A (neo-)classical look at the electron’s anomalous magnetic moment

Introduction

Every student of physics learns that the magnetic moment of the electron is given, to first approximation, by what is referred to as the Dirac value or – the more commonly used term – the Bohr magneton:

μ_e = μ_B = (ħ/2)·(q_e/m_e)

We separate the two factors in it because our (neo-classical) RealQM framework interprets (i) the q_e/m_e factor as a form factor which distinguishes the electron from, say, a proton or the more massive muon-electron (how much charge per mass or energy unit is ‘packed’ into the particle?), and (ii) ħ as the ubiquitous quantum of action that determines how energy, linear or angular momentum, or – in this case – magnetic moments are quantized. Think of it like this: charge comes in lumps (elementary charged particles), and its dynamical properties come in lumps too.

What about the 1/2 factor? We must refer to our Lecture on Quantum Behavior here for a rather common-sense interpretation of the g-factor and the spin-1/2 property of elementary matter-particles. The electron – interpreted as a dynamic oscillation of charge – ‘packs’ one Planck unit of physical action in each cycle of the oscillation but the angular momentum of the orbiting charge only explains half of the energy (and, therefore, the mass of the electron). The other half is in the electromagnetic field it generates, which keeps the charge spinning and effectively explains the magnetic dipole moment. So, just like protons or the more massive muon-electron, the electron can effectively be described as a ‘fermionic’ or ‘spin-1/2’ particle.

So far, so good. Experiments, however, show that the actual (measured) value is slightly larger. The difference is tiny—about one part in a thousand—and is known as the anomaly in the magnetic moment.

Modern physics explains this anomaly through quantum electrodynamics (QED). In that framework, the correction arises from increasingly complicated perturbative calculations involving “loop diagrams.” The first term in the resulting expansion was derived by Julian Schwinger in 1948 and is equal to $\alpha$ /2 $\pi$ . That is a very elegant expression, and it explains the anomaly for about 100.15%. Two quick notes must be made here:

1. We effectively wrote: 100.15%. Not 99.85%. Why? Because Schwinger’s factor overshoots the anomaly effectively by about 0.15%. To be precise, (μ_CODATA-μ_e)/μ_e = 0.00115965… and $\alpha$ _CODATA/2 $\pi$ = 0.0011614… The fact that the Schwinger term slightly overshoots the experimentally measured anomaly, explains why the next correction in QED is actually negative: it must bring the theoretical prediction back into agreement with experiment. This sign swap is rather poorly explained in standard textbooks but, in any case, it is any classical or neo-classical interpretation must probably reproduce such sign patterns. However, in our paper – which is just an introduction to our thinking on this – we do not go into that: we basically just explain – using classical arguments – why α and 2 $\pi$ appear so naturally not only in the leading correction but also higher-order corrections to the Dirac value for the electron’s magnetic moment.

2. Note that we take CODATA values here because these reflect (i) the scientific/academic consensus on the measured magnetic moment (as opposed to the theoretical Dirac value) and (ii) the scientific/academic consensus on the value of the fine-structure constant. The 2019 revision of the system of SI units effectively adds an interesting twist to the debate: the fine-structure constant itself is now defined as being co-determined with the electric and magnetic constant, and the standard relative uncertainty of all three constants is, therefore, now exactly the same: 1.6 $\times$ 10^-10, to be precise.

Let us now go back to the main story line. From the above, it is clear that, while the precision of Schwinger’s factor is higher than 0.15% of one part in a thousand (so that’s a precision of 1.5 parts into a million), we still need higher-order corrections. Why? Because measurements are much more accurate than one part into a million, and some theory should explain all significant digits, isn’t it? 🙂

Using the QED/QFT framework, theoretical physicists have currently computed those higher-order corrections up to what is referred to as the ‘five-loop level’ in QED, with astonishing agreement between theory and experiment. The problem with this ‘astonishing agreement’ is its complexity: the calculations become evermore complicated – as evermore degrees of freedom (cf. the increasing number of Feynman diagrams, which illustrate the various ways in which something might happen) explain less and less, as illustrated in the table below.

In a recently revised working paper on ResearchGate, I therefore explore a different question. Instead of asking how the anomaly emerges from perturbative quantum field theory, I ask whether the structure of the leading correction might also admit a classical interpretation. In the next section, we explain the basics of this classical interpretation, which is based on the ring current model of an electron, which was first advanced by Alfred Lauck Parson in 1915 and, as we explain in our paper on the nature of de Broglie’s matter-wave, also naturally explains Schrödinger’s Zitterbewegung theory. In other words, it is a very classical approach. 🙂

The electron ring current model

The starting point is, effectively, the old ring-current picture of the electron: imagine a localized “blob” of charge circulating around some center at the speed of light. This blob of charge has no other properties (no rest mass or rest energy) but its charge and, yes, some non-finite size. To be precise, from scattering experiments and the formulas that describe photon-electron scattering, we assume its size is of the order of the classical electron radius. As for the assumption of lightspeed, it is only logical to assume that anything with zero rest mass must travel at lightspeed because the slightest force on it will accelerate it to lightspeed.

When we apply this idea to the electron charge, we must conclude that the radius of this orbital motion will be the Compton radius of the electron. Indeed, when we think of the (elementary) wavefunction r = ψ = a·eⁱ^θ as representing the physical position of a pointlike elementary charge – pointlike but not dimensionless – moving at the speed of light around the center of its motion in a space, we must conclude that the radius of this orbital motion – which effectively doubles up as the amplitude of the wavefunction – must be equal to the electron’s Compton radius a = ħ/mc. This can easily be derived from (i) Einstein’s mass-energy equivalence relation, (ii) the Planck-Einstein relation, and (iii) the formula for a tangential velocity:

This electron model also naturally yields the Dirac magnetic moment: we can just calculate it using the standard formula for the magnetic moment of a ring current.

