Paul Ehrenfest and the search for truth

On 25 September 1933, Paul Ehrenfest took his son Wassily, who was suffering from Down syndrome, for a walk in the park. He shot him, and then killed himself. He was only 53. That’s my age bracket. From the letters he left (here is a summary in Dutch), we know his frustration of not being able to arrive at some kind of common-sense interpretation of the new quantum physics played a major role in the anxiety that had brought him to this point. He had taken courses from Ludwig Boltzmann as an aspiring young man. We, therefore, think Boltzmann’s suicide – for similar reasons – might have troubled him too.

His suicide did not come unexpectedly: he had announced it. In one of his letters to Einstein, he complains about ‘indigestion’ from the ‘unendlicher Heisenberg-Born-Dirac-Schrödinger Wurstmachinen-Physik-Betrieb.’ I’ll let you google-translate that. :-/ He also seems to have gone through the trouble of summarizing all his questions on the new approach in an article in what was then one of the top journals for physics: Einige die Quantenmechanik betreffende Erkundigungsfrage, Zeitschrift für Physik 78 (1932) 555-559 (quoted in the above-mentioned review article). This I’ll translate: Some Questions about Quantum Mechanics.


Paul Ehrenfest in happier times (painting by Harm Kamerlingh Onnes in 1920)

A diplomat-friend of mine once remarked this: “It is good you are studying physics only as a pastime. Professional physicists are often troubled people—miserable.” It is an interesting observation from a highly intelligent outsider. To be frank, I understand this strange need to probe things at the deepest level—to be able to explain what might or might not be the case (I am using Wittgenstein’s definition of reality here). Even H.A. Lorentz, who – fortunately, perhaps – died before his successor did what he did, was becoming quite alarmist about the sorry state of academic physics near the end of his life—and he, Albert Einstein, and so many others were not alone. Not then, and not now. All of the founding fathers of quantum mechanics ended up becoming pretty skeptical about the theory they had created. We have documented that elsewhere so we won’t talk too much about it here. Even John Stewart Bell himself – one of the third generation of quantum physicists, we may say – did not like his own ‘No Go Theorem’ and thought that some “radical conceptual renewal”[1] might disprove his conclusions.

The Born-Heisenberg revolution has failed: most – if not all – of contemporary high-brow physicist are pursuing alternative theories—in spite, or because, of the academic straitjackets they have to wear. If a genius like Ehrenfest didn’t buy it, then I won’t buy it either. Furthermore, the masses surely don’t buy it and, yes, truth – in this domain too – is, fortunately, being defined more democratically nowadays. The Nobel Prize Committee will have to do some serious soul-searching—if not five years from now, then ten.

We feel sad for the physicists who died unhappily—and surely for those who took their life out of depression—because the common-sense interpretation they were seeking is so self-evident: de Broglie’s intuition in regard to matter being wavelike was correct. He just misinterpreted its nature: it is not a linear but a circular wave. We quickly insert the quintessential illustration (courtesy of Celani, Vassallo and Di Tommaso) but we refer the reader for more detail to our articles or – more accessible, perhaps – our manuscript for the general public.

aa 2

The equations are easy. The mass of an electron – any matter-particle, really – is the equivalent mass of the oscillation of the charge it carries. This oscillation is, most probably, statistically regular only. So we think it’s chaotic, actually, but we also think the words spoken by Lord Pollonius in Shakespeare’s Hamlet apply to it: “Though this be madness, yet there is method in ‘t.” This means we can meaningfully speak of a cycle time and, therefore, of a frequency. Erwin Schrödinger stumbled upon this motion while exploring solutions to Dirac’s wave equation for free electrons, and Dirac immediately grasped the significance of Schrödinger’s discovery, because he mentions Schrödinger’s discovery rather prominently in his Nobel Prize Lecture:

“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)

Unfortunately, Dirac confuses the concept of the electron as a particle with the concept of the (naked) charge inside. Indeed, the idea of an elementary (matter-)particle must combine the idea of a charge and its motion to account for both the particle- as well as the wave-like character of matter-particles. We do not want to dwell on all of this because we’ve written too many papers on this already. We just thought it would be good to sum up the core of our common-sense interpretation of physics. Why? To honor Boltzmann and Ehrenfest: I think of their demise as a sacrifice in search for truth.


