You may want to have a glance at the other pages before you get started here because we are effectively going to be discussing some fairly advanced concepts here—but we promise we will stick closely to the fundamental formulas, so we hope you can follow. Indeed, we should always try to let the formulas speak for themselves: as far as we know, it is the formulas themselves that describe reality—not our interpretation of them. Let us start with the Planck-Einstein relation—which tells us what we should think of as being discrete, and what may or may not be continuous.

Both the ring current model of matter-particles as well as our photon model tell us particles – matter or light – pack one unit of Planck’s quantum of action as well as energy, and they do so because they are oscillations. The frequency of this oscillation is proportional to the oscillation, and the Planck-Einstein relation tells us the proportionality factor is equal to Planck’s constant:

E = h·f

Matter-particles pack a typical amount of energy, which we get from their mass. They are, effectively, massive and, therefore, we can measure their inertia to a change in their state of motion.

A photon, in contrast, carries no charge and, therefore, has zero rest mass: it is an electromagnetic oscillation without a charge to play with. It can, therefore, pack any amount of energy—theoretically, at least. To give you an idea of what this means, it may be useful to note that, so far, astronomers have detected gamma rays with energies of up to 450 TeV coming out of the Crab Nebula—presumably out of its central pulsar). The equivalent mass of a 450 TeV photon is about 8×10−22 kg. Is this reasonable? It depends on what you want to compare with: if I am not mistaken, I think that’s about 480,000 times the mass of a proton.

Any case, the point is this: what David Hestenes refers to as the clock speed of a particle is variable for the photon. That’s why it can carry energy around and – if we think of energy as the currency of physics – then we should say it can do so in any denomination that you’d want. In contrast, the clock speed of an electron, a muon or a proton is always the same—in the inertial reference frame, that is: when it starts moving, its clock speed goes up. The relation is, effectively, the Planck-Einstein relation:

E = h·f = h/T ⇔ E·T = h

T is the cycle time, and you should think of it as the natural time unit for an electron, a proton or whatever other matter-particle you’re thinking of. It is given by the mass or the energy and it is, therefore, different for each. Think of it like this: we divide Planck’s quantum by the cycle time T to get the energy of our particle. So what is the energy then? It’s the energy of the cycle of the particle. What cycle? The cycle of its oscillation. What oscillation? The oscillation in space and in time of the spinning electric charge inside. For the reader who isn’t familiar with our ring current model, we visualize it below once more.

zbw electron

The question is this: what is the nature of this oscillation? It must be electromagnetic because the centripetal force grabs onto an electric charge: the charge is, effectively, all it can grab onto. We think of it as having no rest mass, which is why it can and does zitter around at lightspeed. However, saying it must be electromagnetic is not enough to fully define what it might or might not be. Our derivation of the Compton radius – which we also copy once again below – does not involve Maxwell’s equations and, therefore, it does not involve the use of the electric and/or magnetic constant. It, therefore, cannot uniquely describe the electron.


To put it simply, the model does not not answer this simple question: why is the mass of an electron – or a proton, muon, etcetera – what it is? Why is the number what it is? The practical answer to this question is, of course, that we would not be living in the world that we are living in if matter-particles would also come in whatever denomination we could possibly think of. No! We have protons and electrons, and together they make neutrons and atoms, etcetera. We can also build all kinds of exotic atoms using muons or anti-matter, but that’s diversity enough, I’d say!

Such common-sense answer is good but, again it is probably not good enough. Let us distinguish two other questions here:

  1. What is the fundamental mechanism of the oscillation?
  2. Our calculation of the forces revealed the force inside of a muon is about 42,753 times stronger than than inside of an electron. For a proton, we get an additional factor—a factor that’s equal to a more modest 19.7 (more or less) and which we relate to its different angular momentum: four units of ħ, instead of just one ! Or so we think, at least. [We need to explain the muon/proton mass factor by something, right? If you think my explanations sounds dodgy, then my question to you is: what’s your guess?]

Let us stay calm and try to tackle the first question first.

