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The difference between a theory, a calculation, and an explanation

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In this paper, we try to show where and why quantum mechanics went wrong-and why and when the job of both the academic physicist as well as the would-be student of quantum mechanics turned into calculating rather than explaining what might or might not be happening. Modern quantum physicists effectively resemble econometrists modeling input-output relations: if they are lucky, they will get some kind of mathematical description of what goes in and what goes out, but the math does not tell them how stuff actually happens. To show what an actual explanation might look like, we bring the Zitterbewegung electron model and our photon model together to provide a classical explanation of Compton scattering of photons by electrons so as to show what electron-photon interference might actually be: two electromagnetic oscillations interfering (classically) with each other. While developing the model, we also offer some reflections on the nature of the Uncertainty Principle. Finally, we also offer a brief history of the bad ideas which led to the current mess in physics.
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The difference between
a theory, a calculation, and an explanation
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil, 25 April 2020
Summary
In this paper, we try to show where and why quantum mechanics went wrong and why and when the
job of both the academic physicist as well as the would-be student of quantum mechanics turned into
calculating rather than explaining what might or might not be happening. Modern quantum physicists
effectively resemble econometrists modeling input-output relations: if they are lucky, they will get some
kind of mathematical description of what goes in and what goes out, but the math does not tell them
how stuff actually happens.
To show what an actual explanation might look like, we bring the Zitterbewegung electron model and
our photon model together to provide a classical explanation of Compton scattering of photons by
electrons so as to show what electron-photon interference might actually be: two electromagnetic
oscillations interfering (classically) with each other. While developing the model, we also offer some
reflections on the nature of the Uncertainty Principle.
Finally, we also offer a brief history of the bad ideas which led to the current mess in physics.
Contents
The characteristics of a good theory ............................................................................................................................. 1
Positive and negative energy states .............................................................................................................................. 1
Positive and negative energy wells ................................................................................................................................ 4
The nature of antimatter ............................................................................................................................................... 6
The Emperor wears no clothes: matter does not dissipate ........................................................................................... 8
The (Un)Certainty Principle ......................................................................................................................................... 12
The meaning of probability amplitudes and state vectors .......................................................................................... 14
The meaning of spin in quantum mechanics ............................................................................................................... 15
The meaning of conjugating amplitudes, wavefunctions and matrices ...................................................................... 20
A classical explanation of Compton scattering ............................................................................................................ 23
Energy and momentum conservation ..................................................................................................................... 23
What might actually be happening ......................................................................................................................... 25
The Compton wavelength as a circumference ........................................................................................................ 26
Probing electrons with photons .............................................................................................................................. 28
The electron-photon excitation: a temporary spin-2 particle? ............................................................................... 30
Planck’s quantum of action as a vector ....................................................................................................................... 31
Conclusions .................................................................................................................................................................. 36
A brief history of bad ideas .......................................................................................................................................... 37
Annex: Light, photons and the Mach-Zehnder interference experiment ................................................................... 39
Polarization states based on the electric field vector only ..................................................................................... 39
The relative strength of the electric and magnetic field ......................................................................................... 40
The geometry of a polarized photon ....................................................................................................................... 42
What happens to the energy when there is destructive interference? .................................................................. 44
1
The difference between
a theory, a calculation, and an explanation
The characteristics of a good theory
The science of physics probes reality physically as well as logically. Physical exploration involves
experiments, measurements and discoveries. Logical progress occurs when a new theory or explanation
replaces previously held beliefs and convictions. Such theory or explanation should be logically and
mathematically consistent. The mathematical description, for example, should respect Occam’s
Razorthe 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
1
should, therefore, be continuously questioned.
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.
The mention of Dirac’s hole theory makes us think we could, perhaps, usefully present the idea of
energy states in a rather intuitive way before requesting the reader to wrestle his or her way through
the nitty-gritty of this paper.
Positive and negative energy states
Dirac’s positron hypothesis is often said to be based on a hole theory. Dirac himself, however, phrased
his theory of antimatter in terms of negative energy states. It is probably useful to quote his original
brief writing on it: “The wave equation for the electron admits of twice as many solutions as it ought to,
half of them referring to states with negative values for the kinetic energy.”
2
The concept of negative energy is, of course, nothing special in physicsbut negative energy will usually
1
We are, obviously, referring to the invention of the concept of strangeness by Murray Gell-Man and Kazuhiko
Nishijima in the 1950s. Feynman’s treatment of it in his 1963 Lectures on physics shows that the concept of
strangeness and the related conservation law is a rather desperate assumption to explain the decay of K-
mesons (kaons). Unfortunately, the concept of strangeness started a strange life of its own and would serve as the
basis for the quark hypothesis which for a reason I 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 chargeor, 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.
2
Paul A.M. Dirac, Principles of Quantum Mechanics, 4th revised edition (1958), section 73: Theory of the positron.
2
be associated with potential energy rather than with kinetic energy.
3
The question, therefore, becomes:
how can kinetic energy be negative? Dirac was very much puzzled by it. Indeed, he refers to his wave
equations as equations of motion and the question, therefore, becomes: how can motion be negative?
The answer is this: motion cannot be negative but its direction is that of the force, so its direction can be
this or that which, using our right-hand rule or Minkowski’s metric signature convention, we call positive
or negative. We will explain this in more detail later but, ultimately, the reader should think of energy as
a force over some distance. The motion may be linear but, as Dirac noted in his Nobel Lecture, “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.”
4
Dirac is talking about the spin property here, and spin is either in this or in the opposite direction. This
explains Dirac’s negative kinetic energy. It is not negative, really. It just appears because of a deep
conceptual flaw in the mainstream interpretation of quantum mechanics. We will come back to this
later, but the gist of the matter is this:
The mainstream interpretation of quantum mechanics does not integrate the concept of particle spin
from the outset because it thinks of the + or sign in front of the imaginary unit (i) in the elementary
wavefunction (a·ei· or a·e+i·) as a mathematical convention only.
Indeed, most introductory courses in quantum mechanics will show that both a·ei· = a·ei·(tkx) and
a·e+i· = a·e+i·(tkx) are acceptable waveforms for a particle that is propagating in a given direction (as
opposed to, say, some real-valued sinusoid). One would then expect that the professors would proceed
to provide some argument showing why one would be better than the other, or some discussion on why
they might be different but, no! That is not the case. The professors usually conclude that “the choice is
a matter of convention” and, that “happily, most physicists use the same convention.”
5
We are totally dumbfounded by this. All of the great physicists knew all particles elementary or not
3
The idea is usually this: the potential energy of a charge at some large distance (infinity, for calculational
purposes) is defined as zero. When bringing that charge closer to some other charge, work is done with or against
the force. Its energy is, accordingly, negative or positive. The potential energy of an electron in an electron orbital,
for example, is, obviously, negative.
4
Paul A.M. Dirac, Theory of electrons and positrons, Nobel Lecture, 12 December 1933. Needless to say, we
associate the elementary wavefunction with this motion. To be precise, we think the a·ei· or a·e+i· function gives
us the spatial position of the oscillating elementary charge at every point in time. In other words: the elementary
wavefunction represents what Erwin Schrödinger referred to as the Zitterbewegung of the electron. The
Zitterbewegung of an electron is a physical application of Wheeler’s ‘mass without mass’ idea: all of the energy
(and, therefore, all of the equivalent mass) of the electron is in the oscillatory motion of the Zitterbewegung (zbw)
or elementary charge. This explains the dual wave-particle character of an electron and, as such, it also explains
diffraction and the interference of an electron with itself when being forced through a single or double slit. As
mentioned, we will come back to this later in the text and the reader should, therefore, just make a mental note of
this and proceed his or her reading.
5
In case you wonder, this is a quote from the MIT’s edX course on quantum mechanics (8.01.1x). We quote this
example for the same reason as why we use Feynman’s Lectures as a standard reference: it’s authoritative and,
more importantly, also available online so the reader can check and explore for himself.
3
have spin.
6
All of them also knew it is spin that generates the magnetic moment. They should, therefore,
have integrated the idea of spin in their mathematical description from the outset. The mistake is
illustrated below.
Figure 1: The meaning of +i and i
It is a very subtle but fundamental blunder. Spin-zero particles do not exist.
7
All real particles
electrons, photons, anything have spin, and spin (a shorthand for angular momentum) is always in one
direction or the other: it is just the magnitude of the spin that differs. It is, therefore, completely odd
that the plus (+) or the minus () sign of the imaginary unit (i) in the a·e±i function was not used to
include the spin direction in the mathematical description from the start.
We do not hesitate to call it a blunder or a mistake rather than just an oddity because it is much more
than just odd: it’s plain wrong, indeed. Why? Because it has a lot of sad consequences. For example, this
non-used degree of freedom in the mathematical description also 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: they think we can just multiply a set of amplitudes
let’s say two amplitudes, to focus our mind (think of a beam splitter or alternative paths here) with 1
and get the same states.
We find it rather obvious that that is not necessarily the case: 1 is not necessarily a common phase
factor. We should think of 1 as a complex number itself: the phase factor may be +π or, alternatively,
6
Even neutrons have a magnetic moment and, therefore, spin! One should also note that photons despite their
spin-one property cannot not come in a zero-spin state. When studying Feynman’s Lectures, I found that to be
one of the weirdest things: Feynman first spends several chapters on explaining spin-one particles to, then, in
some obscure footnote suddenly write this: "The photon is a spin-one particle which has, however, no “zero”
state." The issue is related to another glaring inconsistency: in the first three chapters of his Lectures on physics, he
talks about adding wavefunctions and the basic rules of quantum mechanics, and it all happens with a plus sign.
