For my followers, I should note I’ve published a few more papers on viXra.org as well as on the academia.edu site. My thoughts and reflections need a bit more space than a blog post now! Enjoy! Kindest regards, Jean Louis Van Belle
A young researcher, Oliver Consa, managed to solve a complicated integral: he gave us an accurate calculation of the anomalous magnetic moment based on a (semi-)classical model. Here is the link to his paper, and this is the link to my first-order approach. I admit: he was first. Truth doesn’t need an author. 🙂
This is a great achievement. We now have an electron model that explains all of the mysterious ‘intrinsic’ properties of the electron. It also explains the interference of an electron with itself. Most importantly, the so-called ‘precision test of QED’ (the theoretical and experimental value of the anomalous magnetic moment) also gets a ‘common-sense’ interpretation now. Bye-bye QFT!
So now it’s time for the next step(s). If you have followed this blog, then you know I have a decent photon model too – and other researchers – most are small names but there are one or two big names as well 🙂 – are working to refine it.
The End of Physics is near. Mankind knows everything now. Sadly, this doesn’t solve any of the major issues mankind is struggling with (think of inequality and climate change here).
Post scriptum: When you check the references, it would seem that Consa borrowed a lot of material from the 1990 article he mentions as a reference: David L. Bergman and J. Paul Wesley, Spinning Charged Ring Model of Electron Yielding Anomalous Magnetic Moment, Galilean Electrodynamics, Vol. 1, Sept-Oct 1990, pp. 63–67). It is strange that David Hestenes hadn’t noted this article, because it goes back to the same era during which he tried to launch the Zitterbewegung interpretation of quantum physics ! I really find it very bizarre to see how all these elements for a realist interpretation of quantum physics have been lying around for many decades now. I guess it’s got to do with what Sean Carroll suggested in his 7 Sept 2019 opinion article in the NY Times: mainstream physicists do not want to understand quantum mechanics.
I started to edit and add to the rather useless Wikipedia article on the Zitterbewegung. No mention of Hestenes or more recent electron models (e.g. Burinskii’s Kerr-Newman geometries. No mention that the model only works for electrons or leptons in general – not for non-leptonic fermions. It’s plain useless. But all the edits/changes/additions were erased by some self-appointed ‘censor’. I protested but then I got reported to the administrator ! What can I say? Don’t trust Wikipedia. Don’t trust any ‘authority’. We live in weird times. The mindset of most professional physicists seems to be governed by ego and the Bohr-Heisenberg Diktatur.
For the record, these are the changes and edits I tried to make. You can compare and judge for yourself. Needless to say, I told them I wouldn’t bother to even try to contribute any more. I published my own article on the Vixrapedia e-encyclopedia. Also, as Vixrapedia did not have an entry on realist interpretations of quantum mechanics, I created one: have a look and let me know what you think. 🙂
Zitterbewegung (“trembling” or “shaking” motion in German) – usually abbreviated as zbw – is a hypothetical rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (up to the speed of light) of the position of an electron around the median, with an angular frequency of ω = 2mc2/ħ, or approximately 1.5527×1021 radians per second. Paul Dirac was initially intrigued by it, as evidenced by his rather prominent mention of it in his 1933 Nobel Prize Lecture (it may be usefully mentioned he shared this Nobel Prize with Schrödinger):
“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.”
In light of Dirac’s later comments on modern quantum theory, it is rather puzzling that he did not pursue the idea of trying to understand charged particles in terms of the motion of a pointlike charge, which is what the Zitterbewegung hypothesis seems to offer. Dirac’s views on non-leptonic fermions – which were then (1950s and 1960s) being analyzed in an effort to explain the ‘particle zoo‘ in terms of decay reactions conserving newly invented or ad hoc quantum numbers such as strangeness – may be summed up by quoting the last paragraph in the last edition of his Principles of Quantum Mechanics:
“Now there are other kinds of interactions, which are revealed in high-energy physics. […] These interactions are not at present sufficiently well understood to be incorporated into a system of equations of motion.”
Indeed, in light of this stated preference for kinematic models, it is somewhat baffling that Dirac did not follow up on this or any of the other implications of the Zitterbewegung hypothesis, especially because it should be noted that a reexamination of Dirac theory shows that interference between positive and negative energy states is not a necessary ingredient of Zitterbewegung theories. The Zitterbewegung hypothesis also seems to offer interesting shortcuts to key results of mainstream quantum theory. For example, one can show that, for the hydrogen atom, the Zitterbewegung produces the Darwin term which plays the role in the fine structure as a small correction of the energy level of the s-orbitals. This is why authors such as Hestenes refers to it as a possible alternative interpretation of mainstream quantum mechanics, which may be an exaggerated claim in light of the fact that the zbw hypothesis results from the study of electron behavior only.
Zitterbewegung models have mushroomed and it is, therefore, increasingly difficult to distinguish between them. The key to understanding and distinguishing the various Zitterbewegung models may well be Wheeler‘s ‘mass without mass’ idea, which implies a distinction between the idea of (i) a pointlike electric charge (i.e. the idea of a charge only, with zero rest mass) and (ii) the idea of an electron as an elementary particle whose equivalent mass is the energy of the zbw oscillation of the pointlike charge. The ‘mass without mass’ concept requires a force to act on a charge – and a charge only – to explain why a force changes the state of motion of an object – its momentum p = mγ·v(with γ referring to the Lorentz factor) – in accordance with the (relativistically correct) F = dp/dt force law.
As mentioned above, the zbw hypothesis goes back to Schrödinger’s and Dirac’s efforts to try to explain what an electron actually is. Unfortunately, both interpreted the electron as a pointlike particle with no ‘internal structure’.David Hestenes is to be credited with reviving the Zitterbewegung hypothesis in the early 1990s. While acknowledging its origin as a (trivial) solution to Dirac’s equation for electrons, Hestenes argues the Zitterbewegung should be related to the intrinsic properties of the electron (charge, spin and magnetic moment). He argues that the Zitterbewegung hypothesis amounts to a physical interpretation of the elementary wavefunction or – more boldly – to a possible physical interpretation of all of quantum mechanics: “Spin and phase [of the wavefunction] are inseparably related — spin is not simply an add-on, but an essential feature of quantum mechanics. […] A standard observable in Dirac theory is the Dirac current, which doubles as a probability current and a charge current. However, this does not account for the magnetic moment of the electron, which many investigators conjecture is due to a circulation of charge. But what is the nature of this circulation? […] Spin and phase must be kinematical features of electron motion. The charge circulation that generates the magnetic moment can then be identified with the Zitterbewegung of Schrödinger “ Hestenes’ interpretation amounts to an kinematic model of an electron which can be described in terms of John Wheeler‘s mass without mass concept. The rest mass of the electron is analyzed as the equivalent energy of an orbital motion of a pointlike charge. This pointlike charge has no rest mass and must, therefore, move at the speed of light (which confirms Dirac’s en Schrödinger’s remarks on the nature of the Zitterbewegung). Hestenes summarizes his interpretation as follows: “The electron is nature’s most fundamental superconducting current loop. Electron spin designates the orientation of the loop in space. The electron loop is a superconducting LC circuit. The mass of the electron is the energy in the electron’s electromagnetic field. Half of it is magnetic potential energy and half is kinetic.”
Hestenes‘ articles and papers on the Zitterbewegung discuss the electron only. The interpretation of an electron as a superconducting ring of current (or as a (two-dimensional) oscillator) also works for the muon electron: its theoretical Compton radius rC = ħ/mμc ≈ 1.87 fm falls within the CODATA confidence interval for the experimentally determined charge radius. Hence, the theory seems to offer a remarkably and intuitive model of leptons. However, the model cannot be generalized to non-leptonic fermions (spin-1/2 particles). Its application to protons or neutrons, for example, is problematic: when inserting the energy of a proton or a neutron into the formula for the Compton radius (the rC = ħ/mc formula follows from the kinematic model), we get a radius of the order of rC = ħ/mpc ≈ 0.21 fm, which is about 1/4 of the measured value (0.84184(67) fm to 0.897(18) fm). A radius of the order of 0.2 fm is also inconsistent with the presumed radius of the pointlike charge itself. Indeed, while the pointlike charge is supposed to be pointlike, pointlike needs to be interpreted as ‘having no internal structure’: it does not imply the pointlike charge has no (small) radius itself. The classical electron radius is a likely candidate for the radius of the pointlike charge because it emerges from low-energy (Thomson) scattering experiments (elastic scattering of photons, as opposed to inelastic Compton scattering). The assumption of a pointlike charge with radius re = α·ħ/mpc) may also offer a geometric explanation of the anomalous magnetic moment.
In any case, the remarks above show that a Zitterbewegung model for non-leptonic fermions is likely to be somewhat problematic: a proton, for example, cannot be explained in terms of the Zitterbewegung of a positron (or a heavier variant of it, such as the muon- or tau-positron). This is why it is generally assumed the large energy (and the small size) of nucleons is to be explained by another force – a strong force which acts on a strong charge instead of an electric charge. One should note that both color and/or flavor in the standard quark–gluon model of the strong force may be thought of as zero-mass charges in ‘mass without mass’ kinematic models and, hence, the acknowledgment of this problem does not generally lead zbw theorists to abandon the quest for an alternative realist interpretation of quantum mechanics.