Matter-particles as two-dimensional oscillations

An electromagnetic oscillation is an oscillation in two dimensions—mathematically speaking, at least. Indeed, we may want to think of the electric and magnetic field vectors as being pretty real too but a relativistic analysis shows the electromagnetic force is one force only and that us analyzing it in terms of two forces – the electric or electrostatic force versus the magnetic force – is very useful but also somewhat artificial.

This idea inspired us to analyze the electron – or any matter-particle, really –  as an equivalent oscillation in two dimensions. Indeed, that’s why we think our ring current model adds value in terms of trying to understand what might or might not be going on. We developed the metaphor elsewhere and so we won’t dwell on it here. Let us just look at the basic equations again. The story is like this:

1. An easy geometric analysis confirms we should probably think of the energy as being equally split over (1) the energy of the charge itself (the Zitterbewegung charge as we called it) as it whizzes around at lightspeed and (2) the energy in the force that keeps the charge in its circular orbit or whatever oscillatory motion it might be in. Indeed, at this point we cannot be sure whether the motion is regular or not.

In any case, the point is this: the equivalent mass of the zbw charge, therefore, accounts for only half of the total mass of the electron. We wrote this like this: mγ = m/2.

2. The basic equation for an oscillation in one dimension is this:

F = dp/dt = –kx with p = mvv = γm0v

Multiplying both sides with v = dx/dt and doing a fair amount of rather complicated arithmetic because we want the formulas to be relativistically correct, gives us the following energy conservation equation:

energy conservation 0

We recognize the potential energy term: it is the same kx2/2 formula you’d get for the non-relativistic oscillator. No surprises: potential energy depends on position only, not on velocity, and there is nothing relative about position. However, the m0v2/2 term that we would get when using the non-relativistic formulation of Newton’s Law – which captures the kinetic energy – is now replaced by the mc2 = γm0c2 term.

You should note this mc2 = γm0c2 is not a constant: it varies with time – just like kx2/2 – because of the use of the relativistic mass concept. So how can we calculate the energy? The total energy is constant at any point, so we may equate x to 0 and calculate the energy there. At that point, the potential energy will be zero and, crossing the x = 0 point, our pointlike charge will also reach the speed of light. Using the mγ = mc = me/2 equation, we write this:

energy conservation

3. We can now add the energies in both oscillators so as to arrive at the total energy of the electron:

E = mec2

It is a wonderful result. Why would it be wonderful? We will not be shy here: our two-dimensional oscillator model of matter-particles is a really elegant and intuitive common-sense explanation of Einstein’s mass-energy equivalence relation. Nothing more. Nothing less.

The reader will probably have a lot of questions in regard to the derivation, and rightly so. The analysis is, effectively, quite subtle and implicitly uses Einstein’s distinction between the “longitudinal” and “transverse” mass of a moving charge, which he used in his 1905 article on relativity. This distinction is also related to the rather subtle distinction between the electrostatic and magnetic forces, which are equally relative in the sense that what is electromagnetic and what is magnetic depends on your reference frame! However, we don’t want to get distracted here and so we must refer the reader to earlier papers and analysis.

OK. Let us go back to the question we are tentatively trying to answer here: what is the mechanism here?

The honest answer is: we cannot answer that question. What we are trying to do here is to show that the centripetal force can be analyzed – mathematically, at least – as the sum of a sine and a cosine oscillation. These two oscillations are independent and their energies may, therefore, be added. As to the mechanism, we are not sure. We developed the metaphor of a V-2 engine to explain how potential and kinetic energy might be transferred from one (spatial) dimension to the other but that’s a metaphor only: we don’t see any real pistons, rods or crankshaft here!

Of course, you may think that is a rather poor answer. If that’s the case, think about like this: it is not any better or worse than the mainstream ideas in regard to the electromagnetic force. Try to answer this question: what is an electromagnetic oscillation, exactly? What are the pistons, rods or crankshaft that make sure light keeps moving at… Well… The speed of light. As such, our explanation of matter in terms of a fundamental two-dimensional oscillation in space and in time is probably as good as it gets: please write us if you have any better ideas!