The next step, then, is to acknowledge that charge cannot literally be pointlike. If the circulating charge has a finite spatial extent—of order the classical electron radius—then the current distribution is slightly modified. The mathematical and physical elegance of Schwinger’s factor can then easily be explained by:

(i) noting that the classical electron radius is $\alpha$ times the Compton radius of the electron ħ/m_ec = ħc/E_e, which – in all mainstream accounts of photon-electron scattering experiments – is the scale at which the (free) electron can be localized in a particle-like sense (for more precise academic references (we like LeClair’s (2019) straightforward derivation and textbook explanation of Compton scattering), see our paper on de Broglie’s matter-wave); and

(ii) noting that the 2 $\pi$ factor points at orbital rather than linear motion. Indeed, a division by 2 $\pi$ is all what we need to do whenever we want to relate an orbital velocity or length (a linear wavelength or, when orbital motion rather than linear motion is involved, which is the case here) to radians rather than the SI unit for distance. More generally speaking, the $2\pi$ factor pops up whenever we evaluate some loop integral (i.e, whenever we integrate some quantity over a closed orbit using the phase of the (orbital) oscillation).

This strongly suggests that – from the various physical relations in which the fine-structure constant pops up – we should explore its meaning as a scaling constant. More in particular, the fine-structure constant here is the ratio which relates the classical electron radius and the Compton radius. We do not want to distract the reader too much but it is probably good to immediately point out that this interpretation of the fine-structure constant as a geometric ratio is also valid when examining the relation between the Compton radius (free electron) and the Bohr radius (the radius of the negative charge as it manifest itself in a hydrogen atom):

r_e = α·r_C ( $\approx$ 2.818 $\times$ 10^-15 m), r_B = r_C/α ( $\approx$ 0.529 $\times$ 10^-10 m), and r_C = ħ/m_ec = = ħc/E_e ( $\approx$ 386.53 $\times$ 10^-15 m)

This is intriguing because it shows a 1: 1/α : 1/α² ‘ladder’ or ‘series’ for the value of (i) what we think of the radius of the elementary charge, (ii) the radius of the elementary particle (free electron) in which this charge manifests itself, and (iii) the interaction radius of the bound electron in its most basic state (i.e., bound by a proton in the hydrogen atom). Needless to say, the multitude of physical phenomena in which the fine-structure constant pops up clearly illustrates that measuring its value can be done in a variety of ways. Measuring it through ever more precise measurements of the anomalous magnetic moment is, therefore, just one way to go about it.

To sum it all up, the hypothesis of a pointlike charge with radius r_e = α·r_C is in completely alignment with the suggestion that (i) any first-order correction to the ideal ring current must scale with α and (ii) because the circulating motion is periodic, any cycle-averaged correction would naturally come with a ‘normalization’ over the full ( $2\pi$ ) phase of the orbit. Put these ingredients together and one obtains a leading correction with the same structure as Schwinger’s factor:

Δμ/μ_e ≈ $\alpha$ /2 $\pi$

This does not necessarily replace the first-order QED derivation – especially because we should note that first-order QED calculations are based on the classical electromagnetic equations anyway – but it suggests that the form of the leading term may reflect a simple geometric fact: a finite electromagnetic structure (the ‘naked charge’ inside of a electron, as we call it) undergoing coherent circular motion.

The question, of course, then becomes: what about the higher-order corrections? What geometric or other common-sense arguments could one advance to explain second- or third-order scaling with α. In other words, why would terms with α² or α³ – or, more generally, ( $\alpha$ /2 $\pi$ )²ⁿ – pop up in any theoretical series of powers of α explaining the measured anomaly in the magnetic dipole moment of the electron?

That is what we are discussing in our paper, which we summarize and also comment on this blog post.

Again: why worry about an imprecision of about 1.5 parts in a million?

Again, the casual amateur physics may be tempted to let go of these discussions. The anomaly itself is tiny. The Dirac value already gets the magnetic moment right to about 0.1%, and the famous Schwinger correction then explains almost all of that 1% (about 100.15%, as we wrote). So, the higher-order QED terms only account for a very tiny fraction of the anomaly: something like 0.00015% (so that’s 1.5 parts in a million, indeed). So, why should we look for alternative or more comprehensive interpretations?

The answer is this: because the anomalous magnetic moment is one of the most precise measurements in all of science. Therefore, understanding why i) its leading structure has the form ( $\alpha$ /2 $\pi$ but (2) at the same time, why this leading structure does not fully explain the measured anomaly is not merely a numerical curiosity. It should tell us something about the underlying geometry of electromagnetic interactions. In other words, the tiny discrepancy carries a lot of conceptual weight.

This ‘conceptual weight’ may be illustrated by noting the complexities in QED/QFT calculations. They seem to contradict Occam’s Razor Principle: evermore complicated calculations – which result from allowing evermore degrees of freedom in the higher-order analysis, as illustrated in the table above – seem to explain less and less.

That is probably one of the reasons why there is discontent with the approach even in mainstream academics. We will let the reader google this – we warmly recommend Google’s Gemini assistant in this regard – but, as an example, it looks like lattice theory is currently rapidly emerging as a strong non-perturbative approach to ‘explaining’ the anomaly. [We put ‘explaining’ in brackets because, as far as we can see, lattice theory is, just like QED/QFT a predominantly mathematical approach, in the sense that its first principles are mathematical principles rather than physical laws). Below we briefly highlight this competing approach within the so-called Standard Model, which makes one wonder if there is still a thing such as a ‘standard’ model for explaining physics.

Perturbative QED: Treats interactions as small corrections (powers of the fine-structure constant) using Feynman diagrams. This works for the electron because its coupling is weak, but the complexity grows exponentially at higher orders.
Lattice Theory: Discretizes spacetime into a 4D grid (lattice) of points. It calculates interactions directly from first principles using Monte Carlo simulations on supercomputers, rather than summing infinite series of diagrams.

We must also note to another scientific breakthrough which, in our not-so-humble view, has received insufficient attention, and that is the 2019 revision of SI units. Let us discuss the most salient points of this very significant scientific revision.

The fine-structure constant and the 2019 revision of SI units

The extraordinary precision of the anomaly measurement is very closely connected to the discussion on (i) what the fine-structure constant α. actually is or represents in physics (as mentioned above, it pops up in many relations and equations) and, accordingly, (ii) how we should measure it.

