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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s