To wrap this up, we may want to think about two related questions. We basically that a matter-particle is nothing but a two-dimensional oscillation in space and in time of a charge. This triggers two new questions:

  1. Can we, perhaps, think of this oscillation in space and in time as an oscillation of spacetime?
  2. What is the nature of the charge? If an electron is this weird wavicle-like thing combining a charge and motion, then what’s the nature of the charge?

OK. Two questions, two tentative answers:

1. As for the first question, that is, in fact, what got me started on my very first paper—which I wrote three and a half years ago but which still gets a lot of downloads because of its very sexy title: the quantum-mechanical wavefunction as a gravitational wave. However, I may be mistaken but I know think of such statements as tautologies: yes, we can think of an oscillation in space and in time as an oscillation of spacetime, but what is the added value of saying that? What do you mean by it?

It’s like Einstein’s geometric analysis of gravity, isn’t it? What’s the difference between saying gravity is a force in three-dimensional Cartesian space, or saying that gravity causes spacetime to curve such that a mass always travels on a geodesic (the equivalent of a straight line in curved spacetime)? We just have two different mathematical models describing the same reality here, isn’t it?

I am not saying it may not make a difference. It actually might. In fact, just yesterday I received an email of a friendly reader urging me to re-do the analysis using four-vectors and also incorporating gravity. The honest truth is this: first, I am not all that great in math and I find four-vector notation very confusing. Second, I feel the use of four-vectors may actually decrease our understanding because, when everything is said and done, any intuitive understanding of what might or might not be the case – I am paraphrasing Wittgenstein here – must be based on the elementary concepts of three-dimensional space and time, because I believe – with Hume, Kant, Hegel and other philosophers – that these are fundamental categories of the mind.

In short, we must, of course, use relativistically correct equations, but that doesn’t necessarily involve rocket mathematics!

2. We don’t have any real answer to the second question either. The Zitterbewegung charge is, effectively, a very weird concept. We noted it has zero rest mass, so its only property is its charge, and it acquires an effective relativistic mass because it is effectively photon-like in this regard. The crucial difference is that a photon does not carry charge.

However, we should add something to this: the charge also seems to acquire some spatial dimension as it zitters around. That’s, in fact, how we explained not only Thomson scattering but also the anomaly in the electron radius and magnetic moment: the charge does appear as some hard core charge inside of the electron. The effective radius of this hard core charge is of the order of αħ/mc, so it’s a fraction (α or α/2π—we’re not quite sure here) of the Compton radius.

The point is this: its scale is the femtometer scale, so that’s the same scale as the size of the muon or the proton! Now that is weird, because we also analyze muons and protons as ring currents, so we’re assuming the charge inside is pointlike—relatively speaking. So what happens then? Do we have a charge that shrinks as per the size of the larger particle? We cannot answer that question, but our ring current model seems to imply it does! Dr. Oliver Consa suggests we should, perhaps, think of some kind of fractal structure here. That makes a lot of sense to me, but the truth is: we haven’t even started to think through that, so here also we think your guesses are as good as ours! 🙂

Thinking about the nature of the charge raises another obvious question: what’s the nature of the anti-charge?

What’s the nature of anti-matter?

The phenomenon of electron-positron pair creation and annihilation – or the question in regard to the nature of anti-matter in general – is probably the most puzzling question of all. We really do not have any ideas at all here, but then we don’t feel too dumb because we are in good company here: from all of Dirac’s formal or informal remarks on the state of our knowledge, it’s clear he struggled very much with that too. The gist of the matter is this: our world could be an anti-matter world. We may think of that as a mathematical fiction: who cares if we write q or −q in our equations? No one, right? It’s just a convention, and so we can just swap signs, right?

Well… No. Here you are just talking about some mathematical possibility—which Dirac started exploring from 1928 onwards—so that’s as soon as he had published his equation for the free electron, and which got Carl D. Anderson actually looking for one. He said this about it in his 1933 Nobel Prize Lecture:

“If we accept the view of complete symmetry between positive and negative electric charge so far as concerns the fundamental laws of Nature, we must regard it rather as an accident that the Earth (and presumably the whole solar system), contains a preponderance of negative electrons and positive protons.”