OK. That sounds rather tragic—sorry for that! For the sake of brevity, we will just describe the electron here.

I. Planck’s quantum of action (h) and the speed of light (c) are Nature’s most fundamental constants. Planck’s quantum of action relates the energy of a particle to its cycle time and, therefore, to its frequency:

(1) h = E·T = E/f ⇔ ħ = E/ω

The charge that is whizzing around inside of the electron has zero rest mass, and so it whizzes around at the speed of light: the slightest force on it gives it an infinite acceleration. It, therefore, acquires a relativistic mass which is equal to mγ = me/2 (we refer to our paper(s) for a relativistically correct geometric argument). The momentum of the pointlike charge, in its circular or orbital motion, is, therefore, equal to p = mγ·c = me·c/2.

The (angular) frequency of the oscillation is also given by the formula for the (angular) velocity:

(2) c = a·ω ⇔ ω = c/a

While Eq. (1) is a fundamental law of Nature, Eq. (2) is a simple geometric or mathematical relation only.

II. From (1) and (2), we can now calculate the radius of this tiny circular motion as:

(3a) ħ = E/ω = E·a/c a = (ħ·c)/E

Because we know the mass of the electron is the inertial mass of the state of motion of the pointlike charge, we may use Einstein’s mass-energy equivalence relation to rewrite this as the Compton radius of the electron:

(3b) a = (ħ·c)/E = (ħ·c)/(me·c2) = ħ/(me·c)

Note that we only used two fundamental laws of Nature so far: the Planck-Einstein relation and Einstein’s mass-energy equivalence relation.

III. We must also be able to express the Planck-Einstein quantum as the product of the momentum (p) of the pointlike charge and some length λ:

(4) h = p·λ

The question here is: what length? The circumference of the loop, or its radius? The same geometric argument we used to derive the effective mass of the pointlike charge as it whizzes around at lightspeed around its center, tells us the centripetal force acts over a distance that is equal to two times the radius. Indeed, the relevant formula for the centripetal force is this:

(5) F = (mγ/me)·(E/a) = E/2a

We can therefore reduce Eq. (4) by dividing it by 2π. We then get reduced, angular or circular (as opposed to linear) concepts:

(6) ħ = (p·λ)/(2π) = (me·c/2)·(λ/π) = (me·c/2)·(2a) = me·c·a ⇔ ħ/a = me·c

We can verify the logic of our reasoning by substituting for the Compton radius:

ħ = p·λ = me·c·= me·c·a = me·c·ħ/(me·c) = ħ

IV. We can, finally, re-confirm the logic of our reason by re-deriving Einstein’s mass-energy equivalence relation as well as the Planck-Einstein relation using the ω = c/a and the ħ/a = me·c relations:

(7) ħ·ω = ħ·c/a = (ħ/ac = (me·cc = me·c2 = E

Of course, we note all of the formulas we have derived are interdependent. We, therefore, have no clear separation between axioms and derivations here. If anything, we are only explaining what Nature’s most fundamental laws (the Planck-Einstein relation and Einstein’s mass-energy equivalence relation) actually mean or represent. As such, all we have is a simple description of reality itself—at the smallest scale, of course! Everything that happens at larger scales involves Maxwell’s equations: that’s all electromagnetic in nature. No need for strong or weak forces, or for quarks—who invented that? Ehrenfest, Lorentz and all who suffered with truly understanding the de Broglie’s concept of the matter-wave might have been happier physicists if they would have seen these simple equations!

