Form and substance

Philosophers usually distinguish between form and matter, rather than form and substance. Matter, as opposed to form, is then what is supposed to be formless. However, if there is anything that physics – as a science – has taught us, is that matter is defined by its form: in fact, it is the form factor which explains the difference between, say, a proton and an electron. So we might say that matter combines substance and form.

Now, we all know what form is: it is a mathematical quality—like the quality of having the shape of a triangle or a cube. But what is (the) substance that matter is made of? It is charge. Electric charge. It comes in various densities and shapes – that is why we think of it as being basically formless – but we can say a few more things about it. One is that it always comes in the same unit: the elementary charge—which may be positive or negative. Another is that the concept of charge is closely related to the concept of a force: a force acts on a charge—always.

We are talking elementary forces here, of course—the electromagnetic force, mainly. What about gravity? And what about the strong force? Attempts to model gravity as some kind of residual force, and the strong force as some kind of electromagnetic force with a different geometry but acting on the very same charge, have not been successful so far—but we should immediately add that mainstream academics never focused on it either, so the result may be commensurate with the effort made: nothing much.

Indeed, Einstein basically explained gravity away by giving us a geometric interpretation for it (general relativity theory) which, as far as I can see, confirms it may be some residual force resulting from the particular layout of positive and negative charge in electrically neutral atomic and molecular structures. As for the strong force, I believe the quark hypothesis – which basically states that partial (non-elementary) charges are, somehow, real – has led mainstream physics into the dead end it finds itself in now. Will it ever get out of it?

I am not sure. It does not matter all that much to me. I am not a mainstream scientist and I have the answers I was looking for. These answers may be temporary, but they are the best I have for the time being. The best quote I can think of right now is this one:

‘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.’ (Jacques Lacan, in Jeremy D. Safran (2003), Psychoanalysis and Buddhism: an unfolding dialogue, p. 134)

That says it all, doesn’t it? For the time being, at least. 🙂

Post scriptum: You might think explaining gravity as some kind of residual electromagnetic force should be impossible, but explaining the attractive force inside a nucleus behind like charges was pretty difficult as well, until someone came up with a relatively simple idea based on the idea of ring currents. 🙂

The proton radius and mass

Our alternative realist interpretation of quantum physics is pretty complete but one thing that has been puzzling us is the mass density of a proton: why is it so massive as compared to an electron? We simplified things by adding a factor in the Planck-Einstein relation. To be precise, we wrote it as E = 4·h·f. This allowed us to derive the proton radius from the ring current model:

proton radius This felt a bit artificial. Writing the Planck-Einstein relation using an integer multiple of h or ħ (E = n·h·f = n·ħ·ω) is not uncommon. You should have encountered this relation when studying the black-body problem, for example, and it is also commonly used in the context of Bohr orbitals of electrons. But why is n equal to 4 here? Why not 2, or 3, or 5 or some other integer? We do not know: all we know is that the proton is very different. A proton is, effectively, not the antimatter counterpart of an electron—a positron. While the proton is much smaller – 459 times smaller, to be precise – its mass is 1,836 times that of the electron. Note that we have the same 1/4 factor here because the mass and Compton radius are inversely proportional:

ratii

This doesn’t look all that bad but it feels artificial. In addition, our reasoning involved a unexplained difference – a mysterious but exact SQRT(2) factor, to be precise – between the theoretical and experimentally measured magnetic moment of a proton. In short, we assumed some form factor must explain both the extraordinary mass density as well as this SQRT(2) factor but we were not quite able to pin it down, exactly. A remark on a video on our YouTube channel inspired us to think some more – thank you for that, Andy! – and we think we may have the answer now.

We now think the mass – or energy – of a proton combines two oscillations: one is the Zitterbewegung oscillation of the pointlike charge (which is a circular oscillation in a plane) while the other is the oscillation of the plane itself. The illustration below is a bit horrendous (I am not so good at drawings) but might help you to get the point. The plane of the Zitterbewegung (the plane of the proton ring current, in other words) may oscillate itself between +90 and −90 degrees. If so, the effective magnetic moment will differ from the theoretical magnetic moment we calculated, and it will differ by that SQRT(2) factor.

Proton oscillation

Hence, we should rewrite our paper, but the logic remains the same: we just have a much better explanation now of why we should apply the energy equipartition theorem.

Mystery solved! 🙂

Post scriptum (9 August 2020): The solution is not as simple as you may imagine. When combining the idea of some other motion to the ring current, we must remember that the speed of light –  the presumed tangential speed of our pointlike charge – cannot change. Hence, the radius must become smaller. We also need to think about distinguishing two different frequencies, and things quickly become quite complicated.

