Lectures on Physics
Chapter I : Quantum Behavior
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
3 July 2020
The special problem we try to get at with these lectures is to maintain the interest of the very
enthusiastic and rather smart people trying to understand physics. They have heard a lot about how
interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern
ideas—and spend many years studying textbooks or following online courses. Many are discouraged
because there are really very few grand, new, modern ideas presented to them. Also, when they ask too
many questions in the course, they are usually told to just shut up and calculate. Hence, we were
wondering whether or not we can make a course which would save them by maintaining their
enthusiasm. This paper is a draft first chapter of such course.
Table of Contents
Table of Contents .......................................................................................................................................... 0
Preface .......................................................................................................................................................... 1
1-1 Particle mechanics .................................................................................................................................. 1
1-2 The nature of space and time ................................................................................................................. 3
1-3 Electromagnetism and relativity ............................................................................................................. 5
1-4 Matter-particles ...................................................................................................................................... 8
Electrons ................................................................................................................................................... 8
The anomalous magnetic moment ......................................................................................................... 10
The proton .............................................................................................................................................. 12
The idea of a strong(er) force ................................................................................................................. 14
The neutron ............................................................................................................................................ 17
The form factor ....................................................................................................................................... 20
1-5 Light-particles ....................................................................................................................................... 21
The size and shape of a photon .............................................................................................................. 22
Polarization and photon spin states ....................................................................................................... 24
1-6 Interference experiments ..................................................................................................................... 26
Interference between matter-particles .................................................................................................. 26
The double-slit experiment for electrons ............................................................................................... 27
Interference between light-particles ...................................................................................................... 32
1-7 The meaning of the wavefunction ........................................................................................................ 33
The ring current model and the elementary wavefunction ................................................................... 33
Modelling spin and antimatter ............................................................................................................... 34
The wavefunction in motion ................................................................................................................... 36
Conclusions ................................................................................................................................................. 37
This is an experiment. The special problem we try to get at with these lectures is to maintain the interest
of the very enthusiastic and rather smart people trying to understand physics. They have heard a lot
about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other
modern ideas—and spend many years studying textbooks or following online courses. Many are
discouraged because there are really very few grand, new, modern ideas presented to them. Also, when
they ask too many questions in the course, they are usually told to just shut up and calculate. Hence, we
were wondering whether or not we can make a course which would save them by maintaining their
The lectures here are not in any way meant to be a survey course: they are very serious. I thought it
would be good to re-write Feynman’s Lectures in a way that makes sure most would be able to
encompass (almost) everything that is in the lectures. This is the link to Feynman’s Preface to his
lectures, so you can see how my approach differs from his.
Of course, because we are re-writing Feynman’s lectures on quantum mechanics, we must assume you
have gone through a course in classical mechanics and electromagnetic theory. In fact, we will assume
you are already familiar with the basics of mainstream quantum mechanics. We cannot make any
compromises in this regard: you must cover your bases. Good luck⎯and please do mail your thoughts:
this is the first draft chapter only⎯so we need some encouragement ourselves!
1-1 Particle mechanics
Newton thought that light was made up of particles, but then it was discovered that it behaves like a
wave. Later, however (in the beginning of the twentieth century), it was found that light did indeed
sometimes behave like a particle. Historically, the electron, for example, was thought to behave like a
particle, and then it was found that in many respects it behaved like a wave.
Hence, the challenge is to find a description that takes account both of the wave- as well as of the
particle-like character of both matter- as well as light-particles. We may refer to both as wavicles but –
for historical reasons – this term did not become household language. Light-particles are known as
photons. Photons carry electromagnetic energy, but they do not carry charge. In contrast, matter-
particles always carry charge. If they are neutral – think of a neutron or an atom – they will carry both
positive and negative charges. We should, therefore, think of them as composite particles.
Elementary particles are stable. Composite particles consist of elementary particles and may be stable or
unstable. An atom is an example of a stable composite particle. A neutron is stable inside of the nucleus
but unstable as a free particle: it spontaneously disintegrates into a proton and an electron. This process
involves the emission of a neutrino, which ensures energy is conserved. We think of a neutrino as a
lightlike particle: it also carries energy but no charge.
Electrons and protons are elementary matter-particles. They are stable but not indestructible. High-
energy collisions between protons – or between protons and anti-protons – yield unstable particles
which disintegrate back into stable particles. Because they are unstable, such particles should not be
The nature of this energy is not electromagnetic, however. Electromagnetic energy is related to electromagnetic
forces. We may, therefore, think of the energy of a neutrino as being related to the stronger force inside of a
proton or a neutron.
referred to as particles but as transients or, when very short-lived, as resonances.
The Higgs particle is an example of an extremely short-lived resonance: its lifetime is of the order of
10−22 seconds. Even at the speed of light – which an object with an estimated rest mass of 125 GeV/c2
can never aspire to attain – it cannot travel any further than 0.3 femtometer (0.310−15 m) before it
disintegrates. Such distance is smaller than the radius of a proton, which is in the range of 0.83 to 0.84
fm. Labelling it as a particle is, therefore, hugely misleading. Likewise, quarks have also never been
directly observed or isolated. Their existence is and remains, therefore, a mere hypothesis, which we
will not entertain in these lectures because we have no need for it: high-energy physics studies
disintegration processes, which involve non-equilibrium states⎯and we will not study these in our
These high-energy collisions are interesting though because they show that protons must have some
internal structure. We think of such structure not in terms of quarks or gluons
, but in terms of the
motion of the elementary charge. Paul Dirac wrote the following on that:
“Quantum mechanics may be defined as the application of equations of motion to particles. […]
The domain of applicability of the theory is mainly the treatment of electrons and other charged
particles interacting with the electromagnetic field⎯a domain which includes most of low-
energy physics and chemistry.
Now there are other kinds of interactions, which are revealed in high-energy physics and are
important for the description of atomic nuclei. These interactions are not at present sufficiently
well understood to be incorporated into a system of equations of motion. Theories of them
have been set up and much developed and useful results obtained from them. But in the
absence of equations of motion these theories cannot be presented as a logical development of
the principles set up in this book.
We are effectively in the pre-Bohr era with regard to these other interactions. It is to be hoped
that with increasing knowledge a way will eventually be found for adapting the high-energy
theories into a scheme based on equations of motion, and so unifying them with those of low-
These words were written in 1958 but still ring true today.
If you want to know what we think of the quark hypothesis, we think this hypothesis results from an
unproductive approach to analyzing disintegration processes: Gell-Man and Kazuhiko Nishijima studied
disintegration processes of K-mesons back in the 1950s, and invented new quantities that are supposedly being
conserved in these processes. One of these quantities was referred to as strangeness (see the analysis of K-mesons
in Feynman’s Lectures). These strange new concepts then started to lead an even stranger life of their own.
See our remarks on the quark hypothesis in footnote 2. As for gluons, these are supposed to carry the strong
force. We see no need to invent new particles to carry forces: the concept of fields – electromagnetic or other –
should do. The idea of force-carrying particles resembles 19th century aether theory: there is no need for it, so why
should we entertain it?
Paul A.M. Dirac, The Principles of Quantum Mechanics, 4th edition (1958), p. 312.
1-2 The nature of space and time
When Max Born, Werner Heisenberg, Erwin Schrödinger, Louis de Broglie, and Niels Bohr had presented
their seminal papers at the 1927 Solvay Conference, Hendrik Antoon Lorentz remarked the following:
“I would like to draw your attention to the difficulties in these theories. We are trying to
represent phenomena. We try to form an image of them in our mind. Till now, we always tried
to do so using the ordinary notions of space and time. These notions may be innate; they result,
in any case, from our personal experience, from our daily observations. To me, these notions are
clear, and I admit I am not able to have any idea about physics without those notions. The image
I want to have when thinking physical phenomena has to be clear and well defined, and it seems
to me that cannot be done without these notions of a system defined in space and in time.
To me, the electron is a particle which, at any moment, must be at some specific point in space,
and if I think it should be somewhere else at the next moment, then I need to be able to think of
its trajectory, which is a line in space. And if that electron meets an atom and penetrates it and
if it, after several adventures, leaves that atom, then I need to have some theory in which that
electron conserves its individuality. In other words, I actually think of a trajectory of the same
electron within the atom. It may be difficult to develop such theory but, a priori, this should not
On the Uncertainty Principle in particular, he remarked that all of the actual or theoretical experiments
only proved that we have indeterminism from a practical point of view. He argued that he would like to
keep his belief in determinism as a scientific principle, at least. It is probably useful to literally quote his
last words here: “Can we not keep determinism as an object of faith? Why do we have to elevate
indeterminism to a philosophical principle?”
We take the side of H.A. Lorentz (and Einstein, of course) in these discussions. Heisenberg himself
initially preferred to use the German term Ungenauigkeit to describe the apparent randomness in our
mathematical description of Nature. Ungenauigkeit translates as imprecision and it is a concept that is
valid in classical as well as in quantum mechanics. It is just what it: an imprecision inherent to
measurement⎯as opposed to the weird metaphysical quality which Heisenberg would later claim it to
be and which, without any precise definition, physicists now refer to as ‘Uncertainty’, with a capital
letter that is used like the capital letter in ‘God’. We, therefore, will re-state Lorentz’s question as an
affirmative statement: there is no need whatsoever to elevate indeterminism to a philosophical principle.
The proceedings of the Solvay Conferences were published by the Free University of Brussels. There is, however,
no fully complete English translation of the earlier and most crucial conferences, notably those of 1921 and 1927.
While the reader will be able to find English translations of the papers, we find the questions and answers the
most interesting. We highlighted some of these in our paper on the history of quantum-mechanical ideas.
Scientific lore has it that Einstein lost out to Bohr when discussing a series of hypothetical experiments⎯thought
experiments, as they are usually referred to. We think there is no clear winner in these discussions and that, in any
case, Einstein’s private correspondence indicates he might have gotten tired of them. He probably had better
things to do: there were so many personal and historical events calling for his attention in the 1930s.
These sentences were really one of his last contributions to science: he died a few months later. the 1927
conference proceedings have both the sad announcement of his demise as well his interventions—such was the
practice of actually physically printing stuff at the time.
In fact, without the assumption of determinism at the most elementary level of analysis, science would
basically not be science: we would relegate it to the realm of beliefs. We all need to believe in
something, when doing physics or whatever else, but we should not confuse these beliefs with the
scientific approach. We will not engage in much philosophy here but just make a few remarks.
