We Could Stop Here.

(But the Next Question Is Already Knocking.)

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

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

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

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

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

Just careful rereading.

Why revisit this at all?

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

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

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

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

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

A word about standing waves (and why they appear)

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

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

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

Why this felt worth publishing

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

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

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

And that’s usually a good sign.

Could we stop here?

Yes. Absolutely.

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

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

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

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

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

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

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

— Jean Louis Van Belle

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

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

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

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

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

Most things now feel… settled

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

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

None of this feels revolutionary anymore. It feels readable.

And yet.

Matter–antimatter still feels different

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

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

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

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

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

Annihilation is unsettlingly easy

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

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

That asymmetry matters.

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

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

What happens to charge?

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

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

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

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

And pure vacuum pair creation?

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

That, too, feels telling.

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

So where does that leave us?

It leaves me in an oddly peaceful place.

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

That feels like the right place to stop for tonight.

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

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

— Jean Louis Van Belle

Bad thinking: photons versus the matter wave

Pre-scriptum (dated 26 June 2020): My views on the true nature of light and matter have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics. If you are reading this, then you are probably looking for not-to-difficult reading. In that case, I would suggest you read my re-write of Feynman’s introductory lecture to QM. If you want something shorter, you can also read my paper on what I believe to be the true Principles of Physics.

Original post:

In my previous post, I wrote that I was puzzled by that relation between the energy and the size of a particle: higher-energy photons are supposed to be smaller and, pushing that logic to the limit, we get photons becoming black holes at the Planck scale. Now, understanding what the Planck scale is all about, is important to understand why we’d need a GUT, and so I do want to explore that relation between size and energy somewhat further.

I found the answer by a coincidence. We’ll call it serendipity. 🙂 Indeed, an acquaintance of mine who is very well versed in physics pointed out a terrible mistake in (some of) my reasoning in the previous posts: photons do not have a de Broglie wavelength. They just have a wavelength. Full stop. It immediately reduced my bemusement about that energy-size relation and, in the end, eliminated it completely. So let’s analyze that mistake – which seems to be a fairly common freshman mistake judging from what’s being written about it in some of the online discussions on physics.

If photons are not to be associated with a de Broglie wave, it basically means that the Planck relation has nothing to do with the de Broglie relation, even if these two relations are identical from a pure mathematical point of view:

The Planck relation E = hν states that electromagnetic waves with frequency ν are a bunch of discrete packets of energy referred to as photons, and that the energy of these photons is proportional to the frequency of the electromagnetic wave, with the Planck constant h as the factor of proportionality. In other words, the natural unit to measure their energy is h, which is why h is referred to as the quantum of action.
The de Broglie relation E = hf assigns a de Broglie wave with frequency f to a matter particle with energy E = mc² = γm₀c². [The factor γ in this formula is the Lorentz factor: γ = (1 – v²/c²)^–1/2. It just corrects for the relativistic effect on mass as the velocity of the particle (v) gets closer to the speed of light (c).]

These are two very different things: photons do not have rest mass (which is why they can travel at light speed) and, hence, they are not to be considered as matter particles. Therefore, one should not assign a de Broglie wave to them. So what are they then? A photon is a wave packet but it’s an electromagnetic wave packet. Hence, its wave function is not some complex-valued psi function Ψ(x, t). What is oscillating in the illustration below (let’s say this is a procession of photons) is the electric field vector E. [To get the full picture of the electromagnetic wave, you should also imagine a (tiny) magnetic field vector B, which oscillates perpendicular to E), but that does not make much of a difference. Finally, in case you wonder about these dots: the red and green dot just make it clear that phase and group velocity of the wave are the same: v_g = v_p = v = c.] The point to note is that we have a real wave here: it is not a de Broglie wave. A de Broglie wave is a complex-valued function Ψ(x, t) with two oscillating parts: (i) the so-called real part of the complex value Ψ, and (ii) the so-called imaginary part (and, despite its name, that counts as much as the real part when working with Ψ !). That’s what’s shown in the examples of complex (standing) waves below: the blue part is one part (let’s say the real part), and then the salmon color is the other part. We need to square the modulus of that complex value to find the probability P of detecting that particle in space at point x at time t: P(x, t) = |Ψ(x, t)|². Now, if we would write Ψ(x, t) as Ψ = u(x, t) + iv(x, t), then u(x, t) is the real part, and v(x, t) is the imaginary part. |Ψ(x, t)|²is then equal to u² + u² so that shows that both the blue as well as the salmon amplitude matter when doing the math.

So, while I may have given the impression that the Planck relation was like a limit of the de Broglie relation for particles with zero rest mass traveling at speed c, that’s just plain wrong ! The description of a particle with zero rest mass fits a photon but the Planck relation is not the limit of the de Broglie relation: photons are photons, and electrons are electrons, and an electron wave has nothing to do with a photon. Electrons are matter particles (fermions as physicists would say), and photons are bosons, i.e. force carriers.