The physicist’s worldview

Perhaps I should have titled this post differently: Feynman’s philosophical views. We may, effectively, assume that Richard Feynman’s Lectures on Physics represent mainstream sentiment, and he does get into philosophy—less or more liberally depending on the topic. Hence, yes, Feynman’s worldview is pretty much that of most physicists, I would think. So what is it? One of his more succinct statements is this:

“Often, people in some unjustified fear of physics say you cannot write an equation for life. Well, perhaps we can. As a matter of fact, we very possibly already have an equation to a sufficient approximation when we write the equation of quantum mechanics.” (Feynman’s Lectures, p. II-41-11)

He then jots down that equation which we also find on Schrödinger’s grave (shown below). It is a differential equation: it relates the wavefunction (ψ) to its time derivative through the Hamiltonian coefficients that describe how physical states change with time (Hij), the imaginary unit (i) and Planck’s quantum of action (ħ).

hl_alpb_3453_ptplr

Feynman, and all modern academic physicists in his wake, claim this equation cannot be understood. I don’t agree: the explanation is not easy, and requires quite some prerequisites, but it is not anymore difficult than, say, trying to understand Maxwell’s equations, or the Planck-Einstein relation (E = ħ·ω = h·f).

In fact, a good understanding of both allows you to not only understand Schrödinger’s equation but all of quantum physics. The basics are this: the presence of the imaginary unit tells us the wavefunction is cyclical, and that it is an oscillation in two dimensions. The presence of Planck’s quantum of action in this equation tells us that such oscillation comes in units of ħ. Schrödinger’s wave equation as a whole is, therefore, nothing but a succinct representation of the energy conservation principle. Hence, we can understand it.

At the same time, we cannot, of course. We can only grasp it to some extent. Indeed, Feynman concludes his philosophical remarks as follows:

“The next great era of awakening of human intellect may well produce a method of understanding the qualitative content of equations. Today we cannot. Today we cannot see that the water flow equations contain such things as the barber pole structure of turbulence that one sees between rotating cylinders. We cannot see whether Schrödinger’s equation contains frogs, musical composers, or morality—or whether it does not. We cannot say whether something beyond it like God is needed, or not. And so we can all hold strong opinions either way.” (Feynman’s Lectures, p. II-41-12)

I think that puts the matter to rest—for the time being, at least. 🙂

The geometry of the matter-wave

Yesterday, I was to talk for about 30 minutes to some students who are looking at classical electron models as part of an attempt to try to model what might be happening to an electron when moving through a magnetic field. Of course, I only had time to discuss the ring current model, and even then it inadvertently turned into a two-hour presentation. Fortunately, they were polite and no one dropped out—although it was an online Google Meet. In fact, they reacted quite enthusiastically, and so we all enjoyed it a lot. So much that I adjusted the presentation a bit the next morning (which added even more time to it unfortunately) so as to add it to my YouTube channel. So this is the link to it, and I hope you enjoy it. If so, please like it—and share it! 🙂

Oh! Forgot to mention: in case you wonder why this video is different than others, see my Tweet on Sean Carroll’s latest series of videos hereunder. That should explain it.

Sean Carroll

Post scriptum: I got the usual question, of course: if an electron is a ring current, then why doesn’t it radiate its energy away? The easy answer is: an electron is an electron and it doesn’t—for the same reason that an electron in an atomic orbital or a Cooper pair in a superconducting loop of current does not radiate energy away. The more difficult answer is a bit mysterious: it has got to do with flux quantization and, most importantly, with the Planck-Einstein relation. I cannot be too long here (this is just a footnote in a blog post) but the following elements should be noted:

1. The Planck-Einstein law embodies a (stable) wavicle: a wavicle respects the Planck-Einstein relation (E = h·f) as well as Einstein’s mass-energy equivalence relation (E = mc2). A wavicle will, therefore, carry energy but it will also pack one or more units of Planck’s quantum of action. Both the energy as well as this finite amount of physical action (Wirkung in German) will be conserved—cycle after cycle.

2. Hence, equilibrium states should be thought of as electromagnetic oscillations without friction. Indeed, it is the frictional element that explains the radiation of, say, an electron going up and down in an antenna and radiating some electromagnetic signal out. To add to this rather intuitive explanation, I should also remind you that it is the accelerations and decelerations of the electric charge in an antenna that generate the radio wave—not the motion as such. So one should, perhaps, think of a charge going round and round as moving like in a straight line—along some geodesic in its own space. That’s the metaphor, at least.