The carefully chosen ‘preponderance’ term shows he was aware of the actual existence of anti-matter. Indeed, just the previous year (1932), Carl D. Anderson had actually found what he was looking for: the curved trajectory of a real positron on one of his cloud chamber pictures of what happens to cosmic radiation when it enters… Well… Anderson’s cloud chamber. 🙂 Anderson got his own Nobel Prize for it – and one that’s very well deserved (the reader who’s read our previous posts will know we have serious doubts on the merit of some (other) Nobel Prizes).

So the point is this: we should not think of matter and anti-matter as being ‘separate worlds’ (theoretical and/or physical). No. Pair creation/annihilation are part and parcel of our ‘world’, and there’s only one world, as far as we know. So don’t think about other possible universes here. So what can/should we do with this?

Nothing at all, perhaps. We can, obviously, use our electron and proton models to construct a positron model and an anti-proton model, and then we can combine this to make an anti-neutron[1], and so there is no issue, is there?

Probably not in terms of our models and whatever else we’ve been trying to demonstrate in this paper. We just note that matter-anti-matter pair creation/annihilation out of – out of what, really? – is deeply mysterious. We have explored some of it in various papers[2], in which we noted this, for example:

“Electron-positron pair creation does not happen because gamma-rays happen to spontaneously ‘disintegrate’ into electron-positron pairs. They do not: the presence of a nucleus is required. Plain common-sense tells us the process is likely to be something like this: the photon causes a proton to emit a positron (b+ decay), so the proton turns into a neutron and something else needs to happen now: the atom needs to eject an electron or, more likely, a neutron decays into a proton and emits an electron. Hence, charge is being conserved and we shouldn’t think of it as being a Great Big Mystery.”

I also have a real crackpot theory, but I am hesitant to write about that because you must think I am crazy enough already. It goes a bit like this: Richard Feynman tended to model anti-matter particles (positrons, notably) as particles with their clock turning backwards. Now, I think that’s nonsense: time has to go in one direction only because we live in one world only and time doesn’t go back in this world. [I also have another simple – somewhat related – argument on p. 9-10 of my manuscript: we can’t do physics if the functions describing motion and velocities aren’t well-behaved.]

Having said that, I’ve been thinking that describing anti-matter in terms of negative space (rather than negative time) might make sense. Abusing Minkowski’s notation, we may try this: instead of a (+ +++) signature, we could, perhaps, apply a (+ −−−) four-vector signature when dealing with anti-matter. It is just something that I thought about this morning, so I should probably not be writing about it but then… Well… It’s these kind of niceties that keep us going, isn’t it? [PS: I actually did quickly jot something down: have a look here!]

The idea is less outrageous than it sounds: if we think everything is motion – and if the electron and the zbw charge constitute some fractal structure – then the idea is actually quite simple: a positron then consists of a zbw charge whizzing around in the opposite direction. So, yes, by putting a minus sign in front of the coordinates in space you get something with opposite charge—technically speaking, that is. But then we are still left with the question as to how pair creation and annihilation actually happens.

Hence, let us not speculate too much here and wrap up by noting the obvious: we are very much puzzled by the matter, or… Well… The anti-matter, I should say. 😊

[1] The reader should note that the reality of the antineutron confirms our hypothesis of the neutron being some combination of a proton and an electron (our paper(s) on that are somewhat dated but still useful). If a neutron wouldn’t carry charge (positive and negative), then it wouldn’t have an anti-matter counterpart. But so it has one. The reader can check the Wikipedia article on it, which says the antineutron was discovered in proton–antiproton collisions at the Bevatron (Lawrence Berkeley National Laboratory) by Bruce Cork in 1956, one year after the antiproton was discovered.

[2] See, for example, our paper on conservation laws.

We have to wrap this up. It now time to move to the second of the first set of two questions: why do we have a (much) stronger force inside of a muon and a proton?

The nature of the strong force

How can we be so sure that the centripetal force inside of the muon and the force inside the proton are the same? The honest answer is this: we cannot be sure of that. We just think it is. Why? Our intuition is this: muon disintegration also involves neutrinos—and neutrinos are always present when physicists talk about the strong(er) force.