While Hestenes‘ zbw interpretation (and the geometric calculus approach he developed) is elegant and attractive, he did not seem to have managed to convincingly explain an obvious question of critics of the model: what keeps the pointlike charge in the zbw electron in its circular orbit? To put it simply: one may think of the electron as a superconducting ring but there is no material ring to hold and guide the charge. Of course, one may argue that the electromotive force explains the motion but this raises the fine-tuning problem: the slightest deviation of the pointlike charge from its circular orbit would yield disequilibrium and, therefore, non-stability. [One should note the fine-tuning problem is also present in mainstream quantum mechanics. See, for example, the discussion in Feynman’s Lectures on Physics.] The lack of a convincing answer to these and other questions (e.g. on the distribution of (magnetic) energy within the superconducting ring) led several theorists working on electron models (e.g. Alexander Burinskii) to move on and explore alternative geometric approaches, including Kerr-Newman geometries. Burinskii summarizes his model as follows: “The electron is a superconducting disk defined by an over-rotating black hole geometry. The charge emerges from the Möbius structure of the Kerr geometry.” His advanced modelling of the electron also allows for a conceptual bridge with mainstream quantum mechanics, grand unification theories and string theory: “[…] Compatibility between gravity and quantum theory can be achieved without modifications of Einstein-Maxwell equations, by coupling to a supersymmetric Higgs model of symmetry breaking and forming a nonperturbative super-bag solution, which generates a gravity-free Compton zone necessary for consistent work of quantum theory. Super-bag is naturally upgraded to Wess-Zumino supersymmetric QED model, forming a bridge to perturbative formalism of conventional QED.”
The various geometric approaches (Hestenes’ geometric calculus, Burinskii’s Kerr-Newman model, oscillator models) yield the same results (the intrinsic properties of the electron are derived from what may be referred to as kinematic equations or classical (but relativistically correct) equations) – except for a factor 2 or 1/2 or the inclusion (or not) of variable tuning parameters (Burinskii’s model, for example, allows for a variable geometry) – but the equivalence of the various models that may or may not explain the hypothetical Zitterbewegung still needs to be established.
The continued interest in zbw models may be explained because Zitterbewegung models – in particular Hestenes’ model and the oscillator model – are intuitive and, therefore, attractive. They are intuitive because they combine the Planck-Einstein relation (E = hf) and Einstein’s mass-energy equivalence (E = mc2): each cycle of the Zitterbewegung electron effectively packs (i) the unit of physical action (h) and (ii) the electron’s energy. This allows one to understand Planck’s quantum of action as the product of the electron’s energy and the cycle time: h = E·T = h·f·T = h·f/f = h. In addition, the idea of a centripetal force keeping some zero-mass pointlike charge in a circular orbit also offers a geometric explanation of Einstein’s mass-energy equivalence relation: this equation, therefore, is no longer a rather inexplicable consequence of special relativity theory.
The section below offers a general overview of the original discovery of Schrödinger and Dirac. It is followed by further analysis which may or may not help the reader to judge whether the Zitterbewegung hypothesis might, effectively, amount to what David Hestenes claims it actually is: an alternative interpretation of quantum mechanics.
Theory for a free fermion
[See the article: the author of this section does not seem to know – or does not mention, at least – that the Zitterbewegung hypothesis only applies to leptons (no strong charge).]
The Zitterbewegung may remain theoretical because, as Dirac notes, the frequency may be too high to be observable: it is the same frequency as that of a 0.511 MeV gamma-ray. However, some experiments may offer indirect evidence. Dirac’s reference to electron scattering experiments is also quite relevant because such experiments yield two radii: a radius for elastic scattering (the classical electron radius) and a radius for inelastic scattering (the Compton radius). Zittebewegung theorists think Compton scattering involves electron-photon interference: the energy of the high-energy photon (X- or gamma-ray photons) is briefly absorbed before the electron comes back to its equilibrium situation by emitting another (lower-energy) photon (the difference in the energy of the incoming and the outgoing photon gives the electron some extra momentum). Because of this presumed interference effect, Compton scattering is referred to as inelastic. In contrast, low-energy photons scatter elastically: they seem to bounce off some hard core inside of the electron (no interference).
Some experiments also claim they amount to a simulation of the Zitterbewegung of a free relativistic particle. First, with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation (although the physical situation is different). Then, in 2013, it was simulated in a setup with Bose–Einstein condensates.
The effective mass of the electric charge
The 2m factor in the formula for the zbw frequency and the interpretation of the Zitterbewegung in terms of a centripetal force acting on a pointlike charge with zero rest mass leads one to re-explore the concept of the effective mass of an electron. Indeed, if we write the effective mass of the pointlike charge as mγ = γm0 then we can derive its value from the angular momentum of the electron (L = ħ/2) using the general angular momentum formula L = r × p and equating r to the Compton radius:
This explains the 1/2 factor in the frequency formula for the Zitterbewegung. Substituting m for mγ in the ω = 2mc2/ħ yields an equivalence with the Planck-Einstein relation ω = mγc2/ħ. The electron can then be described as an oscillator (in two dimensions) whose natural frequency is given by the Planck-Einstein relation.
I am going to re-work my manuscript. I am going to restructure it, and also add the QCD analyses I did in recent posts. This is the first draft of the foreword. Let me know what you think of it. 🙂
[…] I had various working titles for this publication. I liked ‘A Bright Shining Lie’ but that title is already taken. The ‘History of a Bad Idea’ was another possibility, but my partner doesn’t like negative words. When I first talked to my new partner about my realist interpretation of quantum mechanics, she spontaneously referred to a story of that wonderful Danish storyteller, Hans Christian Andersen: The Emperor’s New Clothes. She was very surprised to hear I had actually produced a draft manuscript with the above-mentioned title (The Emperor Has No Clothes) on quantum electrodynamics which – after initially positive reactions – got turned down by two major publishers. She advised me to stick to the original title and just give it another go. I might as well because the title is, obviously, also a bit of a naughty wink to one of Roger Penrose’s book.
The ideas in this book are not all that easy to grasp – but they do amount to a full-blown realist interpretation of quantum mechanics, including both quantum electrodynamics (the theory of electrons and photons, and their interactions) and quantum chromodynamics – the theory of what goes on inside of a nucleus. Where is gravity? And what about the weak force, and the new Higgs sector of what is commonly referred to as the Standard Model of physics? Don’t worry. We will talk about these too. Not to make any definite statements because we think science isn’t ready to make any definite statements about them. Why? Because we think it doesn’t make sense to analyze the weak force as a force. It’s just a different beast. Gravity is a different beast too: we will explore Einstein’s geometric interpretation of spacetime. As for the Higgs field, we think it is just an ugly placeholder in an equally ugly theory.
What ugly theory? Isn’t the Standard Model supposed to be beautiful? Sabine Hossenfelder – writes the following about it in her latest book: “The Standard Model, despite its success, doesn’t get much love from physicists. Michio Kaku calls it “ugly and contrived,” Stephen Hawking says it’s “ugly and ad hoc,” Matt Strassler disparages it as “ugly and baroque,” Brian Greene complains that the standard model is “too flexible”, and Paul Davies thinks it “has the air of unfinished business” because “the tentative way in which it bundles together the electroweak and strong forces” is an “ugly feature.” I yet have to find someone who actually likes the standard model.”
You may know Hossenfelder’s name. She recently highlighted work that doubts the rigor of the LIGO detections of gravitational waves. I like it when scientists dare to question the award of a Nobel Prize. If any of what I write is true, then the Nobel Prize Committee has made a few premature awards over the past decades. Hossenfelder’s book explores the discontent with the Standard Model within the scientific community. Of course, the question is: what’s the alternative? That’s what this book is all about. You will be happy to hear that. You will be unhappy to hear that I am not to shy away from formulas and math. However, you should not worry: I am not going to pester you with gauge theory, renormalization, perturbation theory, transformations and what have you. Elementary high-school math is all you need. Reality is beautiful and complicated – but not that complicated: we can all understand it. 😊
 Roger Penrose, The Emperor’s New Mind, 1989.
 Physicists will note this is a rather limited definition of quantum chromodynamics. We will expand on it later.
 You may know her name. She recently highlighted work that doubs the rigor of the LIGO detections of gravitational waves. See: https://www.forbes.com/sites/startswithabang/2017/06/16/was-it-all-just-noise-independent-analysis-casts-doubt-on-ligos-detections. I like it when scientists dare to question a Nobel Prize. If any of what I write is true, then it’s obvious that it wouldn’t be the first time that the Nobel Prize Committee makes a premature award.
 Sabine Hossenfelder, Lost in Math: How Beauty Leads Physics Astray, 2018.
The creation of an electron-positron pair out of a highly energetic photon – the most common example of pair production – is often presented as an example of how energy can be converted into matter. Vice versa, electron-positron annihilation then amounts to the destruction of matter. However, if John Wheeler’s concept of ‘mass without mass’ is correct – or if Schrödinger’s trivial solution to Dirac’s equation for an electron in free space (the Zitterbewegung interpretation of an electron) is correct – then what might actually be happening is probably simpler—but also far more intriguing.
John Wheeler’s intuitive ‘mass without mass’ idea is that matter and energy are just two sides of the same coin. That was Einstein’s intuition too: mass is just a measure of inertia—a measure of the resistance to a change in the state of motion. Energy itself is motion: the motion of a charge. Some force over some distance, and we associate a force with a charge. Not with mass. In this interpretation of physics, an electron is nothing but a pointlike charge whizzing about some center. It’s a charge caught in an electromagnetic oscillation. The pointlike charge itself has zero rest mass, which is why it moves about at the speed of light.
This electron model is easy and intuitive. Developing a similar model for a nucleon – a proton or a neutron – is much more complicated because nucleons are held together by another force, which we commonly refer to as the strong force.
In regard to the latter, the reader should note that I am very hesitant to take the quark-gluon model of this strong force seriously. I entirely subscribe to Dirac’s rather skeptical evaluation of it:
“Now there are other kinds of interactions, which are revealed in high-energy physics and are important for the description of atomic nuclei. These interactions are not at present sufficiently well understood to be incorporated into a system of equations of motion. Theories of them have been set up and much developed and useful results obtained from them. But in the absence of equations of motion these theories cannot be presented as a logical development of the principles set up in this book. We are effectively in the pre-Bohr era with regard to these other interactions.”