The gist of the matter is this: the intuition of Einstein and de Broglie in regard to the wave-nature of matter was, essentially, correct. However, de Broglie’s modeling of it as a wave packet was not: modeling matter-particles as some linear oscillation does not do the trick. It is extremely surprising no one thought of trying to think of some circular oscillation. Indeed, the interpretation of the elementary wavefunction as representing the mentioned Zitterbewegung of the electric charge solves all questions: it amounts to interpreting the real and imaginary part of the elementary wavefunction as the sine and cosine components of the orbital motion of a pointlike charge. We think that, in our 60-odd papers, we’ve shown such easy interpretation effectively does the trick of explaining all of the quantum-mechanical weirdness but, of course, it is up to our readers to judge that. 🙂

[1] See: John Stewart Bell, Speakable and unspeakable in quantum mechanics, pp. 169–172, Cambridge University Press, 1987 (quoted from Wikipedia). J.S. Bell died from a cerebral hemorrhage in 1990 – the year he was nominated for the Nobel Prize in Physics and which he, therefore, did not receive (Nobel Prizes are not awarded posthumously). He was just 62 years old then.

Rutherford’s idea of an electron

Pre-scriptum (dated 27 June 2020): Two illustrations in this post were deleted by the dark force. We will not substitute them. The reference is given and it will help you to look them up yourself. In fact, we think it will greatly advance your understanding if you do so. Mr. Gottlieb may actually have done us a favor by trying to pester us.

Electrons, atoms, elementary particles and wave equations

The New Zealander Ernest Rutherford came to be known as the father of nuclear physics. He was the first to provide a reliable estimate of the order of magnitude of the size of the nucleus. To be precise, in the 1921 paper which we will discuss here, he came up with an estimate of about 15 fm for massive nuclei, which is the current estimate for the size of an uranium nucleus. His experiments also helped to significantly enhance the Bohr model of an atom, culminating – just before WW I started – in the Bohr-Rutherford model of an atom (E. Rutherford, Phil. Mag. 27, 488).

The Bohr-Rutherford model of an atom explained the (gross structure of the) hydrogen spectrum perfectly well, but it could not explain its finer structure—read: the orbital sub-shells which, as we all know now (but not very well then), result from the different states of angular momentum of an electron and the associated magnetic moment.

The issue is probably best illustrated by the two diagrams below, which I copied from Feynman’s Lectures. As you can see, the idea of subshells is not very relevant when looking at the gross structure of the hydrogen spectrum because the energy levels of all subshells are (very nearly) the same. However, the Bohr model of an atom—which is nothing but an exceedingly simple application of the E = h·f equation (see p. 4-6 of my paper on classical quantum physics)—cannot explain the splitting of lines for a lithium atom, which is shown in the diagram on the right. Nor can it explain the splitting of spectral lines when we apply a stronger or weaker magnetic field while exciting the atoms so as to induce emission of electromagnetic radiation.

Schrödinger’s wave equation solves that problem—which is why Feynman and other modern physicists claim this equation is “the most dramatic success in the history of the quantum mechanics” or, more modestly, a “key result in quantum mechanics” at least!

Such dramatic statements are exaggerated. First, an even finer analysis of the emission spectrum (of hydrogen or whatever other atom) reveals that Schrödinger’s wave equation is also incomplete: the hyperfine splitting, the Zeeman splitting (anomalous or not) or the (in)famous Lamb shift are to be explained not only in terms of the magnetic moment of the electron but also in terms of the magnetic moment of the nucleus and its constituents (protons and neutrons)—or of the coupling between those magnetic moments (we may refer to our paper on the Lamb shift here). This cannot be captured in a wave equation: second-order differential equations are – quite simply – not sophisticated enough to capture the complexity of the atomic system here.

Also, as we pointed out previously, the current convention in regard to the use of the imaginary unit (i) in the wavefunction does not capture the spin direction and, therefore, makes abstraction of the direction of the magnetic moment too! The wavefunction therefore models theoretical spin-zero particles, which do not exist. In short, we cannot hope to represent anything real with wave equations and wavefunctions.