Let us try to think this through: we basically think of the strong(er) force as some strong(er) version of the electromagnetic force that keeps an electron together, right? Now, applying the basic equations (see our Matter page) we get this:

force formula

What does this tell us? It tells us we can analyze the difference between the force inside of a muon and a proton − a difference which we express as a ratio here − in terms of two factors:

  1. The difference between their radii: the radius of a muon is about 2.22 times larger than that of a proton.
  2. That ratio then gets multiplied by four, which is the factor which distinguishes their presumed angular momentum: ħ and 4ħ.

Hence, yes, we think it’s the same force: if the angular momentum wouldn’t be different, the proton force − sorry to use shorthand − would just be about 2.22 times stronger than the muon force.

Of course, it still begs the question: so why are their radii different? That amounts to asking this question: why can’t we analyze a proton as an anti-muon, or a muon as an anti-proton? Again, I don’t have a theoretical answer to that question but the practical answer is quite convincing here: if a proton was an anti-muon, or a muon was an anti-proton, then we wouldn’t have protons or muons: they would have annihilated each other and we would not be living in this world.

So, yes, we do think that the muon and the proton are products of the same force, but that their angular momentum is not the same. That’s why a proton is not an anti-muon. It is as simple and as complicated as that, I am afraid.

Having said that, it is rather obvious there is a lot of room for more analysis here. What’s lacking here is some kind of model or theory which would allow us to think of these two different modes of a two-dimensional oscillation of the elementary charge— in space and in time. We don’t have that model yet. :-/ However, we hope that some of what we wrote above may serve as inspiration or, at the very least, amount to some kind of ‘tool box’ to develop such model.

We must now think about yet another question. If we have ħ or ħ/2 particles (the electron and the muon), and we also have a 2ħ or 4ħ particle (the proton), then where are the 2ħ or 3ħ particles?

Where are the 2ħ or 3ħ particles?

It’s a rather obvious question, indeed. If the ratio between the muon’s spin − measured in full or half units of ħ − and the proton’s spin is 1/4, then where is the 1/2 or 1/3 ratio? Skipping niceties about 1, 2 or 1/2 factors here, the question is this: if angular momentum comes in units of ħ and 4ħ, then where are the 2ħ or 3ħ particles?

I am not sure but I think it’s, once again, because of this ħ/2 ratio we get for the orbital angular momentum of our zbw charge. How should I phrase this? Perhaps like this: there can be no 2ħ or 3ħ particles, because they’d imply an orbital angular momentum of ħ or 3ħ/2 particles, and that doesn’t quite fit into our two-dimensional oscillator model. Think of it like this: two modes of a two-dimensional oscillation—the relevant factor must be four rather than two, right?

However, I admit that – from all the tentative or ad hoc answers that I’ve been trying to give to what are, obviously, very difficult questions – this may well be the weakest one. Again, the objective of this site is not to present absolute truth. The objective is to present what we think might be the case. As such, I think of it as my basic version of truth, and I am sharing it exactly because I want to make sure I am not making any logical errors!

In other words, I think I’ve greatly reduced the mystery and weirdness that you are supposed to believe in but – of course – I am not saying there is no mystery left. On the contrary, reducing the number of concepts and exposing fake theories leaves us with a much nicer body of knowledge to wonder about. 🙂

By way of conclusion, I would like to add one more section on the nature of Planck’s constant. Those of you who have read some of my some stuff know it is a bit of a pet topic of mine.

Planck’s constant as a vector?

Going from the E = h·expression of the Planck-Einstein relation to its E = ħ·ω is not innocent. Not in mathematical terms, and not in terms of the physicality of the situation either. Angular momentum and, hence, angular frequency involves the idea of a plane of rotation. They are, therefore, axial vectors. What would it mean to write h as a vector? Nothing much. We would just write it like h, just like we would write L for angular momentum. Look at the wobbling angular momentum of the spinning top below, and then think of the plane of rotation of our pointlike charge.



Nice, isn’t it? Don’t think too much of it. I only inserted the illustration above (borrowed from here) to remind us of the complexities of even the simplest of simple situations involving a spinning object (the top) in some force field (gravity). Just replace the top by our electron, and gravity by the magnetic force field. Then think of the complexities of analyzing a two-dimensional force (electromagnetism) grabbing onto a magneton (the electron as a magnet, but with a particular charge/mass distribution—the shape factor) as compared to the one-dimensional gravitational force. Do you start to get a feel of how complicated the actual situation might be?