Historically, α was measured through a variety of experiments and then used as an input for theoretical calculations. QED/QFT may be said to have inversed that logic: the measurement of the electron’s anomalous magnetic moment are used to then insert it the QED expansion, which is then used to solve for α. The result is, effectively, one of the most precise determinations of the fine-structure constant currently available but, to me, it looks like the 2019 revision of the SI system changed the conceptual landscape. Indeed, the 2019 revision of SI units clearly implies that (i) the fine-structure constant, (ii) the electric constant, and (iii) the magnetic constant must be co-determined because of the following physical equations:

Hence, the 2019 revision of SI units – which, we think, incorporates all of physics – makes it clear that, unlike precisely defined units such as the meter, second, charge, or exactly defined constants of Nature – such as the speed of light, charge, and the quantum of action – neither of these three constants of Nature (electric constant, magnetic constant, and fine-structure constant) have exact values: they must, effectively, be measured in physical experiments and, importantly, their values must be co-determined in such experiments. Indeed, the relations above imply that the relative standard uncertainty in the measures for all three constants, which is currently at 1.6 $\times$ 10^-10, must remain the same in order to ensure conceptual agreement between experiment and theory.

We will not dwell too long on this because we ourselves still need to think through this some more. However, what we write above makes it obvious that this is very important when considering claims about the precision of both the experimental data as well as theoretical arguments on the anomaly: the precision is, indeed, extraordinary – and will probably become even more extraordinary over the coming decades – but, when evaluating any theoretical model explaining this anomaly, it is obvious such theory can no longer be viewed in isolation. Other physical interpretations of the fine-structure constant – such as the geometric interpretation we advance here – must also be considered.

Note: The above probably explains why CODATA remains rather conservative (as compared to the precision of experimental measurements, that is) in its consensus value for the fine-structure constant and the two electromagnetic constants: their value only has roughly ten significant digits. That is about 10000 times better than one part in a million but, still, it effectively does not quite reflect the accuracy of modern-day experiments. Instead, this standard relative uncertainty now probably reflects both theoretical as well as experimental uncertainties, and the theoretical uncertainties should, in our not-so-humble view, also include a critical analysis of the modern-day QED/QFT framework.

Classical or neo-classical explanations of higher-order terms

Again, the aim of our paper is not to fundamentally question or replace quantum electrodynamics, which seems to remain one of the most successful theories ever constructed. Rather, it is to ask whether the leading structure and higher-order corrections in the mainstream explanation of the anomalous magnetic moment might also admit a complementary geometric or other more fundamental interpretation. We think it does. Let us list all obvious elements of such more fundamental interpretation:

1. The Bohr magneton or Dirac’s formula for the magnetic dipole moment emerges naturally from the age-old electron ring current model, which was first advanced by Alfred Lauck Parson in 1915 and, as we explain in our paper on the nature of de Broglie’s matter-wave, also naturally explains Schrödinger’s Zitterbewegung theory.

2. The first-order correction (Schwinger’s factor) emerges, equally naturally, from the assumption that something that is infinitesimally small (i.e, something with zero physical size) does not exist and, hence, that (i) charge must have some size, and (ii) that, for an electron, this size is given by the classical electron radius. The inuuition here was explained above, and the detail of it is described in the paper we want to promote here. [You would not expect us to just copy-paste our paper in a blog article, isn’t it? :-)]

3. To explain the necessity and/or emergence of second- and higher-order terms, we think of the following:

(i) The assumption of an oscillation naked charge – whose size is not infinitesimally small but of the order of the classical electron radius – inside of the electron naturally leads to what we refer to as ‘finite-size’ corrections to the Compton radius. These corrections probably scale just like α but, when allowing for more advanced ideas such as self-interaction (we admit that we are not a fan but, when everything is said and done, self-interaction is an idea which the classical physicists did like to explore), may also be linear in α², α³ or higher-order terms.

(ii) Intrinsic spin is, of course, a very obvious candidate for a correction of the core Dirac value of the magnetic moment of an electron. Indeed, if one thinks of the electron as an oscillation of some naked charge, then this small charge distribution itself should make one rotation around its own axis as it orbits around the center of the electron and, hence, this must result in a combined magnetic dipole moment that is slightly different from the Bohr magneton. We asked AI (ChatGPT) to do a quick calculation and – using the standard formula for the magnetic moment of a uniformly charged solid sphere of radius r_e rotating around its own axis with the same angular velocity ω as its orbital motion – the ratio of (1) this intrinsic magnetic moment and (2) the orbital magnetic moment (i.e., the Bohr magneton) should equal 4α²/5. In other words, this intrinsic spin effects enters at order α². Hence, this is structurally consistent with the observed perturbative hierarchy, in which the leading correction is proportional to α, and higher-order corrections scale with higher powers.

In our paper, we refer to the above alternative explanations as the ‘finite-size’ and ‘intrinsic dipole’ contributions to the magnetic moment, respectively, and we treat them in one and the same chapter because, as mentioned, they are prime candidates for the first- and second-order corrections to the theoretical magnetic moment of the electron.

As for higher-order corrections, we worked with AI to identify a number of additional candidate explanations. Frankly, these convince us somewhat less than the obvious theoretical remarks above but they cannot be dismissed out of hand. We, therefore, list and detail these in a separate chapter in the paper. They include explorations of:

(iii) The precessional motion of the presumed blob of charge in the electron, which one would expect it to have as a result of its intrinsic spin: such and like effects may also be referred to as projection effects resulting from the fact that, in real-life experiments, the free electron is being contained in a Penning trap by an electromagnetic field with which it obviously interacts. The measurement, therefore, must take into account various additional motions – such as complicated precessional or nutational motion – which, again, we will just categorize under the category of ‘projection effects’. [We recommend the reader to dive into the intricacies of what a Penning trap actually is, and look at the amazing technologies involved: a Penning trap combines (i) a homogeneous, strong axial magnetic field for radial confinement as well as (ii) an electrostatic quadrupole field to provide axial confinement.]

(iv) Various interactions which may be categorized as presumed classical ‘self-interaction’ effects. Such effects include interactions between presumed charge elements within the blob, or between the blob and the electromagnetic field sustaining its motion. However, we intuitively feel one can think of many self-interaction effects and, therefore, these theoretical candidate contributions feel quite ad hoc.