I readily admit he wrote this in 1967 (so that’s a very long time ago). He was reacting, most probably, to the invention of a new conservation law (the conservation of strangeness, as proposed by Gell-Mann, Nishijima, Pais and others) and the introduction of many other ad hoc QCD quantum numbers to explain why this or that disintegration path does or does not occur. It was all part of the Great Sense-Making Exercise at the time: how to explain the particle zoo? In short, I am very reluctant to take the quark-gluon model of the strong force seriously.
However, I do acknowledge the experimental discovery of the fact that pairs of matter and anti-matter particles could be created out of highly energetic photons may well be the most significant discovery in post-WW II physics. Dirac’s preface to the 4th edition of the Principles of Quantum Mechanics summarized this as follows:
“In present-day high-energy physics, the creation and annihilation of charged particles is a frequent occurrence. A quantum electrodynamics which demands conservation of the number of charged particles is, therefore, out of touch with physical reality. So I have replaced it by a quantum electrodynamics which includes creation and annihilation of electron-positron pairs. […] It seems that the classical concept of an electron is no longer a useful model in physics, except possibly for elementary theories that are restricted to low-energy phenomena.”
Having said this, I think it’s useful to downplay Dr. Dirac’s excitement somewhat. Our world is governed by low-energy phenomena: if our Universe was created in a Big Bang – some extremely high-energy environment – then it happened 14 billion years or so ago, and the Universe has cooled down since. Hence, these high-energy experiments in labs and colliders are what they are: high-energy collisions followed by disintegration processes. They emulate the conditions of what might have happened in the first second – or the first minute, perhaps (surely not the first day or week or so) – after Creation.
I am, therefore, a bit puzzled by Dr. Dirac’s sentiment. Why would he think the classical concept of an electron is no longer useful? An electron is a permanent fixture. We can create and destroy it in our high-energy colliders, but that doesn’t mean it’s no longer useful as a concept.
Pair production only happens when the photon is fired into a nucleus, and the generalization to ‘other’ bosons ‘spontaneously’ disintegrating into a particle and an anti-particle is outright pathetic. What happens is this: we fire an enormous amount of electromagnetic energy into a nucleus (the equivalent mass of the photon has to match the mass of the electron and the positron that’s being produced) and, hence, we destabilize the stable nucleus. However, Nature is strong. The strong force is strong. Some intermediate energy state emerges but Nature throws out the spanner in the works. The end result is that all can be analyzed, once again, in terms of the Planck-Einstein relation: we have stable particles, once again. [Of course, the positron finds itself in the anti-Universe and will, therefore, quickly disappear in the reverse process: electron-positron annihilation.]
No magic here. And – surely – no need for strange QCD quantum numbers.
Jean Louis Van Belle, 28 July 2019
 Erwin Schrödinger stumbled upon the Zitterbewegung interpretation of an electron when he was exploring solutions to Dirac’s wave equation for free electrons. It’s worth quoting Dirac’s summary of it: “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)
 P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, 4th revised edition, Chapter XII (Quantum Electrodynamics), p. 312.
 Feynman’s 1963 Lecture on K-mesons (http://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S5) is an excellent summary of the state of affairs at the time. The new colliders had, effectively, generated a ‘particle zoo’, and it had to be explained. We think physicists should first have acknowledged that these short-lived particles should, perhaps, not be associated with the idea of a (fundamental) particle: they’re unstable. Transients, at best. Many of them are just resonances.
 I use the term ‘Creation’ as an absolutely non-religious concept here: it’s just a synonym of the presumed ‘Big Bang’. To be very clear on this, I am rather appalled by semi-scientific accounts of the creation of our world in terms of the biblical week.
The nature of the Higgs particle
The images below visualize what is generally referred to as the first ‘evidence’ for the Higgs boson: (1) two gamma rays emerging from the CERN LHC CMS detector, and (2) the tracks of four muons in the CERN LHC ATLAS detector. These tracks result from the collision between two protons that hit each other at a velocity of 99.99999 per cent of the speed of light – which corresponds to a combined energy of about 7 to 8 TeV. That’s huge. After the ‘discovery’ of the Higgs particle, the LHC was shut down for maintenance and an upgrade, and the protons in the LHC can now be accelerated to energies up to 7 TeV – which amounts to 14 TeV when they crash into each other. However, the higher energy level only produced more of the same so far.
We put ‘evidence’ and ‘discovery’ between inverted commas because the Higgs particle is (and, rest assured, will forever remain) a ghost particle only: we cannot directly observe it. Theoretical physicists and experimentalists agree these traces are just signatures of the long-awaited God particle. It was long-awaited indeed: the title of the six-page ‘leaflet’ explaining the award of the 2013 Nobel Prize in Physics to François Englert and Peter Higgs is: “Here, at last!” The long wait for it – and CERN’s excellent media team – may explain why the Nobel Physics Committee and the Royal Swedish Academy of Sciences were so eager to award a Nobel Prize for this ! So we should ask ourselves: what’s the hype, and what are the physics? And do the physics warrant the hype?
The facts are rather simple. We cannot directly observe the Higgs particle because it is just like all of the other ‘particles’ that come out of these collisions: they are too short-lived to leave a permanent trace. Indeed, when two protons hit each other at these incredible velocities, then all that’s left is debris flying around. This debris quickly disintegrates into other more debris – until we’re left with what we’re used: real particles, like electrons or protons. Things that don’t disintegrate.
The energy of the debris (the gamma rays or the muons) coming out of ‘Higgs events’ tells us the energy of the Higgs particle must be about 125 GeV. Besides its mass, it does not seem to have any other properties: no spin, no electric charge. It is, therefore, known as a scalar boson. In everyday language, that means it is just some (real) number. Newton had already told us that mass, as a measure of inertia, is just some real positive number—and Einstein taught us energy and mass are equivalent.
Interpreting the facts is tough. I am just an amateur physicists and so my opinion won’t count for much. However, I can’t help feeling Higg’s theory just confirms the obvious. For starters, we should be very hesitant to use the term ‘particle’ for the Higgs boson because its lifetime is of the order of 10-22 s. Think of it as the time an electron needs to go from electron orbital to another. Even at the speed of light – which an object with a rest mass of 125 GeV/c2 cannot aspire to attain – a particle with such lifetime cannot travel more than a few tenths of a femtometer: about 0.3´10-15 m, to be precise. That’s not something you would associate it with the idea of a particle: a resonance in particle physics has the same lifetime.
That’s why we’ll never see the Higgs boson—just like we’ll never see the W± and Z bosons whose mass it’s supposed to explain. Neither will none of us ever see a quark or a gluon: physicists tell us the signals that come out of colliders such as the LHC or, in the 1970s and 1980s, that came out of the PETRA accelerator in Hamburg, the Positron-Electron Project (PEP) at the Stanford National Accelerator Laboratory (SLAC), and the Super Proton-Antiproton Synchrotron at CERN, are consistent with the hypothesis that the strong and weak forces are mediated through particles known as bosons (force carriers) but – truth be told – the whole idea of forces being mediated by bosons is just what it is: a weird theory.
Are virtual particles the successor to the aether theory?
Maybe we should first discuss the most obvious of all bosons: the photon. Photons are real. Of course, they are. They are, effectively, the particles of light. They are, in fact, the only bosons we can effectively observe. In fact, we’ve got a problem here: the only bosons we can effectively observe – photons – do not have all of the theoretical properties of a boson: as a spin-1 particle, the theoretical values for its angular momentum are ± ħ or 0. However, photons don’t have a zero-spin state. Never. This is one of the things in mainstream quantum mechanics that has always irked me. All courses in quantum mechanics spend like two or three chapters on why bosons and fermions are different (spin-one versus spin-1/2), but when it comes to the specifics – real-life stuff – then the only boson we actually know (the photon) turns out to not be a typical boson because it can’t have zero spin. [Physicists will, of course, say the most important property of bosons is that they you can keep piling bosons on top of bosons, and you can do that with photons. Bosons are supposed to like to be together, because we want to keep adding to the force without limit. But… Well… I have another explanation for that. It’s got to do with the fact that bosons don’t – or shouldn’t – carry charge. But I don’t want to start another digression on that. Not here.]
So photons – the only real-life bosons we’ve ever observed – aren’t typical bosons. More importantly, no course in physics has ever been able to explain why we’d need photons in the role of virtual particles. Why would an electron in some atomic orbital continuously exchange photons with the proton that holds it in its orbit? When you ask that question to a physicist, he or she will start blubbering about quantum field theory and other mathematical wizardry—but he or she will never give you a clear answer. I’ll come back to this in the next section of this paper.
I don’t think there is a clear answer. Worse, I’ve started to think the whole idea of some particle mediating a force is nonsense. It’s like the 19th-century aether theory: we don’t need it. We don’t need it in electromagnetic theory: Maxwell’s Laws – augmented with the Planck-Einstein relation – will do. We also don’t need it to model the strong force. The quark–gluon model – according to which quarks change color all of the time – does not come with any simplification as compared to a simpler parton model:
- The quark-gluon model gives us (at least) two quarks, two anti-quarks and nine gluons, so that adds up to 13 different objects.
- If we just combine the idea of a parton – a pointlike carrier of properties – with… Well… Its properties – the possible electric charges (±2/3 and ±1/3) and the possible color charges (red, green and blue) – we’ve got 12 partons, and such ‘parton model’ explains just as much.
I also don’t think we need it to model the weak force. Let me be very clear about my intuition/sentiment/appreciation—whatever you want to call it:
We don’t need a Higgs theory to explain why W/Z bosons have mass because I think W/Z bosons don’t exist: they’re a figment of our imagination.
Why do we even need the concept of a force to explain why things fall apart? The world of unstable particles – transient particles as I call them – is a different realm altogether. Physicists will cry wolf here: CERN’s Super Proton-Antiproton Synchrotron produced evidence for W+, W– and Z bosons back in 1983, didn’t it?