The illustration above may or may not make some more sense of the (in)famous Uncertainty Principle. Indeed, we know we can also write Planck’s quantum of action as the product of (linear) momentum (p) times a wavelength (the Compton radius multiplied by 2π). So then we have the p·λ = h expression out of the zbw geometry, and then we can re-write the Uncertainty Principle like this, perhaps:

Δp· Δλ = h

So the uncertainty in p is just the wobbling of the h vector. Simple, right? There is only one question left, then: the precession of the spinning top in the illustration above is regular: we can associate it with a precisely defined precession frequency. Can we do that for an electron? In other words, we can sure see there’s going to be statistical uncertainty here, but is there a need to introduce some real uncertainty in our model? Real indeterminacy as opposed to statistical unpredictability?

My instinctive answer to this is the same as Albert Einstein’s: no. But I am happy to get other opinions. In case you are a believer from the other camp, please note that your objections will be philosophical rather than scientific—as philosophical as Einstein’s intuition here. This is, in fact, the fine line where physics ends and where our own beliefs – call it our own intuition about what might or might not be the case – start. The question is very much related to the question I raised a bit earlier: is the motion of our zbw charge chaotic, or regular?

I honestly have no answer to that. If you’ve read my manuscript, you’ll see I’ve become a bit more careful in my assertions now—although I do continue to think Einstein was right all along. However, the motion might effectively be chaotic. In that case, we’d have real indeterminacy, as opposed to statistical unpredictability. We will never be in a position to decide on the matter because, as Dirac noted, we cannot verify this by experiment “since the frequency of the oscillatory motion is so high and its amplitude is so small.” (Paul A.M. Dirac, Theory of Electrons and Positrons, Nobel Lecture, December 12, 1933)

So… Well… I can’t say much about this. Perhaps this is the true Copenhagen interpretation of quantum mechanics: we are not in a position to distinguish between real indeterminacy and statistical unpredictability. For me, this is a game of words, and so I’ll abstain from further comments. But I can imagine it might be an important question for you. Why? Because we’re into physics to understand what might be the case. That will, inevitably, involve strong opinions. I want to stay away from that. My only objective when writing this was to give you some hopefully useful pointers which may or may not enable you to think for yourself: what makes sense, and what doesn’t?

My personal answer is just my personal answer, and so that’s just a bunch of words that don’t matter all that much. Why not? Let me insert a quote from an entirely different line of research:

“We are in the words, and at the same time, apart from them. The words spin out, spin us out, over a void. There, somewhere between us, some words form some answer for some time, allowing us to live more fully in the forgetting face of nonexistence, in the dissolving away of each other.” (Lagan, in Jeremy D. Safran (2003), Psychoanalysis and Buddhism: an unfolding dialogue, p. 134.)

So… Well… We have the E = h·f and E = mc2 relations – which, hopefully, you’ll understand a bit better now – and that’s it! You should just look at them and think for yourself: what do they tell you? Personally, I don’t think there’s any uncertainty in them. But that’s just an non-scientific opinion. 🙂

Jean Louis Van Belle, 25 March 2020

PS: I realize the last section of this page is a hopelessly simple summary of what a more comprehensive new theory of physics would like. It would incorporate Einstein’s guts instinct in his 1905 article on (special) relativity: energy (and, hence, mass) is not just some scalar quantity. The idea of direction is there when moving a charge from here to there (a change in potential energy), and it’s also there when talking kinetic energy: the velocity vector has a specific direction. Of course, we square it and the square of a vector is a scalar quantity, but we should not forget where that scalar quantity came from. We lack space here – you would stop reading anyway – but you should note that Einstein’s distinction between the “longitudinal” and “transverse” mass of a moving charge is crucial to understanding our two-dimensional oscillator interpretation of the ring current model. I know this will sound like rambling to most readers but I am not trying to sell a publication here: I am just trying to share my version of truth. Nothing more. Nothing less. 🙂