(v) Finally, ChatGPT also alludes to complicated corrections that might or should be made as a result of the cycle-averaging or dynamical smearing which is inherent to the ring current model. Indeed, we treat a continuous current distribution just the same as a localized charge in some regular orbital motion. While one might argue we are looking at scales here at which Maxwell’s equations combined with the Planck-Einstein relation might not make sense and that, therefore, cycle-averaging may not be quite legitimate, we are less convinced.

We will end our list here and make two important remarks:

1. From what we write above, it is obvious that the list of classical or neo-classical explanations is sufficiently rich to justify trying non-perturbative theoretical approaches to solve the so-called mystery of the anomaly.

2. That being said, it is not an infinite list (we only have about five items above) and, as mentioned, some explanations make more sense than others. It is, therefore, most likely that the ultimate classical or neo-classical explanation of the anomaly will not be some wonderfully elegant infinite power series. It will likely be a finite series of common-sense terms – each of which embedding one aspect of a system which, in contrast to what is assumed in QED, has very limited degrees of freedom. As such, it should respect Occam’s Razor Principle: the mathematical expression of the explanation should not be more complicated than the physical situation itself.

[…] So, that’s it for this blog. We let AI re-read this blog post, and write the conclusion. We hope it will encourage you to read the full paper itself.

Conclusion

If this exercise shows anything, it is that the fine-structure constant quietly connects several of the most important length scales in electron physics. The classical electron radius, the Compton radius, and the Bohr radius form a simple ladder separated by powers of α. That pattern appears so often in electron physics that it is difficult to believe it is merely accidental.

The ring-current model explored in the working paper uses this observation as a starting point. Once the electron is viewed as a localized charge distribution undergoing coherent circular motion, the Dirac magnetic moment emerges naturally from classical electromagnetism. Introducing a finite charge size of order the classical electron radius then leads to corrections that scale with the ratio (r_e/r_C= α). Because the motion is periodic, these corrections are naturally normalized over all phases within a cycle (2 $\pi$ ), yielding a leading contribution with the same structure as Schwinger’s famous factor.

Whether this geometric interpretation ultimately captures part of the real physical mechanism behind the anomaly remains an open question. What the present work suggests, however, is that the leading structure of the anomalous magnetic moment may not be as mysterious as it sometimes appears. It may reflect a simple interplay between electromagnetic length scales and the geometry of circular motion.

The purpose of this investigation is therefore not to challenge the extraordinary success of quantum electrodynamics. Rather, it is to ask whether the hierarchy of corrections normally obtained through perturbative calculations might also admit a complementary physical interpretation. If the electron is indeed a finite electromagnetic structure rather than an abstract point particle, then at least some features of the anomaly might ultimately have a geometric explanation.

Even if this line of reasoning proves incomplete, it highlights an intriguing fact: the fine-structure constant continues to appear as a scaling parameter linking the structure of the electron to the structure of atoms. Understanding why those scales are related the way they are may still teach us something fundamental about the organization of electromagnetic phenomena.

The full working paper can be found here. As always, comments, questions, and critical feedback are very welcome.

One Equation, Too Many Jobs: Rethinking Schrödinger’s Equation and Wavefunction

I have just republished one of my long-standing papers on de Broglie’s matter-wave concept as a new, standalone publication, with its own DOI:

👉 De Broglie’s matter-wave concept and issues
https://www.researchgate.net/publication/399225854_De_Broglie’s_matter-wave_concept_and_issues
DOI: 10.13140/RG.2.2.30104.25605

The reason for republishing is not cosmetic. A new Annex was added on 31 December 2025 that fundamentally clarified — for me, at least — what Schrödinger’s equation is really doing, and just as importantly, what it is not doing.

This clarification came out of a long and at times uncomfortable dialogue with the most recent version of OpenAI’s GPT model (ChatGPT 5.2). Uncomfortable, because it initially destabilized a view I had held for years. Productive, because it forced a deeper structural distinction that I now believe is unavoidable. Let me explain.

The uncomfortable admission: I was wrong about the $\tfrac{1}{2}$ factor

For a long time, I was convinced that the factor $\tfrac{1}{2}$ factor in Schrödinger’s equation — especially in the hydrogen atom problem — must reflect some deeper pairing mechanism. At times, I even wondered whether the equation was implicitly modeling an electron pair (opposite spin), rather than a single electron.

That intuition was not random. It came from a broader realist programme in which I treat the electron as a structured object, with internal dynamics (zitterbewegung-like orbital motion), not as a point particle. If mass, energy, and phase all have internal structure, why should a simple quadratic kinetic term with a mysterious $\tfrac{1}{2}$ be fundamental?

The hard truth is this: that intuition was misplaced — but it was pointing in the right direction.

The mistake was not questioning the factor $\tfrac{1}{2}$ . The mistake was assuming Schrödinger’s equation was trying to describe everything at once.

The key insight: Schrödinger describes the envelope, not the engine

The decisive realization was structural:

Schrödinger’s wavefunction does not describe the electron’s internal dynamics.
It describes the translational envelope of phase coherence.

Once you see that, several things fall into place immediately:

The hydrogen “orbitals” are not literal orbits, and not internal electron motion.
They are standing-wave solutions of an envelope phase, constrained by a Coulomb potential.
The factor $\tfrac{1}{2}$ is not mysterious at all at this level: it is the natural coefficient that appears in effective, averaged, quadratic envelope dynamics.

In other words:
The $\tfrac{1}{2}$ factor belongs to the envelope layer, not to the internal structure of the electron.

My earlier “electron pair” idea tried to explain a structural feature by inventing new ontology. The correct move was simpler and more radical: separate the layers.

One symbol, too many jobs

Modern quantum mechanics makes a profound — and in my view costly — simplification:

It uses one symbol, ψ, to represent:

internal phase,
translational dynamics,
probability amplitudes,
and experimental observables.

That compression works operationally, but it hides structure.

What the new Annex makes explicit is that Nature almost certainly does not work that way. At minimum, we should distinguish:

Internal phase
Real, physical, associated with internal orbital motion and energy bookkeeping.
Envelope phase
Slow modulation across space, responsible for interference, diffraction, and spectra.
Observables
What experiments actually measure, which are sensitive mainly to envelope-level phase differences.

Once this distinction is made, long-standing confusions dissolve rather than multiply.

Why this does not contradict experiments

This is crucial.