First, it is quite obvious that relativity theory has profoundly changed the meaning of the ideas of space
and time. We now know that they are profoundly related⎯because of the work of Maxwell, Lorentz,
To be precise, they are related through the more fundamental idea of motion. However,
while related, space and time are still very different, and we agree with Lorentz: both are essential when
expressing or explaining an idea in physics. So what can we say about them?
Stating that the concepts of space and time are related because of the more fundamental idea of
motion is stating the obvious: we already had such relation in Galilean relativity, so we should be more
specific. We will use special relativity theory to demonstrate the relativistic invariance of the argument
of the wavefunction at the end of this chapter, so we will not say too much about it at this point. We
only want to make a very simple remark on the arrow of time here⎯a general remark for which you do
not need to understand relativity theory. In fact, we should probably not be making this simple remark
but we feel there is so much humbug around the possibility of time reversal that we feel we should.
It is this: spacetime trajectories – or, to put it more simply
, motions – need to be described by well-
defined functions. That means that for every value of t (time), we should have one, and only one, value
of x (space).
The reverse, of course, is not true: a particle can travel back to where it was (or, if there is
no motion, just stay where it is). Hence, it is easy to see that the concepts of motion and time must be
related because this logic imposes the use of well-behaved functions to describe reality.
This is illustrated below: a pointlike particle which moves like what is show on the right-hand side
cannot exist because there are a few occasions here where the particle occupies multiple positions in
space at the same point in time. Now, some physicists may honestly believe that should actually be
possible, but we do not want to entertain such ideas and, therefore, wish them all the best.
Einstein was a great admirer of H.A. Lorentz, and did not hide his admiration. Conversely, the young Einstein
would not have been invited to the early Solvay Conferences if it were not for Lorentz insisting the organizers
should make him a member of the scientific committee.
We do not like the use of the term spacetime because it is usually not very clearly defined. We may use it as a
shorthand to refer to four-vector algebra.
The x should be a vector in three-dimensional space, of course: x = (x, y, z). However, we may consider motion in
one dimension only, or choose our reference frame such that the direction of motion coincides with the x-axis.
That is done quite often to simplify the calculations. The result can usually be generalized quite easily to also
encompass two- and three-dimensional motion. There is no such thing as four-dimensional physical space.
Mathematical spaces may have any number of dimensions but the notion of physical space is a category of our
mind, and it is three-dimensional: left or right, up or down, front or back. You can try to invent something else but
it will always be some combination of these innate notions. If you find something else, please let me know.
We do not want to use not-so-well-behaved functions to arrive at some kind of description of reality. The matter
is quite serious because it drove some of the brightest minds on Earth to madness.
Figure 1: A well- and a not-well behaved trajectory in spacetime
This shows that time must go in one direction only. We can play a movie backwards, but we cannot
reverse time. Think of this: a movie in which two like charges (say, two electrons⎯or two protons)
would attract rather than repel each other does not make sense. We, therefore, know this would be a
movie which was being played backwards, and we would say it is impossible: time cannot be reversed.
This intuition contrasts with the erroneous suggestion of Richard Feynman that we should, perhaps,
think of antimatter-particles as particles that travel back in time. It is nonsense. We will come back to
this as part of our discussions on the physical meaning of the wavefunction for antimatter particles, so
you should not worry too much about it now.
This rather simple reflection on the arrow of time also provides a fresh perspective on the discussions on
symmetry-breaking. However, these discussions are technical and, therefore, difficult so we will leave
them for the time being.
Instead, we will use your current attention span for another, more important,
technical matter: non-Galilean relativity and electromagnetism.
1-3 Electromagnetism and relativity
Relativity theory is not easy: the ideas of relativistic mass, relativistic length contraction and time
dilation are closely related, very profound, and non-intuitive. We must assume you know the formulas
and that you have done your utmost to try to understand these as best as you can⎯which is all you can
do: more evolved minds might find it easy to work with the formulas but they never become intuitive.
We cannot dwell on this. We just want to think about the relativity of electromagnetic fields. If possible,
you should review your courses on four-vector algebra in the context of electromagnetism!
will only make a few introductory remarks⎯because this is only an introductory chapter, after all!
The fundamental idea is this: we do not want to invent new concepts and, therefore, we will want to
analyze what might be going on inside of the atomic nucleus in terms of electromagnetic interactions
only. We do not say it can be done: we are just saying we want to try it
. Such analysis may be simply
We actually do not like the concept of spacetime very much. It reminds us too much of the old aether idea. Time
and space are surely related (through special and general relativity theory, to be precise) but they are not the
same. Nor are they similar. We do, therefore, not think that some ‘kind of union of the two’ will replace the
separate concepts of space and time any time soon, despite Minkowski’s stated expectations in this regard back in
1908. Grand statements and generalizations are not always useful in physics.
If you are very curious on this, we have some non-technical writing on broken symmetries which you may want
to check out.
If you need a reference, we recommend chapters 25 to 29 of Feynman’s lectures on electromagnetism (Volume
II of his Lectures).
We think mainstream physicists do not try hard enough in this regard. We like the comments of Doris Teplitz:
“The state of the classical electromagnetic theory reminds one of a house under construction that was abandoned
referred to as an electromagnetic theory of nuclear interaction and offer an easy explanation of the
attractive force between protons in terms of the electromagnetic force between ring currents⎯an idea
which is at the core of our (non-mainstream) understanding of quantum mechanics.
We will come back to this idea of ring currents in a moment.
Here, we just want to note that these
analyses are based on a more coherent and complete conceptualization of what might going on inside of
the nucleus by also focusing on the magnetic forces resulting from the regular motion of one or more
elementary (electric) charges inside⎯as opposed to a narrow focus on the electrostatic Coulomb force
resulting from static charges only: the fundamental idea here is that a nuclear lattice structure would
not only arise from the mere presence of the charges inside but also from the pattern of their motion.
Of course, your immediate reaction should be this: we are 2020 now⎯why wasn’t this done before?
Our honest answer is: we do not know. We suspect magnetic forces have traditionally been neglected in
because the magnitude of the magnetic field – and, therefore, of the force – is only 1/c
times that of the electric field. That is a mistake
which becomes obvious when considering the
(1) We can use natural time and distance units to ensure the numerical value of lightspeed does
not distort calculations in regard to the relative strength of both forces. Indeed, the large
numerical value (299,792,458) of c using the second and meter as time and distance units
effectively indicates we may want to think of the second as a rather large unit as compared
to the meter when thinking about the relativity of electric and magnetic fields
by its working workmen upon receiving news of an approaching plague. The plague was in this case, of course,
quantum theory.” (Electromagnetism: Paths to Research, 1982)
Easily accessible references are, for example, Bernard Schaeffer (2016) or Paolo Di Sia (2018). Di Sia relates the
approach to the new nuclear lattice effective field theory (NLEFT) but a good understanding of NLEFT requires very
advanced mathematical skills and a lot of time, which we do not have. The reader may verify this by having a quick
look at Timo Lähde and Ulf-G. Meißner’s new book: Nuclear Lattice Effective Field Theory (2019).
We may not only refer to rather mundane examples here, such as Feynman’s treatment of polarized light, which
considers the electric field vector only. A historically more spectacular example of neglect is Hideki Yukawa’s
invention of a new potential and, therefore, of a new force that is supposed to be stronger than the electrostatic
(repulsive) force between protons. Of course, the invention of a new force and a new potential should also come
with the invention of a new charge so as to explain the origin of this potential. However, physicists preferred the
quark-gluon model instead. We find this illogical: for more details, we may refer to one of our papers.
We think of it as a mistake rather than just an oversight or something incomplete because an analysis of
polarization which would also include the behavior of the magnetic field vector might provide a classical
explanation to one-photon Mach-Zehnder interference.
The definition of the speed of light as being equal to c = 299,792,458 m/s exactly results from the re-definition of
both the meter and the second in terms of the wavelength and cycle time of electromagnetic radiation. Of course,
such re-definition still requires a reference to a real-life electromagnetic oscillation. This reference is the frequency
of the light which is emitted or absorbed as a result of the transition of an electron between the two hyperfine
states that define the unperturbed ground state of the cesium-133 atom, which is defined as f = 9 192 631 770 Hz,
exactly (BIPM, 2019). We may, therefore, think of the cycle time (T = 1/f) and the wavelength ( = c·T = c/f) of this
radiation as the true base units of time and distance. The definition of Planck’s constant as being equal to h =
6.6260701510−34 J·s, exactly, may then be combined with the natural time unit to define a natural unit for energy
which, in turn, may be combined with the natural distance unit to define a natural unit for force (energy is defined
(2) “Magnetism and electricity are not independent things: they should always be taken
together as one complete electromagnetic field”
(3) The idea of a pointlike charge having no other attributes than its charge – a naked charge
with zero rest mass, in other words – implies that it must, theoretically, whizz around at the
speed of light.
This ensures that the β = v/c factor in the transformation formulas we use
when going from one reference frame to another is always equal to 1 and, therefore,
eliminate the need to think about measurement units when discussing the relativity of
electric and magnetic fields. In fact, the idea of the electric or magnetic force being more or
less important than the other completely vanishes here!
The point is this: such electromagnetic theories of nuclear interaction require us to think of the
magnetic moment of protons – and of neutrons too, of course – as being generated by an actual electric
current⎯as opposed to some intrinsic property requiring no further explanation or detail. It is, in fact, at
this point, where such theories start to diverge from the mainstream interpretation of quantum
: they require us to think of electrons and protons as tiny ring currents with an actual radius
and circumference in space, which is what we shall do.
It must also result in a dynamic view of the fields surrounding charged particles. Potential barriers – or
their corollary: potential wells – should, therefore, not be thought of as static fields: they vary in time.
They result from two or more charges moving around and creating some joint or superposed field which
varies in time. Hence, a particle breaking through a ‘potential wall’ or coming out of a potential ‘well’
may just be using a temporary opening corresponding to a very classical trajectory in space and in
That is why we think there is no need to invoke an Uncertainty Principle.
as a force over some distance). Needless to say, if we have a natural unit for energy, we get a natural unit for mass
from Einstein’s mass-energy equivalence relation. We then need to add the defined value of the elementary
charge (1.60217663410−19 C) so as to be able to calculate all other constants that one might need in quantum
Richard Feynman, Lectures on Physics, II-13-6 (the relativity of magnetic and electric fields).