No. The evidence is just the same as the ‘evidence’ for the Higgs boson: we produce a short-lived blob of energy which disintegrates in no time (10-22 s or 10-23 s is no time, really) and, for some reason no one really understands, we think of it as a force carrier: something that’s supposed to be very different from the other blobs of energy that emerge while it disintegrates into jets made up of other transients and/or resonances. The end result is always the same: the various blobs of energy further dis- and reintegrate as stable particles (think of protons, electrons and neutrinos). There is no good reason to introduce a bunch of weird flavor quantum numbers to think of how such processes might actually occur. In reality, we have a very limited number of permanent fixtures (electrons, protons and photons), hundreds of transients (particles that fall apart) and thousands of resonances (excited states of the transient and non-transient stuff).
You’ll ask me: so what’s the difference between them then?
Stable particles respect the E = h·f = ħ·ω relation—and they do so exactly. For non-stable particles – transients – that relation is slightly off, and so they die. They die by falling apart in more stable configurations, until we are left with stable particles only. As for resonances, they are just that: some excited state of a stable or a non-stable particle. Full stop. No magic needed.
Photons as bosons
Photons are real and, yes, they carry energy. When an electron goes from one state to another (read: from one electron orbital to another), it will absorb or emit a photon. Photons make up light: visible light, low-energy radio waves, or high-energy X- and γ-rays. These waves carry energy and – when we look real close – they are made up of photons. So, yes, it’s the photons that carry the energy.
Saying they carry electromagnetic energy is something else than saying they carry electromagnetic force itself. A force acts on a charge: a photon carries no charge. If photons carry no charge, then why would we think of them as carrying the force?
I wrote I’ve always been irked by the fact that photons – again, the only real-life bosons we’ve ever observed – don’t have all of the required properties of the theoretical force-carrying particle physicists invented: the ‘boson’. If bosons exist, then the bosons we associate with the strong and weak force should also not carry any charge: color charge or… Well… What’s the ‘weak’ charge? Flavor? Come on guys ! Give us something we can believe in.
That’s one reason – for me, at least – why the idea of gluons and W/Z bosons is non-sensical. Gluons carry color charge, and W/Z bosons carry electric charge (except for the Z boson – but we may think of it as carrying both positive and negative charge). They shouldn’t. Let us quickly review what I refer to as a ‘classical’ quantum theory of light.
If there is one quantum-mechanical rule that no one never doubts, it is that angular momentum comes in units of ħ: Planck’s (reduced) constant. When analyzing the electron orbitals for the simplest of atoms (the one-proton hydrogen atom), this rule amounts to saying the electron orbitals are separated by a amount of physical action that is equal to h = 2π·ħ. Hence, when an electron jumps from one level to the next – say from the second to the first – then the atom will lose one unit of h. The photon that is emitted or absorbed will have to pack that somehow. It will also have to pack the related energy, which is given by the Rydberg formula:To focus our thinking, let us consider the transition from the second to the first level, for which the 1/12 – 1/22 is equal 0.75. Hence, the photon energy should be equal to (0.75)·ER ≈ 10.2 eV. Now, if the total action is equal to h, then the cycle time T can be calculated as:
This corresponds to a wave train with a length of (3×108 m/s)·(0.4×10-15 s) = 122 nm. That is the size of a large molecule and it is, therefore, much more reasonable than the length of the wave trains we get when thinking of transients using the supposed Q of an atomic oscillator. In fact, this length is the wavelength of the light (λ = c/f = c·T = h·c/E) that we would associate with this photon energy.
We should quickly insert another calculation here. If we think of an electromagnetic oscillation – as a beam or, what we are trying to do here, as some quantum – then its energy is going to be proportional to (a) the square of the amplitude of the oscillation – and we are not thinking of a quantum-mechanical amplitude here: we are talking the amplitude of a physical wave here – and (b) the square of the frequency. Hence, if we write the amplitude as a and the frequency as ω, then the energy should be equal to E = k·a2·ω2, and the k in this equation is just a proportionality factor.
However, relativity theory tells us the energy will have some equivalent mass, which is given by Einstein’s mass-equivalence relation: E = m·c2. This equation tells us the energy of a photon is proportional to its mass, and the proportionality factor is c2. So we have two proportionality relations now, which (should) give us the same energy. Hence, k·a2·ω2 must be equal to m·c2, somehow.
How should we interpret this? It is, obviously, very tempting to equate k and m, but we can only do this if c2 is equal to a2·ω2 or – what amounts to the same – if c = a·ω. You will recognize this as a tangential velocity formula. The question is: the tangential velocity of what? The a in the E = k·a2·ω2 formula that we started off with is an amplitude: why would we suddenly think of it as a radius now? Because our photon is circularly polarized. To be precise, its angular momentum is +ħ or –ħ. There is no zero-spin state. Hence, if we think of this classically, then we will associate it with circular polarization.
However, these properties do not make it a boson or, let me be precise, these properties do not make it a virtual particle. Again, I’ve haven’t seen a textbook – advanced or intermediate level – that answers this simple question: why would an electron in some stable atomic orbital – it does not emit or absorb any energy – continuously exchange virtual photons with the proton that holds it in its orbit?
How would that photon look like? It would have to have some energy, right? And it would have to pack to physical action, right? Why and how would it take that energy – or that action (I like the German Wirkung much better in terms of capturing that concept) – away from the electron orbital? In fact, the idea of an electron orbital combines the idea of the electron and the proton—and their mutual attraction. The physicists who imagine those virtual photons are making a philosophical category mistake. We think they’re making a similar mistake when advancing the hypothesis of gluons and W/Z bosons.
We think the idea of virtual particles, gauge bosons and/or force-carrying particles in general is superfluous. The whole idea of bosons mediating forces resembles 19th century aether theory: we don’t need it. The implication is clear: if that’s the case, then we also don’t need gauge theory and/or quantum field theory.
Jean Louis Van Belle, 21 July 2019
 We took this from the above-mentioned leaflet. A proton has a rest energy of 938,272 eV, more or less. An energy equal to 4 TeV (the tera– prefix implies 12 zeroes) implies a Lorentz factor that is equal to γ = E/E0 = 4´1012/938,272 » 1´106. Now, we know that 1 – β2 = c2/c2 – v2/c2 = 1/γ2 = 1/γ2 » 1´10-12. The square root of that is of the order of a millionth, so we get the same order of magnitude.
 To be fair, the high-energy collisions also resulted in the production of some more short-lived ‘particles’, such as new variants of bottomonium: mesons that are supposed to consist of a bottom quark and its anti-matter counterpart.
 See: https://www.nobelprize.org/uploads/2018/06/popular-physicsprize2013-1.pdf. Higgs’ theory itself – on how gauge bosons can acquire non-zero masses – goes back to 1964 and was put forward by three individual research groups. See: https://en.wikipedia.org/wiki/1964_PRL_symmetry_breaking_papers.
 We write at least because we are only considering u and d quarks here: the constituents of all stable or fairly stable matter (protons and neutrons, basically).
 See: Jean Louis Van Belle, A Realist Interpretation of QCD, 16 July 2019.
 If we think of energy as the currency of the Universe, then you should think of protons and electrons as bank notes, and neutrinos as the coins: they provide the change.
 See: Jean Louis Van Belle, Is the Weak Force a Force?, 19 July 2019.
 This is a very much abbreviated summary. For a more comprehensive analysis, see: Jean Louis Van Belle, A Classical Quantum Theory of Light, 13 June 2019.
 In one of his Lectures (I-32-3), Feynman thinks about the Q of a sodium atom, which emits and absorbs sodium light, of course. Based on various assumptions – assumption that make sense in the context of the blackbody radiation model but not in the context of the Bohr model – he gets a Q of about 5×107. Now, the frequency of sodium light is about 500 THz (500×1012 oscillations per second). Hence, the decay time of the radiation is of the order of 10–8 seconds. So that means that, after 5×107 oscillations, the amplitude will have died by a factor 1/e ≈ 0.37. That seems to be very short, but it still makes for 5 million oscillations and, because the wavelength of sodium light is about 600 nm (600×10–9 meter), we get a wave train with a considerable length: (5×106)·(600×10–9 meter) = 3 meter. Surely You’re Joking, Mr. Feynman! A photon with a length of 3 meter – or longer? While one might argue that relativity theory saves us here (relativistic length contraction should cause this length to reduce to zero as the wave train zips by at the speed of light), this just doesn’t feel right – especially when one takes a closer look at the assumptions behind.
I’ve did what I promised to do – and that is to start posting on my other blog. On quantum chromodynamics, that is. But I think this paper deserves wider distribution. 🙂
The paper below probably sort of sums up my views on quantum field theory. I am not sure if I am going to continue to blog. I moved my papers to an academia.edu site and… Well… I think that’s about it. 🙂
One of the readers of this blog asked me what I thought of the following site: Rational Science (https://www.youtube.com/channel/UC_I_L6pPCwxTgAH7yutyxqA). I watched it – for a brief while – and I must admit I am thoroughly disappointed by it. I think it’s important enough to re-post what I posted on this YouTube channel itself:
“I do believe there is an element of irrationality in modern physics: a realist interpretation of quantum electrodynamics is possible but may not gain acceptance because religion and other factors may make scientists somewhat hesitant to accept a common-sense explanation of things. The mystery needs to be there, and it needs to be protected – somehow. Quantum mechanics may well be the only place where God can hide – in science, that is.
But – in his attempt to do away with the notion of God – Bill Gaede takes things way too far – and so I think he errs on the other side of the spectrum. Mass, energy and spacetime are essential categories of the mind (or concepts if you want) to explain the world. Mass is a measure of inertia to a change in the state of motion of an object, kinetic energy is the energy of motion, potential energy is energy because of an object’s position in spacetime, etcetera. So, yes, these are concepts – and we need these concepts to explain what we human beings refer to as ‘the World’. Space and time are categories of the mind as well – philosophical or mathematical concepts, in other words – but they are related and well-defined.
