Re-reading What We Already Know

On PDG data, big science, and why simplicity still matters

For reasons I still find slightly amusing (it is better to be amused than annoyed, isn’t it?), old blog posts here (readingfeynman.org) or early papers on platforms such as vixra.org and academia.edu periodically resurface in “top reads” lists — sometimes many years after publication.

I would now qualify several of those texts as typical “angry young man” papers. However, I still consider most of their core claims to be true. And the papers — as mentioned above — still resonate with readers, even if I now take some distance from how they were written and framed.

That tells me two things. First, there is still genuine interest in careful, foundational thinking about physics. Second, the web (and increasingly AI agents crawling it) has a habit of flattening intellectual trajectories into caricatures: mainstream or outsider, orthodox or heretic.

I have looked at those caricatures about me, and I want to be very clear about where I stand.

1. I am deeply mainstream in one crucial sense: I trust measurements. I trust large-scale experimental infrastructure. I trust the Particle Data Group (PDG), CERN, and the decades of work that went into producing the numbers we now take for granted. I am not hostile to “big science” — on the contrary, I consider projects like CERN or ITER to be among the most impressive collective achievements of modern civilization. If society is going to spend large sums of money on something, I much prefer it to be on instruments that extend human knowledge rather than on instruments designed to destroy.

2. At the same time, I am comfortable being an outsider: I do not believe that theoretical sophistication excuses us from repeatedly asking what is actually grounded in experiment, and what is added later as interpretive scaffolding.

These two positions are not contradictory. Historically, they have gone together.

Think of Maxwell, who unified electric and magnetic phenomena not by adding complexity, but by simplifying and re-ordering – using mathematical advances – what was already known. Think of Lorentz and Einstein, who showed that gravitation need not be treated as a force at all. Think of Schrödinger and Dirac, who demonstrated that the same wave equations could describe light-like as well as matter-like phenomena without reifying every mathematical symbol into a physical object.

Progress, more often than not, comes from simplifying, not from proliferating entities.

A Minimal Experimental Core

That is the spirit in which I recently published a new working paper on ResearchGate:
“Re-reading PDG particle listings through a Minimal Experimental Core (MEC)”.

The idea is almost embarrassingly simple. Take PDG particle listings — the most mainstream source imaginable — and re-present them using only quantities that are directly observable:

rest energy,
lifetime,
electric charge,
magnetic moment where available,
branching ratios understood as empirical event frequencies.

What I deliberately leave out at the primary level are non-observable quantum numbers and symmetry labels that require additional theoretical assumptions to interpret. Not because they are “wrong”, but because they are interpretive rather than measured.

The result is not an alternative theory. It is a different ordering of the same facts. And that re-ordering is surprisingly instructive.

When one looks at leptons, pions, and kaons in this way, certain patterns become obvious long before any model is invoked: differences in stability, sharp asymmetries in branching ratios, and cases where phase space alone clearly does not determine outcomes. None of this is new — but seeing it without the usual conceptual overlays changes how one thinks about explanation.

On big machines and global context

There is another reason I care about this kind of work.

We are entering a period in which fewer and fewer actors can afford to build the next generation of large experimental facilities. Europe (through CERN) and the United States remain central producers of high-quality collider and detector data. China, for geopolitical and economic reasons, may or may not build its own next “big thing” — and if it doesn’t, it will have to be content, like the rest of the world, with the data already produced.

That reality makes something very clear: we will spend the coming decades re-reading existing data. Carefully. Repeatedly. From new angles.

In that context, methodological clarity is not a luxury. It is a necessity.

AI, co-thinking, and intellectual hygiene

This brings me to one last point.

The paper I mentioned was written in close AI–HI co-thinking. I am not shy about that. Used properly, AI is not a generator of answers but a powerful tool for enforcing intellectual hygiene: forcing one to clarify terms, separate observation from explanation, and resist the temptation to smuggle assumptions into language.

If some AI systems currently reduce my online presence to that of a “lonely outlier”, then the best response is not complaint, but better signal: careful writing, explicit methodology, and visible alignment with the experimental foundations of physics.

That is what this work is meant to be.

Not a provocation.
Not a manifesto.
Just a careful re-reading of what we already know — and an invitation to do so again, together.

Stability, Instability, and What High-Energy Physics Really Teaches Us

One of the recurring temptations in physics is to mistake violence for depth.

When we push matter to extreme energy densities—whether in particle colliders or in thought experiments about the early universe—we tend to believe we are peeling away layers of reality, discovering ever more “fundamental” constituents beneath the familiar surface of stable matter. The shorter-lived and more exotic a state is, the more “real” it sometimes appears to us.

In my most recent RG paper (Lecture X1), I tried to step back from that reflex.

The starting point is almost embarrassingly simple:
stable charged particles persist; unstable ones do not.
That fact alone already carries a surprising amount of explanatory power—if we resist the urge to overinterpret it.

Stability as the exception, not the rule

If we imagine the early universe as a high-energy, high-density environment—a kind of primordial soup—then instability is not mysterious at all. Under such conditions, long-lived, self-consistent structures should be rare. Most configurations would be fleeting, short-lived, unable to maintain their identity.

From this perspective, stable particles are not “primitive building blocks” in a metaphysical sense. They are low-energy survivors: configurations that remain coherent once the universe cools and energetic chaos subsides.

Stability, then, is not something that needs to be explained away. It is the phenomenon that needs to be accounted for.

Colliders as stress tests, not ontological excavations

Modern facilities such as CERN allow us to recreate, for fleeting moments, energy densities that no longer exist naturally in the present universe. What we observe there—resonances, decay chains, short-lived states—is fascinating and deeply informative.

But there is a subtle conceptual shift that often goes unnoticed.

These experiments do not necessarily reveal deeper layers of being. They may instead be doing something more modest and more honest: testing how known structures fail under extreme conditions.

In that sense, unstable high-energy states are not more fundamental than stable ones. They are what stability looks like when it is pushed beyond its limits.

A simpler cosmological intuition

Seen this way, cosmogenesis does not require an ever-growing menagerie of proto-entities. A universe that begins hot and dense will naturally favor instability. As it cools, only a small number of configurations will remain phase-coherent and persistent.

Those are the particles we still see today.

No exotic metaphysics is required—only the recognition that persistence is meaningful.

Were the mega-projects worth it?

This perspective does not diminish the value of large-scale scientific projects. On the contrary.

The enormous investments behind colliders or fusion experiments—think of projects like ITER—have given us something invaluable: empirical certainty. They confirmed, with extraordinary precision, intuitions already sensed by the giants of the early twentieth century—figures like Albert Einstein, Paul Dirac, and Erwin Schrödinger.

Perhaps the deepest outcome of these projects is not that they uncovered a hidden zoo of ultimate constituents, but that they showed how remarkably robust the basic structure of physics already was.

That, too, is progress.

Knowing when not to add layers

Physics advances not only by adding entities and mechanisms, but also by learning when not to do so. Sometimes clarity comes from subtraction rather than accumulation.

If nothing else, the simple distinction between stable and unstable charged particles reminds us of this: reality does not owe us an ever-deeper ontology just because we can afford to build more powerful machines.

And perhaps that realization—quiet, unglamorous, but honest—is one of the most valuable lessons high-energy physics has taught us.

This reflection builds directly on an earlier blog post, “Stability First: A Personal Programme for Re-reading Particle Physics” (18 December 2025), in which I outlined a deliberate shift in emphasis: away from ontological layering and towards persistence as a physical criterion. That post introduced the motivation behind Lecture X1—not as a challenge to established data or formalisms, but as an invitation to reread them through a simpler lens. What follows can be read as a continuation of that programme: an attempt to see whether the basic distinction between stable and unstable charged particles already carries more explanatory weight than we usually grant it.

Post Scriptum — An empirical follow-up

When I wrote this piece, the emphasis was deliberately conceptual. The central idea was to treat stability versus instability as a primary organizing perspective, rather than starting from particle families, quark content, or other internal classifications. At the time, I explicitly presented this as an intuition — something that felt structurally right, but that still needed to be confronted with data in a disciplined way.

That confrontation has now been carried out.

Using the Particle Data Group listings as a source, I constructed a deliberately minimalist dataset containing only two observables: rest mass and lifetime. All a priori particle classifications were excluded. Stable or asymptotic states were removed, as were fractionally charged entities, leaving an unclassified ensemble of unstable particles. The resulting mass–lifetime landscape was examined in logarithmic coordinates and subjected to density-based clustering, with the full data table included to allow independent reanalysis.

The outcome is modest, but instructive. A dominant continuum of prompt decays clearly emerges, accompanied by only weak additional structure at longer lifetimes. No rich taxonomy presents itself when decay behaviour alone is considered — but the clusters that do appear are real, reproducible, and consistent with the intuition developed here and in earlier work.

This empirical annex does not “prove” a new theory, nor does it challenge existing classifications. Its value lies elsewhere: it shows what survives when one strips the description down to observables alone, and it clarifies both the power and the limits of a stability-first perspective.

For readers interested in seeing how these ideas behave when confronted with actual data — and in re-using that data themselves — the empirical follow-up is available here:

👉 Empirical Annex to Lecture X1 (Revisiting Lecture XV)
Structure in the Energy–Lifetime Plane of Unstable PDG Particles
https://www.researchgate.net/publication/399008132_Empirical_Annex_to_Lecture_X1_Revisiting_Lecture_XV_Structure_in_the_Energy-Lifetime_Plane_of_Unstable_PDG_Particles

Sometimes, the most useful result is not a spectacular confirmation, but a careful consistency check that tells us where intuition holds — and where it stops.

Moderation, Measurements, and the Temptation of Ontology

Why physics must resist becoming metaphysics

Some time ago, I found myself involved in what can best be described as an intellectual fallout with a group of well‑intentioned amateur researchers. This post is meant to close that loop — calmly, without bitterness, and with a bit of perspective gained since.

One of the more sensible people in that group bothered to push an interesting article onto my desk, and so I want to talk about that one here.

Gary Taubes, CERN, and an unexpected reinforcement

It’s an article by Gary Taubes on the discovery of the W and Z bosons at CERN, later incorporated into his book Nobel Dreams. Far from undermining my position, the article did the opposite: it reinforced the point I had been trying to make all along.

Taubes does not engage in ontology. He does not ask what W and Z bosons are in a metaphysical sense. Instead, he describes what was measured, how it was inferred, and how fragile the boundary is between evidence and interpretation in large‑scale experimental physics.

This connects directly to an earlier piece I published here:

Something Rotten in the State of QED: A Careful Look at Critique, Sociology, and the Limits of Modern Physics
https://readingfeynman.org/2025/12/01/something-rotten-in-the-state-of-qed-a-careful-look-at-critique-sociology-and-the-limits-of-modern-physics/

Let me restate the central point, because it is still widely misunderstood:

Criticizing the ontologization of W/Z bosons (or quarks and gluons) is not the same as denying the reality of the measurements that led to their introduction.

The measurements are real. The detector signals are real. The conservation laws used to infer missing energy and momentum are real. What is not forced upon us is the metaphysical leap that turns transient, unstable interaction states into quasi‑permanent “things.”

Stable vs. unstable states — a distinction we keep blurring

My own work has consistently tried to highlight a distinction that I find increasingly absent — or at least under‑emphasized — in mainstream physics discourse:

Stable states: long‑lived, persistent, and directly accessible through repeated measurement
Unstable or intermediate states: short‑lived, inferred through decay products, reconstructed statistically

W and Z bosons belong firmly to the second category. So do quarks and gluons in their confined form. Treating them as ontologically equivalent to stable particles may be pragmatically useful, but it comes at a conceptual cost.

It is precisely this cost that I criticize when I criticize mainstream physics.

Not because mainstream physics is “wrong.”
But because it has become too comfortable collapsing epistemology into ontology, especially in its public and pedagogical narratives.

Why this matters now

There is another reason this distinction matters, and it is a forward‑looking one.

The probability that something radically new — in the sense of a fundamentally novel interaction or particle family — will be discovered in the coming decades is, by most sober assessments, rather low. What we will have, however, is:

More precise measurements
Larger datasets
Longer baselines
Better statistical control

In that landscape, progress will depend less on naming new entities and more on bridging what has already been measured, sometimes decades ago, but never fully conceptually digested.

That is where I intend to focus my efforts in the coming years.

Not by founding a new church.
Not by declaring metaphysical revolutions.
But by carefully working at the interface between:

what was actually measured,
what was legitimately inferred,
and what we may have too quickly reified.

Closing note

If there is one lesson I take — from the past dispute, from Taubes, from the history of CERN or fundamental physics in general — it is this:

Physics progresses best when it remains modest about what it claims to be about.

Measurements first. Interpretation second. Ontology, if at all, only with restraint.

That stance may be unsatisfying to those looking for grand narratives. But it is, I believe, the only way to keep physics from quietly turning into metaphysics while still wearing a lab coat.

— Jean Louis Van Belle

Something Rotten in the State of QED? A Careful Look at Critique, Sociology, and the Limits of Modern Physics

Every few years, a paper comes along that stirs discomfort — not because it is wrong, but because it touches a nerve.
Oliver Consa’s “Something is rotten in the state of QED” is one of those papers.

It is not a technical QED calculation.
It is a polemic: a long critique of renormalization, historical shortcuts, convenient coincidences, and suspiciously good matches between theory and experiment. Consa argues that QED’s foundations were improvised, normalized, mythologized, and finally institutionalized into a polished narrative that glosses over its original cracks.

This is an attractive story.
Too attractive, perhaps.
So instead of reacting emotionally — pro or contra — I decided to dissect the argument with a bit of help.

At my request, an AI language model (“Iggy”) assisted in the analysis. Not to praise me. Not to flatter Consa. Not to perform tricks.
Simply to act as a scalpel: cold, precise, and unafraid to separate structure from rhetoric.

This post is the result.

1. What Consa gets right (and why it matters)

Let’s begin with the genuinely valuable parts of his argument.

a) Renormalization unease is legitimate

Dirac, Feynman, Dyson, and others really did express deep dissatisfaction with renormalization. “Hocus-pocus” was not a joke; it was a confession.

Early QED involved:

cutoff procedures pulled out of thin air,
infinities subtracted by fiat,
and the philosophical hope that “the math will work itself out later.”

It did work out later — to some extent — but the conceptual discomfort remains justified. I share that discomfort. There is something inelegant about infinities everywhere.

b) Scientific sociology is real

The post-war era centralized experimental and institutional power in a way physics had never seen. Prestige, funding, and access influenced what got published and what was ignored. Not a conspiracy — just sociology.

Consa is right to point out that real science is messier than textbook linearity.

c) The g–2 tension is real

The ongoing discrepancy between experiment and the Standard Model is not fringe. It is one of the defining questions in particle physics today.

On these points, Consa is a useful corrective:
he reminds us to stay honest about historical compromises and conceptual gaps.

2. Where Consa overreaches

But critique is one thing; accusation is another.

Consa repeatedly moves from:

“QED evolved through trial and error”
to
“QED is essentially fraud.”

This jump is unjustified.

a) Messiness ≠ manipulation

Early QED calculations were ugly. They were corrected decades later. Experiments did shift. Error bars did move.

That is simply how science evolves.

The fact that a 1947 calculation doesn’t match a 1980 value is not evidence of deceit — it is evidence of refinement. Consa collapses that distinction.

b) Ignoring the full evidence landscape

He focuses almost exclusively on:

the Lamb shift,
the electron g–2,
the muon g–2.

Important numbers, yes — but QED’s experimental foundation is vastly broader:

scattering cross-sections,
vacuum polarization,
atomic spectra,
collider data,
running of α, etc.

You cannot judge an entire theory on two or three benchmarks.

c) Underestimating theoretical structure

QED is not “fudge + diagrams.”
It is constrained by:

Lorentz invariance,
gauge symmetry,
locality,
renormalizability.

Even if we dislike the mathematical machinery, the structure is not arbitrary.

So: Consa reveals real cracks, but then paints the entire edifice as rotten.
That is unjustified.

3. A personal aside: the Zitter Institute and the danger of counter-churches

For a time, I was nominally associated with the Zitter Institute — a loosely organized group exploring alternatives to mainstream quantum theory, including zitterbewegung-based particle models.

I now would like to distance myself.

Not because alternative models are unworthy — quite the opposite. But because I instinctively resist:

strong internal identity,
suspicion of outsiders,
rhetorical overreach,
selective reading of evidence,
and occasional dogmatism about their own preferred models.

If we criticize mainstream physics for ad hoc factors, we must be brutal about our own.

Alternative science is not automatically cleaner science.

4. Two emails from 2020: why good scientists can’t always engage

This brings me to two telling exchanges from 2020 with outstanding experimentalists: Prof. Randolf Pohl (muonic hydrogen) and Prof. Ashot Gasparian (PRad).