However, in his chapter on the theoretical distinction between bosons and fermions, he suddenly says we should
be adding the amplitudes of fermions combine with a minus sign. In any case, we cracked our own jokes on the
boson-fermion theory and so we should leave it at that.
We should add one note on the zero-spin state of helium, however: the electrons and protons inside the helium
atom will line up in a way that minimizes energy. The helium atom, as a composite particle, may, therefore, be said
to come in a zero-spin state. One may think of this situation as the exception which confirms the general rule
which applies to all charged particlesincluding composite particles, neutral or not. [The latter idea that of a
neutral charged particle obviously refers to a composite particle in which there are as many positive as negative
electric charges.]
7
We are obviously not talking about the zero-spin state of, say, a spin-one atom here. We talk about elementary
particles only here.
4
π. To put it simply, when going from +1 to 1, it matters how you get there and vice versa as
illustrated below.
8
Figure 2: e+iπ eiπ
We will come back to all of this later as part of a more detailed discussion of the meaning of
quintessential concepts in quantum mechanics so we will let this matter rest for the time being. Before
we really get into the nitty-gritty, we’d just like to make another introductory remark on positive and
negative energy states so as to make sure the reader has a bit of an intuitive grasp of our rather physical
interpretation of what is usually presented as abstract concepts only.
Positive and negative energy wells
The reader should note that the property of the spin being up or down is always defined with reference
to an external electric or magnetic field (or some combination thereof, of course).
9
The energy being
positive or negative can, therefore, only be defined with reference to the direction of the magnetic field.
Positrons should, therefore, not be thought of as holes. Similarly, we may note physicists always think of
particles being trapped in a potential well with lower energy. However, a particle may well be trapped in
a potential well with higher energy. This situation is illustrated in Feynman’s very discussion of a two-
state system, which we summarize by the illustrations below.
10
8
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 eiπ 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’re an amateur physicist, you should
be excited, because it actually is 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 sure think of this as an error in the beginning!
9
Simple experiments such as the Stern-Gerlach experiment involve a magnetic or an electric field only.
However, many experiments and apparatuses such as the Penning trap, for example will involve a combination
of both.
10
The illustration was taken from Feynman’s Lecture on the ammonia maser. We gratefully acknowledge Michael
Gottlieb and Rudolf Pfeiffer, who have worked for decades to get those Lectures online and, therefore, claim
copyright on Feynman now.
5
Figure 3: Two trivial examples of two-state systems in quantum mechanics
The illustration on the left shows the two possible energy states of an ammonia molecule (NH4) in the
context of a discussion of how an ammonia maser works the ammonia molecule basically oscillates
between two states that have an opposite electric dipole moment
11
while, on the right, we have an
illustration of the splitting of energy levels in a magnetic field. We might have added an illustration of
the spin of a proton can flip between an up and a down state in an MRI apparatus
12
or other diagrams
illustrating the basics of a two-state system in quantum mechanics, but we don’t want to be too long
here. Paraphrasing Feynman once again, we may say the point is this:
No matter what the physics is originallythe ammonia molecule in the ammonia maser, laser
technology, the fine or hyperfine splitting of the hydrogen spectrum, proton spin flipping in MRI
technology, or whateveryou can translate it into a corresponding electron problem. So if we can solve
the electron problem in general, we have solved all two-state problems. And we solved the electron
problem. Everything else is, therefore, very easy to understand.
13
In case you wonder what we want you to remember going to the next, it is this: don’t think too much
about the usual hocus-pocus involved in tunneling in and out of potential wells or walls and other
11
The field that separates the two states is, therefore, electric. Other typical quantum-mechanical systems will
typically involve magnetic dipoles which are, therefore, separated by magnetic fields.
12
Indeed, most of us should be very familiar with the proton spin flipping mechanism because of its use in
magnetic resonance imagining. I must assume that if you have ever had an MRI you also googled and watched
one or more YouTube videos explaining the physics underpinning this amazing technology. If not, I warmly
recommend you do so. It will give you a better intuitive grasp of the basics of quantum mechanics than any of the
standard introductory courses. For starters, it gives you a great engineering feel for what might or might not be
going on. MRI machines do not only manipulate the direction of the proton spin but in order to localize the
region that is being scanned, it also manipulates its phase. It is all very sophisticated but beautiful to watch. If
anything, it shows how real spin the motion of the elementary charge within a proton or an electron actually is.
You can google some videos for yourself but, in case you’d appreciate some advice here, we would single out the
one from Doctor Klioze: https://www.youtube.com/watch?v=djAxjtN_7VE. It is almost half an hour long, but well
worth spending the time! Enjoy!
13
We refer the reader to section 7 of Chapter 10 of Feynman’s Lectures on quantum mechanics for a comparison
with Feynman’s original quote. Back in 1963, Feynman modeled proton spin using the Pauli matrices, which is still
very useful, of course!
6
quantum-mechanical blah. Once you have an intuitive understanding of what spin actually is, everything
can be explained in very simple terms.
14
The nature of antimatter
The reader may or may not find the introductory remarks above intuitive and self-evident, but the
question which puzzled Dirac remains, of course: what is the nature of antimatter, really? We have no
definite answer to that question but we do have a few thoughts that may amount to some answer, at
least.
First, you should note that the suggestion of some physicists Richard Feynman, in particular that
anti-particles may be particles travelling back in time is plain nonsense. Time has one direction only, and
we will soon explain why.
15
But what is the alternative? As explained above, Dirac’s suggestion that,
somehow, antimatter may be defined by referring to the concept of negative energy states isn’t much of
an explanation.
The CODATA value
16
of the magnetic moment of an electron is μ 9.2847647043(28)1024 J·T1. Its
theoretical value is μ = qeħ/2m.
17
(its theoretical value). The minus sign is there because of the
negative (elementary) charge of the electron. Indeed: qe is the elementary charge that’s associated with
a proton or a positron the latter being the electron’s antimatter counterpart and so it equal to minus
the electron charge.
Now, consider a particular direction of the elementary current generating the magnetic moment. [In
case you wonder: that’s what spin actually is: it’s an elementary current.
18
] It is then quite easy to see
14
We refer the reader to our papers for what we usually refer to as a classical explanation of various quantum-
mechanical phenomena, including Mach-Zehnder interference and other previously mysterious experimental
results. As for tunneling through a wall, think of this: when you are going to crash a projectile into a wall, it is not
only its momentum that is going to matter. The direction of your projectile is going to matter at least as much,
doesn’t it? Taking a hit directly or sideways make a lot of difference, doesn’t it? 
15
We will do so in one of the next section(s). As for now, the reader may just want to note our argument is
mathematical and philosophical: the very idea of motion includes the idea of moving in time, but in one direction
only. A particle may go back to a place where it was before, or it may stay where it was, but if we’d allow it to go
back in time, we would not be able to capture its motion in a well-behaved function. Well-behaved equations of
motion imply the same particle may re-visit the same place in space, but not the same place in time. We do not
need any complicated arguments in terms of entropy or worse imagined symmetries that cannot exist in
reality.
16
The CODATA value is its measured value as published by the Committee on Data (CODATA) of the International
Science Council. This Committee regularly makes informed decisions on what experiments count (or don’t count)
and, more importantly, to what degree of precision their results are supposed to be valid.
17
You are, of course, aware of the so-called anomaly in the magnetic moment. It is a difference between the
theoretical and measured value. The difference is of the order of Schwinger’s factor (α/2π). Mainstream physicists
explain it using quantum field theory. We explain it by assuming the Zitterbewegung charge has a tiny but non-zero
size itself.
18
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.
7
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.
19
We can, therefore, associate this with the spin property, which is also up or down.
We, therefore, have four possibilities
20
:
Matter-antimatter
Electron
Positron
Table 1: Occam’s Razor: mathematical possibilities versus physical realities (1)
The table shows the ring current model also applies to antimatter. It also shows that 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 + .
21
Of course, we readily admit this cannot be the complete answer: the difference between a positive and
a negative Zitterbewegung charge cannot be reduced to its direction of motion only! Why not? The
answer is obvious: try to bring two electrons together, and then try to bring an electron and a positron
together, and you will see two very different things happening.  Hence, we obviously do have an
unexplained intrinsic property here. Indeed, we need the idea of positive and negative charge here!
Why? Because the magnetic moment alone does not allow one to distinguish between an electron with
spin up and a positron with spin down.
Is this good? We think it is. Is it good enough? We are not sure. Intrinsic properties need an explanation,
don’t they? Yes, most probably.
22
However, we are where we are right now and so we cannot add much
to this discussion for the time being. Having said that, the reader will have to admit we were able to go
way beyond Dirac’s rather native hole theory here: in the electron with spin up, we have the same
current as in the positron with spin down, but it is because we have an opposite charge (negative
instead of positive) spinning in the opposite direction (up instead of down, or right versus
leftwhatever you want to call it). Hence, negative energy states have to do with the direction of the
19
To determine what is up or down, one has to apply the ubiquitous right-hand rule.
20
The use of the subscripts in the magnetic moment may be confusing, but shouldn’t 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).
21
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.
22
If we should summarize our quest for truth in physics, we might say this was our drive: we wanted an
explanation for these so-called intrinsic properties of elementary particles: their mass or energy, their size, and
their magnetic moment. We think we have succeeded in finding that explanation. Having said that, we must admit
the physicality of pair creation and annihilation still puzzles us deeply. We therefore think we greatly reduced the
Mystery, but we sure did not eliminate it completely!
8
force.
23
Antimatter seems to have a different metric signature than matter.
24
That is, for the time being, all we can say about it.
25
Let us get into the meat of the matter nowalmost
literally speaking! 
