy route: humanities, and economics – plus some philosophy and a research degree afterwards. Those rather awkward qualifications (for an author on physics, that is) have served me well – not only because I had a great career abroad, but also because I now realize that physics, as a science, is in a rather sorry state: the academic search for understanding has become a race to get the next nonsensical but conformist theory published.

Why do we want to understand? What is understanding? I am not sure, but my search was fueled by a discontent with the orthodox view that we will never be able to understand quantum mechanics “the way we would like to understand it”, as Richard Feynman puts it. Talking Feynman, I must admit his meandering Lectures are the foundation of my current knowledge, and the reference point from where I started to think for myself. I had been studying them on and off – an original print edition that I had found in a bookshop in Old Delhi – but it was really the 2012 Higgs-Englert experiments in CERN’s LHC accelerator, and the award of the Nobel prize to these two scientists, that made me accelerate my studies. It coincided with my return from Afghanistan – where I had served for five years – and, hence, I could afford to reorient myself. I had married a wonderful woman, Maria, who gave me the emotional and physical space to pursue this intellectual adventure.

I started a blog (readingfeynman.org) as I started struggling through it all – and that helped me greatly. I fondly recall that, back in 2015, Dr. Lloyd N. Trefethen from the Oxford Math Institute reacted to a post in which I had pointed out a flaw in one of Richard Feynman’s arguments. It was on a topic that had nothing to do with quantum mechanics – the rather mundane topic of electromagnetic shielding, to be precise – but his acknowledgement that Feynman’s argument was, effectively, flawed and that he and his colleagues had solved the issue in 2014 only (Chapman, Hewett and Trefethen, The Mathematics of the Faraday Cage) was an eye-opener for me. Trefethen concluded his email as follows: “Most texts on physics and electromagnetism, weirdly, don’t treat shielding at all, neither correctly nor incorrectly. This seems a real oddity of history given how important shielding is to technology.” This resulted in a firm determination to not take any formula for granted – even if they have been written by Richard Feynman! With the benefit of hindsight, I might say this episode provided me with the guts to question orthodox quantum theory.

The informed reader will now wonder: what do I mean with orthodox quantum theory? I should be precise here, and I will. It is the modern theory of quantum electrodynamics (QED) as established by Dyson, Schwinger, Feynman, Tomonaga and other post-World War II physicists. It’s the explanation of the behavior of electrons and photons – and their interactions – in terms of Feynman diagrams and propagators. I instinctively felt their theory might be incomplete because it lacks a good description of what electrons and photons actually are. Hence, all of the weirdness of quantum mechanics is now in this weird description of the fields – as reflected in the path integral formulation of quantum mechanics. Whatever an electron or a photon might be, we cannot really believe that it sort of travels along an infinite number of possible spacetime trajectories all over space simultaneously, can we?

I also found what Brian Hayes refers to as “the tennis match between experiment and theory” – the measurement (experiment) or calculation (theory) of the so-called anomalous magnetic moment – a rather weird business: the complexity in the mathematical framework just doesn’t match the intuition that, if the theory of QED has a simple circle group structure, one should not be calculating a zillion integrals all over space over 891 4-loop Feynman diagrams to explain the magnetic moment of an electron in a Penning trap. There must be some form factor coming out of a decent electron model that can explain it, right?

Of course, all of the above sounds very arrogant, and it is. However, I always felt I was in good company, because I realized that not only Einstein but the whole first generation of quantum physicists (Schrödinger, Dirac, Pauli and Heisenberg) had become skeptical about the theory they had created – if only because perturbation theory yielded those weird diverging higher-order terms. With the benefit of hindsight, we may say that the likes of Dyson, Schwinger, Feynman – the whole younger generation of mainly American scientists who dominated the discourse at the time – lacked a true general: they just kept soldiering on by inventing renormalization and other mathematical techniques to ensure those weird divergences cancel out, but they had no direction.

However, I should not get ahead of myself here. This is just an introduction, after all. Before getting to the meat of the matter, I should just make some remarks and acknowledge all the people who supported me in this rather lonely search. First, whom am I writing for? I am writing for people like me: amateur physicists. Not-so-dummies, that is. People who don’t shy away from calculations. People who understand a differential equation, some complex algebra and classical electromagnetism – all of which are, indeed, necessary, to understand anything at all in this field. I have good news for these people: I have come to the conclusion that we do not need to understand anything about gauges or propagators or Feynman diagrams to understand quantum electrodynamics.