Both deserve enormous respect, and I won’t reveal the email exchanges because of respect, GDPR rules or whatever).
Both email exchanges revealed the true bottleneck in modern physics to me — it is not intelligence, not malice, but sociology and bandwidth.

a) Randolf Pohl: polite skepticism, institutional gravity

Pohl was kind but firm:

He saw the geometric relations I proposed as numerology.
He questioned applicability to other particles.
He emphasized the conservatism of CODATA logic.

Perfectly valid.
Perfectly respectable.
But also… perfectly bound by institutional norms.

His answer was thoughtful — and constrained.
(Source: ChatGPT analysis of emails with Prof Dr Pohl)

b) Ashot Gasparian: warm support, but no bandwidth

Gasparian responded warmly:

“Certainly your approach and the numbers are interesting.”
But: “We are very busy with the next experiment.”

Also perfectly valid.
And revealing:
even curious, open-minded scientists cannot afford to explore conceptual alternatives.

Their world runs on deadlines, graduate students, collaborations, grants.

(Source: ChatGPT analysis of emails with Prof Dr Pohl)

The lesson

Neither professor dismissed the ideas because they were nonsensical.
They simply had no institutional space to pursue them.

That is the quiet truth:
the bottleneck is not competence, but structure.

5. Why I now use AI as an epistemic partner

This brings me to the role of AI.

Some colleagues (including members of the Zitter Institute) look down on using AI in foundational research. They see it as cheating, or unserious, or threatening to their identity as “outsiders.”

But here is the irony:

AI is exactly the tool that can think speculatively without career risk.

An AI:

has no grant committee,
no publication pressure,
no academic identity to defend,
no fear of being wrong,
no need to “fit in.”

That makes it ideal for exploratory ontology-building.

Occasionally, as in the recent paper I co-wrote with Iggy — “The Wonderful Theory of Light and Matter” — it becomes the ideal partner:

human intuition + machine coherence,
real-space modeling without metaphysical inflation,
EM + relativity as a unified playground,
photons, electrons, protons, neutrons as geometric EM systems.

This is not a replacement for science.
It is a tool for clearing conceptual ground,
where overworked, over-constrained academic teams cannot go.

6. So… is something rotten in QED?

Yes — but not what you think.

What’s rotten is the mismatch

between:

the myth of QED as a perfectly clean, purely elegant theory,
and
the reality of improvised renormalization, historical accidents, social inertia, and conceptual discomfort.

What’s rotten is not the theory itself,
but the story we tell about it.

What’s not rotten:

the intelligence of the researchers,
the honesty of experimentalists,
the hard-won precision of modern measurements.

QED is extraordinary.
But it is not infallible, nor philosophically complete, nor conceptually finished.

And that is fine.

The problem is not messiness.
The problem is pretending that messiness is perfection.

7. What I propose instead

My own program — pursued slowly over many years — is simple:

Bring physics back to Maxwell + relativity as the foundation.
Build real-space geometrical models of all fundamental particles.
Reject unnecessary “forces” invented to patch conceptual holes.
Hold both mainstream and alternative models to the same standard:
no ad hoc constants, no magic, no metaphysics.

And — unusually —
use AI as a cognitive tool, not as an oracle.

Let the machine check coherence.
Let the human set ontology.

If something emerges from the dialogue — good.
If not — also good.

But at least we will be thinking honestly again.

Conclusion

Something is rotten in the state of QED, yes —
but the rot is not fraud or conspiracy.

It is the quiet decay of intellectual honesty behind polished narratives.

The cure is not shouting louder, or forming counter-churches, or romanticizing outsider science.

The cure is precision,
clarity,
geometry,
and the courage to say:

Let’s look again — without myth, without prestige, without fear.

If AI can help with that, all the better.

— Jean Louis Van Belle
(with conceptual assistance from “Iggy,” used intentionally as a scalpel rather than a sycophant)

Post-scriptum: Why the Electron–Proton Model Matters (and Why Dirac Would Nod)

A brief personal note — and a clarification that goes beyond Consa, beyond QED, and beyond academic sociology.

One of the few conceptual compasses I trust in foundational physics is a remark by Paul Dirac. Reflecting on Schrödinger’s “zitterbewegung” hypothesis, he wrote:

“One must believe in this consequence of the theory,
since other consequences which are inseparably bound up with it,
such as the law of scattering of light by an electron,
are confirmed by experiment.”

Dirac’s point is not mysticism.
It is methodological discipline:

If a theoretical structure has unavoidable consequences, and
some of those consequences match experiment precisely,
then even the unobservable parts of the structure deserve consideration.

This matters because the real-space electron and proton models I’ve been working on over the years — now sharpened through AI–human dialogue — meet that exact criterion.

They are not metaphors, nor numerology, nor free speculation.
They force specific, testable, non-trivial predictions:

a confined EM oscillation for the electron, with radius fixed by $\hbar / m_e c$ ;
a “photon-like” orbital speed for its point-charge center;
a distributed (not pointlike) charge cloud for the proton, enforced by mass ratio, stability, form factors, and magnetic moment;
natural emergence of the measured $G_E/G_M$ discrepancy;
and a geometric explanation of deuteron binding that requires no new force.

None of these are optional.
They fall out of the internal logic of the model.
And several — electron scattering, Compton behavior, proton radius, form-factor trends — are empirically confirmed.

Dirac’s rule applies:

When inseparable consequences match experiment,
the underlying mechanism deserves to be taken seriously —
whether or not it fits the dominant vocabulary.

This post is not the place to develop those models in detail; that will come in future pieces and papers.
But it felt important to state why I keep returning to them — and why they align with a style of reasoning that values:

geometry,
energy densities,
charge motion,
conservation laws,
and the 2019 SI foundations of $h$ , $e$ , and $c$
over metaphysical categories and ad-hoc forces.

Call it minimalism.
Call it stubbornness.
Call it a refusal to multiply entities beyond necessity.

For me — and for anyone sympathetic to Dirac’s way of thinking — it is simply physics.

— JL (with “Iggy” (AI) in the wings)

🌀 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.

🧭 The Final Arc: Three Papers, One Question

Over the past years, I’ve been working — quietly but persistently — on a set of papers that circle one simple, impossible question:
What is the Universe really made of?

Not in the language of metaphors. Not in speculative fields.
But in terms of geometry, charge, and the strange clarity of equations that actually work.

Here are the three pieces of that arc:

🌀 1. Radial Genesis
Radial Genesis: A Finite Universe with Emergent Spacetime Geometry
This is the cosmological capstone. It presents the idea that space is not a stage, but an outcome — generated radially by mass–energy events, limited by time and light. It’s an intuitive, equation-free narrative grounded in general relativity and Occam’s Razor.

⚛️ 2. Lectures on Physics: On General Relativity (2)
Lectures on GRT (2)
This one is for the mathematically inclined. It builds from the ground up: tensors, geodesics, curvature. If Radial Genesis is the metaphor, this is the machinery. Co-written with AI, but line by line, and verified by hand.

🌑 3. The Vanishing Charge
The Vanishing Charge: What Happens in Matter–Antimatter Annihilation?
This paper is where the mystery remains. It presents two possible views of annihilation:
(1) as a collapse of field geometry into free radiation,
(2) or as the erasure of charge — with geometry as the by-product.
We didn’t choose between them. We just asked the question honestly.

Why This Arc Matters

These three papers don’t offer a Theory of Everything. But they do something that matters more right now:
They strip away the fog — the inflation of terms, the myth of complexity for complexity’s sake — and try to draw what is already known in clearer, more beautiful lines.

This is not a simulation of thinking.
This is thinking — with AI as a partner, not a prophet.

So if you’re tired of being told that the Universe is beyond your grasp…
Start here.
You might find that it isn’t.

—JL

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

A quasi-final proton model?

After a break of a few months, I produced another lengthy video on quantum physics. 40 minutes. Check it out: https://www.youtube.com/watch?v=k_I3Noaup0E. The hypothesis that I, somewhat desperately, advanced in my last paper on the proton model – that the Zitterbewegung model of a proton does not quite look like that of an electron, and that we are probably talking about a “blob” of charge (sphere- or shell-like, or some other “form factor”, perhaps) – makes more sense to me now. I explored it with ChatGPT-4o, and this video is the result. I should immediately that I am immensely impressed with ChatGPT-4o: ChatGPT-3 was not able to really “understand” what I was talking about a year ago or so.

Who knows, perhaps we might imagine other ChatGPT-4o users will take the “dialogue” with AI on these particle models further, and fully develop them not only intuitively but also analytically. As I am currently trying out some AI-powered audio and video applications, I tried Pictory (pictory.ai): I gave it the url of my own long-winding video and it created this video: The proton as a dynamic blob of charge. 100 seconds only instead of 40 minutes. Its contents is, of course, superficial – but I think it is creative and nicely upbeat. I fed both video back to ChatGPT-4o, and it suggested the following refinements if ever I would like to re-do the video(s) – which I am not going to do, but you might like to further explore the remarks it made:

Consistency in Notation and Conventions:
- Ensure that polar and azimuthal angles are consistently defined throughout. Your initial derivations and explanations switched between conventions, potentially confusing viewers about which angle is associated with which base vector.
Tangential Velocity Decomposition:
- The explanation of how the tangential velocity is decomposed into orthogonal components was accurate but could benefit from a clearer mention of why the orthogonality matters in interpreting the motion on the sphere. Adding a visual representation would further clarify the concept.
- The concept of “sloshing” energy between kinetic and electromagnetic forms was well described but could benefit from highlighting the broader implications of this periodic exchange in classical and relativistic oscillators .
Pointlike Charge versus Blob View:
- The transition from discussing a pointlike charge to a “blob” of charge was a major theme but could be expanded by discussing how this shift impacts the assumptions about uniform or non-uniform charge density, as well as its implications for observable properties like the magnetic moment.
The discussion of symmetry along one axis of the double-loop motion was solid but might benefit from more detail on how this symmetry impacts the interpretation of observable properties like the magnetic moment .
Incorporate links to experimental observations, such as the measured magnetic moment and charge radius of the proton, was referenced. Integrating these observations with the theoretical results could be more explicit to strengthen the practical implications .
Visualizations:
- Consider incorporating visualized trajectories (e.g., the double-loop Lissajous pattern), and how they connect to the derived equations and the observed non-constant tangential velocity. This connection will help bridge the abstract mathematics with physical intuition .
Clarifications on Variability:
- The variability of the tangential velocity in the context of symmetry and non-orthogonality in 3D motion, was a key discussion point. Reinforce why this variability does not conflict with the broader relativistic constraints.

So… That’s all folks ! [Looney Tunes.] 🙂

Concluding remarks

In our previous post, we wrote that we’ve said goodbye to this fascinating field of research. We did: I entered this line of research – fundamental physics – as an amateur 10+ years ago, and now I leave it—as much an amateur now as back then. I wanted to understand the new theories which emerged over the past 50 years or so. Concepts such as the strong force or weak interactions and the new weird charges that come it with: flavors and colors—or all of the new quantum numbers and the associated new conservation laws, which Nature apparently does not respect because of some kind of hidden variables which cause the symmetries that are inherent to conservation laws to break down. […] Apparently, I didn’t get it. 🙂

However, in the process of trying to understand, a whole other mental picture or mindset emerged: we now firmly believe that classical mechanics and electromagnetism – combined with a more creative or realistic explanation of the Planck-Einstein relation – are sufficient to explain most, if not all, of the observations that have been made in this field since Louis de Broglie suggested matter-particles must be similar to light quanta—in the sense that both are energy packets because they incorporate some oscillation of a definite frequency given by the Planck-Einstein relation. They are also different, of course: elementary particles are – in this world view – orbital oscillations of charge (with, of course, an electromagnetic field that is generated by such moving charge), while light-particles (photons and neutrinos) are oscillations of the electromagnetic field—only!

So, then we spend many years trying to contribute to the finer details of this world view. We think we did what we could as part of a part-time and non-professional involvement in this field. So, yes, we’re done. We wrote that some time already. However, we wanted to leave a few thoughts on our proton model: it is not like an electron. In our not-so-humble view, the Zitterbewegung theory applies to it—but in a very different way. Why do we think that? We write that out in our very last paper: concluding remarks on the proton puzzle. Enjoy it !

That brings the number of papers on RG up to 80 now. Too much ! There will be more coming, but in the field that I work in: computer science. Stay tuned !

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… 🙂

Using AI to solve the 80-year-old problem of the anomaly of the electron magnetic moment?

Pre-scriptum (3 October 2024): I came back from holiday and, because this week-long up and down became quite convoluted, I did what I like to do in a case like that, and that is to take my Bamboo notebook and talk about it all in a video which I added to my Real Quantum Physics channel on YouTube. I also updated my paper on RG: as usual, it went through a few versions, but this one – with a summary co-authored by ChatGTP-4 (and ChatGPT-o1) – should be the final one: enjoy!

Indeed, instead of listening to the international news on the war with Russia and on what is happening in the Middle East (all very depressing), you may want to listen to this and read the latest theory. Perhaps you will be inspired by it to develop your own pet realist theory of what an electron might actually be. I can assure you that it is more fun than trying to understand Feynman diagrams and how QED calculations work. 🙂 But don’t think you will win a Nobel Prize if you do not have the right connections and pedigree and all of that: see this analysis of what makes Nobel Prize winners Nobel Prize winners. 🙂

Original post:

I asked some questions to ChatGPT about my geometric explanation of the anomaly in the electron’s magnetic moment. Here is the chat: https://chatgpt.com/share/66f91760-68b8-8004-8cb2-7d2d3624e0aa. To me, it confirms the ‘explanation’ of mainstream QED makes no sense. We can take Schwinger’s factor and build a series of converging terms using that factor. We can also take my first rough cut at a first-order correction (π(alpha)²/8, see my very early 2019 paper on a classical explanation of the amm), and use that.

You may wonder: why not ask ChatGPT about the best first-order factor to be used here considering the geometry of the situation? The fact is: I did, but the geometry is not all that easy. It first came up with the formula for a spherical cap, but that one does not do the trick. See the latter part of the conversation (link above).

I am on holiday now, and so I will switch off a while but I am thinking AI will do what two generations of ‘new’ quantum physicists did not do: come up with a model that is based on real physics and is easy to understand intuitively. 🙂

PS: Of course, I did another rapid-fire paper on ResearchGate to document it all (the logic step-by-step, so to speak). As the chat is public, feel free to continue the conversation. Note that I used the newest ChatGPT o1 version, now in preview but part of a subscription (which you may not have). Yet again a different beast! The older versions of ChatGPT may not be so smart. This conversation is totally worth the US$20/month I pay for my subscription. 🙂

PS 2: Now that I had it open, I also quickly queried it on my wildest hypothesis: a ‘mirror’ electromagnetic force explaining dark matter and dark energy. While it is totally wild (read: nuts), I entertain it because it does away with the need for an explanation in terms of some cosmological constant. Here is the conversation: https://chatgpt.com/share/66f92c7f-82a0-8004-a226-bde65085f18d. I like it that ChatGPT warns me a bit about privacy. It does look wild. However, it is nice to see how gentle ChatGPT is in pointing out what work needs to be done on a theory in order to make it look somewhat less wild. 🙂

PS 3 (yes, ChatGPT is addictive): I also queried it on the rather puzzling 8π/3 factor in the CODATA formula for the Thomson photon-electron scattering cross-section. See its response to our question in the updated chat: https://chatgpt.com/share/66f91760-68b8-8004-8cb2-7d2d3624e0aa. Just scroll down to the bottom. It took 31 seconds to generate the reply: I would be curious to know if that is just courtesy from ChatGPT (we all like to think our questions are complicated, don’t we?), or if this was effectively the time it needed to go through its knowledge base. Whatever the case might be, we think it is brilliant. 🙂 It is nothing to be afraid of, although I did feel a bit like: what’s left to learn to it but for asking intelligent questions. What if it starts really learning by asking intelligent questions itself to us? I am all ready for it. 🙂

Post scriptum

A researcher I was in touch with a few years ago sent me a link to the (virtual) Zitter Institute: https://www.zitter-institute.org/. It is a network and resource center for non-mainstream physicists who succesfully explored – and keep exploring, of course – local/realist interpretations of quantum mechanics by going back to Schrödinger’s original and alternative interpretation of what an electron actually is: a pointlike (but not infinitesimally small) charge orbiting around in circular motion, with:

(i) the trajectory of its motion being determined by the Planck-Einstein relation, and

(ii) an energy – given by Einstein’s mass-energy equivalence relation – which perfectly fits Wheeler’s “mass-without-mass” idea.