The Emperor wears no clothes: matter does not dissipate
We already mentioned how science progresses: it does so by probing reality physically as well as
logically. Physical exploration involves experiments, measurements and discoveries. Logical progress
occurs when a new theory or explanation replaces previously held beliefs and convictions.
We said such theory or explanation should be logically and mathematically consistent. We should add
another requirement: it should also be applicable to a great variety of facts. Newton’s force law
26
and
theory of gravitation, Maxwell’s equations
27
, and Einstein’s theory of relativity
28
are obvious examples of
2323
If you find this statement confusing, try to think of it like this: the current that generates the magnetic moment
in an electron versus a proton must be different: one carries a negative charge, and the other carries a positive
charge. The direction of the physical current (the motion of the zbw charge, positive or negative) must, therefore,
be opposite. The question then becomes: what distinguishes the positive and a negative zbw charge inside the zbw
electron and positron? For the latter question, we have no answer: that is why we think of the property of the
charge being positive or negative as being intrinsic. In other words, we have an explanation for the direction of
motion and, therefore, for its fundamental nature. However, we have no explanation for the nature of the charge
itself.
24
We relate this to another proposition in the Annex to this paper: our interpretation of Planck’s constant as a
vector quantityso we would write it as h rather than h. When thinking about natural units and imposing the
condition that the magnetic and electric constant should be each other’s reciprocal, we get the he h2 = 1 condition
(see the section on The relative strength of the electric and magnetic field in the Annex). This effectively
corresponds to two different possibilities: h = 1 or h = 1. The h = 1 possibility corresponds to matter, while the h =
1 corresponds to antimatter. It is nothing but a reversal of spatial directionsboth linear as well as rotational (or
circular) directions.
25
We offer some more very speculative thoughts on the nature of antimatter in our paper on The Ring Current
Model for Antimatter and Other Questions, but we do not want to mention these here because we feel these ideas
do not make all that much sense.
26
Newton’s force law is also referred to as the law of motion. The idea of a force and the idea of motion are
inextricably linked: a force is that what changes the state of motion of an object, and mass is a measure of inertia
or resistance to such change. A force changes the state of motion by acting on a charge. [Of course, you may think
that, in many cases, a force does not necessarily change the state of motion of an object: it may also keep
something together. As Feynman notes, “The Empire State Building swings less than one inch in the wind because
the electrical forces hold every electron and proton more or less in its proper place.” However, even this should be
thought of as motion: Dirac, at least, referred to Schrödinger’s equation as an equation of motion for the
electrons.]
Historically, the idea of a force acting on a (positive or negative) charge came later: it is usually associated with
Benjamin Franklin’s experiments around 1750 and more systematically, perhaps the work of Charles-Augustin
de Coulomb, who published three memoirs on electromagnetism in 1785. Newton’s Philosophiæ Naturalis
Principia Mathematica (Mathematical Principles of Natural Philosophy) were first published in 1687: scientific
revolutions take time. After a hundred years or so, it seems the quantum-mechanical one hasn’t ended yet.
27
Maxwell’s equations show the electric and magnetic forces are one electromagnetic force only.
28
This is a reference to Einstein’s special relativity theory, which may be said to correct Newton’s law of motion so
as to ensure consistency with Maxwell’s equations, which show the velocity of light is absolute: it is always equal
9
such theories or explanations. However, we do not count Heisenberg’s quantum mechanics among
such great theories or explanations.
Why not? First, we think of it as an approach more than like a theorysome kind of problem-solving
algorithm, if you want. Second, even as an approach or an algorithm, it would seem to be based on the
wrong assumptions. Why? We will not ask you to think about some thought experiment here. No. We
ask you to just think for yourself about the following common-sense remarks.
Prof. H. Pleijel, then Chairman of the Nobel Committee for Physics of the Royal Swedish Academy of
Sciences, dutifully notes the following on the nature of the new ‘matter waves’ in the ceremonial speech
for the 1933 Nobel Prize, which was awarded to Heisenberg for nothing less than the creation of
quantum mechanics
29
:
“Matter is formed or represented by a great number of this kind of waves which have somewhat
different velocities of propagation and such phase that they combine at the point in question.
Such a system of waves forms a crest which propagates itself with quite a different velocity from
that of its component waves, this velocity being the so-called group velocity. Such a wave crest
represents a material point which is thus either formed by it or connected with it, and is called a
wave packet. […] As a result of this theory on is forced to the conclusion to conceive of matter as
not being durable, or that it can have definite extension in space. The waves, which form the
matter, travel, in fact, with different velocity and must, therefore, sooner or later separate.
Matter changes form and extent in space. The picture which has been created, of matter being
composed of unchangeable particles, must be modified.
This should sound very familiar to you. If so, you should ask yourself this question: why is this
everything but true? Real-life particles electrons or atoms traveling in space don’t dissipate. Matter
does not change form and extent in space!
30
We trust the academic physicist will switch off and stop reading hereif he or she hasn’t already. That’s
fine. We’re not writing this paper for the academic physicist. We write it for you, so you should think
about it. Show us one experiment in which matter obviously does change form and extent in space.
Matter may disintegrate, of course, but that’s something entirely different, and it only applies to
to cin the inertial as well as in the moving frame of reference. Einstein’s general relativity theory explains gravity
in terms of the geometry of spacetime. We do not consider gravity in this paper and, as such, we do not want to
highlight this theory for the time being.
29
To be precise, Heisenberg got a postponed prize from 1932. Erwin Schrödinger and Paul A.M. Dirac jointly got
the 1933 prize. Prof. Pleijel acknowledges all three in the introduction of his speech: “This year’s Nobel Prizes for
Physics are dedicated to the new atomic physics. The prizes, which the Academy of Sciences has at its disposal,
have namely been awarded to those men, Heisenberg, Schrödinger, and Dirac, who have created and developed
the basic ideas of modern atomic physics.
30
We will slightly nuance this statement later but we will not fundamentally alter it. We think of matter-particles
as an electric charge in motion. The centripetal force that keeps the particle together is electromagnetic in nature,
because it acts on a charge. Matter-particles, therefore, combine wave-particle duality. Of course, it makes a
difference when this electromagnetic oscillation, and the electric charge, move through a slit or in free space. We
will come back to this later. The point to note is: matter-particles do not dissipate. Feynman actually notes that at
the very beginning of his Lectures on quantum mechanics, when describing the double-slit experiment for
electrons: “Electrons always arrive in identical lumps.”
10
composite non-stable particles which for that reason shouldn’t be referred to as particles: call them
resonances or transients instead!
Indeed, we refer to particles as particles because of their stability and integrityand that integrity is
defined by the Planck-Einstein relation: E = h·f.
31
Nothing more. Nothing less. It’s all we need as an
explanation. A rather smart young man recently wrote this rather scathing criticism of his own
professionphysics:
“QED should be the quantized version of Maxwell’s laws, but it is not that at all. QED is a simple
addition to quantum mechanics that attempts to justify two experimental discrepancies in the
Dirac equation: the Lamb shift and the anomalous magnetic moment of the electron.
32
The
reality is that QED is a bunch of fudge factors, numerology, ignored infinities, hocus-pocus,
manipulated calculations, illegitimate mathematics, incomprehensible theories, hidden data,
biased experiments, miscalculations, suspicious coincidences, lies, arbitrary substitutions of
infinite values and budgets of 600 million dollars to continue the game. Maybe it is time to
consider alternative proposals. Winter is coming.”
33
Prof. Dr. John P. Ralston practices the same self-criticism in a recent publication: Quantum mechanics is
the only subject in physics where teachers traditionally present haywire axioms they don’t really believe,
and regularly violate in research.”
34
You think such rebel speech is unconvincingsacrilegious, perhaps? Let us quote a proper Nobel Prize
winner then.
35
Willis E. Lamb got a Nobel Prize in 1955 for, yes, the discovery of the Lamb shift in the
31
The interpretation of the Planck-Einstein relation is another matter, but one doesn’t need matter-waves for it.
The Planck-Einstein relation basically tells us particles have a unique frequency: a clock speed which is equal to the
cycle time T = 1/f = h/E = h/mc2. Combining this idea with the reality of the magnetic moment of elementary
particles, one arrives at the conclusion the frequency must be equal to the frequency of the motion of the electric
charge, which both Erwin Schrödinger as well as Paul A.M. Dirac paid much attention to. We will come back to this.
32
The reader should note that we have very decent classical explanations for both. See our paper on the electron
and the anomalous magnetic moment as well as our paper on the Lamb shift.
33
Oliver Consa, Something is Rotten in the State of QED, February 2020.
34
John P. Ralston, How To Understand Quantum Mechanics, 2017. We must add that, while we find Dr. Ralston’s
criticism of his own profession fairly enlightening, we do not think his own interpretation of quantum mechanics
yet another model, yes is very convincing. Not at all, actually. :-/
35
We could have quoted other founding fathers of quantum mechanics who ended up skeptical about the theory
they had created (or Einstein, of course). Dirac, for example, was rather vocal about it. However, we opted to
quote W.E. Lamb because he is, perhaps, one of the lesser knowntogether with John Stewart Bell. The case of
J.S. Bell is very particular, because he was nominated for a Nobel Prize for the (in)famous No-Go Theorem, which
tells us there are no hidden variables that can explain the interference in some kind of classical way? Bell’s
Theorem is what it is: a mathematical theorem. Nothing more. Nothing less. It, therefore, respects the GIGO
principle: garbage in, garbage out. John Stewart Bell himself a third-generation quantum physicist, we may say
did not like his own ‘proof’ and thought that some “radical conceptual renewal” might disprove his conclusions
(John Stewart Bell, Speakable and unspeakable in quantum mechanics, pp. 169172, Cambridge University Press,
1987). J.S. Bell kept exploring alternative theories including Bohm’s pilot wave theory, which is a hidden variables
theory until his death at a relatively young age (the year when he got nominated for the Nobel Prize, in fact).