More importantly, I would dare to ask this: what use is an ‘explanation’ in terms of a wave equation if we cannot explain what the wave equation actually represents? As Feynman famously writes: “Where did we get it from? Nowhere. It’s not possible to derive it from anything you know. It came out of the mind of Schrödinger, invented in his struggle to find an understanding of the experimental observations of the real world.” Our best guess is that it, somehow, models (the local diffusion of) energy or mass densities as well as non-spherical orbital geometries. We explored such interpretations in our very first paper(s) on quantum mechanics, but the truth is this: we do not think wave equations are suitable mathematical tools to describe simple or complex systems that have some internal structure—atoms (think of Schrödinger’s wave equation here), electrons (think of Dirac’s wave equation), or protons (which is what some others tried to do, but I will let you do some googling here yourself).

We need to get back to the matter at hand here, which is Rutherford’s idea of an electron back in 1921. What can we say about it?

Rutherford’s contributions to the 1921 Solvay Conference

From what you know, and from what I write above, you will understand that Rutherford’s research focus was not on electrons: his prime interest was in explaining the atomic structure and in solving the mysteries of nuclear radiation—most notably the emission of alpha– and beta-particles as well as highly energetic gamma-rays by unstable or radioactive nuclei. In short, the nature of the electron was not his prime interest. However, this intellectual giant was, of course, very much interested in whatever experiment or whatever theory that might contribute to his thinking, and that explains why, in his contribution to the 1921 Solvay Conference—which materialized as an update of his seminal 1914 paper on The Structure of the Atom—he devotes considerable attention to Arthur Compton’s work on the scattering of light from electrons which, at the time (1921), had not even been published yet (Compton’s seminal article on (Compton) scattering was published in 1923 only).

It is also very interesting that, in the very same 1921 paper—whose 30 pages are more than a multiple of his 1914 article and later revisions of it (see, for example, the 1920 version of it, which actually has wider circulation on the Internet)—Rutherford also offers some short reflections on the magnetic properties of electrons while referring to Parson’s ring current model which, in French, he refers to as “l’électron annulaire de Parson.” Again, it is very strange that we should translate Rutherford’s 1921 remarks back in English—as we are sure the original paper must have been translated from English to French rather than the other way around.

However, it is what it is, and so here we do what we have to do: we give you a free translation of Rutherford’s remarks during the 1921 Solvay Conference on the state of research regarding the electron at that time. The reader should note these remarks are buried in a larger piece on the emission of β particles by radioactive nuclei which, as it turns out, are nothing but high-energy electrons (or their anti-matter counterpart—positrons). In fact, we should—before we proceed—draw attention to the fact that the physicists at the time had no clear notion of the concepts of protons and neutrons.

This is, indeed, another remarkable historical contribution of the 1921 Solvay Conference because, as far as I know, this is the first time Rutherford talks about the neutron hypothesis. It is quite remarkable he does not advance the neutron hypothesis to explain the atomic mass of atoms combining what we know think of as protons and neutrons (Rutherford regularly talks of a mix of ‘positive and negative electrons’ in the nucleus—neither the term proton or neutron was in use at the time) but as part of a possible explanation of nuclear fusion reactions in stars or stellar nebulae. This is, indeed, his response to a question during the discussions on Rutherford’s paper on the possibility of nuclear synthesis in stars or nebulae from the French physicist Jean Baptise Perrin who, independently from the American chemist William Draper Harkins, had proposed the possibility of hydrogen fusion just the year before (1919):

“We can, in fact, think of enormous energies being released from hydrogen nuclei merging to form helium—much larger energies than what can come from the Kelvin-Helmholtz mechanism. I have been thinking that the hydrogen in the nebulae might come from particles which we may refer to as ‘neutrons’: these would consist of a positive nucleus with an electron at an exceedingly small distance (“un noyau positif avec un électron à toute petite distance”). These would mediate the assembly of the nuclei of more massive elements. It is, otherwise, difficult to understand how the positively charged particles could come together against the repulsive force that pushes them apart—unless we would envisage they are driven by enormous velocities.”