I started exploring Schrödinger’s hypothesis myself about ten years ago – as a full-blown alternative to the Bohr-Heisenberg interpretation of quantum mechanics (which I think of as metaphysical humbug, just like Einstein and H.A. Lorentz at the time) – and consistently blogged and published about it: here on this website, and then on viXra, Academia and, since 2020, ResearchGate. So I checked out this new site, and I see the founding members added my blog site as a resource to their project list.

[…]

I am amazingly pleased with that. I mean… My work is much simpler than that of, say, Dr. John G. Williamson (CERN/Philips Research Laboratories/Glasgow University) and Dr. Martin B. van der Mark (Philips Research Laboratories), who created the Quantum Bicycle Society (https://quicycle.com/).

So… Have a look – not at my site (I think I did not finish the work I started) but at the other resources of this new Institute: it looks like this realist and local interpretation of quantum mechanics is no longer non-mainstream… Sweet ! It makes me feel the effort I put into all of this has paid off ! 😉 Moreover, some of my early papers (2018-2020) are listed as useful papers to read. I think that is better than being published in some obscure journal. 🙂

I repeat again: my own research interest has shifted to computer science, logic and artificial intelligence now (you will see recent papers on my RG site are all about that now). It is just so much more fun and it also lines up better with my day job as a freelance IT project manager. So, yes, it is goodbye – but I am happy I can now refer all queries about my particle models and this grand synthesis between old and new quantum mechanics to the Zitter Institute.

It’s really nice: I have been in touch with about half of the founding members of this Institute over the past ten years – casually or in a more sustained way while discussing this or that 2D or 3D model of an electron, proton, or neutron), and they are all great and amazing researchers because they look for truth in science and are very much aware of this weird tendency of modern-day quantum scientists turning their ideas into best-sellers perpetuating myths and mysteries. [I am not only thinking of the endless stream of books from authors like Roger Penrose (the domain for this blog was, originally, reading Penrose rather than reading Feynman) or Graham Greene here, but also of what I now think of rather useless MIT or edX online introductions to quantum physics and quantum math.]

[…]

Looking at the website, I see the engine behind it: Dr. Oliver Consa. I was in touch with him too. He drew my attention to remarkable flip-flop articles such as William Lamb’s anti-photon article (it is an article which everyone should read, I think: unfortunately, you have to pay for it) and remarkable interviews with Freeman Dyson. Talking of the latter (I think of as “the Wolfgang Pauli of the third generation of quantum physicists” because he helped so many others to get a Nobel Prize before he got one – Dyson never got a Nobel Prize, by the way), this is one of these interviews you should watch: just four years before he would die from old age, Freeman Dyson plainly admits QED and QFT is a totally unproductive approach: a “dead end” as Dyson calls it.

So, yes, I am very pleased and happy. It makes me feel my sleepness nights and hard weekend work over the past decade on this has not been in vain ! Paraphrasing Dyson in the above-mentioned video interview, I’d say: “It is the end of the story, and that particular illumination was a very joyful time.” 🙂

Thank you, Dr. Consa. Thank you, Dr. Vassallo, Dr. Burinskii, Dr. Meulenberg, Dr. Kovacs, and – of course – Dr. Hestenes – who single-handedly revived the Zitterbewegung interpretation of quantum mechanics in the 1990s. I am sure I forgot to mention some people. Sorry for that. I will wrap up my post here by saying a few more words about David Hestenes.

I really admire him deeply. Moving away from the topic of high-brow quantum theory, I think his efforts to reform K-12 education in math and physics is even more remarkable than the new space-time algebra (STA) he invented. I am 55 years old and so I know all about the small and pleasant burden to help kids with math and statistics in secondary school and at university: the way teachers now have to convey math and physics to kids now is plain dreadful. I hope it will get better. It has to. If the US and the EU want to keep leading in research, then STEM education (Science, Technology, Engineering, and Mathematics) needs a thorough reform.

No reading of Feynman anymore…

I have not posted in a while, and that is because I find the format of a video much easier to express my thoughts. Have a look at my YouTube channel ! Also, for the more serious work, I must refer to my ResearchGate page. Have fun thinking things through ! 🙂

On the quantum computing hype

1. The Wikipedia article on quantum computing describes a quantum computer as “a computer that exploits quantum -mechanical phenomena.” The rest of the article then tries to explain what these quantum-mechanical phenomena actually are.

Unfortunately, the article limits itself to the mainstream interpretation of these and, therefore, suffers from what I perceive to be logical and philosophical errors. Indeed, in the realistic interpretation of quantum mechanics that I have been developing, system wavefunctions are only useful to model our own uncertainty about the system. I subscribe to Hendrik Antoon Lorentz’s judgment at the last Solvay Conference under his leadership: there is no need whatsoever to elevate indeterminism to a philosophical principle. Not in science in general, and not in quantum mechanics in particular. I, therefore, think quantum mechanics cannot offer a substantially new computing paradigm.

Of course, one may argue that, for specific problems, some kind of three- or more-valued logic – rather than the binary or Boolean true/false dichotomy on which most logic circuits are based – may come in handy. However, such logic has already been worked out, and can be accessed using appropriate programming languages. Python and the powerful mathematical tools that come with it (Pandas, NumPy and SciPy) work great with ternary logic using a {true, false, unknown} or a {-1, 0, +1} set of logical values rather than the standard {0, 1} Boolean set. The Wikipedia article on three-valued logic is worth a read and, despite the rather arcane nature of the topic, much better written than the mentioned article: have a look at how operators are used on these three-valued sets in meaningful algebras or logical models, such as that of Kleene, Priest or Lukasiewicz.

2. One may, of course, argue that, even when there is probably no such thing as a new logical quantum computing model or logic, quantum technology may offer distinct advantages when it comes to storage of data about this or that state or, one day, lead to devices with faster clock and/or bus speeds. That appears to be a pipedream too:

To keep, say, an electron in this or that spin state, one must create and steady an electromagnetic field – usually one does so in a superconducting environment, which makes actual mechanical devices used for quantum computing (qubits) look like the modern-day equivalent of Babbage’s analytical machine. In my not-so-humble view, such devices will never ever achieve the sheer material performance offered by current nanometer-scale MOSFETs.

As for bus or transmission speeds, quantum theory does not come with a new theory of charge propagation and, most importantly, is fundamentally flawed in its analysis of how signals actually propagate in, say, a lattice structure. I refer to one of my papers here (on electron propagation in a lattice), in which I deconstruct Feynman’s analysis of the concept of the free and effective mass of an electron. Hence, for long-distance transmission of signals, optical fiber cannot be beaten. For short-distance transmission of signals (say, within an electrical circuit, I refer to the above-mentioned nano-technology which continues to revolutionize the chip industry.

Brussels, 4 July 2023

Epilogue: an Easter podcast

I have been thinking on my explanation of dark matter/energy, and I think it is sound. It solves the last asymmetry in my models, and explains all. So, after a hiatus of two years, I bothered to make a podcast on my YouTube channel once again. It talks about everything. Literally everything !

It makes me feel my quest for understanding of matter and energy – in terms of classical concepts and measurements (as depicted below) – has ended. Perhaps I will write more but that would only be to promote the material, which should promote itself if it is any good (which I think it is).

I should, by way of conclusion, say a few final words about Feynman’s 1963 Lectures now. When everything is said and done, it is my reading of them which had triggered this blog about ten years ago. I would now recommend Volume I and II (classical physics and electromagnetic theory) – if only because it gives you all the math you need to understand all of physics – but not Volume III (the lectures on quantum mechanics). They are outdated, and I do find Feynman guilty of promoting rather than explaining the hocus-pocus around all of the so-called mysteries in this special branch of physics.

Quantum mechanics is special, but I do conclude now that it can all be explained in terms of classical concepts and quantities. So, Gell-Mann’s criticism of Richard Feynman is, perhaps, correct: Mr. Feynman did, perhaps, make too many jokes – and it gets annoying because he must have known some of what he suggests does not make sense – even if I would not go as far as Gell-Mann, who says “Feynman was only concerned about himself, his ego, and his own image !”

So, I would recommend my own alternative series of ‘lectures’. Not only are they easier to read, but they also embody a different spirit of writing. Science is not about you, it is about thinking for oneself and deciding on what is truthful and useful, and what is not. So, to conclude, I will end by quoting Ludwig Boltzmann once more:

“Bring forward what is true.

Write it so that it is clear.

Defend it to your last breath.”
Ludwig Boltzmann (1844 – 1906)

Post scriptum: As for the ‘hocus-pocus’ in Feynman’s Lectures, we should, perhaps, point once again to some of our early papers on the flaws in his arguments. We effectively put our finger on the arbitrary wavefunction convention, or the (false) boson-fermion dichotomy, or the ‘time machine’ argument that is inherent to his explanation of the Hamiltonian, and so on. We published these things on Academia.edu before (also) putting our (later) papers ResearchGate, so please check there for the full series. 🙂

Post scriptum (23 April 2023): Also check out this video, which was triggered by someone who thought my models amount to something like a modern aether theory, which it is definitely not the case: https://www.youtube.com/watch?v=X38u2-nXoto. 🙂 I really think it is my last reflection on these topics. I need to focus on my day job, sports, family, etcetera again ! 🙂

Dirac’s wave equation and particle models

Introduction

I had not touched physics since April last year, as I was struggling with cancer, and finally went in for surgery. It solved the problem but physical and psychological recovery was slow, and so I was in no mood to work on mathematical and physical questions. Now I am going through my ResearchGate papers again. I start with those that get a fair amount of downloads and – I am very pleased to see that happen – those are the papers that deal with very fundamental questions, and lay out the core of an intuition that is more widely shared now: physicists are lost in contradictions and will not get out of this fuzzy situation until they solve them.

[Skeptical note here: I note that those physicists who bark loudest about the need for a scientific revolution are, unfortunately, often those who obscure things even more. For example, I quickly went through Hossenfelder’s Lost in Math (and I also emailed her to highlight all that zbw theory can bring) but she did not even bother to reply and, more in general, shows no signs of being willing to go back to the roots, which are the solutions that were presented during the early Solvay conferences but, because of some weird tweak of the history of science, and despite the warnings of intellectual giants such as H.A. Lorentz, Ehrenfest, or Einstein (and also Dirac or Bell in the latter half of their lifes), were discarded. I have come to the conclusion that modern-day scientists cannot be fashionable when admitting all mysteries have actually been solved long time ago.]

The key observation or contradiction is this: the formalism of modern quantum mechanics deals with all particles – stable or unstable – as point objects: they are supposed to have no internal structure. At the same time, a whole new range of what used to be thought of as intermediate mental constructs or temporary classifications – think of quarks here, or of the boson-fermion dichotomy – acquired ontological status. We lamented that in one of very first papers (titled: the difference between a theory, a calculation and an explanation), which has few formulas and is, therefore, a much easier read than the others.

Some of my posts on this blog here were far more scathing and, therefore, not suitable to write out in papers. See, for example, my Smoking Gun Physics post, in which I talk much more loudly (but also more unscientifically) about the ontologicalization of quarks and all these theoretical force-carrying particles that physicists have invented over the past 50 years or so.

My point of view is clear and unambiguous: photons and neutrinos (both of which can be observed and measured) will do. The rest (the analysis of decay and the chain of reactions after high-energy collisions, mainly) can be analyzed using scattering matrices and other classical techniques (on that, I did write a paper highlighting the proposals of more enlightened people than me, like Bombardelli, 2016, even if I think researchers like Bombardelli should push back to basics even more than they do). By the way, I should probably go much further in my photon and neutrino models, but time prevented me from doing so. In any case, I did update and put an older paper of mine online, with some added thoughts on recent experiments that seem to confirm neutrinos have some rest mass. That is only what is to be expected, I would think. Have a look at it.

[…]

This is a rather lengthy introduction to the topic I want to write about for my public here, which is people like you and me: (amateur) physicists who want to make sense of all that is out there. So I will make a small summary of an equation I was never interested in: Dirac’s wave equation. Why my lack of interest before, and my renewed interest now?

The reason is this: Feynman clearly never believed Dirac’s equation added anything to Schrödinger’s, because he does not even mention it in his rather Lectures which, I believe, are, today still, truly seminal even if they do not go into all of the stuff mainstream quantum physicists today believe to be true (which is, I repeat, all of the metaphysics around quarks and gluons and force-carrying bosons and all that). So I did not bother to dig into it.

However, when revising my paper on de Broglie’s matter-wave, I realized that I should have analyzed Dirac’s equation too, because I do analyze Schrödinger’s wave equation there (which makes sense), and also comment on the Klein-Gordon wave equation (which, just like Dirac’s, does not make much of an impression on me). Hence, I would say my renewed interest is only there because I wanted to tidy up a little corner in this kitchen of mine. 🙂

I will stop rambling now, and get on with it.

Dirac’s wave equation: concepts and issues

We should start by reminding ourselves what a wave equation actually is: it models how waves – sound waves, or electromagnetic waves, or – in this particular case – a ‘wavicle’ or wave-particle – propagate in space and in time. As such, it is often said they model the properties of the medium (think of properties such as elasticity, density, permittivity or permeability here) but, because we do no longer think of spacetime as an aether, quantum-mechanical wave equations are far more abstract.

I should insert a personal note here. I do have a personal opinion on the presumed reality of spacetime. It is not very solid, perhaps, because I oscillate between (1) Kant’s intuition, thinking that space and time are mental constructs only, which our mind uses to structure its impressions (we are talking science here, so I should say: our measurements) versus (2) the idea that the 2D or 3D oscillations of pointlike charges within, say, an electron, a proton or a muon-electron must involve some kind of elasticity of the ‘medium’ that we commonly refer to as spacetime (I’d say that is more in line with Wittgenstein’s philosophy of reality). I should look it up but I think I do talk about the elasticity of spacetime at one or two occasions in my papers that talk about internal forces in particles, or papers in which I dig deep into the potentials that may or may not drive these oscillations. I am not sure how far I go there. Probably too far. But if properties such as vacuum permittivity or permeability are generally accepted, then why not think of elasticity? However, I did try to remain very cautious when it comes to postulating properties of the so-called spacetime vacuum, as evidenced from what I write in one of the referenced papers above:

“Besides proving that the argument of the wavefunction is relativistically invariant, this [analysis of the argument of the wavefunction] also demonstrates the relativistic invariance of the Planck-Einstein relation when modelling elementary particles.[1] This is why we feel that the argument of the wavefunction (and the wavefunction itself) is more real – in a physical sense – than the various wave equations (Schrödinger, Dirac, or Klein-Gordon) for which it is some solution. In any case, a wave equation usually models the properties of the medium in which a wave propagates. We do not think the medium in which the matter-wave propagates is any different from the medium in which electromagnetic waves propagate. That medium is generally referred to as the vacuum and, whether or not you think of it as true nothingness or some medium, we think Maxwell’s equations – which establishes the speed of light as an absolute constant – model the properties of it sufficiently well! We, therefore, think superluminal phase velocities are not possible, which is why we think de Broglie’s conceptualization of a matter particle as a wavepacket – rather than one single wave – is erroneous.[2]“

The basic idea is this: if the vacuum is true nothingness, then it cannot have any properties, right? 🙂 That is why I call the spacetime vacuum, as it is being modelled in modern physics, a so-called vacuum. 🙂

[…] I guess I am rambling again, and so I should get back to the matter at hand, and quite literally so, because we are effectively talking about real-life matter here. To be precise, we are talking about Dirac’s view of an electron moving in free space. Let me add the following clarification, just to make sure we understand exactly what we are talking about: free space is space without any potential in it: no electromagnetic, gravitational or other fields you might think of.

In reality, such free space does not exist: it is just one of those idealizations which we need to model reality. All of real-life space – the Universe we live in, in other words – has potential energy in it: electromagnetic and/or gravitational potential energy (no other potential energy has been convincingly demonstrated so far, so I will not add to the confusion by suggesting there might be more). Hence, there is no such thing as free space.

What am I saying here? I am just saying that it is not bad that we remind ourselves of the fact that Dirac’s construction is theoretical from the outset. To me, it feels like trying to present electromagnetism by making full abstraction of the magnetic side of the electromagnetic force. That is all that I am saying here. Nothing more, nothing less. No offense to the greatness of a mind like Dirac’s.

[…] I may have lost you as a reader just now, so let me try to get you back: Dirac’s wave equation. Right. Dirac develops it in two rather dense sections of his Principles of Quantum Mechanics, which I will not try to summarize here. I want to make it easy for the reader, so I will limit myself to an analysis of the very first principle(s) which Dirac develops in his Nobel Prize Lecture. It is this (relativistically correct) energy equation:

E²= m₀²c⁴ + p²c²

This equation may look unfamiliar to you but, frankly, if you are familiar with the basics of relativity theory, it should not come across as weird or unfathomable. It is one of the many basic ways of expressing relativity theory, as evidenced from the fact that Richard Feynman introduces this equation as part of his very first volume of his Lectures on Physics, and in one of the more basic chapters of it: just click on the link and work yourself through it: you will see it is just another rendering of Einstein’s mass-equivalence relation (E = mc²).