Nothing in this reinterpretation invalidates:

electron diffraction,
hydrogen spectra,
interference experiments,
or the empirical success of standard quantum mechanics.

On the contrary: it explains why Schrödinger’s equation works so well — within its proper domain.

The equation is not wrong.
It is just over-interpreted.

A personal note on changing one’s mind

I’ll be honest: this line of reasoning initially felt destabilizing. It challenged a position I had defended for years. But that discomfort turned out to be a feature, not a bug.

Good theory-building does not preserve intuitions at all costs. It preserves structure, coherence, and explanatory power.

What emerged is a cleaner picture:

internal realism without metaphysics,
Schrödinger demoted from “ultimate truth” to “effective envelope theory”,
and a much clearer map of where different mathematical tools belong.

That, to me, is progress.

Where this opens doors

Once we accept that one wavefunction cannot represent all layers of Nature, new possibilities open up:

clearer interpretations of spin and the Dirac equation,
better realist models of lattice propagation,
a more honest treatment of “quantum mysteries” as category mistakes,
and perhaps new mathematical frameworks that respect internal structure from the start.

Those are not promises — just directions.

For now, I am satisfied that one long-standing conceptual knot has been untied.

And sometimes, that’s enough for a good year’s work. 🙂

Post Scriptum: On AI, Intellectual Sparring, and the Corridor

A final remark, somewhat orthogonal to physics.

The revision that led to this blog post and the accompanying paper did not emerge from a sudden insight, nor from a decisive experimental argument. It emerged from a long, occasionally uncomfortable dialogue with an AI system, in which neither side “won,” but both were forced to refine their assumptions.

At the start of that dialogue, the AI responded in a largely orthodox way, reproducing standard explanations for the factor $\tfrac{1}{2}$ in Schrödinger’s equation. I, in turn, defended a long-held intuition that this factor must point to internal structure or pairing. What followed was not persuasion, but sparring: resistance on both sides, followed by a gradual clarification of conceptual layers. The breakthrough came when it became clear that a single mathematical object — the wavefunction — was being asked to do too many jobs at once.

From that moment on, the conversation shifted from “who is right?” to “which layer are we talking about?” The result was not a victory for orthodoxy or for realism, but a structural separation: internal phase versus translational envelope, engine versus modulation. That separation resolved a tension that had existed for years in my own thinking.

I have explored this mode of human–AI interaction more systematically in a separate booklet on ResearchGate, where I describe such exchanges as occurring within a corridor: a space in which disagreement does not collapse into dominance or deference, but instead forces both sides toward finer distinctions and more mature reasoning.

This episode convinced me that the real intellectual value of AI does not lie in answers, but in sustained resistance without ego — and in the willingness of the human interlocutor to tolerate temporary destabilization without retreating into dogma. When that corridor holds, something genuinely new can emerge.

In that sense, this post is not only about Schrödinger’s equation. It is also about how thinking itself may evolve when humans and machines are allowed to reason together, rather than merely agree.

Readers interested in this kind of human–AI interaction beyond the present physics discussion may want to look at that separate booklet I published on ResearchGate (≈100 pages), in which I try to categorize different modes of AI–human intellectual interaction — from superficial compliance and authority projection to genuine sparring. In that text, exchanges like the one briefly alluded to above are described as a Type-D collapse: a situation in which both human and AI are forced to abandon premature explanatory closure, without either side “winning,” and where progress comes from structural re-layering rather than persuasion.

The booklet is intentionally exploratory and occasionally playful in tone, but it grew out of exactly this kind of experience: moments where resistance, rather than agreement, turns out to be the most productive form of collaboration.

We Could Stop Here.

(But the Next Question Is Already Knocking.)

There is a moment in any long intellectual journey where you could stop.

Not because everything is finished, but because enough has settled to make stopping respectable. The equations close. The concepts line up. Nothing is obviously broken anymore.

This paper — The Photon Wavefunction Revisited — marks one of those moments for me.

👉 The paper is available here on ResearchGate:
https://www.researchgate.net/publication/399111974_The_Photon_Wavefunction_Revisited

It revisits an old and stubborn question — what do we really mean by the photon wavefunction? — using only very old tools: Maxwell’s equations, the Planck–Einstein relation, dimensional analysis, and known scattering results. No new particles. No speculative fields. No hidden dimensions. No “next revolution”.

Just careful rereading.

Why revisit this at all?

Because physics has a habit of answering questions so efficiently that we stop asking what the answers mean. The photon became a “quantum of the electromagnetic field”, calculations worked, experiments agreed — and interpretation quietly retreated.

But interpretation has a way of sneaking back in through the side door.

In this paper, I try to be very explicit about what is being claimed — and what is not:

A photon is treated as a light-like, phase-closed object, not as a little billiard ball and not as a probabilistic smear.
Its wavefunction is not a mystery object “without meaning”, but a compact encoding of phase structure.
Electric and magnetic fields are not competing realities, but orthogonal phase components of a single conserved structure.
Energy and momentum conservation follow cleanly from Maxwell’s equations — even when charge is stripped away.

Nothing here overturns quantum electrodynamics. But some things are, perhaps, put back in their original place.

A word about standing waves (and why they appear)

One appendix uses a standing-wave construction to make something visible that is otherwise hidden: how electric and magnetic field energy exchange internally while total energy remains conserved.

This does not mean photons are standing waves. They propagate in one direction. Momentum has a direction. Energy does not.

The standing wave is simply a diagnostic tool — a way of freezing momentum flow so the bookkeeping of energy becomes transparent. If that sounds almost embarrassingly classical… well, that may be the point.

Why this felt worth publishing

This paper took shape slowly, through many iterations, many dead ends, and many “wait — is that actually true?” moments. Some of it was developed with explicit AI assistance, used not as an oracle but as a very patient consistency checker. That role is openly acknowledged.

What mattered most to me was not novelty, but coherence.

When the dust settled, something quietly reassuring happened: the picture that emerged was simpler than what I started with, not more complicated.

And that’s usually a good sign.

Could we stop here?

Yes. Absolutely.

The paper stands on its own. The equations close. Nothing essential is missing.

But physics has never progressed by stopping at “good enough”. The next question is already there:

How exactly does this phase picture illuminate electron–photon interaction?
What does it really say about the fine-structure constant?
Where does this leave matter–antimatter symmetry?