Newton’s relativistically correct force law F = m·a implies even the smallest of small forces will give it infinite
acceleration. Such charge will, therefore, be photon-like in the sense that all of its mass will result from its motion.
However, photons do not carry charge, and we think of local circular motion here⎯as opposed to the linear
motion of light photons. This distinguishes matter from light.
The Copenhagen interpretation of quantum mechanics is, perhaps, not well defined but we interpret it as having
the Uncertainty Principle at its core, which amounts to saying we should not even try to look for hidden variables
to explain intrinsic properties – most notably inertial mass and magnetic moment – of elementary particles. The
Zitterbewegung interpretation of quantum mechanics explains the mass of an elementary particle as the
equivalent mass of the energy in the oscillation, and the magnetic moment as the magnetic moment of a ring
It also requires us to think of neutrons as, somehow, combining a proton and an electron⎯to explain why they
are electrically neutral but do have a magnetic moment of their own. The non-stability of neutrons outside of the
nucleus and other considerations make this a plausible hypothesis. We will come back to this so the reader should
not worry about it now.
We will talk about this in the next chapter, when we will be talking about the physical meaning of probability
amplitudes in the context of various two-state systems.
All of the considerations above may come across as being rather elementary. Taken together, however,
they amount to a coherent re-formulation of the basic principles of quantum physics – or a common-
sense or realist interpretation of the same – which we will try to summarize below.
The internal structure of the electron was revealed by Arthur Holly Compton’s 1923 scattering
experiment, which showed photons can either scatter elastically or inelastically from an electron. We
have inelastic scattering (Compton scattering) when the outgoing photon has a different frequency than
the incoming photon. The difference in energy is absorbed by the electron as kinetic energy. The
effective area of interference between photons and electrons is defined by the Compton radius of the
electron, which is defined by the motion of the elementary charge. The elementary charge has a radius
itself, which defines the effective area of interference for elastic scattering (Thomson scattering): there
is no frequency change here. Think of the elementary charge as some core within the electron.
The idea of an electron must, therefore, combine the idea of a charge and its motion. This combined
idea effectively accounts for both the particle- as well as the wave-like character of matter-particles. It
also explains the magnetic moment of the electron. In fact, the ring current or magneton model of an
electron was invented by the British chemist and physicist Alfred Lauck Parson (1915) to do exactly
A theoretical local oscillatory motion also emerged from Dirac’s wave equation for an electron,
which he wrote down in 1927. Erwin Schrödinger referred to it as a Zitterbewegung
, and Dirac
highlighted its significance at the occasion of his Nobel Prize lecture:
“It is found that an electron which seems to us to be moving slowly, must actually have a very
high frequency oscillatory motion of small amplitude superposed on the regular motion which
appears to us. As a result of this oscillatory motion, the velocity of the electron at any time
equals the velocity of light. This is a prediction which cannot be directly verified by experiment,
since the frequency of the oscillatory motion is so high, and its amplitude is so small. But one
must believe in this consequence of the theory, since other consequences of the theory which
are inseparably bound up with this one, such as the law of scattering of light by an electron, are
confirmed by experiment.” (Paul A.M. Dirac, Theory of Electrons and Positrons, Nobel Lecture,
December 12, 1933)
Unfortunately, Dirac confuses the idea of the electron and the naked charge here. The Zitterbewegung
charge is a charge only: it has no other properties. It has no rest mass, for example, and it must,
therefore, effectively move at the speed of light: Newton’s law tells us the slightest force on it will give it
The magnetic properties of the electron had just been discovered. We refer to Ernest Rutherford’s remarks on
Parson's ‘électron annulaire’ (ring electron) and the magnetic properties of the electron in his lecture on ‘The
Structure of the Electron’ at the 1921 Solvay Conference. The original version was written in English, but the
conference proceedings have the French-language version only. We warmly recommend a thorough reading of this
paper: although it was written 100 years ago, we think it is still very relevant. Rutherford’s analysis and
interpretation of Compton’s experiment, for example, is very visionary. We wrote about this in our paper on the
history of quantum-mechanical ideas.
Zitter refers to a rapid trembling or shaking motion in German.
The Zitterbewegung of the pointlike charge may be chaotic but can be modelled by the formula for
circular or tangential velocity:
The illustration below provides the simplest of simple visualizations of what might be going on.
Figure 2: The ring current model
We effectively think of an electron (and a proton) as consisting of a pointlike elementary charge –
pointlike but not dimensionless
- moving about at the speed of light around the center of its motion.
This pointlike charge must have some momentum⎯if only by virtue of its motion. This is not the
classical moment of the electron or proton as it is moving about classically⎯with an equally classical
velocity v. No. The momentum in this illustration is equal to p = mc: m is the relativistic mass of the
pointlike charge, whose rest mass itself is zero because the naked charge itself has no other properties
but its charge. We can use the idea of a centripetal force (F) keeping this charge in its orbit to prove this
effective mass will be half of the electron mass. The other half of the mass or energy of the electron is in
the oscillating field which must keep this charge, somehow, in its orbital motion.
Any (regular) oscillation has a frequency and a cycle time T = 1/f = 2π/ω. The Planck-Einstein relation (E
= h·f = ħ·ω) relates f and T to the energy (E) through Planck’s constant h (or, when using the reduced
form of Planck’s equation, ħ). This frequency formula then allows us to use the tangential velocity
formula to calculate the radius of this orbital motion:
This effectively corresponds to what we refer to as the Compton radius of an electron
paraphrasing Prof. Dr. Patrick LeClair, we can now understand as “the scale above which the electron
can be localized in a particle-like sense.”
It is now obvious that Louis de Broglie’s intuition in regard to
We will come back to this when discussing the (anomalous) magnetic moment.
We know this must sound rather fantastical but we ask you to just go along with it for the time being. We will
think about the nature of this force in later developments.
The Compton radius is equal to the reduced Compton wavelength: a = /2π.
See: http://pleclair.ua.edu/PH253/Notes/compton.pdf, p. 10.
the wave nature of matter-particles was correct. However, we should think of de Broglie’s concept of
the matter-wave as corresponding to orbital rather than linear motion.
We can, of course, also measure the scale of the pointlike charge because of the phenomenon of elastic
scattering (Thomson scattering). However, we cannot precisely localize the pointlike charge because of
its lightlike velocity: we can, therefore, only talk about it in terms of energy or mass densities. This scale
is the Thomson radius is the classical electron radius, and it is expressed as a fraction of the Compton
radius: α·a 2.818 fm.
This radius formula introduces the fine-structure constant α 0.0073. The fine-structure constant pops
up in other places too – most notably in the context of the discussions on the (in)famous anomalous
magnetic moment – and we will, therefore, quickly make some calculations on that, which we hope you
will find interesting.
Before we do these calculations, however, you should quickly note that the fine-
structure constant α and the electric and magnetic constants 0 and 0 are related to each other and to
the fine-structure constant as follows:
We mention this because you will need these formulas in subsequent chapters.
You can easily google
these results but do not waste too much time on it now: just make a mental note of it. Also note that we
get the second and third equation from combining the first with the definition of the fine-structure
The anomalous magnetic moment
A ring current generates a magnetic moment. We have a formula for that from electromagnetic theory:
the magnetic moment (μ) is equal to the current (I) times the area of the loop (πa2).
The current here is the elementary charge (qe) times the frequency (f), and the radius of the loop is the
Compton radius (a = ħ/mc), so we can write:
We used the f = c/2πa formula for the frequency: the frequency is the velocity divided by the
circumference of the loop. We can also use the Planck-Einstein relation to get the same result:
De Broglie was also wrong in modeling a particle as a wave packet, rather than as a single wave. See our paper
on de Broglie’s matter-wave for more details.
The measurement and explanation of the anomalous magnetic moment is said to be the ‘high-precision test’ of
quantum electrodynamics. We will show there is a simple classical explanation for it. So much for QED!
The reader should not confuse the magnetic constant μ0 with μ, which is a symbol we will use for the magnetic
moment of an electron in the following section. We will actually also use the μ symbol for the muon-electron but
the context will make clear what is what.
We can now use the values for qe, ħ and m to calculate the magnetic moment.
If you do that, you will
μ = 9.27401…10−24 J·T−1
However, the CODATA value – which represents the actually measured value – for the magnetic
moment is slightly larger:
μCODATA = 9.2847647043(28)10−24 J·T−1
The difference is the so-called anomaly
, which we can easily calculate as follows
The reader may or may not recognize this value: it is, effectively, equal to about 99.85% of Schwinger’s
factor: α/2π = 0.00116141… So why is it that the fine-structure constant pops up here once again?
The formulas for the theoretical value of the magnetic moment ( = qħ/2m) assume all of the charge is
concentrated in one mathematical point. In other words, they assume the Zitterbewegung charge has
no spatial dimension whatsoever. However, we know that is a mathematical idealization only: the
phenomenon of elastic scattering of photons tells us the radius of the charge must be of the order of
α·ħ/mc. We should, therefore, distinguish between a theoretical and an effective radius of the electron.
The illustration below shows why: If the zbw charge is effectively whizzing around at the speed of light,
and we think of it as a charged sphere or shell, then its center of charge will not coincide with a point on
its orbit. The effective radius of the orbit will, therefore, be slightly larger than the theoretical radius a.
Figure 3: The effective and geometric center of a charge in orbital motion
In fact, relativity theory tells us a sphere or shell will appear as a disk or a hoop because of relativistic
You should use the CODATA values as published by the US National Institute of Standards and Technology (NIST).
These take into account the latest measurements as well as the 2018 revision of the system of SI units.
After reading this section, you will understand the anomalous magnetic moment is not ‘anomalous’ at all: one
would expect to see it pop up in any interpretation of quantum mechanics ⎯in any interpretation which does not
assume charges have no dimension whatsoever. The mathematical assumption that a pointlike object has no
dimension is useful (it allows us to calculate stuff and get meaningful results) but it is what it is: a mathematical
You should watch out with the minus signs here – and you may want to think why you put what in the
denominator – but it all works out!
length contraction. Hence, the drawing is actually not correct: the plane of the disk should be
perpendicular to the direction of motion. If we denote the effective radius by r, we should also associate
a velocity v with it⎯and this velocity too will be slightly larger than the theoretical velocity c. We can
now apply the usual formulas for the magnetic moment to get the following result:
There is a crucial step here: we equated the anomaly to 1 + α/2π. Is that a good approximation? In a
first-order approximation, it is. In fact, the reader will probably have heard that Schwinger’s α/2π factor
explains about 99.85% of the anomaly, but it is actually better than. When using the CODATA value for
μr, we get a μr/μa ratio that is equal to 99.99982445% of 1 + α/2π. We think that is good enough to
validate our model. You can also calculate the velocity v from the equation. We do not have an effective
frequency, of course: do not forget we talk about the same electron here, so v and r are not really
different things: they are just a different way of looking at the same thing. Hence, you should be able to
see the logic of the following calculation and be able to explain its result:
Done! Let us move to the next: the proton.