3. Technically, one needs to think in terms of quantized fluxes and Poynting vectors and energy transfers from kinetic to potential (and back) and from ‘electric’ to ‘magnetic’ (and back). In short, the electron really is an electromagnetic perpetuum mobile ! I know that sounds mystical (too) but then I never promised I would take all of the mystery away from quantum physics ! 🙂 If there would be no mystery left, I would not be interested in physics. :wink: On the quantization of flux for superconducting loops: see, for example, http://electron6.phys.utk.edu/qm2/modules/m5-6/flux.htm. There is other stuff you may want to dig into too, like my alternative Principles of Physics, of course ! 🙂  

Revisiting the electron double-slit experiment

We wrote about the significance of the 2012 University of Nebraska-Lincoln double-slit experiment with electrons before—as part of our Reading Feynman blog, to be precise. However, we did not have much of an understanding of matter-waves then. Hence, we talked about the de Broglie wavelength (λL = h/p) and tried to relate it to the interference pattern without any idea of what the concept of the de Broglie wavelength actually means. We, therefore, feel it is appropriate to revisit this subject as one of our very first entries for this new blog, which wants to probe a bit deeper.

Let us recall the basics of the model. We think of an electron as a pointlike charge in perpetual light-like motion (Schrödinger’s Zitterbewegung). The anomaly in the magnetic moment tells us the charge is pointlike but not dimensionless. Indeed, Schwinger’s α/2π factor for the anomaly is consistent with the idea of the classical electron radius being the radius of the pointlike charge, while the radius of its oscillation is equal to the Compton scattering radius of the electron. The two radii are related through the fine-structure constant (α ≈ 0.0073):

re = α·rC = αħ/mc ≈ 0.0073·0.386 pm (10−12 m)  ≈ 2.818 fm (10−15 m)

 It is good to get some sense of the scales here—and of the scale of the slits that were used in the mentioned experiment (shown below).

electron-double-slit-set-up

The insert in the upper-left corner shows the two slits: they are each 50 nanometer wide (50×10–9 m) and 4 micrometer tall (4×10–6 m). The thing in the middle of the slits is just a little support. Please do take a few seconds to contemplate the technology behind this feat: 50 nm is 50 millionths of a millimeter. Try to imagine dividing one millimeter in ten, and then one of these tenths in ten again, and again, and once again, again, and again. You just can’t imagine that, because our mind is used to addition/subtraction and, to some extent, with multiplication/division: our mind is not used to imaging numbers like 10–6 m or 10–15 m. Our mind is not used to imagine (negative) exponentiation because it is not an everyday phenomenon.

The second inset (in the upper-right corner) shows the mask that can be moved to close one or both slits partially, or to close them completely. It gives the interference patterns below (all illustrations here are taken from the original article—we hope the authors do not mind us popularizing their achievements). The inset (upper-left corner) shows the position of the mask vis-á-vis the slits. The electrons are fired one-by-one and, of course, few get through when the slits are closed or partly closed.

Interference 1

The one-by-one firing of the electrons is, without any doubt, the most remarkable thing about the whole experiment. Why do we say that? Because electron interference had already been demonstrated in 1927 (the Davisson-Germer experiment), just a few years after Louis de Broglie had advanced his hypothesis on the matter-wave. However, till this 2012 experiment, it had never been performed in exactly the same way as Feynman describes it in his 1963 Lectures on Quantum Mechanics. The illustration below shows how the interference pattern is being built up as the electrons go through the slit(s), one-by-one.

electron-blobs

The challenge for us is to explain this interference pattern in terms of our electron model, which may be summarized in the illustration below, which we borrow from G. Vassallo and A. Di Tommaso (2019). It shows how the Compton radius of an electron must decrease as it gains linear momentum. Needless to say, the plane of oscillation of the pointlike charge is not necessarily perpendicular to the direction of motion. In fact, it is most likely not perpendicular to the line of motion, which explains why we write the de Broglie relation as a vector equation: λL = h/p. Such vector notation implies h and p can have different directions: h may not even have any fixed direction! It might wobble around in some regular or irregular motion itself!

Celani and Vassallo

The illustration shows that the Compton wavelength (the circumference of the circular motion becomes a linear wavelength as the classical velocity of the electron goes to c. It is now easy to derive the following formula for the de Broglie wavelength:de Broglie wavelengthThe graph below shows how the 1/γβ factor behaves: it is the green curve, which comes down from infinity (∞) to zero (0) as goes from 0 to c (or, what amounts to the same, if β goes from 0 to 1). Illogical? We do not think so: the classical momentum p in the λL = h/p is equal to zero when v = 0, so we have a division by zero. Also note the de Broglie wavelength approaches the Compton wavelength of the electron when v approaches c, and that 1/γβ factor quickly reaches reasonable values: for β = 0.2, for example, 1/γβ is equal to 5, more or less. For higher velocities, the de Broglie wavelength is just three or two times the Compton wavelength—or less. Of course, a = 0.2c velocity is substantial but not uncommon in such experiments.

de Broglie wavelength

These are remarkable relations, based on which it should be possible to derive what we refer to as the equivalent of the Huygens-Fresnel equations for electron interference.