11
spectrum of the hydrogen atom.
36
When he was well over 80, he wrote a rather remarkable paper on
the (non)sense of the theoretical explanations of the phenomenon he had experimentally discovered.
37
The title of the paper is, significantly, Anti-Photon, and, in this paper, he politely blames Bohr rather
than Heisenberg for all of these weird theories we are still struggling to understand:
The Complementarity Principle and the notion of wave-particle duality were introduced by N.
Bohr in 1927. They reflect the fact that he mostly dealt with theoretical and philosophical
concepts, and left the detailed work to postdoctoral assistants. It is very likely that Bohr never,
by himself, made a significant quantum-mechanical calculation after the formulation of
quantum mechanics in 1925-1926.
38
In fact, it is hard to see what Heisenberg’s matrix mechanics actually explain. Pretty much nothing,
perhaps. Indeed, we may want to think of it as a calculation rather than as a theory. We may illustrate
this by referring to Heisenberg’s pioneering of the scattering matrix, which is now more generally
referred to as the S-matrix in quantum mechanics.
39
Scattering is actually a somewhat misleading term
because it may refer to any interaction process. As usual, we can rely on Richard Feynman for a more
vivid description of what an S-matrix actually is:
The high-class theoretical physicist working in high-energy physics considers problems of the
following general nature (because it’s the way experiments are usually done). He starts with a
couple of particles, like a proton and a proton, coming together from infinity. (In the lab, usually
one particle is standing still, and the other comes from an accelerator that is practically at
infinity on atomic level.) The things go crash and out come, say, two K-mesons, six π-mesons,
and two neutrons in certain directions with certain momenta. What’s the amplitude for this to
happen? The mathematics looks like this: The ϕ-state specifies the spins and momenta of the
incoming particles. The χ would be the question about what comes out. For instance, with what
amplitude do you get the six mesons going in such-and-such directions, and the two neutrons
going off in these directions, with their spins so-and-so. In other words, χ would be specified by
giving all the momenta, and spins, and so on of the final products. Then the job of the theorist is
36
We should be precise: Willis Eugene Lamb Jr. got half of the 1955 Nobel Prize in Physics for his discovery. He had
to share it with Polykarp Kusch. To be precise, Lamb got his half of the prize “for his discoveries concerning the fine
structure of the hydrogen spectrum” (which is the Lamb shift), while Polykarp Kusch got it “for his precision
determination of the magnetic moment of the electron” (the so-called anomaly in the magnetic moment). The
reader can look it all up on the website of the Nobel Prize for Physics.
37
One should, indeed, note that both Lamb as well as Kusch had measured the Lamb shift and the anomaly
respectively, but both had left the explaining of it to (other) physicistssome more famous names you probably
are more acquainted with, such as Julian Schwinger and Richard Feynman, who together with Tomonaga got a
Nobel Prize in physics "for their fundamental work in quantum electrodynamics, with deep-ploughing
consequences for the physics of elementary particles"read: their explanations of this and other previously
unexplained measurements.
38
See: W.E. Lamb Jr., Anti-photon, Appl. Phys. B 60, 77-84 (1995).
39
The S in S-matrix refers to scattering, not to the German Spur (the trace of a matrix) or some other term. I am
inserting this rather pedantic note because part of the mystery of quantum mechanics is stuff like this:
abbreviations or concepts that never get a proper definition or an accurate portrayal of the archaeology or
genealogy of the idea. According to the Wikipedia article on it, a proper S-matrix was first introduced by John
Archibald Wheeler in a 1937 article on the composite wavefunction for multi-particle nuclei.
12
to calculate the amplitude χ | S | ϕ .”
40
The reader may or may not recognize the latter admonishment. The job of the theorist or of the
student in quantum mechanics is to calculate rather than to think about what might or might not be
happening.
41
We will come back to this, and in much boring detail, I am afraid. Before we get into the nitty-gritty
and because many readers may not want to venture that far we will first offer some preliminary
remarks on the nature of the Uncertainty Principle.
The (Un)Certainty Principle
What is uncertainty in quantum mechanics? How does it relate to the philosophical concepts of freedom
and determinism? When discussing the philosophical implications of the Uncertainty Principle in
quantum mechanics, Richard Feynman notes that, from a practical point of view”, we have
“indeterminability in classical mechanics as well.
42
He gives the example of water splashing over a dam:
“If water falls over a dam, it splashes. If we stand nearby, every now and then a drop will land on
our nose. This appears to be completely random, yet such a behavior would be predicted by
purely classical laws. The exact position of all the drops depends upon the precise wigglings of
the water before it goes over the dam. How? The tiniest irregularities are magnified in falling, so
that we get complete randomness. Obviously, we cannot really predict the position of the drops
unless we know the motion of the water absolutely exactly.
While it is, effectively, not practically possible to know the motion of the water “absolutely exactly”, the
deeper question is: is it theoretically possible? In classical mechanics, we assume it is. In quantum
mechanics, we are supposed to assume it is not. Quoting Feynman once again, we may say that “the
Uncertainty Principle protects quantum mechanics.” Why should this be so?
40
See: How states change with time, Feynman’s Lectures, Vol. III, Chapter 8, section 4.
41
Lamb’s remark on Niels Bohr leaving the actual calculations to postdocs may or may not reflect a remark of
Feynman I can’t recall if it’s in his Lectures or in his Strange Theory of Light and Matter on the rather
extraordinary amount of calculus one needs to calculate anything based on quantum field theory or Feynman’s
own path integral formulation of quantum mechanics and the fact that, therefore, such calculus is usually left to
doctoral students who, in Feynman’s words, have to “keep going at it for their PhD.” We need to add a lot of the
heavy lifting on the discovery of the Lamb shift was actually done by one of Lamb’s own postdoc students, R.
Retherford, who did not get much recognition for itexcept for a brief mention in Lamb’s Nobel Prize speech.
As for the calculations themselves, we may briefly mention that Stefano Laporta claims in a 2017 article (other
researchers have come up with similar articles in the meanwhile) to have calculated 891 four-loop contributions
to the anomalous magnetic moment, resulting in a precision of ‘up to 1100 digits of precision’ of the electron’s
actual g-ratio (slightly different from a theoretical value of 2, which is in my not-so-humble view simply just is
what it is: a theoretical value based on mathematical idealizations). One gets an uncanny feeling here: if one has to
calculate a zillion integrals all over space using 72 third-order diagrams to calculate the 12th digit in the anomalous
magnetic moment, or 891 fourth-order diagrams to get the next level of precision, then there might something
wrong with the theoryread: our mathematical idealizations. We refer to our own paper(s) for a more classical
explanation of the anomalous magnetic moment.
42
See: Feynman’s Lectures, III-2-6.
13
There is no reason whatsoever to assume the uncertainty in classical mechanics is any different from the
uncertainty in quantum mechanics. The uncertainty in quantum mechanics is not anything more than
the Ungenauigkeitimprecision, which is the term Heisenberg first used to describe the apparent
randomness in our mathematical description of Naturewe encounter in classical mechanics as well. It
is just an imprecision, indeedas 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’.
The concept of ‘uncertainty’ effectively mixes two different connotations:
(1) an imprecision in our measurements that is due to the wave-particle nature of electrons and other
elementary particlesa duality which the ring current model
43
, combining the idea of charge and
its motion, captures from the outset (as opposed to trying to cope with the duality in some wave
equation or other dual mathematical descriptions); and
(2) a rapid magnification of what Feynman refers to as ‘the tiniest irregularities’ but which are, in
practice, purely statistical.
This is, effectively, reflected in what Feynman refers to as the ‘great equation of quantum
mechanics’the wave equation for an electron in free space:




The problem is: it is not a wave equation for an electron. This wavefunction does not model an electron:
it models our rapidly increasing uncertainty about where it might be at a later point in time.
Why does it increase so rapidly? It’s just scale: Planck’s quantum of action can (also) be written as the
product of momentum over some distance (h = p·λ)
44
and, in light of the very tiny distance scale (λ), the
momentum (p) is, effectively, relatively largewhich is why any imprecision in its magnitude or
direction quickly translates into huge uncertainties concerning the actual position of the elementary
particle that we are trying to describe.
We will revisit all of this in very much detail later. In order to do so, we must talk some more about what
quantum mechanics is and isn’t. Before we do so, we briefly want to think a rather weird and totally
non-scientific question: what made Heisenberg change his thoughts about the Ungenauigkeit in
practical experiments? Why did a simple statement about the unavoidable interaction between the
measurement apparatus and that what is being measured turn into a sweeping metaphysical
statement?
We are not sure. We just have this rather uncanny feeling all of these thought experiments are not
about the Uncertainty Principle “protecting” quantum physics (we are quoting Feynman here): we feel
it’s about physicists protecting the Uncertainty Principle instead! Of course, the question then becomes:
why would they do this? They are searching for the truth as well, don’t they? They sure are.
We have no answer here, except this: geniuses like Heisenberg were ultimately religious too, and so
they might have felt quantum-mechanical indeterminism (as opposed to the statistical indeterminism in
43
We will say more about this later.
44
The h = E·T and h = p· λ are complementary but equivalent descriptions of the electron in the ring current model.
14
classical mechanics) was the very last place where God or the Mystery, or whatever we cannot prove
but happen to believe in can hide.
45
Let us turn back to a more scientific and rational discourse. What are those amplitudes, really?