Let’s now re-examine the relationship between the size and the energy of a photon. If the wave packet below would represent an (ideal) photon, what is its energy E as a function of the electric and magnetic field vectors E and B? [Note that the (non-boldface) E stands for energy (i.e. a scalar quantity, so it’s just a number) indeed, while the (italic and bold) E stands for the (electric) field vector (so that’s something with a magnitude (E – with the symbol in italics once again to distinguish it from energy E) and a direction).] Indeed, if a photon is nothing but a disturbance of the electromagnetic field, then the energy E of this disturbance – which obviously depends on E and B – must also be equal to E = hν according to the Planck relation. Can we show that?

Well… Let’s take a snapshot of a plane-wave photon, i.e. a photon oscillating in a two-dimensional plane only. That plane is perpendicular to our line of sight here:

photon

Because it’s a snapshot (time is not a variable), we may look at this as an electrostatic field: all points in the interval Δx are associated with some magnitude E (i.e. the magnitude of our electric field E), and points outside of that interval have zero amplitude. It can then be shown (just browse through any course on electromagnetism) that the energy density (i.e. the energy per unit volume) is equal to (1/2)ε₀E²(ε₀is the electric constant which we encountered in previous posts already). To calculate the total energy of this photon, we should integrate over the whole distance Δx, from left to right. However, rather than bothering you with integrals, I think that (i) the ε₀E²/2 formula and (ii) the illustration above should be sufficient to convince you that:

The energy of a photon is proportional to the square of the amplitude of the electric field. Such E ∝ A²relation is typical of any real wave, be they water waves or electromagnetic waves. So if we would double, triple, or quadruple its amplitude (i.e. the magnitude E of the electric field E), then the energy of this photon with be multiplied with four, nine times and sixteen respectively.
If we would not change the amplitude of the wave above but double, triple or quadruple its frequency, then we would only double, triple or quadruple its energy: there’s no exponential relation here. In other words, the Planck relation E = hν makes perfect sense, because it reflects that simple proportionality: there is nothing to be squared.
If we double the frequency but leave the amplitude unchanged, then we can imagine a photon with the same energy occupying only half of the Δx space. In fact, because we also have that universal relationship between frequency and wavelength (the propagation speed of a wave equals the product of its wavelength and its frequency: v = λf), we would have to halve the wavelength (and, hence, that would amount to dividing the Δx by two) to make sure our photon is still traveling at the speed of light.

Now, the Planck relation only says that higher energy is associated with higher frequencies: it does not say anything about amplitudes. As mentioned above, if we leave amplitudes unchanged, then the same Δx space will accommodate a photon with twice the frequency and twice the energy. However, if we would double both frequency and amplitude, then the photon would occupy only half of the Δx space, and still have twice as much energy. So the only thing I now need to prove is that higher-frequency electromagnetic waves are associated with larger-amplitude E‘s. Now, while that is something that we get straight out of the the laws of electromagnetic radiation: electromagnetic radiation is caused by oscillating electric charges, and it’s the magnitude of the acceleration (written as a in the formula below) of the oscillating charge that determines the amplitude. Indeed, for a full write-up of these ‘laws’, I’ll refer to a textbook (or just download Feynman’s 28th Lecture on Physics), but let me just give the formula for the (vertical) component of E:

You will recognize all of the variables and constants in this one: the electric constant ε₀, the distance r, the speed of light (and our wave) c, etcetera. The ‘a’ is the acceleration: note that it’s a function not of t but of (t – r/c), and so we’re talking the so-called retarded acceleration here, but don’t worry about that.

Now, higher frequencies effectively imply a higher magnitude of the acceleration vector, and so that’s what’s I had to prove and so we’re done: higher-energy photons not only have higher frequency but also larger amplitude, and so they take less space.

It would be nice if I could derive some kind of equation to specify the relation between energy and size, but I am not that advanced in math (yet). 🙂 I am sure it will come.

Post scriptum 1: The ‘mistake’ I made obviously fully explains why Feynman is only interested in the amplitude of a photon to go from point A to B, and not in the amplitude of a photon to be at point x at time t. The question of the ‘size of the arrows’ then becomes a question related to the so-called propagator function, which gives the probability amplitude for a particle (a photon in this case) to travel from one place to another in a given time. The answer seems to involve another important buzzword when studying quantum mechanics: the gauge parameter. However, that’s also advanced math which I don’t master (as yet). I’ll come back on it… Hopefully… 🙂

Post scriptum 2: As I am re-reading some of my post now (i.e. on 12 January 2015), I noted how immature this post is. I wanted to delete it, but finally I didn’t, as it does illustrate my (limited) progress. I am still struggling with the question of a de Broglie wave for a photon, but I dare to think that my analysis of the question at least is a bit more mature now: please see one of my other posts on it.

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