Indeed, as far as we know, that has not been done yet. We are not quite sure if it can be done: an analysis of the interactions between the incoming electron and the electrons in the material of the slits must be hugely complicated, and we need to answer several difficult questions—first and foremost this: how does the pointlike charge – as opposed to the electromagnetic oscillation which keeps the charge in its orbit – go through the slit(s)? It must do so as a single blob—as opposed to the electromagnetic fields, which may or may not split up so as to produce the interference pattern.

What? May or may not split up? They should split up, right? Maybe. Maybe not. We are not so sure. We are not so sure. If we refer to interference in the context of two slits and diffraction when only one slit is open, then it is pretty obvious that the interference pattern that is produced when the two slits are open looks very much like the superposition of the two diffraction patterns that are produced by the electrons coming out of the individual slits. So, no, we do not buy the standard story here. Sorry.

So… What to say? We’ve got good ideas here—a good explanation but, in physics, the question is not (only) how but: how, exactly? The Zitterbewegung interpretation of an electron explains how diffraction and interference of an electron (with itself and/or with other electrons) might work but Zitterbewegung theorists still have some work to do to explain the how exactly. We think it can be done, however, and we therefore hope this post may inspire some smart students! The math is probably quite daunting, but then it is a rather nice PhD topic, isn’t it? And a decent quantitative explanation (as opposed to our qualitative explanation here) would sure make waves! 🙂

Post scriptum: We should, perhaps, also add a few remarks on some of the likely technicalities for the calculations. The shape of the wave combines the characteristics of transverse and longitudinal waves. It may, therefore, be very difficult to model this. The combination of linear and circular motion probably also involves some combination of plane, cylindrical and spherical wave geometry. In our paper on the geometric interpretation of de Broglie wavelength, we actually distinguished three different wavelength concepts which can be related through Menaechmuslatus rectum formula. To this, one should then add the intricacies of diffraction and interference.

Fortunately, there is a lot more quantitative analysis material now: this is a link to a good 2019 article which, in turn, has a good bibliography with links to many other good articles. I find the research by Frabboni, Gazzadi, Grillo and Pozzi particularly interesting. The point is: you are probably not going to produce a decent classical mathematical model of what’s actually going on overnight! 🙂 But it should be possible: the fact that this 1/γβ factor quickly reaches reasonable values, is very encouraging!

At the same time, one has to carefully relate scales and electron energies. The kinetic energy of the electrons in the Nebraska-Lincoln experiment was 600 eV only, so the electrons were quite slow (to accelerate electrons to a velocity of 0.2c, you need to apply something like 11,000 eV). Again, the analysis is not going to be easy, but if you want to be a physicist, you should surely try your hand at it! 🙂

Oh—one more thing: you will say this blog post is all about QED—as opposed to the stated objective of this blog. Well… You are right, of course, but then my thought processes are not exactly linear. 🙂

Gauge theories

Sean Carroll is currently wrapping up a series of videos about the Biggest Ideas in the Universe. All of the usual hocus-pocus around quantum fields and quarks. The last (?) in this series – Idea No. 15 – is about gauge theories. It is one of those things: the multiplication of theoretical and mathematical concepts after WW II has been absolutely mind-boggling !

Any case, as an antidote, it is good to remind ourselves that – unlike other field theories (quantum field theories, to be precise) – we have one gauge only in electromagnetism – the Lorenz gauge – and it is not some weird metaphysical concept resulting from equally weird redundant degrees of freedom in our theory. No. The Lorenz gauge just pops when re-writing Maxwell’s equations in terms of four-vector potentials. That’s all. Nothing more, nothing less.

For a change, the Wikipedia article on it is very readable and straightforward: it also usefully links the unique (Lorenz) gauge for the QED sector to the concept of retarded potentials: traveling fields – and changes in static fields – travel at the speed of light. Any signal, in fact, will travel at the speed of light. We wrote about the implications of this in regard to de Broglie‘s concept of the matter-wave in earlier papers, so we will refer you there. 🙂

So should you or should you not invest in studying gauge theories? I don’t think so, but I’ll keep reading myself. I will keep you informed about what I learn (or not).