We may add that, just to make sure he get this right, Rutherford is immediately requested to elaborate his point by the Danish physicist Martin Knudsen: “What’s the difference between a hydrogen atom and this neutron?”—which Rutherford simply answers as follows: “In a neutron, the electron would be very much closer to the nucleus.” In light of the fact that it was only in 1932 that James Chadwick would experimentally prove the existence of neutrons (and positively charged protons), we are, once again, deeply impressed by the the foresight of Rutherford and the other pioneers here: the predictive power of their theories and ideas is, effectively, truly amazing by any standard—including today’s. I should, perhaps, also add that I fully subscribe to Rutherford’s intuition that a neutron should be a composite particle consisting of a proton and an electron—but that’s a different discussion altogether.

We must come back to the topic of this post, which we will do now. Before we proceed, however, we should highlight one other contextual piece of information here: at the time, very little was known about the nature of α and β particles. We now know that beta-particles are electrons, and that alpha-particles combine two protons and two neutrons. That was not known in the 1920s, however: Rutherford and his associates could basically only see positive or negative particles coming out of these radioactive processes. This further underscores how much knowledge they were able to gain from rather limited sets of data.

Rutherford’s idea of an electron in 1921

So here is the translation of some crucial text. Needless to say, the italics, boldface and additions between [brackets] are not Rutherford’s but mine, of course.

“We may think the same laws should apply in regard to the scattering [“diffusion”] of α and β particles. [Note: Rutherford noted, earlier in his paper, that, based on the scattering patterns and other evidence, the force around the nucleus must respect the inverse square law near the nucleus—moreover, it must also do so very near to it.] However, we see marked differences. Anyone who has carefully studied the trajectories [photographs from the Wilson cloud chamber] of beta-particles will note the trajectories show a regular curvature. Such curved trajectories are even more obvious when they are illuminated by X-rays. Indeed, A.H. Compton noted that these trajectories seem to end in a converging helical path turning right or left. To explain this, Compton assumes the electron acts like a magnetic dipole whose axis is more or less fixed, and that the curvature of its path is caused by the magnetic field [from the (paramagnetic) materials that are used].

Further examination would be needed to make sure this curvature is not some coincidence, but the general impression is that the hypothesis may be quite right. We also see similar curvature and helicity with α particles in the last millimeters of their trajectories. [Note: α-particles are, obviously, also charged particles but we think Rutherford’s remark in regard to α particles also following a curved or helical path must be exaggerated: the order of magnitude of the magnetic moment of protons and neutrons is much smaller and, in any case, they tend to cancel each other out. Also, because of the rather enormous mass of α particles (read: helium nuclei) as compared to electrons, the effect would probably not be visible in a Wilson cloud chamber.]

The idea that an electron has magnetic properties is still sketchy and we would need new and more conclusive experiments before accepting it as a scientific fact. However, it would surely be natural to assume its magnetic properties would result from a rotation of the electron. Parson’s ring electron model [“électron annulaire“] was specifically imagined to incorporate such magnetic polarity [“polarité magnétique“].

A very interesting question here would be to wonder whether such rotation would be some intrinsic property of the electron or if it would just result from the rotation of the electron in its atomic orbital around the nucleus. Indeed, James Jeans usefully reminded me any asymmetry in an electron should result in it rotating around its own axis at the same frequency of its orbital rotation. [Note: The reader can easily imagine this: think of an asymmetric object going around in a circle and returning to its original position. In order to return to the same orientation, it must rotate around its own axis one time too!]

We should also wonder if an electron might acquire some rotational motion from being accelerated in an electric field and if such rotation, once acquired, would persist when decelerating in an(other) electric field or when passing through matter. If so, some of the properties of electrons would, to some extent, depend on their past.”