In fact, space and time define each other also because the primordial idea of motion implies both: the idea of motion implies we imagine something moving in space and in time. So that’s space-time, and it’s a useful idea. That also explains why time goes in one direction only. If we’d allow time to reverse, then we’d also an object to be in two places at the same time (if an object can go back in time, then it can also go back to some other place – and so then it’s in two places at the same time). This is just one example where math makes sense of physical realities – or where our mind meets ‘the World’.
When Bill Gaede quotes Wheeler and other physicists in an attempt to make you feel he’s on the right side of history, he quotes him very selectively. John Wheeler, for example, believed in the idea of ‘mass’ – but it was ‘mass without mass’ for him: the mass of an object was the equivalent mass of the object’s energy. The ideas of Wheeler have been taken forward by a minority of physicists, such as David Hestenes and Alexander Burinskii. They’ve developed a fully-fledged electron model that combines wave and particle characteristics. It effectively does away with all of the hocus-pocus in QED – which Bill Gaede criticizes, and rightly so.
In short, while it’s useful to criticize mainstream physics as hocus-pocus, Bill Gaede is taking it much too far and, unfortunately, gives too much ammunition to critics to think of people like us – amateur physicists or scientists who try to make sense of it all – as wackos or crackpots. Math is, effectively, descriptive but, just like anything else, we need a language to describe stuff, and math is the language in which we describe actual physics. Trying to discredit the mathematical approach to science is at least as bad – much worse, actually – than attaching too much importance to it. Yes, we need to remind ourselves constantly that we are describing something physical, but we need concepts for that – and these concepts are mathematical.
PS: Bill Gaede also has very poor credentials, but you may want to judge these for yourself: https://en.wikipedia.org/wiki/Bill_Gaede. These poor credentials do not imply that his views are automatically wrong, but it does introduce an element of insincerity. In short, watch what you’re watching and always check sources and backgrounds when googling for answers to questions, especially when you’re googling for answers to fundamental questions ! 🙂
This is it, folks ! I am moving on ! It was nice camping out here. 🙂
This has been a very interesting journey for me. I wrote my first post in October 2013, so that’s almost five years ago. As mentioned in the ‘About‘ page, I started writing this blog because — with all those breakthroughs in science (some kind of experimental verification of what is referred to as the Higgs field in July 2012 and, more recently, the confirmation of the reality of gravitational waves in 2016 by Caltech’s LIGO Lab) — I felt I should make an honest effort to try to understand what it was all about.
Despite all of my efforts (including enrolling in MIT’s edX QM course, which I warmly recommend as an experience, especially because it’s for free), I haven’t moved much beyond quantum electrodynamics (QED). Hence, that Higgs field is a still a bit of a mystery to me. In any case, the summaries I’ve read about it say it’s just some scalar field. So that’s not very exciting: mass is some number associated with some position in spacetime. That’s nothing new, right?
In contrast, I am very enthusiastic about the LIGO Lab discovery. Why? Because it confirms Einstein was right all along.
If you have read any of my posts, you will know I actually disagree with Feynman. I have to thank him for his Lectures — and I would, once again, like to thank Michael Gottlieb and Rudolf Pfeiffer, who have worked for decades to get those Lectures online — but my explorations did confirm that guts feeling I had deep inside when starting this journey: the complexity in the quantum-mechanical framework does not match the intuition that, if the theory 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. And the interference of a photon with itself in the Mach-Zehnder interference experiment has a classical explanation too. The ‘zero state’ of a photon – or its zero states (plural), I should say – are the linear components of the circular polarization. In fact, I really wish someone would have gently told me that an actual beam splitter changes the polarization of light. I could have solved the Mach-Zehnder puzzle with that information like a year ago.]
This will probably sound like Chinese to you, so let me translate it: there is no mystery. Not in the QED sector of the Standard Model, at least. All can be explained by simple geometry and the idea of a naked charge: something that has no other property but its electric charge and – importantly – some tiny radius, which is given by the fine-structure constant (the ratio becomes a distance if we think of the electron’s Compton radius as a natural (distance) unit). So the meaning of God’s Number is clear now: there is nothing miraculous about it either. Maxwell’s equations combined with the Planck-Einstein Law (E = h·f) are all we need to explain the whole QED sector. No hocus-pocus needed. The elementary wavefunction exp(±i·θ) = exp(±ω·t) = exp[±(E/ħ)·t] represents an equally elementary oscillation. Physicists should just think some more about the sign convention and, more generally, think some more about Occam’s Razor Principle when modeling their problems. 🙂
Am I a crackpot? Maybe. I must be one, because I think the academics have a problem, not me. So… Well… That’s the definition of a crackpot, isn’t it? 🙂 It feels weird. Almost all physicists I got in touch with – spare two or three (I won’t mention their names because they too don’t quite know what to do with me) – are all stuck in their Copenhagen interpretation of quantum mechanics: reality is some kind of black box and we’ll never understand it the way we would want to understand it. Almost none of them is willing to think outside of the box. I blame vested interests (we’re talking Nobel Prize stuff, unfortunately) and Ivory Tower culture.
In any case, I found the answers to the questions I started out with, and I don’t think the academics I crossed (s)words with have found that peace of mind yet. So if I am a crackpot, then I am a happy one. 😊
The Grand Conclusion is that the Emperor is not wearing any clothes. Not in the QED sector, at least. In fact, I think the situation is a lot worse. The Copenhagen interpretation of quantum mechanics feels like a Bright Shining Lie. [Yes, I know that’s an ugly reference.] But… Yes. Just mathematical gimmicks to entertain students – and academics ! Of course, I can appreciate the fact that Nobel Prizes have been awarded and that academic reputations have to be upheld — posthumously or… I would want to write ‘humously’ here but that word doesn’t exist so I should replace it by ‘humorously’. 🙂 […] OK. Poor joke. 🙂
Frankly, it is a sad situation. Physics has become the domain of hype and canonical nonsense. To the few readers who have been faithful followers (this blog attracted about 154,034 visitors so far which is — of course — close to nothing), I’d say: think for yourself. Honor Boltzman’s spirit: “Bring forward what is true. Write it so that it is clear. Defend it to your last breath.” I actually like another quote of him too: “If you are out to describe the truth, leave elegance to the tailor.” But that’s too rough, isn’t it? And then I am also not sure he really said that. 🙂
Of course, QCD is another matter altogether — because of the non-linearity of the force(s) involved, and the multiplication of ‘colors’ — but my research over the past five years (longer than that, actually) have taught me that there is no ‘deep mystery’ in the QED sector. All is logical – including the meaning of the fine-structure constant: that’s just the radius of the naked charge expressed in natural units. All the rest can be derived. And 99% of what you’ll read or google about quantum mechanics is about QED: perturbation theory, propagators, the quantized field, etcetera to talk about photons and electrons, and their interactions. If you have a good idea about what an electron and a photon actually are, then you do not need anything of that to understand QED.
In short, quantum electrodynamics – as a theory, and in its current shape and form – is 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. All of the weirdness of Nature is, therefore, in this weird description of the fields: gauge theories, Feynman diagrams, quantum field theory, etcetera. And the common-sense is right there: right in front of us. It’s easy and elegant: a plain common-sense interpretation of quantum mechanics — which, I should remind the reader, is based on Erwin Schrödinger’s trivial solution for Dirac’s wave equation for an electron in free space.
So is no one picking this up? Let’s see. Truth cannot be hidden, right? Having said that, I must admit I have been very surprised by the rigidity of thought of academics (which I know all too well from my experience as a PhD student in economics) in this domain. If math is the queen of science, then physics is the king, right? Well… Maybe not. The brightest minds seem to have abandoned the field.
But I will stop my rant here. I want to examine the QCD sector now. What theories do we have for the non-linear force(s) that keep(s) protons together? What explains electron capture by a proton—turning it into a neutron in the process? What’s the nature of neutrinos? How should we think of all these intermediary particles—which are probably just temporary resonances rather than permanent fixtures?
My new readingeinstein.blog will be devoted to that. I think I’ll need some time to post my first posts (pun intended)—but… Well… We’ve started this adventure and so I want to get to the next destination. It’s a mind thing, right? 🙂
My posts on the fine-structure constant – God’s Number as it is often referred to – have always attracted a fair amount of views. I think that’s because I have always tried to clarify this or that relation by showing how and why exactly it pops us in this or that formula (e.g. Rydberg’s energy formula, the ratio of the various radii of an electron (Thomson, Compton and Bohr radius), the coupling constant, the anomalous magnetic moment, etcetera), as opposed to what most seem to try to do, and that is to further mystify it. You will probably not want to search through all of my writing so I will just refer you to my summary of these efforts on the viXra.org site: “Layered Motions: the Meaning of the Fine-Structure Constant.”
However, I must admit that – till now – I wasn’t quite able to answer this very simple question: what is that fine-structure constant? Why exactly does it appear as a scaling constant or a coupling constant in almost any equation you can think of but not in, say, Einstein’s mass-energy equivalence relation, or the de Broglie relations?
I finally have a final answer (pun intended) to the question, and it’s surprisingly easy: it is the radius of the naked charge in the electron expressed in terms of the natural distance unit that comes out of our realist interpretation of what an electron actually is. [For those who haven’t read me before, this realist interpretation is based on Schrödinger’s discovery of the Zitterbewegung of an electron.] That natural distance unit is the Compton radius of the electron: it is the effective radius of an electron as measured in inelastic collisions between high-energy photons and the electron. I like to think of it as a quantum of space in which interference happens but you will want to think that through for yourself.