Those are not answered here. They don’t need to be — yet.

For now, this is a place to pause, look around, and make sure we know where we are.

And then, as always, the next question prompts the next question.

That’s not a problem.
That’s the fun part.

— Jean Louis Van Belle

Post Scriptum: The Last Question That Won’t Let Me Sleep (On matter, antimatter, and why one mystery remains)

There is a strange pattern I’ve noticed over the years.

You work your way through a dense thicket of questions. One by one, they loosen. Concepts that once felt contradictory begin to align. The mathematics stops fighting the intuition. The ontology — cautiously, provisionally — starts to hold.

And then, when almost everything is in place, one question refuses to dissolve.

Tonight, for me, that question is matter–antimatter creation and annihilation.

Most things now feel… settled

After revisiting photons, wavefunctions, phase closure, and electromagnetic energy bookkeeping, I feel unusually calm about many things that once bothered me deeply.

Photons as light-like, phase-closed objects? That works.
Electric and magnetic fields as orthogonal phase components? That works.
Energy conservation without charge? Maxwell already knew how to do that.
Electron–photon interaction as phase reconfiguration rather than “mystical coupling”? That works too.

None of this feels revolutionary anymore. It feels readable.

And yet.

Matter–antimatter still feels different

In low-energy environments, I’m increasingly comfortable with a very unromantic picture.

Pair creation does not happen “out of nothing.” It happens near nuclei, in strong fields, in structured environments. Something must anchor phase. Something must absorb recoil. Something must allow a stable oscillatory configuration to form.

I’ve sometimes called this a Platzwechsel — a change of place, or role — rather than a miraculous transformation of field into charge. The photon doesn’t “become matter”; a charge configuration re-closes in the presence of structure.

That feels honest. And it fits what experiments actually show.

But then there is the “but” question… This is how I phrase now.

Annihilation is unsettlingly easy

Electron–positron annihilation, on the other hand, requires no such help.

Two charged, massive objects meet, and they disappear into light. Cleanly. Elegantly. No nucleus. No lattice. No scaffold.

That asymmetry matters.

Matter → light is easy.
Light → matter is hard.

Quantum field theory encodes this perfectly well, but encoding is not explaining. And pretending the asymmetry isn’t there has never helped.

What happens to charge?

Here is the thought that keeps me awake — and oddly calm at the same time.

If charge is not a substance, but a phase-closed electromagnetic motion, then annihilation is not mysterious at all. The phase closure simply dissolves. What remains is free phase propagation.

Charge doesn’t “go anywhere”.
It stops being a thing because the structure that constituted it no longer exists.

That idea is unsettling only if one insists that charge must persist locally as a substance. I’ve never found good reasons to believe that.

And pure vacuum pair creation?

High-energy photon–photon pair creation is possible, in principle. But it is rare, fragile, and structurally demanding. It requires extreme energies and densities, and often still some form of external assistance.

That, too, feels telling.

Two freely propagating phase objects have no natural way to decide where a charge configuration should live. Without structure, closure is unstable. Nature seems reluctant — not forbidden, but reluctant.

So where does that leave us?

It leaves me in an oddly peaceful place.

Most of the framework now feels coherent. The remaining mystery is not a loose end to be tied up quickly, but a boundary — a place where explanation must slow down instead of speeding up.

That feels like the right place to stop for tonight.

Not because the mystery is solved, but because it is now cleanly stated.

And that, I’ve learned, is often the real precondition for sleep.

— Jean Louis Van Belle

🌀 Two Annexes and a Turtle: Revisiting My Early Lectures on Quantum Physics

Over the past few weeks — and more intensely these past mornings — I’ve returned to two of my earliest texts in the Lectures on Physics series: the first on quantum behavior, and the second on probability amplitudes and quantum interference. Both have now been updated with new annexes, co-authored in dialogue with ChatGPT-4o.

This wasn’t just a consistency check. It was something more interesting: an exercise in thinking with — not through — a reasoning machine.

The first annex (Revisiting the Mystery of the Muon and Tau) tackles the open question I left hanging in Lecture I: how to interpret unstable “generations” of matter-particles like the muon and tau. In the original paper, I proposed a realist model where mass is not an intrinsic property but the result of oscillating charge or field energy — a stance that draws support from the 2019 revision of SI units, which grounded the kilogram in Planck’s constant and the speed of light. That change wasn’t just a technicality; it was a silent shift in ontology. I suspected that much at the time, but now — working through the implications with a well-tuned AI — I can state it more clearly: mass is geometry, inertia is field structure, and the difference between stable and unstable particles might be a matter of topological harmony.

The second annex (Interference, Identity, and the Imaginary Unit) reopens the deeper riddle at the heart of quantum mechanics: why probability amplitudes interfere at all. This annex is the child of years of irritation — visible in earlier, sharper essays I published on academia.edu — with the lazy mysticism that often surrounds “common phase factors.” The breakthrough, for me, was to fully accept the imaginary unit iii not as a mathematical trick but as a rotation operator. When wavefunctions are treated as oriented field objects, not just complex scalars, interference becomes a question of geometric compatibility. Superpositions and spin behavior can then be reinterpreted as topological effects in real space. This is where I think mainstream physics got lost: it started calculating without explaining.

ChatGPT didn’t invent these ideas. But it helped me phrase them, frame them, and press further on the points I had once hesitated to formalize. That’s what I mean when I say this wasn’t just a cleanup job. It was a real act of collaboration — a rare instance of AI not just paraphrasing or predicting, but amplifying and clarifying an unfinished line of human reasoning.

Both revised papers are now live on ResearchGate:

They mark, I think, a modest turning point. From theory and calculation toward something closer to explanation.

And yes — for those following the philosophical side of this project: we did also try to capture all of that in a four-panel comic involving Diogenes, a turtle, and Zeno’s paradox. But that, like all things cartooned by AI, is still a work in progress. 🙂

Post Scriptum (24 June 2025): When You Let the Machine Take the Pen

In the spirit of openness: there’s been one more development since publishing the two annexes above.

Feeling I had taken my analytical skills as far as I could — especially in tackling the geometry of nuclear structure — I decided to do something different. Instead of drafting yet another paper, I asked ChatGPT to take over. Not as a ghostwriter, but as a model builder. The prompt was simple: “Do better than me.”