A proton also carries the elementary charge but with a positive sign. When applying the a = ħ/mc radius
formula to a proton, we get a value which is 1/4 of the measured proton radius: about 0.21 fm, as
opposed to the 0.83-0.84 fm charge radius which was established by Professors Pohl, Gasparan and
others over the past decade.
We get the right radius from using a modified Planck-Einstein relation (E =
4hf = 4ħω) for the orbital frequency of the charge:
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. It is
much smaller (smaller than the radius we calculated for the Zitterbewegung charge) and – somewhat
– its mass is about 1,836 times that of the electron.
For the exact references and contextual information on the (now solved) ‘proton radius puzzle’, see our paper
on it: https://vixra.org/abs/2002.0160.
The a = ħ/mc = ħc/E formula tells us the Zitterbewegung or Compton radius is inversely proportional to the
energy (or the equivalent mass) of the elementary particle. This relation is, obviously, not as intuitive as the easy a
= c/ω or ω = c/a relations.
Why is that so? Why is a proton even smaller than the α·ae radius of the zbw charge? We do not know.
We may offer some reflections, however, which you may or may not find useful.
One set of reflections revolves around the idea of some strong(er) force inside of a proton. Indeed, most
Zitterbewegung theorists think of the ring current as some persistent (superconducting) current, whose
electromagnetic field keeps the motion going⎯an electromagnetic perpetuum mobile, in other words.
That makes sense⎯to some extent, at least. David Hestenes – a physicist who helped to revive what he
refers to as Zitterbewegung interpretation of quantum mechanics – summarized this as follows:
“The electron is nature's most fundamental superconducting current loop. Electron spin
designates the orientation of the loop in space. The electron loop is a superconducting LC
circuit. The mass of the electron is the energy in the electron's electromagnetic field. Half of it is
magnetic potential energy and half is kinetic.”
However, we must highlight a very fundamental problem with this interpretation⎯one that was
pointed out by another bright mind researching electron models. When we first got in touch with him
on these things, he immediately wrote this:
“I know many people who considered the electron as a toroidal photon
and do it up to now. I
also started from this model around 1969 and published an article in JETP in 1974 on it:
"Microgeons with spin". Editor E. Lifschitz prohibited me then to write there about
Zitterbewegung because of ideological reasons, but there is a remnant on this notion. There was
also this key problem: what keeps [the pointlike charge] in its circular orbit?”
Dr. Burinskii also noted that this fundamental flaw was the main reason why had abandoned the simple
Zitterbewegung model in favor of the much more sophisticated Kerr-Newman approaches to the
(possible) geometry of an electron. I am reluctant to make the move he made because I like simple
math, especially when it gives the right answers⎯more or less, at least.
However, his objection makes
a lot of sense: a superconducting current runs in a wire, and there are no such wires in free space. There
are also other problems with the interpretation of an electron as some superconducting current loop:
⎯ First, the theory of superconducting currents assumes electrons pair up. These pairs are referred
to as Cooper pairs and remind us of electron orbitals, which usually also consist of two electrons
with opposite spin so as to lower their joint energy.
Email from Dr. David Hestenes to this author dated 17 March 2019.
Dr. Burinskii agrees we should probably not refer to a photon-like charge. Photons do not carry charge. In fact,
some theorists – including David Hestenes, unfortunately – promote the idea that a photon must, somehow,
combine a positive and a negative charge, but we think there is no need for that. We think of the photon as an
electromagnetic oscillation: nothing more, nothing less. We will come back to that soon enough.
Email from Dr. Burinskii to this author dated 22 December 2018.
One needs very advanced math – including a good grasp of general relativity theory – to understand Kerr-
Newman geometries: these geometries also take the gravitational force into account for modeling particles.
⎯ Second, we have no unique solution here: superconducting loops can come in various sizes
what explains the different sizes of an electron and a proton then?
We think there must be some force inside electrons and protons, whose nature may not be
electromagnetic. To be precise, we think an electron may, perhaps, be modeled using the Planck-
Einstein relation and Maxwell’s equations only⎯but an explanation of protons, neutrons, and other
massive particles (we will briefly mention the muon below, for example) requires a different approach.
We must distinguish two very different ideas in this regard:
1. The idea of a strong(er) force, which we will explore in the next section; and
2. The idea of a different form factor: we have a formula for the radius of a proton which suggests
its angular momentum is four times that of an electron. Its moment of inertia must, therefore,
be quite different.
We will explore the latter idea when discussing the concept of spin. It does not do away with the idea of
a ring current but it substantially modifies it: instead of a circulating pointlike charge, we will want to
think of a rotating sphere or some other shape. You may want to briefly review some formulas you will
remember from your high school classes here: the formulas for the moment of inertia.
For a rotating point mass, or a circular loop or hoop, that formula is I = mr2. For a solid disk, it is I =
mr2/4. This 1/4 factor – this different form factor for a loop and a solid disk – might explain the 1/4
factor in the radius formula for a proton. However, we should not get too much ahead of ourselves. Let
us first look at this concept of a (centripetal) force inside of an elementary particle.
The idea of a strong(er) force
If we think in terms of some force holding the pointlike charge in its orbit, then we calculate this force
for the electron as being equal to about 0.106 N. This is the formula
That is a huge force at the sub-atomic scale: it is equivalent to a force that gives a mass of about 106
gram (1 g = 10−3 kg) an acceleration of 1 m/s per second!
We will now give you a small exercise and something to reflect about. The electron also comes in a more
massive but unstable version: the muon-electron.
The muon energy is about 105.66 MeV, so that is
When you google research on this, you will note some of the research insists such loops should respect the
Planck-Einstein relation or the quantization principle.
We have derived this formula elsewhere. The ½ factor is there because we think of the zbw charge as having an
effective (relativistic) mass that is 1/2 (half) of the total electron mass. We will come back to this.
We could also calculate field strengths and other magnitudes here, but we should not make this first
introductory chapter too long! We have more detail in our paper(s).
You may also have heard about the tau-electron but that is just a resonance with an extremely short lifetime, so
the Planck-Einstein relation does not apply: it is not an equilibrium state. To be precise, the energy of the tau
electron (or tau-particle as it is more commonly referred to
) is about 1776 MeV, so that is almost 3,500 times the
electron mass. Its lifetime, in contrast, is extremely short: 2.910−13 s only. We think the conceptualization of both
about 207 times the electron energy. Its lifetime is much shorter than that of a free neutron
than that of other unstable particles: about 2.2 microseconds (10−6 s). Now that is fairly long as
compared to other non-stable particles – all is relative! – and that may explain why we also get a
sensible result when using the Planck-Einstein relation to calculate its frequency and radius.
The CODATA value for the Compton wavelength of the muon is the following:
1.17344411010−14 m 0.00000002610−14 m
If you divide this by 2 - to get a radius instead of a wavelength, you get the same value: about 1.8710−15
m. Hence, our oscillator model seems to work for a muon as well! Why, then, is it not stable? We think it
is because the oscillation is almost on, but not quite.
In contrast, the exercise for the tau-electron does
not yield such sensible result: the theoretical a = ħ/mc radius does not match the CODATA value.
think this confirms our interpretation of the Planck-Einstein relation as modelling stable particles.
Why this digression? We can also use our model to calculate the centripetal force which must keep the
charge in its orbit for the muon-electron and the ratio of this force for the electron and the muon:
If a force of 0.106 N is pretty humongous, then a force that is 42,753 times as strong, may surely be called
a strong force, right?
What about the force inside of a proton? The proton mass is about 8.88 times that of the muon, and it is
about 2.22 times smaller. Once again, we get this strange 1/4 factor. The Planck-Einstein relation gives us
a frequency, but it also gives us the angular momentum of a (stable) particle. Hence, a proton is also
definitely not some more massive or stable antimatter version of a muon-electron⎯if only the Planck-
Einstein relation suggests its momentum is four times that of an electron or a muon-electron!
the muon- as well as the tau-electron in terms of particle generations is unproductive: stable and unstable
particles are, generally speaking, very different animals!
The mean lifetime of a neutron in the open (outside of the nucleus) is almost 15 minutes!
This presumed longevity of the muon-electron should not be exaggerated, however: the mean lifetime of
charged pions, for example, is about 26 nanoseconds (10−9 s), so that is only 85 times less.
The reader can verify this by re-calculating the Compton wavelength from the radius we obtain and the exact
CODATA values (with or without the last digits for the uncertainties) for the constants and variables. The reader
will see that the value thus obtained falls within the uncertainty interval of the CODATA value for the Compton
wavelength. The reader think about this result and its meaning as part of the exercise.
CODATA/NIST values for the properties of the tau-electron can be found here: https://physics.nist.gov/cgi-
bin/cuu/Results?search_for=tau. To go from wavelength to radius and vice versa, one should divide or multiply by
To calculate those forces, we used an oscillator model which we will come back to later.
We’ll just jot
down the formula for the proton:
A force of 4,532 N inside of a muon and a force of 89,349 N inside of a proton? We get nonsensical results
here, don’t we?
Maybe. Maybe not. A few back-of-the-envelope calculations reveal we should not be too worried we are
modelling a black hole here.
We will come back to these calculations later because – for the time being
– these are the questions to which we have no real answer to them. Here we will just note these
calculations crucially depend on the concept of the effective mass of the charge inside of these particles.
This concept of the effective mass is purely relativistic: it is the mass which the charge acquires because
(and only because) of its velocity (lightspeed). As such, the pointlike charge inside of particles makes one
think of a photon (a photon also acquires an effective mass because of its sheer velocity) but such
comparison does not go far: photons do not carry charge. Hence, you should probably forget about this
comparison for the time being⎯until we talk about the photon, which we will do soon enough.