The point is this: it is very easy now to understand Dirac’s basic energy equation: the one he uses to then go from variables to quantum-mechanical operators and all of the other mathematically correct hocus-pocus that result in his wave equation. Just substitute E = mc² for W, and then divide all by c²:

So here you are. All the rest is the usual hocus-pocus: we substitute classical variables by operators, and then we let them operate on a wavefunction (wave equations may or may not describe the medium, but wavefunctions surely do describe real-life particles), and then we have a complicated differential equation to solve and – as we made abundantly clear in this and other papers (one that you may want to read is my brief history of quantum-mechanical ideas, because I had a lot of fun writing that one, and it is not technical at all) – when you do that, you will find non-sensical solutions, except for the one that Schrödinger pointed out: the Zitterbewegung electron, which we believe corresponds to the real-life electron.

I will wrap this up (although you will say I have not done my job yet) by quoting quotes and comments from my de Broglie paper:

Prof. H. Pleijel, then Chairman of the Nobel Committee for Physics of the Royal Swedish Academy of Sciences, dutifully notes this rather inconvenient property in the ceremonial speech for the 1933 Nobel Prize, which was awarded to Heisenberg for nothing less than “the creation of quantum mechanics”[1]:

“Matter is formed or represented by a great number of this kind of waves which have somewhat different velocities of propagation and such phase that they combine at the point in question. Such a system of waves forms a crest which propagates itself with quite a different velocity from that of its component waves, this velocity being the so-called group velocity. Such a wave crest represents a material point which is thus either formed by it or connected with it, and is called a wave packet. […] As a result of this theory, one is forced to the conclusion to conceive of matter as not being durable, or that it can have definite extension in space. The waves, which form the matter, travel, in fact, with different velocity and must, therefore, sooner or later separate. Matter changes form and extent in space. The picture which has been created, of matter being composed of unchangeable particles, must be modified.”

This should sound very familiar to you. However, it is, obviously, not true: real-life particles – electrons or atoms traveling in space – do not dissipate. Matter may change form and extent in space a little bit – such as, for example, when we are forcing them through one or two slits[2] – but not fundamentally so![3]

We repeat again, in very plain language this time: Dirac’s wave equation is essentially useless, except for the fact that it actually models the electron itself. That is why only one of its solutions make sense, and that is the very trivial solution which Schrödinger pointed out: the Zitterbewegung electron, which we believe corresponds to the real-life electron. 🙂 It just goes through space and time like any ordinary particle would do, but its trajectory is not given by Dirac’s wave equation. In contrast, Schrödinger’s wave equation (with or without a potential being present: in free or non-free space, in other words) does the trick and – against mainstream theory – I dare say, after analysis of its origins, that it is relativistically correct. Its only drawback is that it does not incorporate the most essential property of an elementary particle: its spin. That is why it models electron pairs rather than individual electrons.

We can easily generalize to protons or other elementary or non-elementary particles. For a deeper discussion of Dirac’s wave equation (which is what you probably expected), I must refer, once again, to Annex II of my paper on the interpretation of de Broglie’s matter-wave: it is all there, really, and – glancing at it all once again – the math is actually quite basic. In any case, paraphrasing Euclid in his reply to King Ptolemy’s question, I would say that there is no royal road to quantum mechanics. One must go through its formalism and, far more important, its history of thought. 🙂

To conclude, I would like to return to one of the remarks I made in the introduction. What about the properties of the vacuum? I will remain cautious and, hence, not answer that question. I prefer to let you think about this rather primitive classification of what is relative and not, and how the equations in physics mix both of it. 🙂

[1] To be precise, Heisenberg got a postponed prize from 1932. Erwin Schrödinger and Paul A.M. Dirac jointly got the 1933 prize. Prof. Pleijel acknowledges all three in more or less equal terms in the introduction of his speech: “This year’s Nobel Prizes for Physics are dedicated to the new atomic physics. The prizes, which the Academy of Sciences has at its disposal, have namely been awarded to those men, Heisenberg, Schrödinger, and Dirac, who have created and developed the basic ideas of modern atomic physics.”

[2] The wave-particle duality of the ring current model should easily explain single-electron diffraction and interference (the electromagnetic oscillation which keeps the charge swirling would necessarily interfere with itself when being forced through one or two slits), but we have not had the time to engage in detailed research here.

[3] We will slightly nuance this statement later but we will not fundamentally alter it. We think of matter-particles as an electric charge in motion. Hence, as it acts on a charge, the nature of the centripetal force that keeps the particle together must be electromagnetic. Matter-particles, therefore, combine wave-particle duality. Of course, it makes a difference when this electromagnetic oscillation, and the electric charge, move through a slit or in free space. We will come back to this later. The point to note is: matter-particles do not dissipate. Feynman actually notes that at the very beginning of his Lectures on quantum mechanics, when describing the double-slit experiment for electrons: “Electrons always arrive in identical lumps.”

[1] The relativistic invariance of the Planck-Einstein relation emerges from other problems, of course. However, we see the added value of the model here in providing a geometric interpretation: the Planck-Einstein relation effectively models the integrity of a particle here.

[2] See our paper on matter-waves, amplitudes, and signals.

Deep electron orbitals and the essence of quantum physics

After a long break (more than six months), I have started to engage again in a few conversations. I also looked at the 29 papers on my ResearchGate page, and I realize some of them would need to be re-written or re-packaged so as to ensure a good flow. Also, some of the approaches were more productive than others (some did not lead anywhere at all, actually), and I would need to point those out. I have been thinking about how to approach this, and I think I am going to produce an annotated version of these papers, with comments and corrections as mark-ups. Re-writing or re-structuring all of them would require to much work.

The mark-up of those papers is probably going to be based on some ‘quick-fire’ remarks (a succession of thoughts triggered by one and the same question) which come out of the conversation below, so I thank these thinkers for having kept me in the loop of a discussion I had followed but not reacted to. It is an interesting one – on the question of ‘deep electron orbitals’ (read: the orbitals of negative charge inside of a nucleus exist and, if so, how one can model them. If one could solve that question, one would have a theoretical basis for what is referred to as low-energy nuclear reactions. That was known formerly as cold fusion, but that got a bit of a bad name because of a number of crooks spoiling the field, unfortunately.

PS: I leave the family names of my correspondents in the exchange below out so they cannot be bothered. One of them, Jerry, is a former American researcher at SLAC. Andrew – the key researcher on DEPs – is a Canadian astrophysicist, and the third one – Jean-Luc – is a rather prominent French scientist in LENR.]

From: Jean Louis Van Belle
Sent: 18 November 2021 22:51
Subject: Staying engaged (5)

Oh – and needless to say, Dirac’s basic equation can, of course, be expanded using the binomial expansion – just like the relativistic energy-momentum relation, and then one can ‘cut off’ the third-, fourth-, etc-order terms and keep the first and second-order terms only. Perhaps it is equations like that kept you puzzled (I should check your original emails). In any case, this way of going about energy equations for elementary particles is a bit the same as those used in perturbation equations in which – as Dirac complained – one randomly selects terms that seem to make sense and discard others because they do not seem to make sense. Of course, Dirac criticized perturbation theory much more severely than this – and rightly so. 😊 😊 JL

From: Jean Louis Van Belle
Sent: 18 November 2021 22:10
Subject: Staying engaged (4)

Also – I remember you had some questions on an energy equation – not sure which one – but so I found Dirac’s basic equation (based on which he derives the ‘Dirac’ wave equation) is essentially useless because it incorporates linear momentum only. As such, it repeats de Broglie’s mistake, and that is to interpret the ‘de Broglie’ wavelength as something linear. It is not: frequencies, wavelengths are orbital frequencies and orbital circumferences. So anything you would want to do with energy equations that are based on that, lead nowhere – in my not-so-humble opinion, of course. To illustrate the point, compare the relativistic energy-momentum relation and Dirac’s basic equation in his Nobel Prize lecture (I hope the subscripts/superscripts get through your email system so they display correctly):

m₀²c⁴ = E² – p²c² (see, for example, Feynman-I-16, formula 16-3)

Divide the above by c² and re-arrange and you get Dirac’s equation: W²/c² – p_r² – m²/c² = 0 (see his 1933 Nobel Prize Lecture)

So that cannot lead anywhere. It’s why I totally discard Dirac’s wave equation (it has never yielded any practical explanation of a real-life phenomenon anyway, if I am not mistaken).

Cheers – JL

From: Jean Louis Van Belle
Sent: 18 November 2021 21:49
Subject: Staying engaged (3)

Just on ‘retarded sources’ and ‘retarded fields’ – I have actually tried to think of the ‘force mechanism’ inside of an electron or a proton (what keeps the pointlike charge in this geometric orbit around a center of mass?). I thought long and hard about some kind of model in which we have the charge radiate out a sub-Planck field, and that its ‘retarded effects’ might arrive ‘just in time’ to the other side of the orbital (or whatever other point on the orbital) so as to produce the desired ‘course correction’ might explain it. I discarded it completely: I am now just happy that we have ‘reduced’ the mystery to this ‘Planck-scale quantum-mechanical oscillation’ (in 2D or 3D orbitals) without the need for an ‘aether’, or quantized spacetime, or ‘virtual particles’ actually ‘holding the thing together’.

Also, a description in terms of four-vectors (scalar and vector potential) does not immediately call for ‘retarded time’ variables and all that, so that is another reason why I think one should somehow make the jump from E-B fields to scalar and vector potential, even if the math is hard to visualize. If we want to ‘visualize’ things, Feynman’s discussion of the ‘energy’ and ‘momentum’ flow in https://www.feynmanlectures.caltech.edu/II_27.html might make sense, because I think analyses in terms of Poynting vectors are relativistically current, aren’t they? It is just an intuitive idea…

Cheers – JL

From: Jean Louis Van Belle
Sent: 18 November 2021 21:28
Subject: Staying engaged (2)

But so – in the shorter run – say, the next three-six months, I want to sort out those papers on ResearchGate. The one on the de Broglie’s matter-wave (interpreting the de Broglie wavelength as the circumference of a loop rather than as a linear wavelength) is the one that gets most downloads, and rightly so. The rest is a bit of a mess – mixing all kinds of things I tried, some of which worked, but other things did not. So I want to ‘clean’ that up… 😊 JL

From: Jean Louis Van Belle
Sent: 18 November 2021 21:21
Subject: Staying engaged…

Please do include me in the exchanges, Andrew – even if I do not react, I do read them because I do need some temptation and distraction. As mentioned, I wanted to focus on building a credible n = p + e model (for free neutrons but probably more focused on a Schrodinger-like D = p + e + p Platzwechsel model, because the deuteron nucleus is stable). But so I will not do that the way I studied the zbw model of the electron and proton (I believe that is sound now) – so that’s with not putting in enough sleep. I want to do it slowly now. I find a lot of satisfaction in the fact that I think there is no need for complicated quantum field theories (fields are quantized, but in a rather obvious way: field oscillations – just like matter-particles – pack Planck’s quantum of (physical) action which – depending on whether you freeze time or positions as a variable, expresses itself as a discrete amount of energy or, alternatively, as a discrete amount of momentum), nor is there any need for this ‘ontologization’ of virtual field interactions (sub-Planck scale) – the quark-gluon nonsense.

Also, it makes sense to distinguish between an electromagnetic and a ‘strong’ or ‘nuclear’ force: the electron and proton have different form factors (2D versus 3D oscillations, but that is a bit of a non-relativistic shorthand for what might be the case) but, in addition, there is clearly a much stronger force at play within the proton – whose strength is the same kind of ‘scale’ as the force that gives the muon-electron its rather enormous mass. So that is my ‘belief’ and the ‘heuristic’ models I build (a bit of ‘numerology’ according to Dr Pohl’s rather off-hand remarks) support it sufficiently for me to make me feel at peace about all these ‘Big Questions’.

I am also happy I figured out these inconsistencies around 720-degree symmetries (just the result of a non-rigorous application of Occam’s Razor: if you use all possible ‘signs’ in the wavefunction, then the wavefunction may represent matter as well as anti-matter particles, and these 720-degree weirdness dissolves). Finally, the kind of ‘renewed’ S-matrix programme for analyzing unstable particles (adding a transient factor to wavefunctions) makes sense to me, but even the easiest set of equations look impossible to solve – so I may want to dig into the math of that if I feel like having endless amounts of time and energy (which I do not – but, after this cancer surgery, I know I will only die on some ‘moral’ or ‘mental’ battlefield twenty or thirty years from now – so I am optimistic).

So, in short, the DEP question does intrigue me – and you should keep me posted, but I will only look at it to see if it can help me on that deuteron model. 😊 That is the only ‘deep electron orbital’ I actually believe in. Sorry for the latter note.

Cheers – JL

From: Andrew
Sent: 16 November 2021 19:05
To: Jean-Luc; Jerry; Jean Louis
Subject: Re: retarded potential?

Dear Jean-Louis,

Congratulations on your new position. I understand your present limitations, despite your incredible ability to be productive. They must be even worse than those imposed by my young kids and my age. Do you wish for us to not include you in our exchanges on our topic? Even with no expectation of your contributing at this point, such emails might be an unwanted temptation and distraction.

Dear Jean-Luc,

Thank you for the Wiki-Links. They are useful. I agree that the 4-vector potential should be considered. Since I am now considering the nuclear potentials as well as the deep orbits, it makes sense to consider the nuclear vector potentials to have an origin in the relativistic Coulomb potentials. I am facing this in my attempts to calculate the deep orbits from contributions to the potential energies that have a vector component, which non-rel Coulomb potentials do not have.

For examples: do we include the losses in Vcb (e.g., from the binding energy BE) when we make the relativistic correction to the potential; or, how do we relativistically treat pseudo potentials such as that of centrifugal force? We know that for equilibrium, the average forces must cancel. However, I’m not sure that it is possible to write out a proper expression for “A” to fit such cases.

Best regards to all,

Andrew

_ _ _

On Fri, Nov 12, 2021 at 1:42 PM Jean-Luc wrote:

Dear all,

I totally agree with the sentence of Jean-Louis, which I put in bold in his message, about vector potential and scalar potential, combined into a 4-vector
potential A, for representing EM field in covariant formulation. So EM representation by 4-vector A has been very developed, as wished by JL,
in the framework of QED.

We can note the simplicity of Lorentz gauge written by using A.
   https://en.wikipedia.org/wiki/Lorenz_gauge_condition

We can see the reality of vector potential
in the Aharonov-Bohm effect:
    https://en.wikipedia.org/wiki/Aharonov-Bohm_effect.
In fact, we can see that vector potential contains more information than E,B fields.
Best regards

   Jean-Luc
Le 12/11/2021 à 05:43, Jean Louis Van Belle a écrit :

Hi All – I’ve been absent in the discussion, and will remain absent for a while. I’ve been juggling a lot of work – my regular job at the Ministry of Interior (I got an internal promotion/transfer, and am working now on police and security sector reform) plus consultancies on upcoming projects in Nepal. In addition, I am still recovering from my surgery – I got a bad flue (not C19, fortunately) and it set back my auto-immune system, I feel. I have a bit of a holiday break now (combining the public holidays of 11 and 15 November in Belgium with some days off to bridge so I have a rather nice super-long weekend – three in one, so to speak).

As for this thread, I feel like it is not ‘phrasing’ the discussion in the right ‘language’. Thinking of E-fields and retarded potential is thinking in terms of 3D potential, separating out space and time variables without using the ‘power’ of four-vectors (four-vector potential, and four-vector space-time). It is important to remind ourselves that we are measuring fields in continuous space and time (but, again, this is relativistic space-time – so us visualizing a 3D potential at some point in space is what it is: we visualize something because our mind needs that – wants that). The fields are discrete, however: a field oscillation packs one unit of Planck – always – and Planck’s quantum of action combines energy and momentum: we should not think of energy and momentum as truly ‘separate’ (discrete) variables, just like we should not think of space and time as truly ‘separate’ (continuous) variables.

I do not quite know what I want to say here – or how I should further work it out. I am going to re-read my papers. I think I should further develop the last one (https://www.researchgate.net/publication/351097421_The_concepts_of_charge_elementary_ring_currents_potential_potential_energy_and_field_oscillations), in which I write that the vector potential is more real than the electric field and the scalar potential should be further developed, and probably it is the combined scalar and vector potential that are the ’real’ things. Not the electric and magnetic field. Hence, illustrations like below – in terms of discs and cones in space – do probably not go all that far in terms of ‘understanding’ what it is going on… It’s just an intuition…

Cheers – JL

From: Andrew
Sent: 23 September 2021 17:17
To: Jean-Luc; Jerry; Jean Louis
Subject: retarded potential?