Indeed, rather than “using his renormalized QED to calculate the one loop electron vertex function in an external magnetic field”, Schwinger should, perhaps, have listened to Oppenheimer’s predecessor on the Manhattan project, Gregory Breit, who wrote a number of letters to both fellow scientists as well as the editors of the Physical Review journal suggesting that the origin of the so-called discrepancy might be due to an ”intrinsic magnetic moment of the electron of the order of αµB.” In other words, I do not think Breit was acting schizophrenic when complaining about the attitude of Kusch and Lamb when they got the 1955 Nobel Prize for Physics for their work on the anomalous magnetic moment. I think he was just making a very sensible suggestion – and that is that one should probably first try investing in a good theory of the electron before embarking on mindless quantum field calculations.

My search naturally led me to the Zitterbewegung hypothesis. Zitter is German for shaking or trembling. It refers to a presumed local oscillatory motion – which I now believe to be true, whatever that means. Erwin Schrödinger found this Zitterbewegung as he was exploring solutions to Dirac’s wave equation for free electrons. In 1933, he shared the Nobel Prize for Physics with Paul Dirac for “the discovery of new productive forms of atomic theory”, and it is worth quoting Dirac’s summary of Schrödinger’s discovery:

“The variables give rise to some rather unexpected phenomena concerning the motion of the electron. These have been fully worked out by Schrödinger. It is found that an electron which seems to us to be moving slowly, must actually have a very high frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us. As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light. This is a prediction which cannot be directly verified by experiment, since the frequency of the oscillatory motion is so high and its amplitude is so small. But one must believe in this consequence of the theory, since other consequences of the theory which are inseparably bound up with this one, such as the law of scattering of light by an electron, are confirmed by experiment.” (Paul A.M. Dirac, Theory of Electrons and Positrons, Nobel Lecture, December 12, 1933)

Dirac obviously refers to the phenomenon of Compton scattering of light by an electron. Indeed, as we shall see, the Zitterbewegung model naturally yields the Compton radius of an electron and – as such – effectively provides some geometric explanation of what might be happening. It took me a while to figure out that some non-mainstream physicists had actually continued to further explore this concept, and the writings of David Hestenes from the Arizona State University of Arizona who – back in 1990 – proposed a whole new interpretation of quantum mechanics based on the Zitterbewegung concept (Hestenes, 1990, The Zitterbewegung Interpretation of Quantum Mechanics) made me realize there was sort of a parallel universe of research out there – but it is not being promoted by the likes of MIT, Caltech or Harvard University – and, even more importantly, their friends who review and select articles for scientific journals.

I reached out to Hestenes, but he is 85 by now – and I don’t have his private email, so I never got any reply to the one or two emails I sent him on his ASU address. In contrast, Giorgio Vassallo – one of the researchers of an Italian group centered around Francesco Celani – who followed up on the Schrödinger-Hestenes zbw model of an electron – politely directed me towards Dr. Alex Burinskii (I should have put a Prof. and/or Dr. title in front of every name mentioned above, because they all are professors and/or doctors in science). Both have been invaluable – not because they would want to be associated with any of our ideas – but because they gave me the benefit of the doubt in their occasional but consistent communications. Hence, I would like to thank them here for reacting and encouraging me for at least trying to understand.

I think Mr. Burinskii deserves a Nobel Price, but he will probably never get one – because it would question not one but two previously awarded Nobel Prizes (1955 and 1965). We feel validated because, in his latest communication, Dr. Burinskii wrote he takes my idea of trying to corroborate his Dirac-Kerr-Newman electron model by inserting it into models that involve some kind of slow orbital motion of the electron – as it does in the Penning trap – seriously. [He is working on an article right now, and I am sure it is going to take a lot of people out of their comfort zone – which is always a good thing.]

It is now time to start the book. However, before we do so, I should wrap up the acknowledgments section, so let us do that here. I have also been in touch with Prof. Dr. John P. Ralston, who wrote one of a very rare number of texts that, at the very least, tries to address some of the honest questions of amateur physicists and philosophers upfront. I was not convinced by his interpretation of quantum mechanics, but I loved the self-criticism of the profession: “Quantum mechanics is the only subject in physics where teachers traditionally present haywire axioms they don’t really believe, and regularly violate in research.” We exchanged some messages, but then concluded that our respective interpretations of the wavefunction are very different and, hence, that we should not “waste any electrons” (his expression) on trying to convince each other. In the same vein, I should mention some other seemingly random exchanges – such as those with the staff and fellow students when going through the MIT’s edX course on quantum mechanics which – I admit – I did not fully complete because, while I don’t mind calculations in general, I do mind mindless calculations.

I am also very grateful to my brother, Prof. Dr. Jean Paul Van Belle, for totally unrelated discussions on his key topic of research (which is information systems and artificial intelligence), which included discussions on Roger Penrose’s books – mainly The Emperor’s New Mind and The Road to Reality. These discussions actually provided the inspiration for the earlier draft title of this book: The Emperor has no clothes: the sorry state of Quantum Physics. We will go for another mountainbike or mountain-climbing adventure when this project is over.