Each and every sentence in these very brief remarks is wonderfully consistent with modern-day modelling of electron behavior. We should add, of course, non-mainstream modeling of electrons but the addition is superfluous because mainstream physicists stubbornly continue to pretend electrons have no internal structure, and nor would they have any physical dimension. In light of the numerous experimental measurements of the effective charge radius as well as of the dimensions of the physical space in which photons effectively interfere with electrons, such mainstream assumptions seem completely ridiculous. However, such is the sad state of physics today.

Thinking backward and forward

We think that it is pretty obvious that Rutherford and others would have been able to adapt their model of an atom to better incorporate the magnetic properties not only of electrons but also of the nucleus and its constituents (protons and neutrons). Unfortunately, scientists at the time seem to have been swept away by the charisma of Bohr, Heisenberg and others, as well as by the mathematical brilliance of the likes of Sommerfeld, Dirac, and Pauli.

The road then was taken then has not led us very far. We concur with Oliver Consa’s scathing but essentially correct appraisal of the current sorry state of physics:

“QED should be the quantized version of Maxwell’s laws, but it is not that at all. QED is a simple addition to quantum mechanics that attempts to justify two experimental discrepancies in the Dirac equation: the Lamb shift and the anomalous magnetic moment of the electron. The reality is that QED is a bunch of fudge factors, numerology, ignored infinities, hocus-pocus, manipulated calculations, illegitimate mathematics, incomprehensible theories, hidden data, biased experiments, miscalculations, suspicious coincidences, lies, arbitrary substitutions of infinite values and budgets of 600 million dollars to continue the game. Maybe it is time to consider alternative proposals. Winter is coming.”

I would suggest we just go back where we went wrong: it may be warmer there, and thinking both backward as well as forward must, in any case, be a much more powerful problem solving technique than relying only on expert guessing on what linear differential equation(s) might give us some S-matrix linking all likely or possible initial and final states of some system or process. 🙂

Post scriptum: The sad state of physics is, of course, not limited to quantum electrodynamics only. We were briefly in touch with the PRad experimenters who put an end to the rather ridiculous ‘proton radius puzzle’ by re-confirming the previously established 0.83-0.84 range for the effective charge radius of a proton: we sent them our own classical back-of-the-envelope calculation of the Compton scattering radius of a proton based on the ring current model (see p. 15-16 of our paper on classical physics), which is in agreement with these measurements and courteously asked what alternative theories they were suggesting. Their spokesman replied equally courteously:

“There is no any theoretical prediction in QCD. Lattice [theorists] are trying to come up [with something] but that will take another decade before any reasonable  number [may come] from them.”

This e-mail exchange goes back to early February 2020. There has been no news since. One wonders if there is actually any real interest in solving puzzles. The physicist who wrote the above may have been nominated for a Nobel Prize in Physics—I surely hope so because, in contrast to some others, he and his team surely deserve one— but I think it is rather incongruous to finally firmly establish the size of a proton while, at the same time, admit that protons should not have any size according to mainstream theory—and we are talking the respected QCD sector of the equally respected Standard Model here!

We understand, of course! As Freddy Mercury famously sang: The Show Must Go On.

Mainstream QM: A Bright Shining Lie

Yesterday night, I got this email from a very bright young physicist: Dr. Oliver Consa. He is someone who – unlike me – does have the required Dr and PhD credentials in physics (I have a drs. title in economics) – and the patience that goes with it – to make some more authoritative statements in the weird world of quantum mechanics. I recommend you click the link in the email (copied below) and read the paper. Please do it! 

It is just 12 pages, and it is all extremely revealing. Very discomforting, actually, in light of all the other revelations on fake news in other spheres of life.

Many of us – and, here, I just refer to those who are reading my post – all sort of suspected that some ‘inner circle’ in the academic circuit had cooked things up:the Mystery Wallahs, as I refer to them now. Dr. Consa’s paper shows our suspicion is well-founded.