The point is: that’s it. That’s all. All the other calculations follow from it. Why? It would take me a while to explain that but, if you carefully look at the logic in my classical calculations of the anomalous magnetic moment, then you should be able to understand why these calculations are somewhat more fundamental than the others and why we can, therefore, get everything else out of them. 🙂
Post scriptum: I quickly checked the downloads of my papers on Phil Gibbs’ site, and I am extremely surprised my very first paper (the quantum-mechanical wavefunction as a gravitational wave) of mine still gets downloads. To whomever is interested in this paper, I would say: the realist interpretation we have been pursuing – based on the Zitterbewegung model of an electron – is based on the idea of a naked charge (with zero rest mass) orbiting around some center. The energy in its motion – a perpetual current ring, really – gives the electron its (equivalent) mass. That’s just Wheeler’s idea of ‘mass without mass’. But the force is definitely not gravitational. It cannot be. The force has to grab onto something, and all it can grab onto here is that naked charge. The force is, therefore, electromagnetic. It must be. I now look at my very first paper as a first immature essay. It did help me to develop some basic intuitive ideas on what any realist interpretation of QM should look like, but the quantum-mechanical wavefunction has nothing to do with gravity. Quantum mechanics is electromagnetics: we just add the quantum. The idea of an elementary cycle. Gravity is dealt with by general relativity theory: energy – or its equivalent mass – bends spacetime. That’s very significant, but it doesn’t help you when analyzing the QED sector of physics. I should probably pull this paper of the site – but I won’t. Because I think it shows where I come from: very humble origins. 🙂
My book is moving forward. I just produced a very first promotional video. Have a look and let me know what you think of it ! 🙂
This post explores the limits of the physical interpretation of the wavefunction we have been building up in previous posts. It does so by examining if it can be used to provide a hidden-variable theory for explaining quantum-mechanical interference. The hidden variable is the polarization state of the photon.
The outcome is as expected: the theory does not work. Hence, this paper clearly shows the limits of any physical or geometric interpretation of the wavefunction.
This post sounds somewhat academic because it is, in fact, a draft of a paper I might try to turn into an article for a journal. There is a useful addendum to the post below: it offers a more sophisticated analysis of linear and circular polarization states (see: Linear and Circular Polarization States in the Mach-Zehnder Experiment). Have fun with it !
Duns Scotus wrote: pluralitas non est ponenda sine necessitate. Plurality is not to be posited without necessity. And William of Ockham gave us the intuitive lex parsimoniae: the simplest solution tends to be the correct one. But redundancy in the description does not seem to bother physicists. When explaining the basic axioms of quantum physics in his famous Lectures on quantum mechanics, Richard Feynman writes:
“We are not particularly interested in the mathematical problem of finding the minimum set of independent axioms that will give all the laws as consequences. Redundant truth does not bother us. We are satisfied if we have a set that is complete and not apparently inconsistent.”
Also, most introductory courses on quantum mechanics will show that both ψ = exp(iθ) = exp[i(kx-ωt)] and ψ* = exp(-iθ) = exp[-i(kx-ωt)] = exp[i(ωt-kx)] = -ψ are acceptable waveforms for a particle that is propagating in the x-direction. Both have the required mathematical properties (as opposed to, say, some real-valued sinusoid). We would then think some proof should follow of why one would be better than the other or, preferably, one would expect as a discussion on what these two mathematical possibilities might represent¾but, no. That does not happen. The physicists conclude that “the choice is a matter of convention and, happily, most physicists use the same convention.”
Instead of making a choice here, we could, perhaps, use the various mathematical possibilities to incorporate spin in the description, as real-life particles – think of electrons and photons here – have two spin states (up or down), as shown below.
Table 1: Matching mathematical possibilities with physical realities?
|Spin and direction||Spin up||Spin down|
|Positive x-direction||ψ = exp[i(kx-ωt)]||ψ* = exp[i(ωt-kx)]|
|Negative x-direction||χ = exp[i(ωt-kx)]||χ* = exp[i(kx+ωt)]|
That would make sense – for several reasons. First, theoretical spin-zero particles do not exist and we should therefore, perhaps, not use the wavefunction to describe them. More importantly, it is relatively easy to show that the weird 720-degree symmetry of spin-1/2 particles collapses into an ordinary 360-degree symmetry and that we, therefore, would have no need to describe them using spinors and other complicated mathematical objects. Indeed, the 720-degree symmetry of the wavefunction for spin-1/2 particles is based on an assumption that the amplitudes C’up = -Cup and C’down = -Cdown represent the same state—the same physical reality. As Feynman puts it: “Both amplitudes are just multiplied by −1 which gives back the original physical system. It is a case of a common phase change.”
In the physical interpretation given in Table 1, these amplitudes do not represent the same state: the minus sign effectively reverses the spin direction. Putting a minus sign in front of the wavefunction amounts to taking its complex conjugate: -ψ = ψ*. But what about the common phase change? There is no common phase change here: Feynman’s argument derives the C’up = -Cup and C’down = -Cdown identities from the following equations: C’up = eiπCup and C’down = e–iπCdown. The two phase factors are not the same: +π and -π are not the same. In a geometric interpretation of the wavefunction, +π is a counterclockwise rotation over 180 degrees, while -π is a clockwise rotation. We end up at the same point (-1), but it matters how we get there: -1 is a complex number with two different meanings.
We have written about this at length and, hence, we will not repeat ourselves here. However, this realization – that one of the key propositions in quantum mechanics is basically flawed – led us to try to question an axiom in quantum math that is much more fundamental: the loss of determinism in the description of interference.
The reader should feel reassured: the attempt is, ultimately, not successful—but it is an interesting exercise.
The standard MIT course on quantum physics vaguely introduces Bell’s Theorem – labeled as a proof of what is referred to as the inevitable loss of determinism in quantum mechanics – early on. The argument is as follows. If we have a polarizer whose optical axis is aligned with, say, the x-direction, and we have light coming in that is polarized along some other direction, forming an angle α with the x-direction, then we know – from experiment – that the intensity of the light (or the fraction of the beam’s energy, to be precise) that goes through the polarizer will be equal to cos2α.
But, in quantum mechanics, we need to analyze this in terms of photons: a fraction cos2α of the photons must go through (because photons carry energy and that’s the fraction of the energy that is transmitted) and a fraction 1-cos2α must be absorbed. The mentioned MIT course then writes the following:
“If all the photons are identical, why is it that what happens to one photon does not happen to all of them? The answer in quantum mechanics is that there is indeed a loss of determinism. No one can predict if a photon will go through or will get absorbed. The best anyone can do is to predict probabilities. Two escape routes suggest themselves. Perhaps the polarizer is not really a homogeneous object and depending exactly on where the photon is it either gets absorbed or goes through. Experiments show this is not the case.
A more intriguing possibility was suggested by Einstein and others. A possible way out, they claimed, was the existence of hidden variables. The photons, while apparently identical, would have other hidden properties, not currently understood, that would determine with certainty which photon goes through and which photon gets absorbed. Hidden variable theories would seem to be untestable, but surprisingly they can be tested. Through the work of John Bell and others, physicists have devised clever experiments that rule out most versions of hidden variable theories. No one has figured out how to restore determinism to quantum mechanics. It seems to be an impossible task.”
The student is left bewildered here. Are there only two escape routes? And is this the way how polarization works, really? Are all photons identical? The Uncertainty Principle tells us that their momentum, position, or energy will be somewhat random. Hence, we do not need to assume that the polarizer is nonhomogeneous, but we need to think of what might distinguish the individual photons.
Considering the nature of the problem – a photon goes through or it doesn’t – it would be nice if we could find a binary identifier. The most obvious candidate for a hidden variable would be the polarization direction. If we say that light is polarized along the x-direction, we should, perhaps, distinguish between a plus and a minus direction? Let us explore this idea.
The simple experiment above – linearly polarized light going through a polaroid – involves linearly polarized light. We can easily distinguish between left- and right-hand circular polarization, but if we have linearly polarized light, can we distinguish between a plus and a minus direction? Maybe. Maybe not. We can surely think about different relative phases and how that could potentially have an impact on the interaction with the molecules in the polarizer.
Suppose the light is polarized along the x-direction. We know the component of the electric field vector along the y-axis will then be equal to Ey = 0, and the magnitude of the x-component of E will be given by a sinusoid. However, here we have two distinct possibilities: Ex = cos(ω·t) or, alternatively, Ex = sin(ω·t). These are the same functions but – crucially important – with a phase difference of 90°: sin(ω·t) = cos(ω·t + π/2).
Figure 1: Two varieties of linearly polarized light?
Would this matter? Sure. We can easily come up with some classical explanations of why this would matter. Think, for example, of birefringent material being defined in terms of quarter-wave plates. In fact, the more obvious question is: why would this not make a difference?
Of course, this triggers another question: why would we have two possibilities only? What if we add an additional 90° shift to the phase? We know that cos(ω·t + π) = –cos(ω·t). We cannot reduce this to cos(ω·t) or sin(ω·t). Hence, if we think in terms of 90° phase differences, then –cos(ω·t) = cos(ω·t + π) and –sin(ω·t) = sin(ω·t + π) are different waveforms too. In fact, why should we think in terms of 90° phase shifts only? Why shouldn’t we think of a continuum of linear polarization states?