Among other academics, I would like to single out Dr. Ines Urdaneta. Her independent research is very similar to ours. She has, therefore, provided much-needed moral support and external validation. We also warmly thank Jason Hise, whose wonderful animations of 720-degree symmetries did not convince me that electrons – as spin-1/2 particles – actually have such symmetries – but whose communications stimulated my thinking on the subject-object relation in quantum mechanics.

Finally, I would like to thank all of my friends (my university friends, in particular (loyal as ever), and I will also single out Soumaya Hasni, who has provided me with a whole new fan club here here in Brussels) and, of course, my family, for keeping me sane. I would like to thank, in particular, my children – Hannah and Vincent – and my wife, Maria, for having given me the emotional, intellectual and financial space to grow into the person I am right now.

So, now we should really start the book. Its structure is simple. In the first chapters, I’ll just introduce the most basic math – Euler’s function, basically – and then we’ll take it from there. I will regularly refer to a series of papers I published on what I refer to as the Los Alamos Site for Spacetime Rebels: vixra.org. The site is managed by Phil Gibbs. I would like to acknowledge and thank him here for providing a space for independent thinkers. You can find my papers on http://vixra.org/author/jean_louis_van_belle. They are numbered, and I will often refer to those papers by mentioning their number between square brackets. In fact, this very first version of this book follows the structure of paper [17]. Click on the link above, have a look, and you’ll understand. 😊

Or so I hope. This brings me to the final point in my introduction. This is just the first version of this book. It is rather short – cryptic, I’d say. As such, you might give up after a few pages and say: this may be a classical interpretation but it is not an easy one. You are right. But let me say two things to you:

  1. It may not be easy, but it is definitely easier than whatever else you’ll read when exploring the more serious stuff.
  2. To get my degree in philsophy, I had to study Wittgenstein’s Tractatus Logico-Philosophicus. I hated that booklet – not because it is dense but because it is nonsense. Wittgenstein wasn’t even aware of the scientific revolution that was taking place while he was writing it. Still, it became a bestseller. Why? Because it was so abstruse it made people think for themselves.

The first version of this book is going to be dense but – hopefully – you will find it is full of sense. If so (I’ll find out from the number of copies sold), I might go through the trouble of unpacking it in the second edition. 🙂

Jean Louis Van Belle,  7 January 2019

[START OF THE BOOK]

[FIRST CHAPTERS: EXPLAIN EULER’S FORMULA AND BASIC THEORY.]

[OTHER CHAPTERS: SEE VIXRA.ORG]

My new book project

Dear readers of this blog – As you may or may not know, I had already published two or three books on amazon.com with some of the ideas on the geometric of physical interpretation of the wavefunction that I have been promoting on this blog. These books sold some copies but – all in all – were not a huge success. That’s fine – because I just wanted to try things out.

I will soon come up with an entirely new book. Its working title is what is mentioned in the current draft of the acknowledgments – copied below. The e-book will be published in a few weeks from now. It may – by some magic 🙂 – coincide with the publication of a convincing classical explanation of the anomalous magnetic moment of an electron – not written by me, of course, but by one of the foremost experts on quantum gravity (and QED in general). 🙂 It would upset the orthodox/mainstream/Copenhagen interpretation of quantum electrodynamics, and that will be a good thing: it will bring more reality to the interpretation (read: just a much easier way to truly understand everything).

If so, my book should sell – if only because it will document a history of scientific discovery. 🙂

The Emperor has no clothes:

The sorry state of Quantum Physics.

Acknowledgements

Although Dr. Alex Burinskii, Dr. Giorgio Vassallo and Dr. Christoph Schiller would probably prefer not to be associated with anything we write, they gave us the benefit of the doubt in their occasional, terse, but consistent communications and, hence, we would like to thank them here – not for believing in anything we write but for encouraging us for at least trying to understand.