Dear fellow scientist,

I send you this mail because you have been skeptical about Foundations of Physics. I think that this new paper will be of your interest. Feel free to share it with your colleagues or publish it on the web. I consider it important that this paper serves to open a public debate on this subject.

Something is Rotten in the State of QED

“Quantum electrodynamics (QED) is considered the most accurate theory in the history of science. However, this precision is based on a single experimental value: the anomalous magnetic moment of the electron (g-factor). An examination of QED history reveals that this value was obtained using illegitimate mathematical traps, manipulations and tricks. These traps included the fraud of Kroll & Karplus, who acknowledged that they lied in their presentation of the most relevant calculation in QED history. As we will demonstrate in this paper, the Kroll & Karplus scandal was not a unique event. Instead, the scandal represented the fraudulent manner in which physics has been conducted from the creation of QED through today.”  (12 pag.)

Best Regards,
Oliver Consa


The Mystery Wallahs

I’ve been working across Asia – mainly South Asia – for over 25 years now. You will google the exact meaning but my definition of a wallah is a someone who deals in something: it may be a street vendor, or a handyman, or anyone who brings something new. I remember I was one of the first to bring modern mountain bikes to India, and they called me a gear wallah—because they were absolute fascinated with the number of gears I had. [Mountain bikes are now back to a 2 by 10 or even a 1 by 11 set-up, but I still like those three plateaux in front on my older bikes—and, yes, my collection is becoming way too large but I just can’t do away with it.]

Any case, let me explain the title of this post. I stumbled on the work of the research group around Herman Batelaan in Nebraska. Absolutely fascinating ! Not only did they actually do the electron double-slit experiment, but their ideas on an actual Stern-Gerlach experiment with electrons are quite interesting:

I also want to look at their calculations on momentum exchange between electrons in a beam:

Outright fascinating. Brilliant ! […]

It just makes me wonder: why is the outcome of this 100-year old battle between mainstream hocus-pocus and real physics so undecided?

I’ve come to think of mainstream physicists as peddlers in mysteries—whence the title of my post. It’s a tough conclusion. Physics is supposed to be the King of Science, right? Hence, we shouldn’t doubt it. At the same time, it is kinda comforting to know the battle between truth and lies rages everywhere—including inside of the King of Science.


A common-sense interpretation of (quantum) physics

This is my summary of what I refer to as a common-sense interpretation of quantum physics. It’s a rather abstruse summary of the 40 papers I wrote over the last two years.

1. A force acts on a charge. The electromagnetic force acts on an electric charge (there is no separate magnetic charge) and the strong force acts on a strong charge. A charge is a charge: a pointlike ‘thing’ with zero rest mass. The idea of an electron combines the idea of a charge and its motion (Schrödinger’s Zitterbewegung). The electron’s rest mass is the equivalent mass of the energy in its motion (mass without mass). The elementary wavefunction represents this motion.

2. There is no weak force: a force theory explaining why charges stay together must also explain when and how they separate. A force works through a force field: the idea that forces are mediated by virtual messenger particles resembles 19th century aether theory. The fermion-boson dichotomy does not reflect anything real: we have charged and non-charged wavicles (electrons versus photons, for example).

3. The Planck-Einstein law embodies a (stable) wavicle. A stable wavicle respects the Planck-Einstein relation (E = hf) and Einstein’s mass-energy equivalence relation (E = m·c2). A wavicle will, therefore, carry energy but it will also pack one or more units of Planck’s quantum of action. Planck’s quantum of action represents an elementary cycle in Nature. An elementary particle embodies the idea of an elementary cycle.

4. The ‘particle zoo’ is a collection of unstable wavicles: they disintegrate because their cycle is slightly off (the integral of the force over the distance of the loop and over the cycle time is not exactly equal to h).

5. An electron is a wavicle that carries charge. A photon does not carry charge: it carries energy between wavicle systems (atoms, basically). It can do so because it is an oscillating field.