Various calculations and arguments – both geometric as well as calculational – show that this effective
mass is equal to half of the total mass of the particle. Hence, the effective mass of the charge in an electron
is half of the mass of an electron, and it is half of the mass of the muon inside of a muon, and half of the
mass inside of a proton. Why is that so? The intuitive argument here is the energy equipartition theorem:
half of the energy of an elementary particle is in the kinetic energy of this charge, and the other half must
then be in the oscillation – electromagnetic, strong, or whatever it might be – which keeps this charge in
its orbital motion. Does this sound mysterious? It should⎯because it is!
We need to move on, but we want to make one final remark on this concept of an effective mass. You
may or may not have come across the idea of Wheeler’s idea of ‘mass without mass’. If so, one of such
‘mass without mass’ models is the idea of the mass of an electron being all electromagnetic. The basic
idea is to ‘assemble’ the elementary charge by bringing infinitesimally small charge fractions together.
The calculation involves an easy integral which we will not repeat here.
We will just present the grand
result. When calculating the electromagnetic mass or energy of a sphere of charge with radius a, you get
the following interesting formula
Feynman was puzzled by that ½ factor: where is the other half of the energy (or the mass) of the
electron? Our ring current model shows the ½ factor is quite logical: Feynman is assembling the zbw
If you cannot wait, you can read our paper on this oscillator model of an electron.
The more informed reader should calculate the Schwarzschild radius of a proton. He or she may also calculate
the related energy densities and electromagnetic field strength. They are humongous but not impossible.
We may refer to one of Feynman’s lectures on electromagnetism here.
You need the
charge here⎯not the electron as a whole. Hence, the missing mass is in the Zitterbewegung or
orbital/circular motion of the zbw charge. We can now derive the classical electron radius from the
As mentioned, we will come back to all of this. For the time being, we just want to plant these ideas in
your head so you can start thinking through them for yourself.
We think of photons, electrons, and protons – and neutrinos – as elementary particles. Elementary
particles are, obviously, stable. They would not be elementary, otherwise. Again, we repeat that the
difference between photons and neutrinos on the one hand, and electrons, protons, and other matter-
particles on the other, is that we think all matter-particles carry charge⎯even if they are neutral. A
neutron is an example of a neutral matter-particle. We know it is unstable outside of the nucleus but its
longevity – as compared to other non-stable particles – is quite remarkable: it survives about 15
minutes⎯for other unstable particles, we usually talk about micro- or nano-seconds, or worse!
Let us explore what it might be⎯if only to provide some kind of model for analyzing other unstable
particle, perhaps. We should first note that the neutron radius is about the same as that of a proton.
How do we know this? NIST only gives the rms charge radius for a proton based on the various proton
radius measurements. There is only a CODATA value for the Compton wavelength for a neutron, which
is more or less the same as that for the proton. To be precise, the two values are this.
neutron = 1.31959090581(75)10-15 m
proton = 1.32140985539(40)10-15 m
These values are just mechanical calculations based on the mass or energy of protons and neutrons
respectively: the Compton wavelength is, effectively, calculated as = h/mc.
However, you should, of
course, not only rely on CODATA values only: you should google for experiments measuring the size of a
neutron directly or indirectly to get an idea of what is going on here.
Let us look at the energies. The neutron’s energy is about 939,565,420 eV. The proton energy is about
938,272,088 eV. Hence, the difference is about 1,293,332 eV. This mass difference, combined with the
fact that neutrons spontaneously decay into protons but – conversely – there is no such thing as
spontaneous proton decay
, makes us think a neutron must, somehow, combine a proton and an
electron. The mass of an electron is 0.511 MeV/c2, so that is only about 40% of the energy difference,
but the kinetic and binding energy could make up for the remainder.
The reader should note that the Compton wavelength and, therefore, the Compton radius is inversely
proportional to the mass: a more massive particle is, therefore, associated with a smaller radius. This is somewhat
counterintuitive but it is what it is.
None of the experiments (think of the Super-Kamiokande detector here) found any evidence of proton decay so
The reader should note that the mass of a proton and an electron add up to less than the mass of a neutron,
So, yes, we will want to think of a neutron as carrying both positive and negative charge inside. These
charges balance each other out (there is no net electric charge) but their respective motion still yields a
small magnetic moment, which we think of as some net result from the motion of the positive and
negative charge inside.
We will come back to this later. As mentioned above, we want to limit ourselves to rather short
introductory descriptions of the stuff we are going to look at in much more detail later. Just make a
mental note of what you instinctively may or may not agree with, and why, and then you will see it will
all fall into place later. We should probably also say something about other composite matter-particles
here – like a nucleus or an atom – but we will also save that for later chapters: we cannot do everything
here, right? So let us move to the next.
Before we do so, however, we should quickly say a few words about the concept of spin. It is one of
those ill-defined concepts in physics and we should, therefore, tell you what we think about it.
The concepts of spin, magnetic moment and angular momentum and the gyromagnetic ratio are closely
related. Indeed, if we think of our charge having some effective or relativistic mass, then the particle as a
whole will have some real angular momentum. The gyromagnetic ratio can then be defined as the ratio
between the magnetic moment () and this angular moment (L). The same kind of geometric argument
we have used to derive the force can then be used to calculate this g-ratio.
Of course, we need to get
the units right, so what is it this unit for the g-ratio? If you do a dimensional analysis
, you will find it is
the charge per unit mass. We can, therefore, define the g-ratio like this:
We can now use the theoretical values we found for the electron to calculate this g-ratio for an electron.
It turns out to be equal to 1/2. You can quickly verify this:
We can repeat this exercise for the muon and the proton and – despite that weird 1/4 or 4 factor for the
proton – you will find you get the same ratio. Let us put them all together here.
which is why it is only logical that a neutron should decay into a proton and an electron. Binding energies – think of
Feynman’s calculations of the radius of the hydrogen atom, for example – are usually negative.
The reader should note this simpler definition of the g-ratio differs from the mainstream definition by a factor ½
because of the definition of the Bohr magneton, which also has this ½ factor. The origin of this factor is, once
again, related to the concept of the effective or relativistic mass of the charge, which is half of the mass of the
particle. In the context of this discussion (spin and g-ratios), we think this factor confuses rather than clarifies the
matter – literally – which is why we omit it. The mainstream physicist will cry wolf here but he should not: the
Bohr magneton will pop up again in a moment and we will see that it is, effectively, a meaningful unit. We just feel
one should not insert the Bohr magneton at the start of the analysis but present it as a result of the analysis
This may sound formidable but you should just write out the units for and L and see what you are missing to
be able to write g as a number without any physical dimension.
We may, therefore, say that the only meaningful g-factor that can be defined is really this: q/2m. It is,
effectively, the ratio between the magnetic moment and the angular momentum for all of the matter-
particles we looked at here, which are the electron, the muon, and the proton. That is why matter-
particles are referred to as spin-1/2 particles. It is weird but not as weird as some would like you to
believe: there is no magic here!
We should note something else that is oft forgotten: spin is an axial vector. Read this again: it is a vector,
and it is an axial vector.
This is related to another remark: we tend to think of the Planck-Einstein relation as a scalar equation
but we are actually missing something when we do so. Indeed, the E = h·f equation involves scalar
quantities only – notably energy
and frequency – but, in their reduced form, we should probably think
of Planck’s quantum and (angular) frequency as vectors, so we get a vector dot product
E = ħ·ω
This brings us, in turn, to the relation between the Planck-Einstein relation (E = h·f = ħ·ω) and the de
Broglie relation ( = h/p h = p·). We can, effectively, write Planck’s quantum of action as the product
of some energy and a cycle time (E = h·f h = E/f = E·T) but we can also write it as the product of some
There is enough magic elsewhere so we would rather tell you upfront what we do and what we do not
The reader should be familiar with the concepts of axial and polar vectors. Polar vectors (think of a position or
radius vector, for example) are sometimes referred to as real vectors, while axial vectors (think of angular
momentum or the magnetic moment) are often referred to as pseudovectors, but they are equally real in a
physical sense, of course! Their mathematical behavior is different, however. That is because they can be written
as the cross-product of two other vectors. Real vectors reverse sign when a coordinate axis is reversed.
Pseudovectors do not. We will let the reader google the nitty-gritty here. The definition crucially matters because
it is used in core quantum-mechanical arguments. We may refer, for example, to Feynman’s chapter on
symmetries and conservation laws.
With some imagination, you may think of potential energy as a quantity also involving the idea of direction
because we measure or define it as an energy difference between two points: we move a charge with or against a
force. However, this idea is not very productive in this context.
The boldface notation is subtle but powerful!
momentum (p) and some length⎯linear or circular, perhaps: think of the wavelength of light, or the
circumference of the orbital motion of the pointlike charge. If we denote such length by and, keeping
in mind that (linear) momentum is a vector too, we can write the de Broglie relation as a vector
In a similar vein, we will actually want to think of the reduced Planck constant (ħ = h/2π) as a proper
angular momentum, which can and should be written as ħ = I·ω: the product of an angular mass (the
rotational inertia I) and an orbital angular frequency (ω). This, then, also gives meaning to the concept of
spin (which is either up or down).
You should not there is no uncertainty in these concepts except for the uncertainty in regard to the
plane of oscillation (which is given by the direction of ħ and ω) in the absence of an external
electromagnetic field. Indeed, the oscillatory motion of the charge generates a classical magnetic
moment which – equally classically – will precess in an external electromagnetic field. Hence, it is only in
the absence of an electromagnetic field that we cannot know what the plane of oscillation will be. This is
quite consistent from an epistemological point of view: how would we define up or down, left or right,
back and front – space itself, actually – in the absence of an electromagnetic field?
The form factor
When we talked about the radius of a proton, we promised you we would talk some more about the
form factor. The idea is very simple: an angular momentum (L) can always be written as the product of a
moment of inertia (I) and an angular frequency (ω). We also know that the moment of inertia for a
rotating mass or a hoop is equal to I = mr2, while it is equal to I = mr2/4 for a solid disk. So you might
think this explains the 1/4 factor: a proton is just an anti-muon but in disk version, right? It is like a muon
because of the strong force inside, but it is even smaller because it packs its charge differently, right?
Maybe. Maybe not. We think probably not. Maybe you will have more luck when playing with the
formulas but we could not demonstrate this. First, we must note, once again, that the radius of a muon
(about 1.87 fm) and a proton (0.83-0.84 fm) are both smaller than the radius of the pointlike charge
inside of an electron (α·ħ/mec 2.818 fm). Hence, we should start by suggesting how we would pack
that into a muon first! Second, we noted this proton mass being 8.88 times that of the muon while the
radius is only 2.22 times smaller – so, yes, that 1/4 ratio once more – but these numbers are still weird:
even if we would manage to, somehow, make abstraction of this form factor by accounting for the
different angular momentum of a muon and a proton, we would probably still be left with a mass
difference we cannot explain in terms of a unique force geometry.