The meaning of probability amplitudes and state vectors
The χ | S | ϕ amplitude which, according to Feynman, is all that matters in quantum physics is a
probability amplitude associated with the in-state ϕ and the out-state χ. These states are referred to as
state vectors
46
and they are described in terms of base states which are written as i | and | j for an
out- and an in-state respectively. Hence, the S-matrix is actually written as i | S | j and, assuming we
know what it is, we will be able to resolve and calculate the χ | S | ϕ amplitude as Σij χ | i  i | S | j 
j | ϕ .
This may all look a bit mind-boggling and rightly so, because it actually is mind-boggling. Just google, for
example the S-matrix for Compton scattering
47
and let me know how inspiring you think these so-called
explanation of a physical process actually are. For me, they are not an explanation at all. I think it is way
out of whack. It is like economists modeling input-output relations: we get a mathematical description of
what goes in and what goes out but it doesn’t tell us how stuff actually happens.
48
Furthermore, when
the process involves the creation or annihilation of elementary or non-elementary particles (think of
short-lived mesons here), even the math becomes ugly. Paraphrasing Dirac, one is then tempted to think
that it may be “more important to have beauty in one's equations than to have them fit experiment.
49
The question, of course, runs deeper than that: we can and should not doubt that the likes of Bohr,
Pauli, Heisenberg, Schrödinger, Dirac did what we are all doing, and that is to, somehow, try to make
sense of it all. Hence, I am sure that even Heisenberg did not initially think of his interpretation of these
relations as some kind of surrender of reasonwhich is, effectively, what we think it actually amounts
to: stating that it is a law of Nature that even experts cannot possibly understand Nature “the way they
would like to”, as Richard Feynman put it
50
, relegates science to the realm of religious belief.
45
In case the reader would be interested in our own convictions here, we will just say this: we think it’s a ridiculous
argument. If God is hiding Him- or Herself, He (or She) hides in much better places. Think of what we refer to as
love here or more mundane, perhaps this strange order that comes out of disorder.
46
Of course, these vectors are a bit different than the usual Cartesian coordinate vectors that you are used to but
the idea of base vectors (base states) and combining or adding them remains the same. Physicists will solemnly
talk of a Hilbert space but that is just jargon meant to impress you.
47
I did this randomly just now and these two recent articles came out on top: https://arxiv.org/abs/1512.06681
and https://arxiv.org/abs/1810.11455. You may get others. This input-output calculus approach allows for endless
refinements and, hence, endless researchwhich is always nice from an academic point of view, of course.
48
The reader may be interested to know the author of this paper is an economist who has, effectively, dabbled in
both macro- as well as micro-economic modeling and the training in statistics that comes with it. We are,
therefore, very familiar with Marc Twain’s 1907 quote: "There are three kinds of lies: lies, damned lies, and
statistics."
49
The quote is, apparently, from a May 1963 paper or article in Scientific American. We didn’t analyze the context
in which Dirac is supposed to have said this.
50
See: Feynman’s Lectures, III-1-1.
15
To be sure, the accounting rules and the properties of the S-matrix (as well as those of operator and
Hamiltonian matrices) do incorporate some basic physical principles. One of them is the principle of
reversibility, which is related to CPT-symmetry. We will say more about this later. As for now, the reader
should note that the property of reversibility in Nature is logical but somewhat less straightforward than
it appears to be at first. If we reverse all signs the direction of time, the directions in space
51
, and the
sign of the charges then, yes, Nature seems to respect the principle of reversibility.
52
However, we may
see what physicists gravely refer to as symmetry-breaking if we only reverse signs of, say, the directions
in space and the sign of the charge: some processes violate what is referred to as CP-symmetry, which
involves swapping spatial directions and charges only.
We are not worried about that: if anything, it shows physical time has one direction only. However,
because we wrote at length about these things elsewhere, we do not want to bore the reader with
unnecessary philosophical reflections here.
53
We do want to highlight a few points here though, so as to
give the reader a bit of an impression of what quantum-mechanical descriptions of physical processes by
talking about amplitudes and states may actually represent.
The meaning of spin in quantum mechanics
We must now turn back to what we referred to as a deep conceptual flaw in the mainstream
interpretation of quantum mechanics: the mainstream interpretation does not integrate the concept of
particle spin from the outset because it thinks of the + or sign in front of the imaginary unit (i) in the
elementary wavefunction (a·ei· or a·e+i·) as a mathematical convention only. Indeed, most
introductory courses in quantum mechanics will show that both a·ei· = a·ei·(tkx) and a·e+i· = a·e+i·(tkx)
are acceptable waveforms for a particle that is propagating in a given direction (as opposed to, say,
some real-valued sinusoid). Hence, one would, effectively, expect that the professors mindful of
Occam’s Razor for a good theory would then proceed to provide some argument showing why one
would be better than the other, or some discussion on why they might be different but, no! That is not
the case. The professors usually conclude that “the choice is a matter of convention” and, that “happily,
most physicists use the same convention.”
54
We are really dumbfounded by this.
55
All of the great physicists knew all particles elementary or not
have spin.
56
All of them also knew it is spin that generates the magnetic moment and that we should,
51
We are talking parity-symmetry here: think of the mirror image of things, or of turning space inside out.
52
As mentioned, we will make a critical note here later.
53
See, for example, our paper on the physical meaning of the quantum-mechanical wavefunction (Euler’s
wavefunction: the double life of
1) and our blog post(s) on CPT-symmetry.
54
In case you wonder, this is a quote from the MIT’s edX course on quantum mechanics (8.01.1x). We quote this
example for the same reason as why we use Feynman’s Lectures as a standard reference: it’s authoritative and,
more importantly, also available online so the reader can check and explore for himself.
55
The reader should not think we did not ask the question in the mentioned edX course on quantum mechanics.
We did. We asked many other questions as well. However, we were effectively kindly instructed to “shut up and
calculate” so as to prepare ourselves for the examination. We will be honest and admit we did not do the exam.
We don’t mind calculations. We do mind mindless calculations, however.
56
Even photons despite their spin-one property cannot not come in a zero-spin state. When studying
Feynman’s Lectures, I found that to be one of the weirdest things: Feynman first spends several chapters on
16
therefore, consider the idea of thinking of elementary particles as a ring current. Indeed, the ring
current model is not applicable to electrons (and positrons only). Indeed, Dirac starts his derivation of
his famous wave equation for free electrons in his 1933 Nobel Prize speech with the following remark
57
:
“This procedure is successful in the case of electrons and positrons. It is to be hoped that in the
future some such procedure will be found for the case of the other particles. I should like here to
outline the method for electrons and positrons, showing how one can deduce the spin properties
of the electron, and then how one can infer the existence of positrons with similar spin
properties.
This should suffice to remove any doubt in regard to the importance of the concept of spin in quantum
mechanics: it is the lynchpin of everything. As mentioned, the introduction above serves as an
introduction to Dirac’s derivation of his wave equation for an electron in free space
58
which we will not
write down but which involves four operators, denoted as αr pr (r = 1, 2, 3) and α0mc respectively
59
and
says the following about it:
The new variables αr which we have to introduce to get a relativistic wave equation linear in
W
60
, give rise to the spin of the electron. From the general principles of quantum mechanics one
can easily deduce that these variables a give the electron a spin angular momentum of half a
quantum and a magnetic moment of one Bohr magneton in the reverse direction to the angular
momentum. These results are in agreement with experiment. They were, in fact, first obtained
from the experimental evidence provided-by spectroscopy and afterwards confirmed by the
theory.”
He then continues by leaving us an excellent historical summary of the ring current or Zitterbewegung
model of the electron
61
:
explaining spin-one particles to, then, in some obscure footnote suddenly writes this: "The photon is a spin-one
particle which has, however, no “zero” state." The issue is related to another glaring inconsistency: in the first
three chapters of his Lectures on physics, he talks about adding wavefunctions and the basic rules of quantum
mechanics, and it all happens with a plus sign. However, in his chapter on the theoretical distinction between
bosons and fermions, he suddenly says we should be adding the amplitudes of fermions combine with a minus
sign. In any case, we cracked our own jokes on the boson-fermion theory and so we should leave it at that.
57
Needless to say, all italics in the quote are ours.
58
The concept of free space should not confuse the reader: it makes abstraction of any external fields or forces,
such as the force between a positively charged nucleus and the electron in some orbital.
59
Dirac’s wave equation can be written in various equivalent ways. We refer to Dirac’s Principles of Quantum
Mechanics or, for the reader who can follow Dirac’s very succinct summary of it, the above-mentioned Nobel Prize
speech.
60
As mentioned, we don’t want to get into the detail of (the derivation of) Dirac’s equation but the reader should
note Dirac only considers the kinetic energy of the electron which is, therefore, denoted as W rather than as E =
mc2. Our ring current model of an electron is more holistic in the sense that it integrates potential and kinetic
energy.
61
The term Zitterbewegung was, effectively, coined by Erwin Schrödinger in a 1930 paper which analyzed the
solutions to Dirac’s equation (Dirac had derived his equation in 1928 but the whole theoretical framework that
accompanied it his Principles to Quantum Mechanics was also published in 1930 as well.
17
“The variables α also give rise to some rather unexpected phenomena concerning the motion of
the electron. These have been fully worked out by Schrödinger. 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.”
We wrote about this elsewhere
62
and so we will not dwell on it much longer. The question here is: why
did such great intellects fail to fully exploit the power of Euler’s ubiquitous ψ = a·ei function? They knew
about Occam’s Razor as a criterion for a good scientific theory too, didn’t they? We are not sure.
Perhaps Schrödinger and Dirac were too obsessed by their wave equation as opposed to the
wavefunction that is its solution. Whatever the reason might have been, the fact is this: they did not
integrate spin in their mathematical descriptionnot from the outset, at least. The mistake is illustrated
below.