6. An atom is a wavicle system. A wavicle system has an equilibrium energy state. This equilibrium state packs one unit of h. Higher energy states pack two, three,…, n units of h. When an atom transitions from one energy state to another, it will emit or absorb a photon that (i) carries the energy difference between the two energy states and (ii) packs one unit of h.

7. Nucleons (protons and neutrons) are held together because of a strong force. The strong force acts on a strong charge, for which we need to define a new unit: we choose the dirac but – out of respect for Yukawa, we write one dirac as 1 Y. If Yukawa’s function models the strong force correctly, then the strong force – which we denote as FN – can be calculated from the Yukawa potential:


This function includes a scale parameter a and a nuclear proportionality constant υ0. Besides its function as an (inverse) mathematical proportionality constant, it also ensures the physical dimensions on the left- and the right-hand side of the force equation are the same. We can choose to equate the numerical value of υ0 to one.

8. The nuclear force attracts two positive electric charges. The electrostatic force repels them. These two forces are equal at a distance r = a. The strong charge unit (gN) can, therefore, be calculated. It is equal to:


9. Nucleons (protons or neutrons) carry both electric as well as strong charge (qe and gN). A kinematic model disentangling both has not yet been found. Such model should explain the magnetic moment of protons and neutrons.

10. We think of a nucleus as wavicle system too. When going from one energy state to another, the nucleus emits or absorbs neutrinos. Hence, we think of the neutrino as the photon of the strong force. Such changes in energy states may also involve the emission and/or absorption of an electric charge (an electron or a positron).

Does this make sense? I look forward to your thoughts. 🙂


Because the above is all very serious, I thought it would be good to add something that will make you smile. 🙂


Polarization states as hidden variables?

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 !

A physical interpretation of the wavefunction

Duns Scotus wrote: pluralitas non est ponenda sine necessitate. Plurality is not to be posited without necessity.[1] And William of Ockham gave us the intuitive lex parsimoniae: the simplest solution tends to be the correct one.[2] 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.”[3]

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.”[4]

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[5] (up or down), as shown below.

Table 1: Matching mathematical possibilities with physical realities?[6]

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.[7] 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.”[8]

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 = eCup and C’down = eCdown. 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.[9] 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 loss of determinism in quantum mechanics

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.”[10]

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.

Linear polarization states

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[11] 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?[12]


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.[13] 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.”[14]

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.

Polarization states as hidden variables in the Mach-Zehnder 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[15]

Mach Zehnder

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:

Final result Probability
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).

[1] Duns Scotus, Commentaria.

[2] See:

[3] Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 5, Section 5.

[4] See, for example, the MIT’s edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 4, Section 3.

[5] Photons are spin-one particles but they do not have a spin-zero state.

[6] Of course, the formulas only give the elementary wavefunction. The wave packet will be a Fourier sum of such functions.

[7] See, for example,, accessed on 30 October 2018

[8] Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 6, Section 3.

[9] Jean Louis Van Belle, Euler’s wavefunction (, accessed on 30 October 2018)

[10] See: MIT edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 1, Section 3 (Loss of determinism).

[11] 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.

[12] Source of the illustration:

[13] 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.

[14] Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 1, Section 1.

[15] Source of the illustration: MIT edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 1, Section 4 (Quantum Superpositions).

The speed of light as an angular velocity

Over the weekend, I worked on a revised version of my paper on a physical interpretation of the wavefunction. However, I forgot to add the final remarks on the speed of light as an angular velocity. I know… This post is for my faithful followers only. It is dense, but let me add the missing bits here:


Post scriptum (29 October): Einstein’s view on aether theories probably still holds true: “We may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity, space without aether is unthinkable – for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.”

The above quote is taken from the Wikipedia article on aether theories. The same article also quotes Robert Laughlin, the 1998 Nobel Laureate in Physics, who said this about aether in 2005: “It is ironic that Einstein’s most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed. […] The word ‘aether’ has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum. […]The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic aether. But we do not call it this because it is taboo.”

I really love this: a relativistic aether. My interpretation of the wavefunction is very consistent with that.