The result is here:
👉 ChatGPT Trying to Do Better Than a Human Researcher

It’s dense, unapologetically geometric, and proposes a full zbw-based model for the neutron and deuteron — complete with energy constraints, field equations, and a call for numerical exploration. If the earlier annexes were dialogue, this one is delegation.

I don’t know if this is the end of the physics path for me. But if it is, I’m at peace with it. Not because the mystery is gone — but because I finally believe the mystery is tractable. And that’s enough for now.

Taking Stock: Zitterbewegung, Electron Models, and the Role of AI in Thinking Clearly

Over the past few years, I’ve spent a fair amount of time exploring realist interpretations of quantum mechanics, particularly the ring-current or Zitterbewegung (zbw) model of the electron. I’ve written many posts about it here — and also tried to help to promote the online “Zitter Institute”, which brings a very interesting group of both amateur and professional researchers together, as well as a rather impressive list of resources and publications which help to make sense of fundamental physics – especially on theories regarding the internal structure of the electron.

The goal — or at least my goal — was (and still is) to clarify what is real and what is not in the quantum-electrodynamic zoo of concepts. That is why I try to go beyond electron models only. I think the electron model is complete as for now: my most-read paper (on a physical interpretation of de Broglie’s matter-wave) settles the question not only for me but, I judge based on its many views, for many others as well. The paper shows how the magnetic moment of the electron, its wavefunction, and the notion of a quantized “packet of energy” can easily be grounded in Maxwell’s equations, special relativity, and geometry. They do not require speculative algebra, nor exotic ontologies.

In that light, I now feel the need to say something — brief, but honest — about where I currently stand in my research journey — which is not on the front burner right now but, yes, I am still thinking about it all. 🙂

On the term “Zitterbewegung” itself

Originally coined by Schrödinger and later mentioned by Dirac, “Zitterbewegung” translates as “trembling motion.” It was meant to capture the high-frequency internal oscillation predicted by Dirac’s wave equation.

But here lies a subtle issue: I no longer find the term entirely satisfying.

I don’t believe the motion is “trembling” in the sense of randomness or jitter. I believe it is geometrically structured, circular, and rooted in the relativistic dynamics of a massless point charge — leading to a quantized angular momentum and magnetic moment. In this view, there is nothing uncertain about it. The electron has an internal clock, not a random twitch.

So while I still value the historical connection, I now prefer to speak more plainly: an electromagnetic model of the electron, based on internal motion and structure, not spooky probabilities.

On tone and openness in scientific dialogue

Recent internal exchanges among fellow researchers have left me with mixed feelings. I remain grateful for the shared curiosity that drew us together, but I was disappointed by the tone taken toward certain outside critiques and tools.

I say this with some personal sensitivity: I still remember the skepticism I faced when I first shared my own interpretations. Papers were turned down not for technical reasons, but because I lacked the “right” institutional pedigree. I had degrees, but no physics PhD. I was an outsider.

Ridicule — especially when directed at dissent or at new voices — leaves a mark. So when I see similar reactions now, I feel compelled to say: we should be better than that.

If we believe in the integrity of our models, we should welcome critique — and rise to the occasion by clarifying, refining, or, if necessary, revising our views. Defensive posturing only weakens our case.

On the use of AI in physics

Some recent comments dismissed AI responses as irrelevant or superficial. I understand the concern. But I also believe this reaction misses the point.

I didn’t try all available platforms, but I did prompt ChatGPT, and — with the right framing — it offered a coherent and balanced answer to the question of the electron’s magnetic moment. Here’s a fragment:

“While the ‘definition’ of the intrinsic magnetic moment may be frame-invariant in the Standard Model, the observable manifestation is not. If the moment arises from internal circular motion (Zitterbewegung), then both radius and frequency are affected by boosts. Therefore, the magnetic moment, like momentum or energy, becomes frame-dependent in its effects.”

The jury is still out, of course. But AI — if guided by reason — might help us unravel what makes sense and what does not.

It is not a substitute for human thinking. But it can reflect it back to us — sometimes more clearly than we’d expect.

A final reflection

I’ll keep my older posts online, including those that reference the Zitter Institute. They reflected what I believed at the time, and I still stand by their substance.

But moving forward, I’ll continue my work independently — still fascinated by the electron, still curious about meaning and structure in quantum mechanics, but less interested in labels, echo chambers, or theoretical tribalism.

As always, I welcome criticism and dialogue. As one business management guru once said:

“None of us is as smart as all of us.” — Kenneth Blanchard

But truth and clarity come first.

— Jean Louis Van Belle

The failure of physics as a science?

It is a coincidence but Sabine Hossenfelder just produced a new video in which she talks once again about the problems of academic physics, while I did what I said what I would not do – and that is to write out why the discovery of new rare kaon decay modes is a problem for the Standard Model. I think the video and the paper complement each other nicely, although Sabine Hossenfelder probably still believes the strong force and weak interactions are, somehow, still real. [I did not read her book, so I don’t know: I probably should buy her book but then one can only read one book at a time, isn’t it?]

The paper (on ResearchGate – as usual: link here) does what Sabine Hossenfelder urges her former colleagues to do: if a hypothesis or an ad hoc theory doesn’t work, then scientists should be open and honest about that and go back to the drawing board. Indeed, in my most-read paper – on de Broglie’s matter-wave – I point out how de Broglie’s original thesis was misinterpreted and how classical quantum theory suddenly makes sense again when acknowledging that mistake: it probably explains why I am getting quite a lot of reads as an amateur physicist. So what’s this new paper of mine all about?

I go back to the original invention of the concept of strangeness, as documented by Richard Feynman in his 1963 Lectures on quantum physics (Vol. III, Chapter 11-5) and show why and how it does not make all that much sense. In fact, I always thought these new quantum conservation laws did not make sense theoretically and that, at best, they were or are what Dr. Kovacs and Dr. Vassallo refer to as phenomenological models rather than sound physical theories (see their chapter on superconductivity in their latest book). However, now it turns out these fancy new concepts do not even do what they are supposed to do, and that is to correctly describe the phenomenology of high-energy particle reactions.