Perhaps we should introduce other hypotheses: a muon is, after all, unstable, and so there may be
another factor there: excited states of electrons are unstable too and involve an n = 2 or some other
number, so perhaps we can play with that too. Our answer to such musings is: yes, you can. And please
The symbol is obvious in the context of a linear wavelength but much less so when denoting a
circumference⎯which we want you to think of as a circular wavelength!
do let us know if you have more luck then us when playing with these formulas: it is the key to the
mystery of the strong force, and we did not find it⎯so we hope you do!
We should now talk about the nature of light-particles. In fact, we only know one: the photon. We
effectively think neutrinos are light-like too: we think that, just like photons, they carry energy too, but
the nature of this energy is apparently related to some strong force of which we know very little (we
refer to the discussion of forces inside particles as part of our proton model here).
Dirac was of the opinion that a lot of what we write above could not be verified directly by experiment
because “the frequency of the oscillatory motion is so high, and its amplitude is so small.” We are
effectively reflecting on very small dimensions, both in space as well as in time: distances as short as 0.2
or 0.3 femtometer (10−15 m), and times as short as 10−22 or 10−23 seconds. You will often read that we
cannot imagine how short such distances or time intervals actually are. We disagree. Imaging a particle
with no dimension whatsoever is an impossible task, but thinking of very short time or distance intervals
should not be a problem: if you can, somehow, imagine a radio wave, then you can also imagine the
extreme frequencies of gamma-rays or the Zitterbewegung of the electric charge.
So that is what we ask you do: think about extremely small or extremely large values, but do not try to
imagine zero-dimensional or infinite stuff. The finite speed of light (c) probably tells us the mathematical
concept of infinity is useful as a limiting idea but that, in Nature, actual infinities do probably not exist.
Likewise, forget about the idea of the electric charge having no dimension whatsoever. We already
talked about this. In fact, if you wonder what you should remember from this lecture, it is this: matter-
particles carry electric charge and they are not pointlike. If they have a shape, then it is probably disk-
But we have light-particles also, and if there is one particle which we should probably think of as being
pointlike, then it is the photon (and the neutrino⎯which we think of as the photon of the strong
), although you will soon see why we may think of them as being linear rather than pointlike
So let us quickly tell you how we think of the photon as a model of a light-particle⎯the light-
The reader may wonder why we do not say anything about the weak force. We do not think very highly of the
hypothesis of a weak force. We think a force theory that explains why charges stay together must also explain
when and how they separate. More fundamentally, we think a force works through a force field: the idea that
forces are mediated by virtual messenger particles sounds medieval to us: it reflects earlier aether theories, which
died because of Occam’s Razor principle: a theory should be parsimonious. The idea of a medium turned out to be
superfluous at the end of the 19th century. We think the idea of gluons and W/Z bosons are headed the same way:
no one has ever observed these things so why should we assume they actually exist?
You may think this contradicts our earlier statements on the idea of force-carrying particles and you are
right⎯to some extent, at least. We do not believe in W and Z bosons and gluons, but we do believe photons carry
electromagnetic energy: carrying energy and carrying force are slightly different concepts. Likewise, we think the
concept of a strong(er) force inside the nucleus is useful and, therefore, we feel the concept of neutrinos carrying
some strong(er) energy is useful too.
A photon has a (linear) wavelength, and that is not some mysterious mathematical quantity but an actual
physical length. However, relativity theory implies such lengths contract when moving at the speed of light: there
is, therefore, no contradiction in thinking of photons as being linear and pointlike at the same time!
particle, in fact. We refer to it as the one-cycle photon model and the argument goes like this.
The size and shape of a photon
1. Photons are very real and carry equally real energy: electromagnetic energy. When an electron goes
from one state to another – from one electron orbital to another, for example
– it will absorb or emit a
photon. Photons make up light: visible light, low-energy radio waves, or high-energy X- and γ-rays. These
waves carry energy and, yes, when we look really closely, these waves are made up of photons.
yes, the energy in the light is carried by the photons.
Saying they carry electromagnetic energy is something else than saying they carry electromagnetic force
itself. A force acts on a charge: a photon carries no charge. So what are they then? How should we think
of them? Think of it like this: a photon is an oscillating electromagnetic field. We describe this field by an
electric and a magnetic field vector E and B. Field vectors do not take up any space: think of them as a
force without a charge to act on. Indeed, a non-zero field at some point in space and time – which we
describe using the (x, y, x, t) coordinates – tell us what the force would be if we would happen to have a
unit charge at the same point in space and in time. You know the formula for the electromagnetic force:
it is the Lorentz force F = q·(E + vB). Hence, the electromagnetic force is the sum of two (orthogonal)
component vectors: q·E and q·vB.
The velocity vector v in the equation shows both of these two component force vectors depend on our
frame of reference. Hence, we should think of the separation of the electromagnetic force into an
‘electric’ (or electrostatic) and a ‘magnetic’ force component as being somewhat artificial: the
electromagnetic force is (very) real – because it determines the motion of the charge – but our cutting-
up of it in two separate components depends on our frame of reference and is, therefore, (very)
relative. We should refer to our remarks on the relative strength of the electric and magnetic field,
however: the reader should not think in terms of the electric or magnetic force being more or less
important in the analysis and always analyze both as aspects of one and the same reality.
Let us get back to our photon: we think the photon is pointlike because the E and B vectors that describe
it will be zero at each and every point in time and in space except if our photon happens to be at the (x,
In most cases, we will be talking about atoms emitting or absorbing light but that is not always the case. We
think Compton scattering may be explained conceptually by accepting the incoming and outgoing photon are
different photons (they have different wavelengths so it should not be too difficult to accept this as a logical
statement: the wavelength pretty much defines the photon⎯so if it is different, you have a different photon). This,
then, leads us to think of an excited electron state, which briefly combines the energy of the stationary electron
and the photon it has just absorbed. The electron then returns to its equilibrium state by emitting a new photon.
The energy difference between the incoming and outgoing photon then gets added to the kinetic energy of the
electron through the following law:
This physical law can be easily derived from first principles (see, for example, Patrick R. Le Clair, 2019): the energy
and momentum conservation laws, to be precise. More importantly, however, it has been confirmed
We know this because a zillion experiments did confirm the reality of the photoelectric effect.
y, z) location at time t.
[…] Please read the above again: our photon is pointlike because the electric and magnetic field
vectors that describe it are zero everywhere except where our photon happens to be.
2. At the same time, we know a photon is defined by its wavelength. So how does that work? What is
the physical meaning of the wavelength? It is, quite simply, the distance over which the electric and
magnetic field vectors will go through a full cycle of their oscillation. That is all there is to it: nothing
more, nothing less.
That distance is, of course, a linear distance: to be precise, it is the distance s between two points (x1,
y1, z1) and (x2, y2, z2) where the E and B vectors have the same value. The photon will need some time t
to travel between these two points, and these intervals in time and space are related through the
(constant) velocity of the wave, which is also the velocity of the pointlike photon. That velocity is, of
course, the speed of light, and the time interval is the cycle time T = 1/f. We, therefore, get the equation
that will be familiar to you:
We can now relate this to the Planck-Einstein relation.
3. Any (regular) oscillation has a frequency and a cycle time T = 1/f = 2π/ω. The Planck-Einstein relation
relates f and T to the energy (E) through Planck’s constant (h):
The Planck-Einstein relation does not only apply to matter-particles but also to a photon. In fact, it was
first applied to a photon.
Think of the photon as packing not only the energy E but also an amount of
physical action that is equal to h.
We have not talked much about the meaning of h so far, so let us do that now. Physical action is a
concept that is not used all that often in physics: physicists will talk about energy or momentum rather
than about physical action.
However, we find the concept as least as useful. Physical action can express
itself in two ways: as some energy over some time (E·T) or – alternatively – as some momentum over
some distance (p·). For example, we know the (pushing) momentum of a photon
will be equal to p =
E/c. We can, therefore, write the Planck-Einstein relation for the photon in two equivalent ways:
The application of the Planck-Einstein relation to matter-particles is implicit in the de Broglie relation.
Unfortunately, Louis de Broglie imagined the matter-wave as a linear instead of a circular or orbital oscillation. He
also made the mistake of thinking of a particle as a wave packet, rather than as a single wave! The latter mistake
then led Bohr and Heisenberg to promote uncertainty to a metaphysical principle.
We think the German term for physical action – Wirkung – describes the concept much better than English.
For an easily accessible treatment and calculation of the formula, see: Feynman’s Lectures, Vol. I, Chapter 34,
We could jot down many more relations, but we should not be too long here.
We said the photon
packs an energy that is given by its frequency (or its wavelength or cycle time through the c = f
relation) through the Planck-Einstein relation. We also said it packs an amount of physical action that is
equal to h. So how should we think of that? Let us connect all of the dots now.
4. The Planck-Einstein relation does not only apply to a photon, but it also applies to electron
orbitals⎯but in a different way. When analyzing the electron orbitals for the simplest of atoms (the
one-proton hydrogen atom), the Planck-Einstein rule amounts to saying the electron orbitals are
separated by an amount of physical action that is equal to h = 2π·ħ.
Hence, when an electron jumps
from one level to the next – say from the second to the first – then the atom will lose one unit of h. The
photon that is emitted or absorbed will have to pack that somehow. It will also have to pack the related
energy, which is given by the Rydberg formula:
To focus our thinking, let us consider the transition from the second to the first level, for which the 1/12
– 1/22 is equal 0.75. Hence, the photon energy should be equal to (0.75)·ER ≈ 10.2 eV. Now, if the total
action is equal to h, then the cycle time T can be calculated as:
This corresponds to a wave train with a length of (3×108 m/s)·(0.4×10-15 s) = 122 nm. It is, in fact, the
wavelength of the light (λ = c/f = c·T = h·c/E) that we would associate with this photon energy.
The reader may think all of the above is rather trivial. If so, then that is good: the reader should just
consider it as a warm-up for the math that follows. If not, then it is also good: it then means it was
useful to take you through this. Before we move on to the next, we should make a final remark in regard
to the spin of a photon.