More importantly, they made me realize that QED, as a theory, is probably incomplete: it is all about electrons and photons, and the interactions between the two – but the theory lacks a good description of what electrons and photons actually are. Hence, all of the weirdness of Nature is now, somehow, in this weird description of the fields: perturbation theory, gauge theories, Feynman diagrams, quantum field theory, etcetera. This complexity in the mathematical framework does not match the intuition that, if the theory has a simple circle group structure[1], one should not be calculating a zillion integrals all over space over 891 4-loop Feynman diagrams to explain the magnetic moment of an electron in a Penning trap.[2] We feel validated because, in his latest communication, Dr. Burinskii wrote he takes our idea of trying to corroborate his Dirac-Kerr-Newman electron model by inserting it into models that involve some kind of slow orbital motion of the electron – as it does in the Penning trap – seriously.[3]

There are some more professors who may or may not want to be mentioned but who have, somehow, been responsive and, therefore, encouraging. I fondly recall that, back in 2015, Dr. Lloyd N. Trefethen from the Oxford Math Institute reacted to a blog article on mine[4] – in which I pointed out a potential flaw in one of Richard Feynman’s arguments. It was on a totally unrelated topic – the rather mundane topic of shielding, to be precise – but his acknowledgement that Feynman’s argument was, effectively, flawed and that he and his colleagues had solved the issue in 2014 only (Chapman, Hewett and Trefethen, The Mathematics of the Faraday Cage) was an eye-opener for me. Trefethen concluded his email as follows: “Most texts on physics and electromagnetism, weirdly, don’t treat shielding at all, neither correctly nor incorrectly. This seems a real oddity of history given how important shielding is to technology.” When I read this, it made me think: how is it possible that engineers, technicians, physicists just took these equations for granted? How is it possible that scientists, for almost 200 years,[5], worked with a correct formula based on the wrong argument? This, too, resulted in a firm determination to not take any formula for granted but re-visit its origin instead.[6]

We have also been in touch with Dr. John P. Ralston, who wrote one of a very rare number of texts that address the honest questions of amateur physicists and philosophers upfront. I love the self-criticism of the profession: “Quantum mechanics is the only subject in physics where teachers traditionally present haywire axioms they don’t really believe, and regularly violate in research.”[7] We both concluded that our respective interpretations of the wavefunction are very different and, hence, that we should not  waste any electrons on trying to convince each other. However, the discussions were interesting.

I am grateful to my brother, Dr. Jean Paul Van Belle, for totally unrelated discussions on his key topic of research (which is information systems and artificial intelligence), which included discussions on Roger Penrose’s books – mainly The Emperor’s New Mind and The Road to Reality. These books made me think of a working title for the book: The Emperor has no clothes: the sorry state of Quantum Physics. We should go for another mountainbike or mountain-climbing adventure when this project is over.

Among other academics, I would like to single out Dr. Ines Urdaneta who – benefiting from more academic freedom than other researchers, perhaps – has just been plain sympathetic and, as such, provided great moral support. I also warmly thank Jason Hise, whose wonderful animations of 720-degree symmetries did not convince me that electrons (or spin-1/2 particles in general) actually have such symmetries – but whose communications stimulated my thinking on the subject-object relation in quantum mechanics.

Finally, I would like to thank all my friends and my family for keeping me sane. I would like to thank, in particular, my children – Hannah and Vincent – and my wife, Maria, for having given me the emotional, intellectual and financial space to pursue this intellectual adventure.

[1] QED is an Abelian gauge theory with the symmetry group U(1). This sounds extremely complicated – and it is. However, it can be translated as: its mathematical structure is basically the same as that of classical electromagnetics.

[2] We refer to the latest theoretical explanation of the anomalous magnetic moment here: Stefano Laporta, High-precision calculation of the 4-loop contribution to the electron g-2 in QED, 10 July 2017, https://arxiv.org/abs/1704.06996.

[3] Prof. Dr. Burinskii, email communication, 29 December 2018 2.13 pm (Brussels time). To be precise, he just wrote me to say he is ‘working on the magnetic moment’. I interpret this as saying he is looking at his model again to calculate the magnetic moment of the Dirac-Kerr-Newman electron so we will be in a position to show how the Kerr-Newman geometry – which I refer to as the (neglected) form factor in QED – might affect it. To be fully transparent, Dr. Burinskii made it clear his terse reactions do not amount to any endorsement or association of the ideas expressed in this and other papers. It only amounts to an admission our logic may have flaws but no fatal errors – not at first reading, at least.

[4] Jean Louis Van Belle, The field from a grid, 31 August 2015, https://readingfeynman.org/2015/08/.

[5] We should not be misunderstood here: the formulas – the conclusions – are fully correct, but the argument behind was, somehow, misconstrued. As Faraday performed his experiment with a metal mesh (instead of a metal shell) in 1836, we may say it took mankind 2014 – 1836 = 178 years to figure this out. In fact, the original experiments on Faraday’s cage were done by Benjamin Franklin – back in 1755, so that is 263 years ago!

[6] We reached out to Dr. Trefethen and some of his colleagues again to solicit comments on our more recent papers, but we received no reply. Only Dr. André Weideman wrote us back saying that this was completely out of his field and that he would, therefore, not invest in it.

[7] John P. Ralston, How to understand quantum mechanics (2017), p. 1-10.