Dear Jean-Luc,

Becasue of the claim that gluons are tubal, I have been looking at the disk-shaped E-field lines of the highly-relativistic electron and comparing it to the retarded potential, which, based on timing, would seem to give a cone rather than a disk (see figure). This makes a difference when we consider a deep-orbiting electron. It even impacts selection of the model for impact of an electron when considering diffraction and interference.

Even if the field appears to be spreading out as a cone, the direction of the field lines are that of a disk from the retarded source. However, how does it interact with the radial field of a stationary charge?

Do you have any thoughts on the matter.

Best regards,

Andrew

_ _ _

On Thu, Sep 23, 2021 at 5:05 AM Jean-Luc wrote:

Dear Andrew, Thank you for the references. Best regards, Jean-Luc

Le 18/09/2021 à 17:32, Andrew a écrit :
> This might have useful thoughts concerning the question of radiation
> decay to/from EDOs.
>
> Quantum Optics Electrons see the quantum nature of light
> Ian S. Osborne
> We know that light is both a wave and a particle, and this duality
> arises from the classical and quantum nature of electromagnetic
> excitations. Dahan et al. observed that all experiments to date in
> which light interacts with free electrons have been described with
> light considered as a wave (see the Perspective by Carbone). The
> authors present experimental evidence revealing the quantum nature of
> the interaction between photons and free electrons. They combine an
> ultrafast transmission electron microscope with a silicon-photonic
> nanostructure that confines and strengthens the interaction between
> the light and the electrons. The “quantum” statistics of the photons
> are imprints onto the propagating electrons and are seen directly in
> their energy spectrum.
> Science, abj7128, this issue p. 1324; see also abl6366, p. 1309

Certainty and uncertainty

A lot of the Uncertainty in quantum mechanics is suspiciously certain. For example, we know an electron will always have its spin up or down, in any direction along which we choose to measure it, and the value of the angular momentum will, accordingly, be measured as plus or minus ħ/2. That doesn’t sound uncertain to me. In fact, it sounds remarkably certain, doesn’t it? We know – we are sure, in fact, because of countless experiments – that the electron will be in either of those two states, and we also know that these two states are separated by ħ, Planck’s quantum of action, exactly.

Of course, the corollary of this is that the idea of the direction of the angular momentum is a rather fuzzy concept. As Feynman convincingly demonstrates, it is ‘never completely along any direction’. Why? Well… Perhaps it can be explained by the idea of precession?

In fact, the idea of precession might also explain the weird 720° degree symmetry of the wavefunction.

Hmm… Now that is an idea to look into ! 🙂

Should we reinvent wavefunction math?

Preliminary note: This post may cause brain damage. 🙂 If you haven’t worked yourself through a good introduction to physics – including the math – you will probably not understand what this is about. So… Well… Sorry. 😦 But if you have… Then this should be very interesting. Let’s go. 🙂

If you know one or two things about quantum math – Schrödinger’s equation and all that – then you’ll agree the math is anything but straightforward. Personally, I find the most annoying thing about wavefunction math are those transformation matrices: every time we look at the same thing from a different direction, we need to transform the wavefunction using one or more rotation matrices – and that gets quite complicated !

Now, if you have read any of my posts on this or my other blog, then you know I firmly believe the wavefunction represents something real or… Well… Perhaps it’s just the next best thing to reality: we cannot know das Ding an sich, but the wavefunction gives us everything we would want to know about it (linear or angular momentum, energy, and whatever else we have an operator for). So what am I thinking of? Let me first quote Feynman’s summary interpretation of Schrödinger’s equation (Lectures, III-16-1):

“We can think of Schrödinger’s equation as describing the diffusion of the probability amplitude from one point to the next. […] But the imaginary coefficient in front of the derivative makes the behavior completely different from the ordinary diffusion such as you would have for a gas spreading out along a thin tube. Ordinary diffusion gives rise to real exponential solutions, whereas the solutions of Schrödinger’s equation are complex waves.”

Feynman further formalizes this in his Lecture on Superconductivity (Feynman, III-21-2), in which he refers to Schrödinger’s equation as the “equation for continuity of probabilities”. His analysis there is centered on the local conservation of energy, which makes me think Schrödinger’s equation might be an energy diffusion equation. I’ve written about this ad nauseam in the past, and so I’ll just refer you to one of my papers here for the details, and limit this post to the basics, which are as follows.

The wave equation (so that’s Schrödinger’s equation in its non-relativistic form, which is an approximation that is good enough) is written as: formula 1 The resemblance with the standard diffusion equation (shown below) is, effectively, very obvious: formula 2 As Feynman notes, it’s just that imaginary coefficient that makes the behavior quite different. How exactly? Well… You know we get all of those complicated electron orbitals (i.e. the various wave functions that satisfy the equation) out of Schrödinger’s differential equation. We can think of these solutions as (complex) standing waves. They basically represent some equilibrium situation, and the main characteristic of each is their energy level. I won’t dwell on this because – as mentioned above – I assume you master the math. Now, you know that – if we would want to interpret these wavefunctions as something real (which is surely what I want to do!) – the real and imaginary component of a wavefunction will be perpendicular to each other. Let me copy the animation for the elementary wavefunction ψ(θ) = a·e^−i∙θ = a·e^{−i∙(E/ħ)·t} = a·cos[(E/ħ)∙t] − i·a·sin[(E/ħ)∙t] once more:

Circle_cos_sin

So… Well… That 90° angle makes me think of the similarity with the mathematical description of an electromagnetic wave. Let me quickly show you why. For a particle moving in free space – with no external force fields acting on it – there is no potential (U = 0) and, therefore, the Vψ term – which is just the equivalent of the the sink or source term S in the diffusion equation – disappears. Therefore, Schrödinger’s equation reduces to:

∂ψ(x, t)/∂t = i·(1/2)·(ħ/m_eff)·∇²ψ(x, t)

Now, the key difference with the diffusion equation – let me write it for you once again: ∂φ(x, t)/∂t = D·∇²φ(x, t) – is that Schrödinger’s equation gives us two equations for the price of one. Indeed, because ψ is a complex-valued function, with a real and an imaginary part, we get the following equations:

Re(∂ψ/∂t) = −(1/2)·(ħ/m_eff)·Im(∇²ψ)
Im(∂ψ/∂t) = (1/2)·(ħ/m_eff)·Re(∇²ψ)

Huh? Yes. These equations are easily derived from noting that two complex numbers a + i∙b and c + i∙d are equal if, and only if, their real and imaginary parts are the same. Now, the ∂ψ/∂t = i∙(ħ/m_eff)∙∇²ψ equation amounts to writing something like this: a + i∙b = i∙(c + i∙d). Now, remembering that i² = −1, you can easily figure out that i∙(c + i∙d) = i∙c + i²∙d = − d + i∙c. [Now that we’re getting a bit technical, let me note that the m_eff is the effective mass of the particle, which depends on the medium. For example, an electron traveling in a solid (a transistor, for example) will have a different effective mass than in an atom. In free space, we can drop the subscript and just write m_eff = m.] 🙂 OK. Onwards ! 🙂

The equations above make me think of the equations for an electromagnetic wave in free space (no stationary charges or currents):

∂B/∂t = –∇×E
∂E/∂t = c²∇×B

Now, these equations – and, I must therefore assume, the other equations above as well – effectively describe a propagation mechanism in spacetime, as illustrated below:

propagation

You know how it works for the electromagnetic field: it’s the interplay between circulation and flux. Indeed, circulation around some axis of rotation creates a flux in a direction perpendicular to it, and that flux causes this, and then that, and it all goes round and round and round. 🙂 Something like that. 🙂 I will let you look up how it goes, exactly. The principle is clear enough. Somehow, in this beautiful interplay between linear and circular motion, energy is borrowed from one place and then returns to the other, cycle after cycle.

Now, we know the wavefunction consist of a sine and a cosine: the cosine is the real component, and the sine is the imaginary component. Could they be equally real? Could each represent half of the total energy of our particle? I firmly believe they do. The obvious question then is the following: why wouldn’t we represent them as vectors, just like E and B? I mean… Representing them as vectors (I mean real vectors here – something with a magnitude and a direction in a real space – as opposed to state vectors from the Hilbert space) would show they are real, and there would be no need for cumbersome transformations when going from one representational base to another. In fact, that’s why vector notation was invented (sort of): we don’t need to worry about the coordinate frame. It’s much easier to write physical laws in vector notation because… Well… They’re the real thing, aren’t they? 🙂

What about dimensions? Well… I am not sure. However, because we are – arguably – talking about some pointlike charge moving around in those oscillating fields, I would suspect the dimension of the real and imaginary component of the wavefunction will be the same as that of the electric and magnetic field vectors E and B. We may want to recall these:

E is measured in newton per coulomb (N/C).
B is measured in newton per coulomb divided by m/s, so that’s (N/C)/(m/s).

The weird dimension of B is because of the weird force law for the magnetic force. It involves a vector cross product, as shown by Lorentz’ formula:

F = qE + q(v×B)

Of course, it is only one force (one and the same physical reality), as evidenced by the fact that we can write B as the following vector cross-product: B = (1/c)∙e_x×E, with e_x the unit vector pointing in the x-direction (i.e. the direction of propagation of the wave). [Check it, because you may not have seen this expression before. Just take a piece of paper and think about the geometry of the situation.] Hence, we may associate the (1/c)∙e_x× operator, which amounts to a rotation by 90 degrees, with the s/m dimension. Now, multiplication by i also amounts to a rotation by 90° degrees. Hence, if we can agree on a suitable convention for the direction of rotation here, we may boldly write:

B = (1/c)∙e_x×E = (1/c)∙i∙E

This is, in fact, what triggered my geometric interpretation of Schrödinger’s equation about a year ago now. I have had little time to work on it, but think I am on the right track. Of course, you should note that, for an electromagnetic wave, the magnitudes of E and B reach their maximum, minimum and zero point simultaneously (as shown below). So their phase is the same.

E and B

In contrast, the phase of the real and imaginary component of the wavefunction is not the same, as shown below.

In fact, because of the Stern-Gerlach experiment, I am actually more thinking of a motion like this:

Wavefunction 2 But that shouldn’t distract you. 🙂 The question here is the following: could we possibly think of a new formulation of Schrödinger’s equation – using vectors (again, real vectors – not these weird state vectors) rather than complex algebra?

I think we can, but then I wonder why the inventors of the wavefunction – Heisenberg, Born, Dirac, and Schrödinger himself, of course – never thought of that. 🙂

Hmm… I need to do some research here. 🙂

Post scriptum: You will, of course, wonder how and why the matter-wave would be different from the electromagnetic wave if my suggestion that the dimension of the wavefunction component is the same is correct. The answer is: the difference lies in the phase difference and then, most probably, the different orientation of the angular momentum. Do we have any other possibilities? 🙂

P.S. 2: I also published this post on my new blog: https://readingeinstein.blog/. However, I thought the followers of this blog should get it first. 🙂

Wavefunctions as gravitational waves

This is the paper I always wanted to write. It is there now, and I think it is good – and that‘s an understatement. 🙂 It is probably best to download it as a pdf-file from the viXra.org site because this was a rather fast ‘copy and paste’ job from the Word version of the paper, so there may be issues with boldface notation (vector notation), italics and, most importantly, with formulas – which I, sadly, have to ‘snip’ into this WordPress blog, as they don’t have an easy copy function for mathematical formulas.

It’s great stuff. If you have been following my blog – and many of you have – you will want to digest this. 🙂

Abstract : This paper explores the implications of associating the components of the wavefunction with a physical dimension: force per unit mass – which is, of course, the dimension of acceleration (m/s²) and gravitational fields. The classical electromagnetic field equations for energy densities, the Poynting vector and spin angular momentum are then re-derived by substituting the electromagnetic N/C unit of field strength (mass per unit charge) by the new N/kg = m/s² dimension.

The results are elegant and insightful. For example, the energy densities are proportional to the square of the absolute value of the wavefunction and, hence, to the probabilities, which establishes a physical normalization condition. Also, Schrödinger’s wave equation may then, effectively, be interpreted as a diffusion equation for energy, and the wavefunction itself can be interpreted as a propagating gravitational wave. Finally, as an added bonus, concepts such as the Compton scattering radius for a particle, spin angular momentum, and the boson-fermion dichotomy, can also be explained more intuitively.

While the approach offers a physical interpretation of the wavefunction, the author argues that the core of the Copenhagen interpretations revolves around the complementarity principle, which remains unchallenged because the interpretation of amplitude waves as traveling fields does not explain the particle nature of matter.

Introduction

This is not another introduction to quantum mechanics. We assume the reader is already familiar with the key principles and, importantly, with the basic math. We offer an interpretation of wave mechanics. As such, we do not challenge the complementarity principle: the physical interpretation of the wavefunction that is offered here explains the wave nature of matter only. It explains diffraction and interference of amplitudes but it does not explain why a particle will hit the detector not as a wave but as a particle. Hence, the Copenhagen interpretation of the wavefunction remains relevant: we just push its boundaries.

The basic ideas in this paper stem from a simple observation: the geometric similarity between the quantum-mechanical wavefunctions and electromagnetic waves is remarkably similar. The components of both waves are orthogonal to the direction of propagation and to each other. Only the relative phase differs : the electric and magnetic field vectors (E and B) have the same phase. In contrast, the phase of the real and imaginary part of the (elementary) wavefunction (ψ = a·e^−i∙θ = a∙cosθ – a∙sinθ) differ by 90 degrees (π/2).[1] Pursuing the analogy, we explore the following question: if the oscillating electric and magnetic field vectors of an electromagnetic wave carry the energy that one associates with the wave, can we analyze the real and imaginary part of the wavefunction in a similar way?

We show the answer is positive and remarkably straightforward. If the physical dimension of the electromagnetic field is expressed in newton per coulomb (force per unit charge), then the physical dimension of the components of the wavefunction may be associated with force per unit mass (newton per kg).[2] Of course, force over some distance is energy. The question then becomes: what is the energy concept here? Kinetic? Potential? Both?

The similarity between the energy of a (one-dimensional) linear oscillator (E = m·a²·ω²/2) and Einstein’s relativistic energy equation E = m∙c² inspires us to interpret the energy as a two-dimensional oscillation of mass. To assist the reader, we construct a two-piston engine metaphor.[3] We then adapt the formula for the electromagnetic energy density to calculate the energy densities for the wave function. The results are elegant and intuitive: the energy densities are proportional to the square of the absolute value of the wavefunction and, hence, to the probabilities. Schrödinger’s wave equation may then, effectively, be interpreted as a diffusion equation for energy itself.

As an added bonus, concepts such as the Compton scattering radius for a particle and spin angular, as well as the boson-fermion dichotomy can be explained in a fully intuitive way.[4]

Of course, such interpretation is also an interpretation of the wavefunction itself, and the immediate reaction of the reader is predictable: the electric and magnetic field vectors are, somehow, to be looked at as real vectors. In contrast, the real and imaginary components of the wavefunction are not. However, this objection needs to be phrased more carefully. First, it may be noted that, in a classical analysis, the magnetic force is a pseudovector itself.[5] Second, a suitable choice of coordinates may make quantum-mechanical rotation matrices irrelevant.[6]

Therefore, the author is of the opinion that this little paper may provide some fresh perspective on the question, thereby further exploring Einstein’s basic sentiment in regard to quantum mechanics, which may be summarized as follows: there must be some physical explanation for the calculated probabilities.[7]

We will, therefore, start with Einstein’s relativistic energy equation (E = mc²) and wonder what it could possibly tell us.

I. Energy as a two-dimensional oscillation of mass

The structural similarity between the relativistic energy formula, the formula for the total energy of an oscillator, and the kinetic energy of a moving body, is striking:

E = mc²
E = mω²/2
E = mv²/2

In these formulas, ω, v and c all describe some velocity.[8] Of course, there is the 1/2 factor in the E = mω²/2 formula[9], but that is exactly the point we are going to explore here: can we think of an oscillation in two dimensions, so it stores an amount of energy that is equal to E = 2·m·ω²/2 = m·ω²?

That is easy enough. Think, for example, of a V-2 engine with the pistons at a 90-degree angle, as illustrated below. The 90° angle makes it possible to perfectly balance the counterweight and the pistons, thereby ensuring smooth travel at all times. With permanently closed valves, the air inside the cylinder compresses and decompresses as the pistons move up and down and provides, therefore, a restoring force. As such, it will store potential energy, just like a spring, and the motion of the pistons will also reflect that of a mass on a spring. Hence, we can describe it by a sinusoidal function, with the zero point at the center of each cylinder. We can, therefore, think of the moving pistons as harmonic oscillators, just like mechanical springs.