Polarization and photon spin states
You may have heard that photons are part of a more general category of particles which are referred to
as bosons, while the above-mentioned matter-particles are generally referred to as fermions. The
difference between these two is either described in terms of their amplitudes having to be added with a
We may refer the reader to our manuscript, our paper on the meaning of the fine-structure constant, or various
others papers in which we explore the nature of light. We just like to point out one thing that is quite particular for
the photon: the reader should note that the E = mc2 mass-energy equivalence relation and the p = mc = E/c can be
very easily related when discussing photons. There is an easy mathematical equivalence here. That is not the case
for matter-particles: the de Broglie wavelength can be interpreted geometrically but the analysis is somewhat
more complicated⎯not impossible (not at all, actually) but just a bit more convoluted because of its circular (as
opposed to linear) nature.
The model of the atom here is the Bohr model. It does not take incorporate the finer structure of electron
orbitals and energy states. That finer structure is explained by differences in magnetic energies due to the spin
(angular momentum) of the electron. We will come back to this. Also note we take the most general of cases: a
photon being emitted or absorbed by an atom. Photons can also be emitted by free electrons in an excited state.
The basics of the analysis remain the same.
plus or a minus sign
or, else, in terms of their spin number being 1 or 1/2. In addition, mainstream
quantum physicists will tell you bosons mediate forces and should, therefore, be thought of as “force-
carriers.” Finally, you will also have heard about (1) the Pauli exclusion principle, which applies to
fermions only and which states two fermions can only be in the same quantum state if their spin is
opposite and (2) its opposite, which states any number of identical bosons can occupy the same state.
Let us go over these statements one by one.
1. We think the concept of force-carriers is superfluous at best, and plain nonsense at worst. We think
Feynman’s description of fermions and bosons in terms of adding amplitudes with a plus or a minus sign
is nonsense too: we should not expect electrons and photons to perform some kind of a handshake
when they meet each other to decide what sign they should apply to their interaction.
You should also
note that – even if we could imagine some mechanism for such handshakes – the situation becomes
extremely complicated in the context of three or more particles.
2. In contrast, the distinction in terms of their spin state makes sense⎯to some extent, at least. Indeed,
we have shown that the g-ratio for matter-particles is effectively equal to 1/2, even if their angular
momentum can come as a multiple of ħ (think of the proton here or, if you prefer not to trust our
proton model, the n = 2, 3,… electron orbital states). So what is that spin-one state of a photon? It has to
do with (1) the polarization of light and (2) the fact that, with light, we get two waves for the prices of
one, so to speak: one should analyze the behavior of both the electric as well as the magnetic field
Such analysis requires a bit of imagination but we trust we will be able to google a few things here.
one thing you should note, however, is that the mainstream definition of a photon as a spin-one particle
is not very consistent. A spin-one particle should have three possible states: 1 (up), −1 (down), or zero
(no spin). However, photons – which, we should remind you, are the only bosons known to exist from
real-life experiments – do not have a spin-zero state. Never ever. Their spin is always up or down. It is
never zero. Hence, we think there is no use for this fermion-boson distinction because the only boson we
know – the photon – does not behave like it should behave according to this theory.
3. The only thing that is left to explain now, is this Pauli exclusion principle and its opposite. Here, we
should note that electrons with opposite spin states are not identical: their magnetic moment will be
opposite and, hence, depending on their orientation, these magnetic moments will attract or repel each
other. We can, therefore, imagine two electrons with opposite spin states may lower their joint energy
in some geometric arrangement involving some alignment of their magnetic moments.
What about photons? Photons do not carry charge, and the superposition principle tells us we can
simply add fields. There is, therefore, no physical reason why one photon could not occupy the same
position as another one. It is not like two electric charges which must repel each other or – in case of an
See, for example, Feynman, Lectures on Quantum Mechanics, Chapter IV.
We could point out some more contradictions but we do not want to be lengthy here: we may refer the reader
to a more detailed analysis of Feynman’s Chapter 4.
If you need a reference, we will refer you, once again, to Feynman’s lectures, although the lectures on polarized
light and polarization states are a bit scattered.
electron and a positron coming together – which must annihilate each other because the charges are
Is that all there is? For the time being, yes. Again, we have to leave some of the analysis for later
, or we
would not have anything left to write about!
1-6 Interference experiments
Interference between matter-particles
Most of Feynman’s first chapter of lectures on quantum mechanics talks about the double-slit
experiment for electrons. When Feynman wrote this, back in 1963, such interference experiments with
matter-particles were a thought experiment only, and Feynman could not image technology would ever
advance enough to actually do this experiment. Feynman did coin the word nanotechnology,
however⎯and nanotechnology eventually allowed for such experiments to be actually demonstrated.
While previous experiments had provided sufficient indirect evidence for electron interference
first experiment which was done exactly as Feynman had imagined it, was carried out in 2012 only. It
was done by a now rather famous research group at the University of Nebraska–Lincoln (Bach, Pope,
Liou and Batelaan, 2012) and we will let you google its results and the subsequent analysis of these
results. Such analysis is complicated because it is done in terms of quantum-mechanical amplitudes⎯as
opposed to the wave-particle duality as modelled in our realist interpretation of quantum mechanics.
Hence, the equivalent of the Huygens-Fresnel equations
for electron interference is not yet available.
Because an analysis of the interactions between the incoming electron and the electrons in the material
of the slits is hugely complicated, we doubt whether such analysis will soon be available. However, we
do not doubt that our conceptualization of matter-particles – which combines both their wave- as well
particle-like character of both matter – fits the bill.
We are not saying it is easy, however. It all is and remains somewhat of a mystery: it is and remains hard
to explain what is going where exactly. Let us walk over the double-slit experiment for electrons to show
The nature of anti-matter is another mystery. We can clarify it, to some extent, by an analysis in terms of
opposite spacetime signatures (we will briefly say something about that in the last section of this chapter) but such
analysis does not provide a coherent explanation of the mystery of matter-antimatter pair creation and
annihilation. It is good some mystery is left⎯otherwise physics would be dead as a science!
We will talk some more about bosonic behavior in the next lecture or chapter of our little alternative course, in
the context of quantum-mechanical amplitudes in two-state systems (such as a maser).
Bach, Pope, Liou and Batelaan (2012) usefully highlight the experiments of Jönsson (1961), Merli, Missiroli and
Pozzi (1976), and Tonomura, Matsuda, Kawasaki and Ezawa (1989). However, these experiments did not involve
the build-up of the double-slit diffraction pattern by measuring (the impact of) one electron at a time.
We use this term rather loosely to refer to the principles and equations describing interference, diffraction, and
wave propagation both in the far- as well as in the near-field. The near-field refers to distances within which one
should apply an analysis in terms of light particles, while the far-field allows for a purely classical analysis in terms
of wave mechanics only.
The double-slit experiment for electrons
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 (Figure 4).
Figure 4: The actual double-slit experiment with electrons
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
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 meant. We, therefore, felt it was appropriate to revisit this subject as one of
our very first entries for our new Reading Einstein blog, which wants to probe a bit deeper. This section
of our paper largely reflects that entry.
This and other illustrations are taken from the original 2012 University of Nebraska-Lincoln double-slit
experiment with electrons article.
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.
Figure 5: Diffraction and interference patterns (double-slit experiment with electrons)
The one-by-one firing of the electrons is, without any doubt, the most remarkable thing about the whole
experiment. Why? 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
This and other illustrations are taken from the original 2012 University of Nebraska-Lincoln double-slit
experiment with electrons article.
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.
Figure 6: Diffraction and interference patterns (double-slit experiment with electrons)
So the challenge is to explain this interference pattern in terms of our electron model, which may be
summarized in the illustration below (). It shows how the Compton radius of an electron must decrease
as it gains linear momentum. Why is that so? Because the speed of light is the speed of light: the
pointlike charge cannot travel any faster if we are adding a linear component to its motion. Hence, some
of its lightlike velocity is now linear instead of circular and it can, therefore, no longer do the original
orbit in the same cycle time.
This and other illustrations are taken from the original 2012 University of Nebraska-Lincoln double-slit
experiment with electrons article.
Figure 7: The Compton radius must decrease with increasing velocity
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!
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
The graph below shows how the 1/γβ factor behaves: it is the green curve, which comes down from
infinity (∞) to zero (0) as v 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 only if v approaches c.
We borrow this illustrations from G. Vassallo and A. Di Tommaso (2019).
You should do some calculations here. They are fairly easy. If you do not find what you are looking for, you can
always have a look at Chapter VI of our manuscript.
Figure 8: The 1/γ, 1/β and 1/γβ graphs
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. As far as we know, that has not
been done yet. We think it can be done, but it is not going to be easy: 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
should split up so as to produce the interference pattern.
In short, the Zitterbewegung interpretation of an electron explains the diffraction and interference of an
electron (with itself and/or with other electrons) but Zitterbewegung theorists still have some work to
do to explain how exactly. Again, we are optimistic, and so we think it can be done. We therefore hope
this lecture 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! :-)
We used the free desmos.com graphing tool for these and other graphs.
Interference between light-particles
The equivalent of the double-slit experiments for electrons going through the slit(s) one at a time, is the
one-photon Mach-Zehnder interference experiment. This experiment typically involves two beam
splitters and two mirrors, as illustrated below.
Figure 9: The Mach-Zehnder interferometer
Both popular as well as academic accounts of this experiments wax lyrical about the mystery of the
results of this experiment but usually forget to mention the obvious: when beam splitters split a beam,
they produce two linearly polarized waves. The energy of the incoming beam is, therefore, also split.
There is no reason whatsoever to assume such energy split cannot possibly happen when one single
photon goes through the splitter. This provides a rather classical explanation of what might be going on
in these experiments:
1. The incoming photon is circularly polarized (left- or right-handed).
2. The first beam splitter splits our photon into two linearly polarized waves.
3. The mirrors reflect those waves and the second beam splitter recombines the two linear
waves back into a circularly polarized wave.
4. The positive or negative interference then explains the binary outcome of the Mach-
Zehnder experiment – at the level of a photon – in classical terms.
Are we sure about this? Our honest answer is this: no, but we are quite sure. Why are we so sure?
Because of the energy conservation law. However, because we still have more than one chapter to
write, we will save the full-blown development of this argument for a later lecture. In the meanwhile,
you may want to think about it yourself.
In case our reader would not be able to wait till we release the next chapter(s) – or would need some inspiration
to start thinking about it – we can refer him or her to our paper on Mach-Zehnder interference, which already
sketches the contours of the argument.