The alternative – a realist interpretation of quantum physics – is there. It is just not mainstream – yet! 🙂

Post scriptum (8 November 2024): For those who do not like to read, you can also watch what I think of my very last video on the same topic: what makes sense and what does not in academic or mainstream physics? Enjoy and, most importantly, do not take things too seriously ! Life family and friends – and work or action-oriented engagement are far more important than personal philosophy or trying to finding truth in science… 🙂

The ultimate zbw electron model

Just after finishing a rather sober and, probably, overly pessimistic reflection on where the Zitterbewegung interpretation of quantum theory stands, I am excited to see a superbly written article by Dr. Kovacs and Dr. Vassallo on what I now think of as the ultimate electron model: Rethinking electron statistics rules (10 September 2024). I think it is great because it addresses several points in my rather depressing description of the state of zbw theory:

Multiple Zitterbewegung interpretations of what an electron actually is, currently coexist. Indeed, both mainstream and non-mainstream physicists have now been going back and forth for about 100 years on this or that electron model: the referenced Kovacs/Vassallo article effectively appeared in a special journal issue titled: “100 Years of Quantum Matter Waves: Celebrating the Work of Louis De Broglie.” 100+ years of discussion have basically led us back to Parson’s 1915 ring current model, which Joseph Larmor presented so well at the 1921 Solvay Conference. We do not think that is a good situation: it looks a bit like 100 years of re-inventing the wheel – or, perhaps, I should say: wheels within wheels. 🙂 I could write more about this but I am happy to see the discussion on – just one example of differing views here – whether or not there should be a 1/2 factor in the electron’s frequency may be considered to be finally solved: de Broglie’s matter-wave frequency is just the same as the Planck-Einstein frequency in this paper. This factor 2 or 1/2 pops up when considering ideas such as the effective mass of the zbw charge or – in the context of Schrödinger’s equation – because we’re modeling the motion of electron pairs rather than electrons (see the annexes to my paper on de Broglie’s matter-wave concept). In short: great! Now we can, finally, leave those 100+ years of discussions behind us. 🙂
Dr. Kovacs and Dr. Vassallo also explore the nature of superconductivity and Bose-Einstein statistics, and not only does their analysis away with the rather mystical explanation in Feynman’s last and final chapter of his lectures on quantum mechanics but it also offers a very fine treatment of n-electron systems. Their comments on ‘bosonic’ and ‘fermionic’ properties of matter-particles also tie in with my early assessment that the boson-fermion dichotomy has no ontological basis.

The hundreds of downloads of their article since it was published just two weeks ago also shows new and old ways of thinking and modelling apparently come nicely together in this article: if your articles get hundreds of reads as soon as published, then you are definitely not non-mainstream any more: both Dr. Kovacs and Dr. Vassallo have an extraordinary talent for rephrasing old questions in the new “language” of modern quantum theory. That is to be lauded. Hopefully, work on a proton and a neutron model will now complement what I think of as the ultimate electron model based on a local and realist interpretation of what de Broglie’s matter-wave actually is. Indeed, critics of modern quantum theory often quote the following line from Philip Pearle’s Classical Electron Models [1]:

“The state of the classical electron theory reminds one of a house under construction that was abandoned by its workmen upon receiving news of an approaching plague. The plague in this case, of course, was quantum theory. As a result, classical electron theory stands with many interesting unsolved or partially solved problems.”

I think Dr. Kovacs and Dr. Vassallo may have managed to finish this “abandoned construction” – albeit with an approach which differs significantly from that of Pearle: that is good because I think there were good reasons for the “workmen” to leave the construction site (see footnote [1]). 🙂 So, yes, I hope they will be able – a few years from now – to also solve the questions related to a Zitterbewegung proton and neutron model.

In fact, they already have a consistent proton model (see: the proton and Occam’s Razor, May 2023), but something inside of me says that they should also explore different topologies, such as this Lissajous-like trajectory which intrigues me more than helical/toroidal approaches – but then who am I? I am the first to recognize my limitations as an amateur and it is, therefore, great to see professionals such as Dr. Kovacs and Dr. Vassallo applying their formidable skills and intuition to the problem. 🙂

[1] Pearle’s paper is the seventh in a volume of eight chapters. The book’s title is, quite simply, titled Electromagnetism (1982), and it was put together and edited by Doris Teplitz (1982). Few who quote this famous line, bother to read the Philip Pearle paper itself. This paper effectively presents what Pearle refers to as classical electron models: all of them are based on “rigid or flexible shell surfaces” of charge, which is why we think they did not “cut it” for the many “workmen” (read: the mainstream scientists who thought the Bohr-Heisenberg amplitude math and the probability theory that comes with it) who left the then unfinished construction.

We think the approach taken by Dr. Kovacs and Dr. Vassallo is more productive when it comes to bringing mainstream and Zitterbewegung theorists together around a productive mathematical framework in which the probabibilities are explained based on a plain interpretation of Schrödinger’s ‘discovery’ – which is that the elementary wavefunction represents a real equation of motion of a pointlike but not infinitesimally charge inside of an electron.

As for trying out different topologies, we understand Dr. Kovacs and Dr. Vassallo are working very hard on that, so all we can do is to wish them the best of luck. Godspeed! 🙂

All of physics

This five-pager has it: all you ever wanted to know about the Universe. Electron mass and proton mass are seen as input to the model. To the most famous failed experiment in all of classical physics – the 1887 Michelson-Morley experiment, which disproved aether theories and established the absoluteness of lightspeed – we should add the Kamioka Nucleon Decay Experiment, which firmly established that protons do not decay. All the rest is history. 🙂

Post scriptum (26 April): I added another five-pager on fundamental concepts on ResearchGate, which may or may not help to truly understand what might be the case (I am paraphrasing Wittgenstein’s definition of reality here). It is on potentials, and it explains why thinking in terms of neat 1/r or 1/r² functions is not all that helpful: reality is fuzzier than that. Even a simple electrostatic potential may be not very simple. The fuzzy concept of near and far fields remains useful.

I am actually very happy with the paper, because it sort of ‘completes’ my thinking on elementary particles in terms of ring currents. It made me feel like it is the first time I truly understand the complementarity/uncertainty principle – and that I invoke it to make an argument.