We have no sensible answer to that question. We can only say: this is quantum mechanics. We think of a photon as a spin-one particle and, for that matter, as a rather particular one, because it misses the zero state: it is either up, or down. We may now also assume two (linear) polarization states for the molecules in our polarizer and suggest a basic theory of interaction that may or may not explain this very basic fact: a photon gets absorbed, or it gets transmitted. The theory is that if the photon and the molecule are in the same (linear) polarization state, then we will have constructive interference and, somehow, a photon gets through. If the linear polarization states are opposite, then we will have destructive interference and, somehow, the photon is absorbed. Hence, our hidden variables theory for the simple situation that we discussed above (a photon does or does not go through a polarizer) can be summarized as follows:
|Linear polarization state||Incoming photon up (+)||Incoming photon down (-)|
|Polarizer molecule up (+)||Constructive interference: photon goes through||Destructive interference: photon is absorbed|
|Polarizer molecule down (-)||Destructive interference: photon is absorbed||Constructive interference: photon goes through|
Nice. No loss of determinism here. But does it work? The quantum-mechanical mathematical framework is not there to explain how a polarizer could possibly work. It is there to explain the interference of a particle with itself. In Feynman’s words, this is the phenomenon “which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics.”
So, let us try our new theory of polarization states as a hidden variable on one of those interference experiments. Let us choose the standard one: the Mach-Zehnder interferometer experiment.
The setup of the Mach-Zehnder interferometer is well known and should, therefore, probably not require any explanation. We have two beam splitters (BS1 and BS2) and two perfect mirrors (M1 and M2). An incident beam coming from the left is split at BS1 and recombines at BS2, which sends two outgoing beams to the photon detectors D0 and D1. More importantly, the interferometer can be set up to produce a precise interference effect which ensures all the light goes into D0, as shown below. Alternatively, the setup may be altered to ensure all the light goes into D1.
Figure 2: The Mach-Zehnder interferometer
The classical explanation is easy enough. It is only when we think of the beam as consisting of individual photons that we get in trouble. Each photon must then, somehow, interfere with itself which, in turn, requires the photon to, somehow, go through both branches of the interferometer at the same time. This is solved by the magical concept of the probability amplitude: we think of two contributions a and b (see the illustration above) which, just like a wave, interfere to produce the desired result¾except that we are told that we should not try to think of these contributions as actual waves.
So that is the quantum-mechanical explanation and it sounds crazy and so we do not want to believe it. Our hidden variable theory should now show the photon does travel along one path only. If the apparatus is set up to get all photons in the D0 detector, then we might, perhaps, have a sequence of events like this:
|Photon polarization||At BS1||At BS2||Final result|
|Up (+)||Photon is reflected||Photon is reflected||Photon goes to D0|
|Down (–)||Photon is transmitted||Photon is transmitted||Photon goes to D0|
Of course, we may also set up the apparatus to get all photons in the D1 detector, in which case the sequence of events might be this:
|Photon polarization||At BS1||At BS2||Final result|
|Up (+)||Photon is reflected||Photon is transmitted||Photon goes to D1|
|Down (–)||Photon is transmitted||Photon is reflected||Photon goes to D1|
This is a nice symmetrical explanation that does not involve any quantum-mechanical weirdness. The problem is: it cannot work. Why not? What happens if we block one of the two paths? For example, let us block the lower path in the setup where all photons went to D0. We know – from experiment – that the outcome will be the following:
|Photon is absorbed at the block||0.50|
|Photon goes to D0||0.25|
|Photon goes to D1||0.25|
How is this possible? Before blocking the lower path, no photon went to D1. They all went to D0. If our hidden variable theory was correct, the photons that do not get absorbed should also go to D0, as shown below.
|Photon polarization||At BS1||At BS2||Final result|
|Up (+)||Photon is reflected||Photon is reflected||Photon goes to D0|
|Down (–)||Photon is absorbed||Photon was absorbed||Photon was absorbed|
Our hidden variable theory does not work. Physical or geometric interpretations of the wavefunction are nice, but they do not explain quantum-mechanical interference. Their value is, therefore, didactic only.
Jean Louis Van Belle, 2 November 2018
This paper discusses general principles in physics only. Hence, references were limited to references to general textbooks and courses and physics textbooks only. The two key references here are the MIT introductory course on quantum physics and Feynman’s Lectures – both of which can be consulted online. Additional references to other material are given in the text itself (see footnotes).
 Duns Scotus, Commentaria.
 Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 5, Section 5.
 See, for example, the MIT’s edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 4, Section 3.
 Photons are spin-one particles but they do not have a spin-zero state.
 Of course, the formulas only give the elementary wavefunction. The wave packet will be a Fourier sum of such functions.
 See, for example, https://warwick.ac.uk/fac/sci/physics/staff/academic/mhadley/explanation/spin/, accessed on 30 October 2018
 Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 6, Section 3.
 See: MIT edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 1, Section 3 (Loss of determinism).
 The z-direction is the direction of wave propagation in this example. In quantum mechanics, we often define the direction of wave propagation as the x-direction. This will, hopefully, not confuse the reader. The choice of axes is usually clear from the context.
 Source of the illustration: https://upload.wikimedia.org/wikipedia/commons/7/71/Sine_cosine_one_period.svg..
 Classical theory assumes an atomic or molecular system will absorb a photon and, therefore, be in an excited state (with higher energy). The atomic or molecular system then goes back into its ground state by emitting another photon with the same energy. Hence, we should probably not think in terms of a specific photon getting through.
 Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 1, Section 1.
 Source of the illustration: MIT edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 1, Section 4 (Quantum Superpositions).
I think I cracked the nut. Academics always throw two nasty arguments into the discussion on any geometric or physical interpretations of the wavefunction:
- The superposition of wavefunctions is done in the complex space and, hence, the assumption of a real-valued envelope for the wavefunction is, therefore, not acceptable.
- The wavefunction for spin-1/2 particles cannot represent any real object because of its 720-degree symmetry in space. Real objects have the same spatial symmetry as space itself, which is 360 degrees. Hence, physical interpretations of the wavefunction are nonsensical.
Well… I’ve finally managed to deconstruct those arguments – using, paradoxically, Feynman’s own arguments against him. Have a look: click the link to my latest paper ! Enjoy !
I realized that my last posts were just some crude and rude soundbites, so I thought it would be good to briefly summarize them into something more coherent. Please let me know what you think of it.
The Uncertainty Principle: epistemology versus physics
Anyone who has read anything about quantum physics will know that its concepts and principles are very non-intuitive. Several interpretations have therefore emerged. The mainstream interpretation of quantum mechanics is referred to as the Copenhagen interpretation. It mainly distinguishes itself from more frivolous interpretations (such as the many-worlds and the pilot-wave interpretations) because it is… Well… Less frivolous. Unfortunately, the Copenhagen interpretation itself seems to be subject to interpretation.
One such interpretation may be referred to as radical skepticism – or radical empiricism: we can only say something meaningful about Schrödinger’s cat if we open the box and observe its state. According to this rather particular viewpoint, we cannot be sure of its reality if we don’t make the observation. All we can do is describe its reality by a superposition of the two possible states: dead or alive. That’s Hilbert’s logic: the two states (dead or alive) are mutually exclusive but we add them anyway. If a tree falls in the wood and no one hears it, then it is both standing and not standing. Richard Feynman – who may well be the most eminent representative of mainstream physics – thinks this epistemological position is nonsensical, and I fully agree with him:
“A real tree falling in a real forest makes a sound, of course, even if nobody is there. Even if no one is present to hear it, there are other traces left. The sound will shake some leaves, and if we were careful enough we might find somewhere that some thorn had rubbed against a leaf and made a tiny scratch that could not be explained unless we assumed the leaf were vibrating.” (Feynman’s Lectures, III-2-6)
So what is the mainstream physicist’s interpretation of the Copenhagen interpretation of quantum mechanics then? To fully answer that question, I should encourage the reader to read all of Feynman’s Lectures on quantum mechanics. But then you are reading this because you don’t want to do that, so let me quote from his introductory Lecture on the Uncertainty Principle: “Making an observation affects the phenomenon. The point is that the effect cannot be disregarded or minimized or decreased arbitrarily by rearranging the apparatus. When we look for a certain phenomenon we cannot help but disturb it in a certain minimum way.” (ibidem)
It has nothing to do with consciousness. Reality and consciousness are two very different things. After having concluded the tree did make a noise, even if no one was there to hear it, he wraps up the philosophical discussion as follows: “We might ask: was there a sensation of sound? No, sensations have to do, presumably, with consciousness. And whether ants are conscious and whether there were ants in the forest, or whether the tree was conscious, we do not know. Let us leave the problem in that form.” In short, I think we can all agree that the cat is dead or alive, or that the tree is standing or not standing¾regardless of the observer. It’s a binary situation. Not something in-between. The box obscures our view. That’s all. There is nothing more to it.
Of course, in quantum physics, we don’t study cats but look at the behavior of photons and electrons (we limit our analysis to quantum electrodynamics – so we won’t discuss quarks or other sectors of the so-called Standard Model of particle physics). The question then becomes: what can we reasonably say about the electron – or the photon – before we observe it, or before we make any measurement. Think of the Stein-Gerlach experiment, which tells us that we’ll always measure the angular momentum of an electron – along any axis we choose – as either +ħ/2 or, else, as -ħ/2. So what’s its state before it enters the apparatus? Do we have to assume it has some definite angular momentum, and that its value is as binary as the state of our cat (dead or alive, up or down)?
We should probably explain what we mean by a definite angular momentum. It’s a concept from classical physics, and it assumes a precise value (or magnitude) along some precise direction. We may challenge these assumptions. The direction of the angular momentum may be changing all the time, for example. If we think of the electron as a pointlike charge – whizzing around in its own space – then the concept of a precise direction of its angular momentum becomes quite fuzzy, because it changes all the time. And if its direction is fuzzy, then its value will be fuzzy as well. In classical physics, such fuzziness is not allowed, because angular momentum is conserved: it takes an outside force – or torque – to change it. But in quantum physics, we have the Uncertainty Principle: some energy (force over a distance, remember) can be borrowed – so to speak – as long as it’s swiftly being returned – within the quantitative limits set by the Uncertainty Principle: ΔE·Δt = ħ/2.