Figure 4: The meaning of +i and i
It is a very subtle but fundamental blunder. Spin-zero particles do not exist.
63
All real particles
electrons, photons, anything have spin, and spin (a shorthand for angular momentum) is always in one
direction or the other: it is just the magnitude of the spin that differs. It is, therefore, completely odd
that the plus (+) or the minus () sign of the imaginary unit (i) in the a·e±i function is not being used to
include the spin direction in the mathematical description.
In fact, we call it a blunder because it is more than just odd: it’s plain wrong because this non-used
degree of freedom in the mathematical description leads to the false argument that the wavefunction of
spin-½ particles have a 720-degree symmetry. Indeed, physicists treat 1 as a common phase factor in
the argument of the wavefunction: they think we can just multiply a set of amplitudes let’s say two
amplitudes, to focus our mind (think of a beam splitter or alternative paths here) with 1 and get the
same states. We find it rather obvious that that is not necessarily the case: 1 is not necessarily a
common phase factor. We should think of 1 as a complex number itself: the phase factor may be +π or,
alternatively, π. To put it simply, when going from +1 to 1, it matters how you get there and vice
62
See our Explanation of the Electron and Its Wavefunction.
63
We are obviously not talking about the zero-spin state of, say, a spin-one atom here. We talk about elementary
particles only here.
18
versa as illustrated below.
64
Figure 5: e+iπ eiπ
The reader may be very tired by now
65
, so let us ask the primordial question for him or her: what’s the
point here? It is this: if we exploit the full descriptive power of Euler’s function, then all weird
symmetries disappear and we just talk standard 360-degree symmetries in space. Also, weird
mathematical conditions such as the Hermiticity of quantum-mechanical operators can easily be
explained as embodying some common-sense physical law: energy and momentum conservation, for
example, or physical reversibility: we need to be able to play the movie backwards.
Here we must note something funny with charges, however. Combining the + and sign for the
imaginary unit with the direction of travel, we get four possible structures for the wavefunction for an
electron:
Spin and direction of travel
Spin up (J = +ħ/2)
Spin down (J = ħ/2)
Positive x-direction
= exp[i(kx−t)]
* = exp[i(kx−t)] = exp[i(tkx)]
Negative x-direction
χ = exp[i(kx+t)] = exp[i(tkx)]
χ* = exp[i(kx+t)]
Table 2: Occam’s Razor: mathematical possibilities versus physical realities (2)
However, this triggers the obvious question: how do we know it’s an electron or a negative charge, as
opposed to a positron, or a positive charge? Indeed, consider a particular direction of the elementary
current generating the magnetic moment. It is then very 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 associate this with
the spin property, which is also up or down. We, therefore, have another set of four possibilities
reflecting one another:
64
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 eiπ 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’re an amateur physicist, you should
be excited, because it actually is 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 sure think of this as an error in the beginning!
65
We should also apologize for the (partial) overlap between this and our introductory discussions. We will try to
rationalize and shorten the next version of this paper.
19
Matter-antimatter
Spin up (J = +ħ/2)
Spin down (J = ħ/2)
Electron
μe = qeħ/2m
μe = +qeħ/2m
Positron
μe+ = +qeħ/2m
μe+ = qeħ/2m
Table 3: Electron versus positron spin
So how do we distinguish them? It is a deep philosophical question which we cannot fully answer for the
time being. We can only offer a few thoughts here:
1. The idea of time reversal is a wonderful invention of our mind but we think it does not correspond to
anything real. When charges are involved, you will see two like charges repel each other and, therefore,
you will see them move away from each other. However, when playing the movie backward, you will
know it the movie cannot be real because the same two charges now move towards each other. The
funny thing is that a reversal of the sign of the charges doesn’t help to fix the situation because, say,
reversing the charge of two electrons now gives us two positrons moving towards each other, which
doesn’t make sense physically either!
The reader may think we did, perhaps, not think of P-symmetry: all left-handed things are now right-
handed and vice versa, right? Yes, but that doesn’t solve the problem here.
2. So what about CPT-symmetry then? Frankly, we think the concept of CPT-symmetry is not all that
useful and we, therefore, wonder why physicists are so enthralled about it. We think the concept of
motion itself implies that time can have one direction only. If it wouldn’t, then we would not be able to
describe trajectories in spacetime by a well-behaved function. In other words, the concept of motion
itself would become meaningless. The diagrams below illustrate the point. The spacetime trajectory in
the diagram on the right is not kosher, because our object travels back in time in not less than three
sections of the graph. Spacetime trajectories or, to put it more simply, motions need to be described
by well-defined functions: for every value of t, we should have one, and only one, value of x. 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 are related to us using
well-behaved functions to describe reality.
66
Figure 6: A well- and a not-well behaved trajectory in spacetime
67
66
We wish the reader who would want to try using not-so-well-behaved functions to arrive at some kind of
description of reality the best of luck.
67
We actually do not like the concept of spacetime very much: time and space are 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
20
We must now come back to the matter at hand, which is the meaning of conjugates in quantum physics.
The meaning of conjugating amplitudes, wavefunctions and matrices
The ubiquitous concept of a state in quantum mechanics is very abstract and general and, therefore,
adds to the mystery of the quantum-mechanical description of things. We may start by noting that the
concept of an initial and a final state or an in- and an out-state are very intimately linked: they
correspond to the bra and the ket in Dirac’s bra-ket notation which we should not remind the reader
must be read from right to left. As such, we like to think the concepts of the initial and final state simply
separate the present from the (possible) the future.
In practice, we will use matrix mechanics when states are discrete, while the wavefunction approach
allows us to model how a particle may evolve which, in practice, usually means what and where we
expect its momentum and position to be. One of Feynman’s very first examples of a state is the spin
state of spin-one atoms
68
, which may be +1, 0 or 1. These are referred to as base states. Base states
need to be independent, which is captured by the usual Kronecker delta rule:
i | j = ij
This basically says a base state is a base state: i | j = 1 if, and only if, i = j. Otherwise i | j = 0. It
obviously also means a particle can be in one state only at any point in time. The evolution from one
state to another involves a process or another oft-used terms in introductory courses an apparatus,
such as a filter, which may also be referred to as polarizer.
69
Now, the amplitude for a particle to go from
state ϕ to state χ will be χ | ϕ . An amplitude is, of course, a probability amplitude here: it is a complex
number or a complex-valued wavefunction. There is a most remarkable law in quantum mechanics, and
it is this: the amplitude to go from the out-state χ to the in-state ϕ is the (complex) conjugate of the
amplitude to go from the in-state ϕ to the out-state χ and vice versa, of course! It is more easily
written like this: ϕ | χ = χ | ϕ * and χ | ϕ = ϕ | χ *
Feynman proves this rule for a three-state system and a simple filter apparatus
70
but we find it easier to
think of it as a simple consequence of this theoretical possibility of reversibility: a sign reversal of the
imaginary unit i. Instead of +i (or, conversely, i), we now use i (or, conversely, +i) in our wavefunction.
It is probably easiest to show what this means in terms of actual probabilities some easily
understandable number between 0 and 1, that is rather than in terms of probability amplitudes, which
are complex-valued numbers evolving in space and in time. Indeed, we get the actual probability from
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 often
useful in physics.
68
While a photon is generally considered to be a spin-one particle, it does not have a zero state. In contrast, spin-
one atoms are composite particles and, hence, its component particles may effectively line up in a way that
produces a spin-zero state.
69
We find the term somewhat confusing but a polarized beam is a beam of particles with the same spin.
70
See: Feynman’s Lectures, III-5-5 (Interfering Amplitudes) and III-5-6 (The Machinery of Quantum Mechanics).
21
taking the absolute square
71
of the complex-valued amplitude.
In our realist interpretation, we understand this quantum-mechanical rule in terms of mass or energy
densities: we think of the oscillating or Zitterbewegung charge as passing more time here than there
and, hence, the associated energy density will be higher here than there.
72
However, that interpretation
does not matter here. We just want to show you that we can use this quantum-mechanical rule for
calculating probabilities to derive the above-mentioned quantum-mechanical reversibility law. Indeed,
the absolute square of a complex number is the product of the same number with its complex
conjugate. We, therefore, get this
73
:
| ϕ | χ |2 = ϕ | χ  ϕ | χ * = χ | ϕ * χ | ϕ = χ | ϕ * χ | ϕ = | χ | ϕ |2
At this point, the reader may wonder what we want to say here. We are just saying that the quantum-
mechanical ϕ | χ = χ | ϕ * rule only states physical processes should be reversible in space and in
time, and that it amounts to stating that the probability of going from the out-state χ to the in-state ϕ
is the same as going from the in-state ϕ to the out-state χ.
The question is: does this rule or law make sense? In other words, is it true, always? In mechanics
think of elastic collisions of particles or particles moving in force fields it should be the case. However,
when disintegration processes are involved, one should not expect this law to apply. Why not?
Disintegration involves unstable systems moving to some kind of stable state. Energy, angular
momentum, and linear momentum will be conserved, but that’s about all we can say about it. Inventing
new quantities to be conserved such as strangeness
74
or invoking other consequences of so-called
symmetry-breaking processes does not add much explanatory power in our view.
[…]
Let us wrap up herethis section of this rather long and winding paper, at least! Let’s summarize the
basics of our discussion above: if we can go from state ϕ to state χ, then we must be able to go back to
state ϕ from state χ in space and in time. This amounts to playing a movie backwards: an exploding
suitcase bomb will, in practice, not suddenly un-explode and get back into the suitcase but, theoretically
(read: making abstraction of arguments involving entropy and other macroscopic considerations), such
71
The absolute square is, obviously, a shorthand for the absolute value of the square.