1-7 The meaning of the wavefunction
The ring current model and the elementary wavefunction
We will talk a lot about wavefunctions and probability amplitudes in the next section, so we will be brief
here. When looking at Figure 2, it is obvious that we can use the elementary wavefunction (Euler’s
formula) to represents the motion of the pointlike charge by interpreting r = a·eiθ = a·ei·(E·t − k·x)/ħ as its
position vector. The coefficient a is then, equally obviously, nothing but the Compton radius a = ħ/mc.
The relativistic invariance of the argument of the wavefunction is then easily demonstrated by noting
that the position of the pointlike particle in its own reference frame will be equal to x’(t’) = 0 for all t’.
We can then relate the position and time variables in the reference frame of the particle and in our
frame of reference by using Lorentz’s equations
When denoting the energy and the momentum of the electron in our reference frame as Ev and p =
m0v, the argument of the (elementary) wavefunction a·ei can be re-written as follows
E0 is, obviously, the rest energy and, because p’ = 0 in the reference frame of the electron, the
argument of the wavefunction effectively reduces to E0t’/ħ in the reference frame of the electron itself.
Besides proving that the argument of the wavefunction is relativistically invariant, this calculation also
demonstrates the relativistic invariance of the Planck-Einstein relation when modelling elementary
This is why we feel that the argument of the wavefunction (and the wavefunction itself) is
When discussing the concept of probability amplitudes, we will talk about the need to normalize them because
the sum of all probabilities – as per our conventions – has to add up to 1. However, the reader may already
appreciate we will want to talk about normalization based on physical realities⎯as opposed to unexplained
mathematical conventions or quantum-mechanical rules.
We can use these simplified Lorentz equations if we choose our reference frame such that the (classical) linear
motion of the electron corresponds to our x-axis.
One can use either the general E = mc2 or – if we would want to make it look somewhat fancier – the pc = Ev/c
relation. The reader can verify they amount to the same.
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.
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
Modelling spin and antimatter
A good theory should respect Occam’s Razor⎯the lex parsimoniae: one should not multiply concepts
without necessity. The need for new concepts or new principles – such as the conservation of
strangeness, or postulating the existence of a new force or a new potential
– should, therefore, be
continuously questioned. Conversely, when postulating the existence of the positron in 1928 – which
directed experimental research to a search for it and which, about five years later, was effectively found
to exist – Paul Dirac unknowingly added another condition for a good theory: all of the degrees of
freedom in the mathematical description should map to a physical reality.
It is, therefore, surprising that the mainstream interpretation of quantum mechanics does not integrate
the concept of particle spin from the outset because the + or − sign in front of the imaginary unit (i) in
the elementary wavefunction (a·e−i· or a·e+i·) is thought as a mathematical convention only. This non-
used degree of freedom in the mathematical description then leads to the false argument that the
wavefunction of spin-½ particles has a 720-degree symmetry. Indeed, physicists treat −1 as a common
phase factor in the argument of the wavefunction.
However, we should think of −1 as a complex
number itself: the phase factor may be +π or, alternatively, −π: when going from +1 to −1 (or vice versa),
See our paper on matter-waves, amplitudes and signals.
We think of the invention of the concept of strangeness by Murray Gell-Man and Kazuhiko Nishijima in the 1950s
here. This concept started a rather strange life of its own and would later serve as the basis for the quark
hypothesis which – for a reason we find even stranger than the concept of strangeness itself – was officially
elevated to the status of a scientific dogma by the Nobel Prize Committee for Physics.
As for the invention of a new force or a new potential, we are, obviously, referring to the Yukawa potential. This
hypothesis – which goes back to 1935 – might actually have been productive if it would have led to a genuine
exploration of a stronger short-range force on an electric charge⎯or, if necessary, the invention of a new charge.
Indeed, if the electromagnetic force acts on an electric charge, it would be more consistent to postulate some new
charge – or some new wave equation, perhaps – matching the new force. Unfortunately, theorists took a whole
different route. They invented a new aether theory instead: it is based on the medieval idea of messenger or
virtual particles mediating forces. The latter led to the invention of gluons which – as a concept – we find at least
as weird as quarks.
Mainstream physicists therefore think one can just multiply a set of amplitudes – let us say two amplitudes, to
focus our mind (think of a beam splitter or alternative paths here) – with −1 and get the same physical states.
it matters how you get there⎯as illustrated below.
Figure 10: e+iπ e−iπ
Combining the + and − sign for the imaginary unit with the direction of travel, we get four mutually
exclusive structures for the electron wavefunction:
Spin and direction of travel
Spin up (J = +ħ/2)
Spin down (J = −ħ/2)
* = exp[−i(kx−t)] = exp[i(t−kx)]
χ = exp[−i(kx+t)] = exp[i(t−kx)]
χ* = exp[i(kx+t)]
Table 1: Occam’s Razor: mathematical possibilities versus physical realities (1)
We may now combine these four possibilities with the properties of anti-matter. Indeed, we think
antimatter is different from matter because of its opposite spacetime signature. The logic here is the
following. Consider a particular direction of the elementary current generating the magnetic moment
(we effectively define spin as an (elementary) current
). It is then quite easy to see that the magnetic
moment of an electron (μ = −qeħ/2m) and that of a positron (μ = +qeħ/2m) would be opposite. We may,
therefore, associate a particular direction of rotation with an angular frequency vector ω which −
depending on the direction of the current − will be up or down with regard to the plane of rotation.
We can, therefore, associate this with the spin property, which is also up or down.
We, therefore, have another table with four mutually exclusive possibilities, which we should combine
with the possible directions of travel in Table 1
The quantum-mechanical argument is technical, and I did not reproduce it in this book. I encourage the reader
to glance through it, though. See: Euler’s Wavefunction: The Double Life of – 1. Note that the e+iπ e−iπ expression
may look like horror to a mathematician! However, if he or she has a bit of a sense for geometry and the difference
between identity and equivalence relations, there should be no surprise. If you are an amateur physicist, you
should be excited: it is, effectively, the secret key to unlocking the so-called mystery of quantum mechanics.
Remember Aquinas’ warning: quia parvus error in principio magnus est in fine. A small error in the beginning can
lead to great errors in the conclusions, and we think of this as a rather serious error in the beginning!
We are aware this may sound shocking to those who have been brainwashed in the old culture. If so, make the
switch. It should not be difficult: a magnetic moment – any magnetic moment, really – is generated by a current.
The magnetic moment of elementary particles is no exception.
To determine what is up or down, one has to apply the ubiquitous right-hand rule.
The use of the subscripts in the magnetic moment may be confusing, but should not be: we use −e for an
electron and +e for a positron. We do so to preserve the logic of denoting the (positive) elementary charge as qe
(without a + or a − in the subscript here).
μ−e = −qeħ/2m
μ−e = +qeħ/2m
μ+e = +qeħ/2m
μ+e = −qeħ/2m
Table 2: Occam’s Razor: mathematical possibilities versus physical realities (2)
Table 2 shows that (1) the ring current model also applies to antimatter but that (2) antimatter has a
different spacetime signature. Abusing Minkowski’s notation, we may say the spacetime signature of an
electron would be + + + + while that of a positron would be + − − −.
Table 1 and Table 2, therefore,
complement each other.
The wavefunction in motion
The geometry of a matter-particle can be related to the geometry of the photon using the following
illustration, which shows how the Compton wavelength (the circumference of the Zitterbewegung of the
pointlike charge) becomes a linear wavelength as the classical velocity goes from 0 to c: the radius of the
circulatory motion must effectively diminish as the electron gains speed.
Figure 11: The Compton radius must decrease with increasing velocity
The illustration above is, of course, didactic only: there is no reason to assume perfect perpendicularity
between the plane of the ring current and the linear motion of the electron: the plane of oscillation
should be taken to be random or – when an external magnetic field is applied – to correspond to
Larmor’s precessional motion of the angular momentum vector.
We should also reiterate that the fundamental Zitterbewegung is probably be chaotic or irregular:
In case the reader wonders why we associate the + +++ signature with the positron rather than with the
electron, the answer is: convention. Indeed, if I am not mistaken (which may or may not be the case), it is the + +++
metric signature which is the one which defines the usual righthand rule when dealing with the direction of electric
currents and magnetic forces.
We thank Prof. Dr. Giorgio Vassallo and his publisher to let us re-use this diagram. It originally appeared in an
article by Francesco Celani, Giorgio Vassallo and Antonino Di Tommaso (Maxwell’s equations and Occam’s Razor,
November 2017). Once again, however, we should warn the reader that he or she should imagine the plane of
oscillation to rotate or oscillate itself. He should not think of it of being static – unless we think of the electron
moving in a magnetic field, in which case we should probably think of the plane of oscillation as being parallel to
the direction of propagation. To be precise, he should think of it as precessing in the external field.
perfect circularity or perfect linearity may exist in our mind only. However, our measurements give
statistically regular results and, hence, “though this be madness, yet there is method in ‘t”
and we can,
therefore, meaningfully associate a frequency ω = c/a and a radius a = c/ω to the Zitterbewegung
motion – whether it be perfectly or not-so-perfectly circular or regular – which, using natural time and
distance units (c = 1) are nothing but the reciprocal of each other.
This is not a mainstream introduction to quantum physics.
We just connected some dots in a clearly
emerging picture of what Paul Ehrenfest – in one of the letters he wrote to Einstein a few months before
he ended his life – referred to as the ‘unendlicher Heisenberg-Born-Dirac-Schrödinger Wurstmachinen-
We hope you have as much fun reading as we had while writing. Let me know what makes sense to you
and what does not. It may motivate us to further develop this into an actual (text)book.
The reader will recognize the quote from Polonius in The Tragedy of Hamlet (Shakespeare), Act 2 Scene 2.
The a = ħ/mc = ħc/E formula effectively tells us the Zitterbewegung or Compton radius is inversely proportional
to the energy (or the mass) of the elementary particle. This inverse proportionality is a simple but profound
However, most of what we write about is actively being researched by other scientists⎯and in much more
detail than what we presented here. We could, once again, refer to Schaeffer (2016), Di Sia (2018) and Celani,
Vassallo and Di Tommaso (2017) but you may also want to google the research of David Hestenes (the
Zitterbewegung interpretation of quantum physics) or, far more advanced, Alex Burinskii (Dirac-Kerr-Newman
models of the electron).
We will let you find a good translation. For the context, including Ehrenfest’s tragic final act, we may refer to
one of our blog pieces.