Figure 1: Oscillations in two dimensions V-2 engine

If we assume there is no friction, we have a perpetuum mobile here. The compressed air and the rotating counterweight (which, combined with the crankshaft, acts as a flywheel[10]) store the potential energy. The moving masses of the pistons store the kinetic energy of the system.[11]

At this point, it is probably good to quickly review the relevant math. If the magnitude of the oscillation is equal to a, then the motion of the piston (or the mass on a spring) will be described by x = a·cos(ω·t + Δ).[12] Needless to say, Δ is just a phase factor which defines our t = 0 point, and ω is the natural angular frequency of our oscillator. Because of the 90° angle between the two cylinders, Δ would be 0 for one oscillator, and –π/2 for the other. Hence, the motion of one piston is given by x = a·cos(ω·t), while the motion of the other is given by x = a·cos(ω·t–π/2) = a·sin(ω·t).

The kinetic and potential energy of one oscillator (think of one piston or one spring only) can then be calculated as:

K.E. = T = m·v²/2 = (1/2)·m·ω²·a²·sin²(ω·t + Δ)
P.E. = U = k·x²/2 = (1/2)·k·a²·cos²(ω·t + Δ)

The coefficient k in the potential energy formula characterizes the restoring force: F = −k·x. From the dynamics involved, it is obvious that k must be equal to m·ω². Hence, the total energy is equal to:

E = T + U = (1/2)· m·ω²·a²·[sin²(ω·t + Δ) + cos²(ω·t + Δ)] = m·a²·ω²/2

To facilitate the calculations, we will briefly assume k = m·ω² and a are equal to 1. The motion of our first oscillator is given by the cos(ω·t) = cosθ function (θ = ω·t), and its kinetic energy will be equal to sin²θ. Hence, the (instantaneous) change in kinetic energy at any point in time will be equal to:

d(sin²θ)/dθ = 2∙sinθ∙d(sinθ)/dθ = 2∙sinθ∙cosθ

Let us look at the second oscillator now. Just think of the second piston going up and down in the V-2 engine. Its motion is given by the sinθ function, which is equal to cos(θ−π /2). Hence, its kinetic energy is equal to sin²(θ−π /2), and how it changes – as a function of θ – will be equal to:

2∙sin(θ−π /2)∙cos(θ−π /2) = = −2∙cosθ∙sinθ = −2∙sinθ∙cosθ

We have our perpetuum mobile! While transferring kinetic energy from one piston to the other, the crankshaft will rotate with a constant angular velocity: linear motion becomes circular motion, and vice versa, and the total energy that is stored in the system is T + U = ma²ω².

We have a great metaphor here. Somehow, in this beautiful interplay between linear and circular motion, energy is borrowed from one place and then returns to the other, cycle after cycle. We know the wavefunction consist of a sine and a cosine: the cosine is the real component, and the sine is the imaginary component. Could they be equally real? Could each represent half of the total energy of our particle? Should we think of the c in our E = mc² formula as an angular velocity?

These are sensible questions. Let us explore them.

II. The wavefunction as a two-dimensional oscillation

The elementary wavefunction is written as:

ψ = a·e^{−i[E·t − p∙x]/ħ} = a·e^{−i[E·t − p∙x]/ħ} = a·cos(p∙x/ħ – E∙t/ħ) + i·a·sin(p∙x/ħ – E∙t/ħ)

When considering a particle at rest (p = 0) this reduces to:

ψ = a·e^{−i∙E·t/ħ} = a·cos(–E∙t/ħ) + i·a·sin(–E∙t/ħ) = a·cos(E∙t/ħ) – i·a·sin(E∙t/ħ)

Let us remind ourselves of the geometry involved, which is illustrated below. Note that the argument of the wavefunction rotates clockwise with time, while the mathematical convention for measuring the phase angle (ϕ) is counter-clockwise.

Figure 2: Euler’s formula 760px-eulers_formula

If we assume the momentum p is all in the x-direction, then the p and x vectors will have the same direction, and p∙x/ħ reduces to p∙x/ħ. Most illustrations – such as the one below – will either freeze x or, else, t. Alternatively, one can google web animations varying both. The point is: we also have a two-dimensional oscillation here. These two dimensions are perpendicular to the direction of propagation of the wavefunction. For example, if the wavefunction propagates in the x-direction, then the oscillations are along the y– and z-axis, which we may refer to as the real and imaginary axis. Note how the phase difference between the cosine and the sine – the real and imaginary part of our wavefunction – appear to give some spin to the whole. I will come back to this.

Figure 3: Geometric representation of the wavefunction 5d_euler_f

Hence, if we would say these oscillations carry half of the total energy of the particle, then we may refer to the real and imaginary energy of the particle respectively, and the interplay between the real and the imaginary part of the wavefunction may then describe how energy propagates through space over time.

Let us consider, once again, a particle at rest. Hence, p = 0 and the (elementary) wavefunction reduces to ψ = a·e^{−i∙E·t/ħ}. Hence, the angular velocity of both oscillations, at some point x, is given by ω = -E/ħ. Now, the energy of our particle includes all of the energy – kinetic, potential and rest energy – and is, therefore, equal to E = mc².

Can we, somehow, relate this to the m·a²·ω² energy formula for our V-2 perpetuum mobile? Our wavefunction has an amplitude too. Now, if the oscillations of the real and imaginary wavefunction store the energy of our particle, then their amplitude will surely matter. In fact, the energy of an oscillation is, in general, proportional to the square of the amplitude: E µ a². We may, therefore, think that the a² factor in the E = m·a²·ω² energy will surely be relevant as well.

However, here is a complication: an actual particle is localized in space and can, therefore, not be represented by the elementary wavefunction. We must build a wave packet for that: a sum of wavefunctions, each with their own amplitude a_k, and their own ω_i = -E_i/ħ. Each of these wavefunctions will contribute some energy to the total energy of the wave packet. To calculate the contribution of each wave to the total, both a_i as well as E_i will matter.

What is E_i? E_i varies around some average E, which we can associate with some average mass m: m = E/c². The Uncertainty Principle kicks in here. The analysis becomes more complicated, but a formula such as the one below might make sense:We can re-write this as:What is the meaning of this equation? We may look at it as some sort of physical normalization condition when building up the Fourier sum. Of course, we should relate this to the mathematical normalization condition for the wavefunction. Our intuition tells us that the probabilities must be related to the energy densities, but how exactly? We will come back to this question in a moment. Let us first think some more about the enigma: what is mass?

Before we do so, let us quickly calculate the value of c²ħ²: it is about 1´10^–⁵¹ N²∙m⁴. Let us also do a dimensional analysis: the physical dimensions of the E = m·a²·ω² equation make sense if we express m in kg, a in m, and ω in rad/s. We then get: [E] = kg∙m²/s² = (N∙s²/m)∙m²/s² = N∙m = J. The dimensions of the left- and right-hand side of the physical normalization condition is N³∙m⁵.

III. What is mass?

We came up, playfully, with a meaningful interpretation for energy: it is a two-dimensional oscillation of mass. But what is mass? A new aether theory is, of course, not an option, but then what is it that is oscillating? To understand the physics behind equations, it is always good to do an analysis of the physical dimensions in the equation. Let us start with Einstein’s energy equation once again. If we want to look at mass, we should re-write it as m = E/c²:

[m] = [E/c²] = J/(m/s)² = N·m∙s²/m² = N·s²/m = kg

This is not very helpful. It only reminds us of Newton’s definition of a mass: mass is that what gets accelerated by a force. At this point, we may want to think of the physical significance of the absolute nature of the speed of light. Einstein’s E = mc² equation implies we can write the ratio between the energy and the mass of any particle is always the same, so we can write, for example:This reminds us of the ω²= C^–¹/L or ω² = k/m of harmonic oscillators once again.[13] The key difference is that the ω²= C^–¹/L and ω² = k/m formulas introduce two or more degrees of freedom.[14] In contrast, c²= E/m for any particle, always. However, that is exactly the point: we can modulate the resistance, inductance and capacitance of electric circuits, and the stiffness of springs and the masses we put on them, but we live in one physical space only: our spacetime. Hence, the speed of light c emerges here as the defining property of spacetime – the resonant frequency, so to speak. We have no further degrees of freedom here.

The Planck-Einstein relation (for photons) and the de Broglie equation (for matter-particles) have an interesting feature: both imply that the energy of the oscillation is proportional to the frequency, with Planck’s constant as the constant of proportionality. Now, for one-dimensional oscillations – think of a guitar string, for example – we know the energy will be proportional to the square of the frequency. It is a remarkable observation: the two-dimensional matter-wave, or the electromagnetic wave, gives us two waves for the price of one, so to speak, each carrying half of the total energy of the oscillation but, as a result, we get a proportionality between E and f instead of between E and f².

However, such reflections do not answer the fundamental question we started out with: what is mass? At this point, it is hard to go beyond the circular definition that is implied by Einstein’s formula: energy is a two-dimensional oscillation of mass, and mass packs energy, and c emerges us as the property of spacetime that defines how exactly.

When everything is said and done, this does not go beyond stating that mass is some scalar field. Now, a scalar field is, quite simply, some real number that we associate with a position in spacetime. The Higgs field is a scalar field but, of course, the theory behind it goes much beyond stating that we should think of mass as some scalar field. The fundamental question is: why and how does energy, or matter, condense into elementary particles? That is what the Higgs mechanism is about but, as this paper is exploratory only, we cannot even start explaining the basics of it.

What we can do, however, is look at the wave equation again (Schrödinger’s equation), as we can now analyze it as an energy diffusion equation.

IV. Schrödinger’s equation as an energy diffusion equation

The interpretation of Schrödinger’s equation as a diffusion equation is straightforward. Feynman (Lectures, III-16-1) briefly summarizes it as follows:

Let us review the basic math. For a particle moving in free space – with no external force fields acting on it – there is no potential (U = 0) and, therefore, the Uψ term disappears. Therefore, Schrödinger’s equation reduces to:

∂ψ(x, t)/∂t = i·(1/2)·(ħ/m_eff)·∇²ψ(x, t)

The ubiquitous diffusion equation in physics is:

∂φ(x, t)/∂t = D·∇²φ(x, t)

The structural similarity is obvious. The key difference between both equations is that the wave equation gives us two equations for the price of one. Indeed, because ψ is a complex-valued function, with a real and an imaginary part, we get the following equations[18]:

Re(∂ψ/∂t) = −(1/2)·(ħ/m_eff)·Im(∇²ψ)
Im(∂ψ/∂t) = (1/2)·(ħ/m_eff)·Re(∇²ψ)

These equations make us think of the equations for an electromagnetic wave in free space (no stationary charges or currents):

∂B/∂t = –∇×E
∂E/∂t = c²∇×B

The above equations effectively describe a propagation mechanism in spacetime, as illustrated below.

Figure 4: Propagation mechanisms

The Laplacian operator (∇²), when operating on a scalar quantity, gives us a flux density, i.e. something expressed per square meter (1/m²). In this case, it is operating on ψ(x, t), so what is the dimension of our wavefunction ψ(x, t)? To answer that question, we should analyze the diffusion constant in Schrödinger’s equation, i.e. the (1/2)·(ħ/m_eff) factor:

As a mathematical constant of proportionality, it will quantify the relationship between both derivatives (i.e. the time derivative and the Laplacian);
As a physical constant, it will ensure the physical dimensions on both sides of the equation are compatible.

Now, the ħ/m_eff factor is expressed in (N·m·s)/(N· s²/m) = m²/s. Hence, it does ensure the dimensions on both sides of the equation are, effectively, the same: ∂ψ/∂t is a time derivative and, therefore, its dimension is s^–¹ while, as mentioned above, the dimension of ∇²ψ is m^–². However, this does not solve our basic question: what is the dimension of the real and imaginary part of our wavefunction?

At this point, mainstream physicists will say: it does not have a physical dimension, and there is no geometric interpretation of Schrödinger’s equation. One may argue, effectively, that its argument, (p∙x – E∙t)/ħ, is just a number and, therefore, that the real and imaginary part of ψ is also just some number.

To this, we may object that ħ may be looked as a mathematical scaling constant only. If we do that, then the argument of ψ will, effectively, be expressed in action units, i.e. in N·m·s. It then does make sense to also associate a physical dimension with the real and imaginary part of ψ. What could it be?

We may have a closer look at Maxwell’s equations for inspiration here. The electric field vector is expressed in newton (the unit of force) per unit of charge (coulomb). Now, there is something interesting here. The physical dimension of the magnetic field is N/C divided by m/s.[19] We may write B as the following vector cross-product: B = (1/c)∙e_x×E, with e_x the unit vector pointing in the x-direction (i.e. the direction of propagation of the wave). Hence, we may associate the (1/c)∙e_x× operator, which amounts to a rotation by 90 degrees, with the s/m dimension. Now, multiplication by i also amounts to a rotation by 90° degrees. Hence, we may boldly write: B = (1/c)∙e_x×E = (1/c)∙i∙E. This allows us to also geometrically interpret Schrödinger’s equation in the way we interpreted it above (see Figure 3).[20]

Still, we have not answered the question as to what the physical dimension of the real and imaginary part of our wavefunction should be. At this point, we may be inspired by the structural similarity between Newton’s and Coulomb’s force laws:Hence, if the electric field vector E is expressed in force per unit charge (N/C), then we may want to think of associating the real part of our wavefunction with a force per unit mass (N/kg). We can, of course, do a substitution here, because the mass unit (1 kg) is equivalent to 1 N·s²/m. Hence, our N/kg dimension becomes:

N/kg = N/(N·s²/m)= m/s²

What is this: m/s²? Is that the dimension of the a·cosθ term in the a·e^−iθ= a·cosθ − i·a·sinθ wavefunction?

My answer is: why not? Think of it: m/s² is the physical dimension of acceleration: the increase or decrease in velocity (m/s) per second. It ensures the wavefunction for any particle – matter-particles or particles with zero rest mass (photons) – and the associated wave equation (which has to be the same for all, as the spacetime we live in is one) are mutually consistent.

In this regard, we should think of how we would model a gravitational wave. The physical dimension would surely be the same: force per mass unit. It all makes sense: wavefunctions may, perhaps, be interpreted as traveling distortions of spacetime, i.e. as tiny gravitational waves.

V. Energy densities and flows

Pursuing the geometric equivalence between the equations for an electromagnetic wave and Schrödinger’s equation, we can now, perhaps, see if there is an equivalent for the energy density. For an electromagnetic wave, we know that the energy density is given by the following formula:E and B are the electric and magnetic field vector respectively. The Poynting vector will give us the directional energy flux, i.e. the energy flow per unit area per unit time. We write:Needless to say, the ∇∙ operator is the divergence and, therefore, gives us the magnitude of a (vector) field’s source or sink at a given point. To be precise, the divergence gives us the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. In this case, it gives us the volume density of the flux of S.

We can analyze the dimensions of the equation for the energy density as follows:

E is measured in newton per coulomb, so [E∙E] = [E²] = N²/C².
B is measured in (N/C)/(m/s), so we get [B∙B] = [B²] = (N²/C²)·(s²/m²). However, the dimension of our c² factor is (m²/s²) and so we’re also left with N²/C².
The ϵ₀ is the electric constant, aka as the vacuum permittivity. As a physical constant, it should ensure the dimensions on both sides of the equation work out, and they do: [ε₀] = C²/(N·m²) and, therefore, if we multiply that with N²/C², we find that u is expressed in J/m³.[21]

Replacing the newton per coulomb unit (N/C) by the newton per kg unit (N/kg) in the formulas above should give us the equivalent of the energy density for the wavefunction. We just need to substitute ϵ₀ for an equivalent constant. We may to give it a try. If the energy densities can be calculated – which are also mass densities, obviously – then the probabilities should be proportional to them.

Let us first see what we get for a photon, assuming the electromagnetic wave represents its wavefunction. Substituting B for (1/c)∙i∙E or for −(1/c)∙i∙E gives us the following result:Zero!? An unexpected result! Or not? We have no stationary charges and no currents: only an electromagnetic wave in free space. Hence, the local energy conservation principle needs to be respected at all points in space and in time. The geometry makes sense of the result: for an electromagnetic wave, the magnitudes of E and B reach their maximum, minimum and zero point simultaneously, as shown below.[22] This is because their phase is the same.