Mainstream physicists – including Feynman – do not try to think about this. For them, the Stern-Gerlach apparatus is just like Schrödinger’s box: it obscures the view. The cat is dead or alive, and each of the two states has some probability – but they must add up to one – and so they will write the state of the electron before it enters the apparatus as the superposition of the up and down states. I must assume you’ve seen this before:
|ψ〉 = Cup|up〉 + Cdown|down〉
It’s the so-called Dirac or bra-ket notation. Cup is the amplitude for the electron spin to be equal to +ħ/2 along the chosen direction – which we refer to as the z-direction because we will choose our reference frame such that the z-axis coincides with this chosen direction – and, likewise, Cup is the amplitude for the electron spin to be equal to -ħ/2 (along the same direction, obviously). Cup and Cup will be functions, and the associated probabilities will vary sinusoidally – with a phase difference so as to make sure both add up to one.
The model is consistent, but it feels like a mathematical trick. This description of reality – if that’s what it is – does not feel like a model of a real electron. It’s like reducing the cat in our box to the mentioned fuzzy state of being alive and dead at the same time. Let’s try to come up with something more exciting. 😊
 Academics will immediately note that radical empiricism and radical skepticism are very different epistemological positions but we are discussing some basic principles in physics here rather than epistemological theories.
 The reference to Hilbert’s logic refers to Hilbert spaces: a Hilbert space is an abstract vector space. Its properties allow us to work with quantum-mechanical states, which become state vectors. You should not confuse them with the real or complex vectors you’re used to. The only thing state vectors have in common with real or complex vectors is that (1) we also need a base (aka as a representation in quantum mechanics) to define them and (2) that we can make linear combinations.
The ‘flywheel’ electron model
Physicists describe the reality of electrons by a wavefunction. If you are reading this article, you know how a wavefunction looks like: it is a superposition of elementary wavefunctions. These elementary wavefunctions are written as Ai·exp(-iθi), so they have an amplitude Ai and an argument θi = (Ei/ħ)·t – (pi/ħ)·x. Let’s forget about uncertainty, so we can drop the index (i) and think of a geometric interpretation of A·exp(-iθ) = A·e–iθ.
Here we have a weird thing: physicists think the minus sign in the exponent (-iθ) should always be there: the convention is that we get the imaginary unit (i) by a 90° rotation of the real unit (1) – but the rotation is counterclockwise rotation. I like to think a rotation in the clockwise direction must also describe something real. Hence, if we are seeking a geometric interpretation, then we should explore the two mathematical possibilities: A·e–iθ and A·e+iθ. I like to think these two wavefunctions describe the same electron but with opposite spin. How should we visualize this? I like to think of A·e–iθ and A·e+iθ as two-dimensional harmonic oscillators:
A·e–iθ = cos(-θ) + i·sin(-θ) = cosθ – i·sinθ
A·e+iθ = cosθ + i·sinθ
So we may want to imagine our electron as a pointlike electric charge (see the green dot in the illustration below) to spin around some center in either of the two possible directions. The cosine keeps track of the oscillation in one dimension, while the sine (plus or minus) keeps track of the oscillation in a direction that is perpendicular to the first one.
Figure 1: A pointlike charge in orbit
So we have a weird oscillator in two dimensions here, and we may calculate the energy in this oscillation. To calculate such energy, we need a mass concept. We only have a charge here, but a (moving) charge has an electromagnetic mass. Now, the electromagnetic mass of the electron’s charge may or may not explain all the mass of the electron (most physicists think it doesn’t) but let’s assume it does for the sake of the model that we’re trying to build up here. The point is: the theory of electromagnetic mass gives us a very simple explanation for the concept of mass here, and so we’ll use it for the time being. So we have some mass oscillating in two directions simultaneously: we basically assume space is, somehow, elastic. We have worked out the V-2 engine metaphor before, so we won’t repeat ourselves here.
Figure 2: A perpetuum mobile?
Previously unrelated but structurally similar formulas may be related here:
- The energy of an oscillator: E = (1/2)·m·a2ω2
- Kinetic energy: E = (1/2)·m·v2
- The rotational (kinetic) energy that’s stored in a flywheel: E = (1/2)·I·ω2 = (1/2)·m·r2·ω2
- Einstein’s energy-mass equivalence relation: E = m·c2
Of course, we are mixing relativistic and non-relativistic formulas here, and there’s the 1/2 factor – but these are minor issues. For example, we were talking not one but two oscillators, so we should add their energies: (1/2)·m·a2·ω2 + (1/2)·m·a2·ω2 = m·a2·ω2. Also, one can show that the classical formula for kinetic energy (i.e. E = (1/2)·m·v2) morphs into E = m·c2 when we use the relativistically correct force equation for an oscillator. So, yes, our metaphor – or our suggested physical interpretation of the wavefunction, I should say – makes sense.
If you know something about physics, then you know the concept of the electromagnetic mass – its mathematical derivation, that is – gives us the classical electron radius, aka as the Thomson radius. It’s the smallest of a trio of radii that are relevant when discussing electrons: the other two radii are the Bohr radius and the Compton scattering radius respectively. The Thomson radius is used in the context of elastic scattering: the frequency of the incident particle (usually a photon), and the energy of the electron itself, do not change. In contrast, Compton scattering does change the frequency of the photon that is being scattered, and also impacts the energy of our electron. [As for the Bohr radius, you know that’s the radius of an electron orbital, roughly speaking – or the size of a hydrogen atom, I should say.]
Now, if we combine the E = m·a2·ω2 and E = m·c2 equations, then a·ω must be equal to c, right? Can we show this? Maybe. It is easy to see that we get the desired equality by substituting the amplitude of the oscillation (a) for the Compton scattering radius r = ħ/(m·c), and ω (the (angular) frequency of the oscillation) by using the Planck relation (ω = E/ħ):
a·ω = [ħ/(m·c)]·[E/ħ] = E/(m·c) = m·c2/(m·c) = c
We get a wonderfully simple geometric model of an electron here: an electric charge that spins around in a plane. Its radius is the Compton electron radius – which makes sense – and the radial velocity of our spinning charge is the speed of light – which may or may not make sense. Of course, we need an explanation of why this spinning charge doesn’t radiate its energy away – but then we don’t have such explanation anyway. All we can say is that the electron charge seems to be spinning in its own space – that it’s racing along a geodesic. It’s just like mass creates its own space here: according to Einstein’s general relativity theory, gravity becomes a pseudo-force—literally: no real force. How? I am not sure: the model here assumes the medium – empty space – is, somehow, perfectly elastic: the electron constantly borrows energy from one direction and then returns it to the other – so to speak. A crazy model, yes – but is there anything better? We only want to present a metaphor here: a possible visualization of quantum-mechanical models.
However, if this model is to represent anything real, then many more questions need to be answered. For starters, let’s think about an interpretation of the results of the Stern-Gerlach experiment.
A spinning charge is a tiny magnet – and so it’s got a magnetic moment, which we need to explain the Stern-Gerlach experiment. But it doesn’t explain the discrete nature of the electron’s angular momentum: it’s either +ħ/2 or -ħ/2, nothing in-between, and that’s the case along any direction we choose. How can we explain this? Also, space is three-dimensional. Why would electrons spin in a perfect plane? The answer is: they don’t.
Indeed, the corollary of the above-mentioned binary value of the angular momentum is that the angular momentum – or the electron’s spin – is never completely along any direction. This may or may not be explained by the precession of a spinning charge in a field, which is illustrated below (illustration taken from Feynman’s Lectures, II-35-3).
Figure 3: Precession of an electron in a magnetic field
So we do have an oscillation in three dimensions here, really – even if our wavefunction is a two-dimensional mathematical object. Note that the measurement (or the Stein-Gerlach apparatus in this case) establishes a line of sight and, therefore, a reference frame, so ‘up’ and ‘down’, ‘left’ and ‘right’, and ‘in front’ and ‘behind’ get meaning. In other words, we establish a real space. The question then becomes: how and why does an electron sort of snap into place?
The geometry of the situation suggests the logical angle of the angular momentum vector should be 45°. Now, if the value of its z-component (i.e. its projection on the z-axis) is to be equal to ħ/2, then the magnitude of J itself should be larger. To be precise, it should be equal to ħ/√2 ≈ 0.7·ħ (just apply Pythagoras’ Theorem). Is that value compatible with our flywheel model?
Maybe. Let’s see. The classical formula for the magnetic moment is μ = I·A, with I the (effective) current and A the (surface) area. The notation is confusing because I is also used for the moment of inertia, or rotational mass, but… Well… Let’s do the calculation. The effective current is the electron charge (qe) divided by the period (T) of the orbital revolution: : I = qe/T. The period of the orbit is the time that is needed for the electron to complete one loop. That time (T) is equal to the circumference of the loop (2π·a) divided by the tangential velocity (vt). Now, we suggest vt = r·ω = a·ω = c, and the circumference of the loop is 2π·a. For a, we still use the Compton radius a = ħ/(m·c). Now, the formula for the area is A = π·a2, so we get:
μ = I·A = [qe/T]·π·a2 = [qe·c/(2π·a)]·[π·a2] = [(qe·c)/2]·a = [(qe·c)/2]·[ħ/(m·c)] = [qe/(2m)]·ħ
In a classical analysis, we have the following relation between angular momentum and magnetic moment:
μ = (qe/2m)·J
Hence, we find that the angular momentum J is equal to ħ, so that’s twice the measured value. We’ve got a problem. We would have hoped to find ħ/2 or ħ/√2. Perhaps it’s because a = ħ/(m·c) is the so-called reduced Compton scattering radius…
Maybe we’ll find the solution one day. I think it’s already quite nice we have a model that’s accurate up to a factor of 1/2 or 1/√2. 😊
Post scriptum: I’ve turned this into a small article which may or may not be more readable. You can link to it here. Comments are more than welcome.