72
You may think of this like follows. The energy in an oscillation any physical oscillation is always proportional
to the square of the (maximum) amplitude of the oscillation, so we can write this: E a2. To be precise, the energy
of an electron in our ring current model is equal to E = m·a2·ω2. We get this from (1) Einstein’s mass-energy
equivalence relation (E = m·c2) and (2) our interpretation of c (lightspeed) as a tangential velocity (c = a·ω). We
think the mass of the electron is just the equivalent mass of the energy of the Zitterbewegung charge in its
oscillation. We, therefore, think it makes sense to think that the probability of actually finding the charge here or
there will be proportional to the energy density here and there.
73
The reader should carefully note what is commutative and what not here!
74
A careful analysis of the processes involving K-mesons in Feynman’s rather enigmatic Lecture on it reveals that
the quark hypothesis results from a rather unproductive approach to analyzing disintegration processes: inventing
new quantities that are supposedly being conserved, such as strangeness, is… Well… As strange as it sounds. We,
therefore, think the concept of quarks confuses rather than illuminates the search for a truthful theory of matter.
See our introduction on Smoking Gun Physics in our general paper on the power of classical physics. This text is
based on an earlier critical appraisal of the theory of virtual particles.
22
reversed processes may be imagined.
75
However, instead of imaging an exploding or un-exploding
bomb, let us stick to the more abstract business of states and vectors and so we should try to think of an
apparatus or a process that will be operating on some state |ψ to produce some other state |ϕ. The
Great Dirac taught us to write that like this: ϕ|A|ψ
We can now take the complex conjugate:
ϕ|A|ψ* = ψ|A†|ϕ
A† is, of course, the conjugate transpose of A – we write: A†ij=(Aji)* and we will call the operator (and
the matrix) Hermitian if the conjugate transpose of this operator (or the matrix) gives us the same
operator matrix, so that is if A† = A. Many quantum-mechanical operators are Hermitian, and we will
also often impose that condition on the S-matrix. Why? Because you should think of an operator or an S-
matrix as a symmetric apparatus or a reversible process. It’s as simple as that. Hence, the Hermiticity
condition amounts to a simple reversibility condition too!
Needless to say, we may be mistaken, of course! We, therefore, invite the professional reader to
challenge this interpretation! The professional reader may also note we forgot to specify the second of
the three basic rules of quantum math, and that’s the resolution of amplitudes into base stateswhich
is written like this: χ | ϕ = Σall i χ | i  i | ϕ
This, too, can be shown to be true based on a thought experiment
76
or, else, one can use a much more
abstract argument (as Dirac does as part of his formalization of the theoretical framework for working
with amplitudes) and simply write:
| = Σall i | i  i |
This, then allows us to do what we did, and that’s to write the χ | S | ϕ amplitude as Σij χ | i  i | S |
j  j | ϕ .
Any case, this is all very abstract and, therefore, very tiring. Furthermore, it does not amount to what we
think of as any real explanation for real-life physical processes: it merely describes them. As mentioned,
this is more like economists modeling input-output relations: we get a mathematical description of what
goes in and what goes out but it doesn’t tell us how stuff actually happens. So how would or might a real
explanation then look like? Let us illustrate this using the example of Compton scattering.
75
This rather extreme example of before and after states is, perhaps, less extreme than it appears to be: the
proton-proton collisions in CERN’s LHC colliders are quite violent too, albeit at a much smaller scale. However, as
mentioned, one should not necessarily expect that the disintegration processes involved in the creation of such
disequilibrium situations would or should be reversible.
76
Feynman provides such argument based on placing an open filter (no masks) in-between two other filters. See
the above-mentioned reference to Feynman’s Lectures (III-5-5).
23
A classical explanation of Compton scattering
Energy and momentum conservation
Compton scattering is referred to as inelastic because the frequency of the incoming and outgoing
photon are different. The situation is illustrated below.
Figure 7: The scattering of a photon from an electron (Compton scattering)
The reader will be familiar with the formulas and, most probably, also with how these formulas can be
derived from two very classical laws only: (1) energy conservation and (2) momentum conservation. Let
us quickly do this to remind the reader of the elegance of classical reasoning:
1. The energy conservation law tells us that the total (relativistic) energy of the electron (E = mec2) and
the incoming photon must be equal to the total energy of the outgoing photon and the electron, which
is now moving and, hence, includes the kinetic energy from its (linear) motion. We use a prime (‘) to
designate variables measured after the interaction. Hence, Ee’ and Eγ’ are the energy of the moving
electron (e’) and the outgoing photon (γ’) in the state after the event. We write:

We can now use (i) the mass-energy equivalence relation (E = mc2), (ii) the Planck-Einstein relation for a
photon (E = h·f) and (iii) the relativistically correct relation (E2 p2c2 = m2c4) between energy and
momentum for any particle charged or non-charged, matter-particles or photons or whatever other
distinction one would like to make
77
to re-write this as
78
:




2. This looks rather monstrous but things will fall into place soon enough because we will now derive
another equation based on the momentum conservation law. Momentum is a vector, and so we have a
77
This is, once again, a standard textbook equation but if the reader would require a reminder of how this
formula comes out of special relativity theory we may refer him to the online Lectures of Richard Feynman.
Chapters 15 and 16 offer a concise but comprehensive overview of the basics of relativity theory and section 5 of
Chapter 6 gives the reader the formula he needs here. It should be noted that we dropped the 0 subscript for the
rest mass or energy: m0 = m. The prime symbol (‘) takes care of everything here and so you should carefully
distinguish between primed and non-primed variables.
78
We realize we are cutting some corners. We trust the reader will be able to google the various steps in-between.
24
vector equation here
79
:



For reasons that will be obvious later it is just the usual thing: ensuring we can combine two equations
into one, as we did with our formulas for the radius we square this equation and multiply with
Einstein’s constant c2 to get this
80
:



3. We can now combine equations (1) and (2):


The reader will be able to do the horrible stuff of actually squaring the expression between the brackets
and verifying only cross-products remain. We get:

Multiplying both sides of the equation by the 1/hmeff’ constant yields the formula we were looking for:




The formulas allow us also to calculate the angle in which the electron is going to recoil. It is equal to:



The h/mc factor on the left-hand side of the right-hand side of the formula for the difference between
the wavelengths is, effectively, a distance: about 2.426 picometer (1012 m). The 1 cosθ factor goes
from 0 to 2 as θ goes from 0 to π. Hence, the maximum difference between the two wavelengths is
79
We could have used boldface to denote vectors, but the calculations make the arrow notation more convenient
here. So as to make sure our reader stays awake, we note that the objective of the step from the first to the
second equation is to derive a formula for the (linear) momentum of the electron after the interaction. As
mentioned, the linear momentum of the electron before the interaction is zero, because its (linear) velocity is zero:
pe = 0.
80
We do not want to sound disrespectful when referring to c2 as Einstein’s constant. It has a deep meaning, in fact.
Einstein does not have any fundamental constant or unit named after him. Nor does Dirac. We think c2 would be
an appropriate ‘Einstein constant’. Also, in light of Dirac’s remarks on the nature of the strong force, we would
suggest naming the unit of the strong charge after him. More to the point, note these steps finally !
incorporated the directional aspect we needed for the analysis. Note that we also use the rather obvious E = pc
relation for photons in the transformation of formulas here.
25
about 4.85 pm. This corresponds, unsurprisingly, to half the (relativistic) energy of an electron.
81
Hence,
a highly energetic photon could lose up to 255 keV.
82
That sounds enormous, but Compton scattering is
usually done with highly energetic X- or gamma-rays.
Could we imagine that a photon loses all of its energy to the electron? No. We refer to Prof. Dr. Patrick
LeClair’s course notes on Compton scattering
83
for a very interesting and more detailed explanation of
what actually happens to energies and frequencies, and what depends on what exactly. He shows that
the electron’s kinetic energy will always be a fraction of the incident photon’s energy, and that fraction
may approach but will never actually reach unity. In his words: “This means that there will always be
some energy left over for a scattered photon. Put another way, it means that a stationary, free electron
cannot absorb a photon! Scattering must occur. Absorption can only occur if the electron is bound to a
nucleus.”
What might actually be happening
A photon interacts with an electron, so we actually think of the photon as being briefly absorbed, before
the electron emits another photon of lower energy. The energy difference between the incoming and
outgoing photon then gets added to the kinetic energy of the electron according to the law we just
derived:


Now, we think of the interference as a process during which temporarily an unstable wavicle is
created. This unstable wavicle does not respect the integrity of Planck’s quantum of action (E = h·f). The
equilibrium situation is then re-established as the electron emits a new photon and moves away. Both
the electron and the photon respect the integrity of Planck’s quantum of action again and they are,
therefore, stable.
The geometry of the whole thing is simple and difficult at the same time. There is, for example, also a
formula for the angle of the outgoing photon, which uses the angle for the incoming photon, but that’s
all stuff which the reader can look up. The question is: how does this happen, exactly? And what
determines the plane which is formed by the outgoing photon and the recoiling electron?
None of the standard textbooks will try to answer that question, because they don’t think of electrons
and photons as having some internal structure which may explain all of the formulas they get out of
their arguments, which are based on the conservation of (1) energy and (2) linear and angular
momentum. In contrast, we believe our geometrical models may not only show why but also how all of
this happens. Let us walk over the basics of that.
81
The energy is inversely proportional to the wavelength: E = h·f = hc/λ.
82
The electron’s rest energy is about 511 keV.
83
See the exposé of Prof. Dr. Patrick R. LeClair on Compton scattering.