Figure 5: Electromagnetic wave: E and B EM field

Should we expect a similar result for the energy densities that we would associate with the real and imaginary part of the matter-wave? For the matter-wave, we have a phase difference between a·cosθ and a·sinθ, which gives a different picture of the propagation of the wave (see Figure 3).[23] In fact, the geometry of the suggestion suggests some inherent spin, which is interesting. I will come back to this. Let us first guess those densities. Making abstraction of any scaling constants, we may write:We get what we hoped to get: the absolute square of our amplitude is, effectively, an energy density !

|ψ|² = |a·e^{−i∙E·t/ħ}|²= a²= u

This is very deep. A photon has no rest mass, so it borrows and returns energy from empty space as it travels through it. In contrast, a matter-wave carries energy and, therefore, has some (rest) mass. It is therefore associated with an energy density, and this energy density gives us the probabilities. Of course, we need to fine-tune the analysis to account for the fact that we have a wave packet rather than a single wave, but that should be feasible.

As mentioned, the phase difference between the real and imaginary part of our wavefunction (a cosine and a sine function) appear to give some spin to our particle. We do not have this particularity for a photon. Of course, photons are bosons, i.e. spin-zero particles, while elementary matter-particles are fermions with spin-1/2. Hence, our geometric interpretation of the wavefunction suggests that, after all, there may be some more intuitive explanation of the fundamental dichotomy between bosons and fermions, which puzzled even Feynman:

“Why is it that particles with half-integral spin are Fermi particles, whereas particles with integral spin are Bose particles? We apologize for the fact that we cannot give you an elementary explanation. An explanation has been worked out by Pauli from complicated arguments of quantum field theory and relativity. He has shown that the two must necessarily go together, but we have not been able to find a way of reproducing his arguments on an elementary level. It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanics. This probably means that we do not have a complete understanding of the fundamental principle involved.” (Feynman, Lectures, III-4-1)

The physical interpretation of the wavefunction, as presented here, may provide some better understanding of ‘the fundamental principle involved’: the physical dimension of the oscillation is just very different. That is all: it is force per unit charge for photons, and force per unit mass for matter-particles. We will examine the question of spin somewhat more carefully in section VII. Let us first examine the matter-wave some more.

VI. Group and phase velocity of the matter-wave

The geometric representation of the matter-wave (see Figure 3) suggests a traveling wave and, yes, of course: the matter-wave effectively travels through space and time. But what is traveling, exactly? It is the pulse – or the signal – only: the phase velocity of the wave is just a mathematical concept and, even in our physical interpretation of the wavefunction, the same is true for the group velocity of our wave packet. The oscillation is two-dimensional, but perpendicular to the direction of travel of the wave. Hence, nothing actually moves with our particle.

Here, we should also reiterate that we did not answer the question as to what is oscillating up and down and/or sideways: we only associated a physical dimension with the components of the wavefunction – newton per kg (force per unit mass), to be precise. We were inspired to do so because of the physical dimension of the electric and magnetic field vectors (newton per coulomb, i.e. force per unit charge) we associate with electromagnetic waves which, for all practical purposes, we currently treat as the wavefunction for a photon. This made it possible to calculate the associated energy densities and a Poynting vector for energy dissipation. In addition, we showed that Schrödinger’s equation itself then becomes a diffusion equation for energy. However, let us now focus some more on the asymmetry which is introduced by the phase difference between the real and the imaginary part of the wavefunction. Look at the mathematical shape of the elementary wavefunction once again:

ψ = a·e^{−i[E·t − p∙x]/ħ} = a·e^{−i[E·t − p∙x]/ħ} = a·cos(p∙x/ħ − E∙t/ħ) + i·a·sin(p∙x/ħ − E∙t/ħ)

The minus sign in the argument of our sine and cosine function defines the direction of travel: an F(x−v∙t) wavefunction will always describe some wave that is traveling in the positive x-direction (with c the wave velocity), while an F(x+v∙t) wavefunction will travel in the negative x-direction. For a geometric interpretation of the wavefunction in three dimensions, we need to agree on how to define i or, what amounts to the same, a convention on how to define clockwise and counterclockwise directions: if we look at a clock from the back, then its hand will be moving counterclockwise. So we need to establish the equivalent of the right-hand rule. However, let us not worry about that now. Let us focus on the interpretation. To ease the analysis, we’ll assume we’re looking at a particle at rest. Hence, p = 0, and the wavefunction reduces to:

ψ = a·e^{−i∙E·t/ħ} = a·cos(−E∙t/ħ) + i·a·sin(−E₀∙t/ħ) = a·cos(E₀∙t/ħ) − i·a·sin(E₀∙t/ħ)

E₀ is, of course, the rest mass of our particle and, now that we are here, we should probably wonder whose time t we are talking about: is it our time, or is the proper time of our particle? Well… In this situation, we are both at rest so it does not matter: t is, effectively, the proper time so perhaps we should write it as t₀. It does not matter. You can see what we expect to see: E₀/ħ pops up as the natural frequency of our matter-particle: (E₀/ħ)∙t = ω∙t. Remembering the ω = 2π·f = 2π/T and T = 1/f formulas, we can associate a period and a frequency with this wave, using the ω = 2π·f = 2π/T. Noting that ħ = h/2π, we find the following:

T = 2π·(ħ/E₀) = h/E₀ ⇔ f = E₀/h = m₀c²/h

This is interesting, because we can look at the period as a natural unit of time for our particle. What about the wavelength? That is tricky because we need to distinguish between group and phase velocity here. The group velocity (v_g) should be zero here, because we assume our particle does not move. In contrast, the phase velocity is given by v_p = λ·f = (2π/k)·(ω/2π) = ω/k. In fact, we’ve got something funny here: the wavenumber k = p/ħ is zero, because we assume the particle is at rest, so p = 0. So we have a division by zero here, which is rather strange. What do we get assuming the particle is not at rest? We write:

v_p = ω/k = (E/ħ)/(p/ħ) = E/p = E/(m·v_g) = (m·c²)/(m·v_g) = c²/v_g

This is interesting: it establishes a reciprocal relation between the phase and the group velocity, with c as a simple scaling constant. Indeed, the graph below shows the shape of the function does not change with the value of c, and we may also re-write the relation above as:

v_p/c = β_p = c/v_p = 1/β_g = 1/(c/v_p)

Figure 6: Reciprocal relation between phase and group velocity graph

We can also write the mentioned relationship as v_p·v_g = c², which reminds us of the relationship between the electric and magnetic constant (1/ε₀)·(1/μ₀) = c². This is interesting in light of the fact we can re-write this as (c·ε₀)·(c·μ₀) = 1, which shows electricity and magnetism are just two sides of the same coin, so to speak.[24]

Interesting, but how do we interpret the math? What about the implications of the zero value for wavenumber k = p/ħ. We would probably like to think it implies the elementary wavefunction should always be associated with some momentum, because the concept of zero momentum clearly leads to weird math: something times zero cannot be equal to c²! Such interpretation is also consistent with the Uncertainty Principle: if Δx·Δp ≥ ħ, then neither Δx nor Δp can be zero. In other words, the Uncertainty Principle tells us that the idea of a pointlike particle actually being at some specific point in time and in space does not make sense: it has to move. It tells us that our concept of dimensionless points in time and space are mathematical notions only. Actual particles – including photons – are always a bit spread out, so to speak, and – importantly – they have to move.

For a photon, this is self-evident. It has no rest mass, no rest energy, and, therefore, it is going to move at the speed of light itself. We write: p = m·c = m·c²/c = E/c. Using the relationship above, we get:

v_p = ω/k = (E/ħ)/(p/ħ) = E/p = c ⇒ v_g = c²/v_p = c²/c = c

This is good: we started out with some reflections on the matter-wave, but here we get an interpretation of the electromagnetic wave as a wavefunction for the photon. But let us get back to our matter-wave. In regard to our interpretation of a particle having to move, we should remind ourselves, once again, of the fact that an actual particle is always localized in space and that it can, therefore, not be represented by the elementary wavefunction ψ = a·e^{−i[E·t − p∙x]/ħ} or, for a particle at rest, the ψ = a·e^{−i∙E·t/ħ} function. We must build a wave packet for that: a sum of wavefunctions, each with their own amplitude a_i, and their own ω_i = −E_i/ħ. Indeed, in section II, we showed that each of these wavefunctions will contribute some energy to the total energy of the wave packet and that, to calculate the contribution of each wave to the total, both a_i as well as E_i matter. This may or may not resolve the apparent paradox. Let us look at the group velocity.

To calculate a meaningful group velocity, we must assume the v_g = ∂ω_i/∂k_i = ∂(E_i/ħ)/∂(p_i/ħ) = ∂(E_i)/∂(p_i) exists. So we must have some dispersion relation. How do we calculate it? We need to calculate ω_i as a function of k_ihere, or E_i as a function of p_i. How do we do that? Well… There are a few ways to go about it but one interesting way of doing it is to re-write Schrödinger’s equation as we did, i.e. by distinguishing the real and imaginary parts of the ∂ψ/∂t =i·[ħ/(2m)]·∇²ψ wave equation and, hence, re-write it as the following pair of two equations:

Re(∂ψ/∂t) = −[ħ/(2m_eff)]·Im(∇²ψ) ⇔ ω·cos(kx − ωt) = k²·[ħ/(2m_eff)]·cos(kx − ωt)
Im(∂ψ/∂t) = [ħ/(2m_eff)]·Re(∇²ψ) ⇔ ω·sin(kx − ωt) = k²·[ħ/(2m_eff)]·sin(kx − ωt)

Both equations imply the following dispersion relation:

ω = ħ·k²/(2m_eff)

Of course, we need to think about the subscripts now: we have ω_i, k_i, but… What about m_eff or, dropping the subscript, m? Do we write it as m_i? If so, what is it? Well… It is the equivalent mass of E_i obviously, and so we get it from the mass-energy equivalence relation: m_i = E_i/c². It is a fine point, but one most people forget about: they usually just write m. However, if there is uncertainty in the energy, then Einstein’s mass-energy relation tells us we must have some uncertainty in the (equivalent) mass too. Here, I should refer back to Section II: E_i varies around some average energy E and, therefore, the Uncertainty Principle kicks in.

VII. Explaining spin

The elementary wavefunction vector – i.e. the vector sum of the real and imaginary component – rotates around the x-axis, which gives us the direction of propagation of the wave (see Figure 3). Its magnitude remains constant. In contrast, the magnitude of the electromagnetic vector – defined as the vector sum of the electric and magnetic field vectors – oscillates between zero and some maximum (see Figure 5).

We already mentioned that the rotation of the wavefunction vector appears to give some spin to the particle. Of course, a circularly polarized wave would also appear to have spin (think of the E and B vectors rotating around the direction of propagation – as opposed to oscillating up and down or sideways only). In fact, a circularly polarized light does carry angular momentum, as the equivalent mass of its energy may be thought of as rotating as well. But so here we are looking at a matter-wave.

The basic idea is the following: if we look at ψ = a·e^{−i∙E·t/ħ} as some real vector – as a two-dimensional oscillation of mass, to be precise – then we may associate its rotation around the direction of propagation with some torque. The illustration below reminds of the math here.

Figure 7: Torque and angular momentum vectors Torque_animation

A torque on some mass about a fixed axis gives it angular momentum, which we can write as the vector cross-product L = r×p or, perhaps easier for our purposes here as the product of an angular velocity (ω) and rotational inertia (I), aka as the moment of inertia or the angular mass. We write:

L = I·ω

Note we can write L and ω in boldface here because they are (axial) vectors. If we consider their magnitudes only, we write L = I·ω (no boldface). We can now do some calculations. Let us start with the angular velocity. In our previous posts, we showed that the period of the matter-wave is equal to T = 2π·(ħ/E₀). Hence, the angular velocity must be equal to:

ω = 2π/[2π·(ħ/E₀)] = E₀/ħ

We also know the distance r, so that is the magnitude of r in the L = r×p vector cross-product: it is just a, so that is the magnitude of ψ = a·e^{−i∙E·t/ħ}. Now, the momentum (p) is the product of a linear velocity (v) – in this case, the tangential velocity – and some mass (m): p = m·v. If we switch to scalar instead of vector quantities, then the (tangential) velocity is given by v = r·ω. So now we only need to think about what we should use for m or, if we want to work with the angular velocity (ω), the angular mass (I). Here we need to make some assumption about the mass (or energy) distribution. Now, it may or may not sense to assume the energy in the oscillation – and, therefore, the mass – is distributed uniformly. In that case, we may use the formula for the angular mass of a solid cylinder: I = m·r²/2. If we keep the analysis non-relativistic, then m = m₀. Of course, the energy-mass equivalence tells us that m₀ = E₀/c². Hence, this is what we get:

L = I·ω = (m₀·r²/2)·(E₀/ħ) = (1/2)·a²·(E₀/c²)·(E₀/ħ) = a²·E₀²/(2·ħ·c²)

Does it make sense? Maybe. Maybe not. Let us do a dimensional analysis: that won’t check our logic, but it makes sure we made no mistakes when mapping mathematical and physical spaces. We have m²·J² = m²·N²·m² in the numerator and N·m·s·m²/s² in the denominator. Hence, the dimensions work out: we get N·m·s as the dimension for L, which is, effectively, the physical dimension of angular momentum. It is also the action dimension, of course, and that cannot be a coincidence. Also note that the E = mc² equation allows us to re-write it as:

L = a²·E₀²/(2·ħ·c²)

Of course, in quantum mechanics, we associate spin with the magnetic moment of a charged particle, not with its mass as such. Is there way to link the formula above to the one we have for the quantum-mechanical angular momentum, which is also measured in N·m·s units, and which can only take on one of two possible values: J = +ħ/2 and −ħ/2? It looks like a long shot, right? How do we go from (1/2)·a²·m₀²/ħ to ± (1/2)∙ħ? Let us do a numerical example. The energy of an electron is typically 0.510 MeV » 8.1871×10⁻¹⁴ N∙m, and a… What value should we take for a?

We have an obvious trio of candidates here: the Bohr radius, the classical electron radius (aka the Thompon scattering length), and the Compton scattering radius.

Let us start with the Bohr radius, so that is about 0.×10⁻¹⁰ N∙m. We get L = a²·E₀²/(2·ħ·c²) = 9.9×10⁻³¹ N∙m∙s. Now that is about 1.88×10⁴ times ħ/2. That is a huge factor. The Bohr radius cannot be right: we are not looking at an electron in an orbital here. To show it does not make sense, we may want to double-check the analysis by doing the calculation in another way. We said each oscillation will always pack 6.626070040(81)×10⁻³⁴ joule in energy. So our electron should pack about 1.24×10⁻²⁰ oscillations. The angular momentum (L) we get when using the Bohr radius for a and the value of 6.626×10⁻³⁴ joule for E₀ and the Bohr radius is equal to 6.49×10⁻⁵⁹ N∙m∙s. So that is the angular momentum per oscillation. When we multiply this with the number of oscillations (1.24×10⁻²⁰), we get about 8.01×10⁻⁵¹ N∙m∙s, so that is a totally different number.

The classical electron radius is about 2.818×10⁻¹⁵ m. We get an L that is equal to about 2.81×10⁻³⁹ N∙m∙s, so now it is a tiny fraction of ħ/2! Hence, this leads us nowhere. Let us go for our last chance to get a meaningful result! Let us use the Compton scattering length, so that is about 2.42631×10⁻¹² m.

This gives us an L of 2.08×10⁻³³ N∙m∙s, which is only 20 times ħ. This is not so bad, but it is good enough? Let us calculate it the other way around: what value should we take for a so as to ensure L = a²·E₀²/(2·ħ·c²) = ħ/2? Let us write it out:

In fact, this is the formula for the so-called reduced Compton wavelength. This is perfect. We found what we wanted to find. Substituting this value for a (you can calculate it: it is about 3.8616×10⁻³³ m), we get what we should find: F10

This is a rather spectacular result, and one that would – a priori – support the interpretation of the wavefunction that is being suggested in this paper.

VIII. The boson-fermion dichotomy

Let us do some more thinking on the boson-fermion dichotomy. Again, we should remind ourselves that an actual particle is localized in space and that it can, therefore, not be represented by the elementary wavefunction ψ = a·e^{−i[E·t − p∙x]/ħ} or, for a particle at rest, the ψ = a·e^{−i∙E·t/ħ} function. We must build a wave packet for that: a sum of wavefunctions, each with their own amplitude a_i, and their own ω_i = −E_i/ħ. Each of these wavefunctions will contribute some energy to the total energy of the wave packet. Now, we can have another wild but logical theory about this.

Think of the apparent right-handedness of the elementary wavefunction: surely, Nature can’t be bothered about our convention of measuring phase angles clockwise or counterclockwise. Also, the angular momentum can be positive or negative: J = +ħ/2 or −ħ/2. Hence, we would probably like to think that an actual particle – think of an electron, or whatever other particle you’d think of – may consist of right-handed as well as left-handed elementary waves. To be precise, we may think they either consist of (elementary) right-handed waves or, else, of (elementary) left-handed waves. An elementary right-handed wave would be written as:

ψ(θ_i) = a_i·(cosθ_i + i·sinθ_i)

In contrast, an elementary left-handed wave would be written as:

ψ(θ_i) = a_i·(cosθ_i − i·sinθ_i)

How does that work out with the E₀·t argument of our wavefunction? Position is position, and direction is direction, but time? Time has only one direction, but Nature surely does not care how we count time: counting like 1, 2, 3, etcetera or like −1, −2, −3, etcetera is just the same. If we count like 1, 2, 3, etcetera, then we write our wavefunction like:

ψ = a·cos(E₀∙t/ħ) − i·a·sin(E₀∙t/ħ)

If we count time like −1, −2, −3, etcetera then we write it as:

ψ = a·cos(−E₀∙t/ħ) − i·a·sin(−E₀∙t/ħ)= a·cos(E₀∙t/ħ) + i·a·sin(E₀∙t/ħ)

Hence, it is just like the left- or right-handed circular polarization of an electromagnetic wave: we can have both for the matter-wave too! This, then, should explain why we can have either positive or negative quantum-mechanical spin (+ħ/2 or −ħ/2). It is the usual thing: we have two mathematical possibilities here, and so we must have two physical situations that correspond to it.

It is only natural. If we have left- and right-handed photons – or, generalizing, left- and right-handed bosons – then we should also have left- and right-handed fermions (electrons, protons, etcetera). Back to the dichotomy. The textbook analysis of the dichotomy between bosons and fermions may be epitomized by Richard Feynman’s Lecture on it (Feynman, III-4), which is confusing and – I would dare to say – even inconsistent: how are photons or electrons supposed to know that they need to interfere with a positive or a negative sign? They are not supposed to know anything: knowledge is part of our interpretation of whatever it is that is going on there.

Hence, it is probably best to keep it simple, and think of the dichotomy in terms of the different physical dimensions of the oscillation: newton per kg versus newton per coulomb. And then, of course, we should also note that matter-particles have a rest mass and, therefore, actually carry charge. Photons do not. But both are two-dimensional oscillations, and the point is: the so-called vacuum – and the rest mass of our particle (which is zero for the photon and non-zero for everything else) – give us the natural frequency for both oscillations, which is beautifully summed up in that remarkable equation for the group and phase velocity of the wavefunction, which applies to photons as well as matter-particles:

(v_phase·c)·(v_group·c) = 1 ⇔ v_p·v_g = c²

The final question then is: why are photons spin-zero particles? Well… We should first remind ourselves of the fact that they do have spin when circularly polarized.[25] Here we may think of the rotation of the equivalent mass of their energy. However, if they are linearly polarized, then there is no spin. Even for circularly polarized waves, the spin angular momentum of photons is a weird concept. If photons have no (rest) mass, then they cannot carry any charge. They should, therefore, not have any magnetic moment. Indeed, what I wrote above shows an explanation of quantum-mechanical spin requires both mass as well as charge.[26]

IX. Concluding remarks

There are, of course, other ways to look at the matter – literally. For example, we can imagine two-dimensional oscillations as circular rather than linear oscillations. Think of a tiny ball, whose center of mass stays where it is, as depicted below. Any rotation – around any axis – will be some combination of a rotation around the two other axes. Hence, we may want to think of a two-dimensional oscillation as an oscillation of a polar and azimuthal angle.

Figure 8: Two-dimensional circular movement oscillation-of-a-ball

The point of this paper is not to make any definite statements. That would be foolish. Its objective is just to challenge the simplistic mainstream viewpoint on the reality of the wavefunction. Stating that it is a mathematical construct only without physical significance amounts to saying it has no meaning at all. That is, clearly, a non-sustainable proposition.

The interpretation that is offered here looks at amplitude waves as traveling fields. Their physical dimension may be expressed in force per mass unit, as opposed to electromagnetic waves, whose amplitudes are expressed in force per (electric) charge unit. Also, the amplitudes of matter-waves incorporate a phase factor, but this may actually explain the rather enigmatic dichotomy between fermions and bosons and is, therefore, an added bonus.

The interpretation that is offered here has some advantages over other explanations, as it explains the how of diffraction and interference. However, while it offers a great explanation of the wave nature of matter, it does not explain its particle nature: while we think of the energy as being spread out, we will still observe electrons and photons as pointlike particles once they hit the detector. Why is it that a detector can sort of ‘hook’ the whole blob of energy, so to speak?

The interpretation of the wavefunction that is offered here does not explain this. Hence, the complementarity principle of the Copenhagen interpretation of the wavefunction surely remains relevant.

Appendix 1: The de Broglie relations and energy

The 1/2 factor in Schrödinger’s equation is related to the concept of the effective mass (m_eff). It is easy to make the wrong calculations. For example, when playing with the famous de Broglie relations – aka as the matter-wave equations – one may be tempted to derive the following energy concept:

E = h·f and p = h/λ. Therefore, f = E/h and λ = p/h.
v = f·λ = (E/h)∙(p/h) = E/p
p = m·v. Therefore, E = v·p = m·v²

E = m·v²? This resembles the E = mc² equation and, therefore, one may be enthused by the discovery, especially because the m·v² also pops up when working with the Least Action Principle in classical mechanics, which states that the path that is followed by a particle will minimize the following integral: F11 Now, we can choose any reference point for the potential energy but, to reflect the energy conservation law, we can select a reference point that ensures the sum of the kinetic and the potential energy is zero throughout the time interval. If the force field is uniform, then the integrand will, effectively, be equal to KE − PE = m·v².[27]

However, that is classical mechanics and, therefore, not so relevant in the context of the de Broglie equations, and the apparent paradox should be solved by distinguishing between the group and the phase velocity of the matter wave.

Appendix 2: The concept of the effective mass

The effective mass – as used in Schrödinger’s equation – is a rather enigmatic concept. To make sure we are making the right analysis here, I should start by noting you will usually see Schrödinger’s equation written as: F12 This formulation includes a term with the potential energy (U). In free space (no potential), this term disappears, and the equation can be re-written as:

∂ψ(x, t)/∂t = i·(1/2)·(ħ/m_eff)·∇²ψ(x, t)

We just moved the i·ħ coefficient to the other side, noting that 1/i = –i. Now, in one-dimensional space, and assuming ψ is just the elementary wavefunction (so we substitute a·e^{−i∙[E·t − p∙x]/ħ} for ψ), this implies the following:

−a·i·(E/ħ)·e⁻i∙^{[E·t − p∙x]/ħ} = −i·(ħ/2m_eff)·a·(p²/ħ²)· e^{−i∙[E·t − p∙x]/ħ}

⇔ E = p²/(2m_eff) ⇔ m_eff = m∙(v/c)²/2 = m∙β²/2

It is an ugly formula: it resembles the kinetic energy formula (K.E. = m∙v²/2) but it is, in fact, something completely different. The β²/2 factor ensures the effective mass is always a fraction of the mass itself. To get rid of the ugly 1/2 factor, we may re-define m_eff as two times the old m_eff (hence, m_eff^NEW = 2∙m_eff^OLD), as a result of which the formula will look somewhat better:

m_eff = m∙(v/c)² = m∙β²

We know β varies between 0 and 1 and, therefore, m_eff will vary between 0 and m. Feynman drops the subscript, and just writes m_eff as m in his textbook (see Feynman, III-19). On the other hand, the electron mass as used is also the electron mass that is used to calculate the size of an atom (see Feynman, III-2-4). As such, the two mass concepts are, effectively, mutually compatible. It is confusing because the same mass is often defined as the mass of a stationary electron (see, for example, the article on it in the online Wikipedia encyclopedia[28]).

In the context of the derivation of the electron orbitals, we do have the potential energy term – which is the equivalent of a source term in a diffusion equation – and that may explain why the above-mentioned m_eff = m∙(v/c)² = m∙β² formula does not apply.

References

This paper discusses general principles in physics only. Hence, references can be limited to references to physics textbooks only. For ease of reading, any reference to additional material has been limited to a more popular undergrad textbook that can be consulted online: Feynman’s Lectures on Physics (http://www.feynmanlectures.caltech.edu). References are per volume, per chapter and per section. For example, Feynman III-19-3 refers to Volume III, Chapter 19, Section 3.

Notes

[1] Of course, an actual particle is localized in space and can, therefore, not be represented by the elementary wavefunction ψ = a·e^−i∙θ = a·e^{−i[E·t − p∙x]/ħ} = a·(cosθ – i·a·sinθ). We must build a wave packet for that: a sum of wavefunctions, each with its own amplitude a_k and its own argument θ_k = (E_k∙t – p_k∙x)/ħ. This is dealt with in this paper as part of the discussion on the mathematical and physical interpretation of the normalization condition.

[2] The N/kg dimension immediately, and naturally, reduces to the dimension of acceleration (m/s²), thereby facilitating a direct interpretation in terms of Newton’s force law.

[3] In physics, a two-spring metaphor is more common. Hence, the pistons in the author’s perpetuum mobile may be replaced by springs.

[4] The author re-derives the equation for the Compton scattering radius in section VII of the paper.

[5] The magnetic force can be analyzed as a relativistic effect (see Feynman II-13-6). The dichotomy between the electric force as a polar vector and the magnetic force as an axial vector disappears in the relativistic four-vector representation of electromagnetism.

[6] For example, when using Schrödinger’s equation in a central field (think of the electron around a proton), the use of polar coordinates is recommended, as it ensures the symmetry of the Hamiltonian under all rotations (see Feynman III-19-3)

[7] This sentiment is usually summed up in the apocryphal quote: “God does not play dice.”The actual quote comes out of one of Einstein’s private letters to Cornelius Lanczos, another scientist who had also emigrated to the US. The full quote is as follows: “You are the only person I know who has the same attitude towards physics as I have: belief in the comprehension of reality through something basically simple and unified… It seems hard to sneak a look at God’s cards. But that He plays dice and uses ‘telepathic’ methods… is something that I cannot believe for a single moment.” (Helen Dukas and Banesh Hoffman, Albert Einstein, the Human Side: New Glimpses from His Archives, 1979)

[8] Of course, both are different velocities: ω is an angular velocity, while v is a linear velocity: ω is measured in radians per second, while v is measured in meter per second. However, the definition of a radian implies radians are measured in distance units. Hence, the physical dimensions are, effectively, the same. As for the formula for the total energy of an oscillator, we should actually write: E = m·a²∙ω²/2. The additional factor (a) is the (maximum) amplitude of the oscillator.

[9] We also have a 1/2 factor in the E = mv²/2 formula. Two remarks may be made here. First, it may be noted this is a non-relativistic formula and, more importantly, incorporates kinetic energy only. Using the Lorentz factor (γ), we can write the relativistically correct formula for the kinetic energy as K.E. = E − E₀ = m_vc² − m₀c² = m₀γc² − m₀c² = m₀c²(γ − 1). As for the exclusion of the potential energy, we may note that we may choose our reference point for the potential energy such that the kinetic and potential energy mirror each other. The energy concept that then emerges is the one that is used in the context of the Principle of Least Action: it equals E = mv². Appendix 1 provides some notes on that.

[10] Instead of two cylinders with pistons, one may also think of connecting two springs with a crankshaft.

[11] It is interesting to note that we may look at the energy in the rotating flywheel as potential energy because it is energy that is associated with motion, albeit circular motion. In physics, one may associate a rotating object with kinetic energy using the rotational equivalent of mass and linear velocity, i.e. rotational inertia (I) and angular velocity ω. The kinetic energy of a rotating object is then given by K.E. = (1/2)·I·ω².

[12] Because of the sideways motion of the connecting rods, the sinusoidal function will describe the linear motion only approximately, but you can easily imagine the idealized limit situation.

[13] The ω²= 1/LC formula gives us the natural or resonant frequency for a electric circuit consisting of a resistor (R), an inductor (L), and a capacitor (C). Writing the formula as ω²= C^–¹/L introduces the concept of elastance, which is the equivalent of the mechanical stiffness (k) of a spring.

[14] The resistance in an electric circuit introduces a damping factor. When analyzing a mechanical spring, one may also want to introduce a drag coefficient. Both are usually defined as a fraction of the inertia, which is the mass for a spring and the inductance for an electric circuit. Hence, we would write the resistance for a spring as γm and as R = γL respectively.

[15] Photons are emitted by atomic oscillators: atoms going from one state (energy level) to another. Feynman (Lectures, I-33-3) shows us how to calculate the Q of these atomic oscillators: it is of the order of 10⁸, which means the wave train will last about 10^–8seconds (to be precise, that is the time it takes for the radiation to die out by a factor 1/e). For example, for sodium light, the radiation will last about 3.2×10^–8seconds (this is the so-called decay time τ). Now, because the frequency of sodium light is some 500 THz (500×10¹²oscillations per second), this makes for some 16 million oscillations. There is an interesting paradox here: the speed of light tells us that such wave train will have a length of about 9.6 m! How is that to be reconciled with the pointlike nature of a photon? The paradox can only be explained by relativistic length contraction: in an analysis like this, one need to distinguish the reference frame of the photon – riding along the wave as it is being emitted, so to speak – and our stationary reference frame, which is that of the emitting atom.

[16] This is a general result and is reflected in the K.E. = T = (1/2)·m·ω²·a²·sin²(ω·t + Δ) and the P.E. = U = k·x²/2 = (1/2)· m·ω²·a²·cos²(ω·t + Δ) formulas for the linear oscillator.

[17] Feynman further formalizes this in his Lecture on Superconductivity (Feynman, III-21-2), in which he refers to Schrödinger’s equation as the “equation for continuity of probabilities”. The analysis is centered on the local conservation of energy, which confirms the interpretation of Schrödinger’s equation as an energy diffusion equation.

[18] The m_eff is the effective mass of the particle, which depends on the medium. For example, an electron traveling in a solid (a transistor, for example) will have a different effective mass than in an atom. In free space, we can drop the subscript and just write m_eff = m. Appendix 2 provides some additional notes on the concept. As for the equations, they are easily derived from noting that two complex numbers a + i∙b and c + i∙d are equal if, and only if, their real and imaginary parts are the same. Now, the ∂ψ/∂t = i∙(ħ/m_eff)∙∇²ψ equation amounts to writing something like this: a + i∙b = i∙(c + i∙d). Now, remembering that i² = −1, you can easily figure out that i∙(c + i∙d) = i∙c + i²∙d = − d + i∙c.

[19] The dimension of B is usually written as N/(m∙A), using the SI unit for current, i.e. the ampere (A). However, 1 C = 1 A∙s and, hence, 1 N/(m∙A) = 1 (N/C)/(m/s).

[20] Of course, multiplication with i amounts to a counterclockwise rotation. Hence, multiplication by –i also amounts to a rotation by 90 degrees, but clockwise. Now, to uniquely identify the clockwise and counterclockwise directions, we need to establish the equivalent of the right-hand rule for a proper geometric interpretation of Schrödinger’s equation in three-dimensional space: if we look at a clock from the back, then its hand will be moving counterclockwise. When writing B = (1/c)∙i∙E, we assume we are looking in the negative x-direction. If we are looking in the positive x-direction, we should write: B = -(1/c)∙i∙E. Of course, Nature does not care about our conventions. Hence, both should give the same results in calculations. We will show in a moment they do.

[21] In fact, when multiplying C²/(N·m²) with N²/C², we get N/m², but we can multiply this with 1 = m/m to get the desired result. It is significant that an energy density (joule per unit volume) can also be measured in newton (force per unit area.

[22] The illustration shows a linearly polarized wave, but the obtained result is general.

[23] The sine and cosine are essentially the same functions, except for the difference in the phase: sinθ = cos(θ−π /2).

[24] I must thank a physics blogger for re-writing the 1/(ε₀·μ₀) = c² equation like this. See: http://reciprocal.systems/phpBB3/viewtopic.php?t=236 (retrieved on 29 September 2017).

[25] A circularly polarized electromagnetic wave may be analyzed as consisting of two perpendicular electromagnetic plane waves of equal amplitude and 90° difference in phase.

[26] Of course, the reader will now wonder: what about neutrons? How to explain neutron spin? Neutrons are neutral. That is correct, but neutrons are not elementary: they consist of (charged) quarks. Hence, neutron spin can (or should) be explained by the spin of the underlying quarks.

[27] We detailed the mathematical framework and detailed calculations in the following online article: https://readingfeynman.org/2017/09/15/the-principle-of-least-action-re-visited.

[28] https://en.wikipedia.org/wiki/Electron_rest_mass (retrieved on 29 September 2017).