26
The Compton wavelength as a circumference
According to our ring current model of an electron, the wavefunction ψ(x, t) = ψ(x, y, z, t) describes the
actual position of the pointlike Zitterbewegung charge in its oscillatory motion around some center.
84
We may therefore paraphrase Prof. Dr. Patrick LeClair and identify the Compton wavelength with a
distance scale within which we can localize the electron in a particle-like sense.”
85
Why are we so sure
of that?
We should first discuss the 2π factor. Indeed, we have h, not ħ, in the equation. Hence, should we think
of the Compton wavelength or the Compton radius of an electron? 2π is a sizable factor a factor equal
to about 6.28 so that is large enough to matter when discussing size.
86
Of course, we know it is the
factor which relates the circumference of a circle with its radius but a wavelength is something linear,
isn’t it? If we should think of the wavelength of an electron, then what should we imagine it to be
anyway? We all know that Louis de Broglie associated a wavelength with the classical momentum p =
mv of a particle but, again, how should we imagine this wavelength? The de Broglie relation says λ = h/p
= h/mv goes to infinity () for v going to 0 and m going to m0 (the rest mass of the electron). Hence, the
de Broglie and the Compton wavelength of an electron are very different concepts:
λC = h/mc λ = h/p = h/mv
The illustration below (for which credit goes to an Italian group of zbw theorists
87
) helps to make an
84
We are often tempted to use a semicolon to separate the time variable from the space coordinates. Hence, we
would prefer to write ψ(x, t) = ψ(x, y, z; t) instead of ψ(x, t) = ψ(x, y, z; t). The semicolon then functions as a serial
comma. Of course, we are well aware of Minkowski’s view on the relativity of space and time: “Space by itself, and
time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an
independent reality.” However, we feel space and time are related but also very separate categories of our mind:
perhaps some very advanced minds may claim they can effectively understand the Universe in terms of four-vector
algebra, but we surely do not. We understand things in terms of motion, and any equation of motion implies the
idea of a motion in space and in timenot in some ‘kind of union of the two’. What I am saying is that the oft-used
concept of spacetime is not something we can easily imagine: space and time are fundamentally different
categories of the mind. We cannot go backward in time, for example. Not in our reality, at least. Why not? Because
the x = (x, y, z) = (fx(t), fy(t), fz(t) = f(t) the fundamental equation of motion would no longer be a proper
function: a physical object cannot be at two different places at the same point in time. It can return to a place
where it was (or it can simply stay where it is), but it will then be there at a different point of time. That’s why time
goes by in one direction only. Why do we highlight this point? Because some physicists including Feynman
seem to suggest we should think of antimatter as particles traveling back in time. We think that is plain
nonsensical.
85
See the reference above.
86
The nature of a 2π factor is definitely very different from that of a 2 or 1/2 factor, which we often got when
analyzing something using non-relativistic equations. A 2π factor is associated with something circular, so we need
to explain what circular feature, and not in approximate but in exact termswhich is not easy in this particular
case.
87
Vassallo, G., Di Tommaso, A. O., and Celani, F, The Zitterbewegung interpretation of quantum mechanics as
theoretical framework for ultra-dense deuterium and low energy nuclear reactions, in: Journal of Condensed
Matter Nuclear Science, 2017, Vol 24, pp. 32-41. Don’t worry about the rather weird distance scale (1106 eV1).
Time and distance can be expressed in inverse energy units when using so-called natural units (c = ħ = 1). We are
not very fond of using natural units because we think they may hide rather than clarify or simplify some of the
more fundamental relations. Just note that 1109 eV1 = 1 GeV1 0.19751015 m. As you can see, the zbw radius
27
interesting point. Think of the black circle (in the illustration on the left-hand side below) as
circumscribing the Zitterbewegung of the pointlike charge. Think of it as the electron at rest: its radius is
the radius of the oscillation of the oscillation of the zbw charge a = ħ/mc. Note that we actually do ask
you to make abstraction of the two-dimensional plane of the oscillation of the Zitterbewegung (zbw)
charge, which need not be perpendicular to the direction of motion of the electron as a whole: it can be
in the same plane or, most likely, it may be zittering around itself. The point is this: we can introduce yet
another definition of a wavelength herethe distance between two crests or two troughs of the wave,
as shown in the illustration on the right-hand side.
88
Figure 8: The wavelength(s) of an electron
We should now present a rather particular geometric property of the Zitterbewegung (zbw) motion: the
a = ħ/mc radius the Compton radius of our electron must decrease as the (classical) velocity of our
electron increases. That is what is visualized in the illustration on the left-hand side (above). What
happens here is quite easy to understand. If the tangential velocity remains equal to c, and the pointlike
charge has to cover some horizontal distance as well, then the circumference of its rotational motion
must decrease so it can cover the extra distance. How can we analyze this more precisely? This rather
remarkable thing should be consistent with the use of the relativistic mass concept in our formula for
the zbw radius a, which we write as:


The λC is the Compton wavelength, so that’s the circumference of the circular motion.
89
How can it
decrease? If the electron moves, it will have some kinetic energy, which we must add to the rest energy.
Hence, the mass m in the denominator (mc) increases and, because ħ and c are physical constants, a
(for v = 0) is of the order of 2106 eV1 in the diagram, so that’s about 0.41012 m, which is more or less in
agreement with the Compton radius as calculated (av=0 0.3861012 m).
88
Because it is a wave in two or three dimensions, we cannot really say there are crests or troughs, but the
terminology might help you with the interpretation of the rather particular geometry here, which is that of an
Archimedes screw, but that’s only because of the rather particular orientation of the plane of the zbw oscillation,
which we ask you not to accept for granted: you should, instead, imagine the plane of oscillation itself is probably
not stable: the (in)famous uncertainty in quantum mechanics may actually be related to our lack of knowledge in
regard to the plane of the zbw oscillation: it may itself be zittering around.
89
Needless to say, the C subscript stands for the C of Compton, not for the speed of light (c).
28
must decrease.
90
How does that work with the frequency? The frequency is proportional to the energy
(E = ħ·ω = h·f = h/T) so the frequency in whatever way you want to measure it must also increase.
The cycle time T, therefore, must decrease. What happens, really, is that we are stretching our
Archimedes’ screw, so to speak. It is quite easy to see that we get the following formula for our new λ
wavelength:


Can the (classical or linear) velocity go to c? In theory, yes, but, in practice, no. The m in the formula is
not the mass of the zbw charge but the mass of the electron as a whole. That is non-zero for v = 0, unlike
the rest mass of the zbw charge, which only acquires mass because of its motion. We calculated this
relativistic mass of the zbw charge as equal to 1/2 of the electron (rest) mass. The point is this: we are
not moving a zero-mass thing here. The energy that is, therefore, required to bring v up to c will be
infinitely large: think of the enormous energies that are required to speed electrons up to near-
lightspeed in accelerators.
The point is this: an electron does not become photon-like when moving at near-lightspeed velocities.
However, we do see that the circumference of the circle that circumscribes the two-dimensional zbw
oscillation of the zbw charges does seem to transform into some linear wavelength when v goes to c!
Of course, we immediately admit this still does not explain what’s going on, exactly. It only shows we
should not necessarily think of the Compton wavelength as a purely linear feature.
Another remark that we should make here is that, while we emphasize that we should not think of a
photon as a charge travelling at lightspeed it is notand I mean not at all: photons do not carry
charge
91
the analysis above does relate the geometry of our zbw electron to the geometry of our
photon model. Let us quickly introduce that now as part of a larger reflection of what may or may not be
going on in photon-electron interactions.
Probing electrons with photons
The relation between what we think of as the radius of the Zitterbewegung oscillation of the electric
charge (a = ħ/mec), the Compton wavelength of an electron (λC = h/mec), and the wavelength of a
photon (λ = c/f) is not very obvious. However, we should not be discouraged because we immediately
note on thing, at least: the wavelength of a photon is the same as its Compton wavelength and its de
Broglie wavelength. Furthermore, because v = c and, therefore, β = 1, it is also equal to that third
wavelength we introduced above: λ = βλC. In short, the wavelength of a photon is its wavelength, so we
write:




But so what is that wavelength, really? Indeed, we should probably start by recognizing this: when
90
We advise the reader to always think about proportional (y = kx) and inversely proportional (y = x/k) relations in
our exposé, because they are not always intuitive.
91
This is, in fact, the quintessential difference between matter-particles and energy carriers such as photons (for
the electromagnetic force) and neutrinos (for the strong(er) force).
29
probing the size of an electron with photons, we had better have some idea of what a photon actually is.
Indeed, almost any textbook will tell you that, because of the wave nature of light, there is a limitation
on how close two spots can be and still be seen as two separate spots: that distance is of the order of
the wavelength of the light that we are using.
92
There is a reason for that, of course, and it’s got as much
to do with the wave as with the particle nature of light. Light comes in lumps too: photons. These
photons pack energy but they also pack one (natural) unit of physical action (h) or what amounts to
the same one unit of angular momentum (ħ).
Let us recall the basics of what we have actually presented a few times in previous papers already.
93
Angular momentum comes in units of ħ. When analyzing the electron orbitals for the simplest of atoms
(the one-proton hydrogen atom), this rule amounts to saying the electron orbitals are separated by a
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, we considered the transition from the second to the first level, for which the 1/12
1/22 factor is equal 0.75. Hence, the energy of the photon that is being emitted will 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 wavelength of (3×108 m/s)·(0.4×1015 s) = 122 nm, which is the wavelength of the
light (λ = c/f = c·T = h·c/E) that we would associate with this photon energy.
94
This rather simple
calculation is sufficient to illustrate our photon model: we think of a photon as being pointlike but, at the
same time, the Planck-Einstein relation tells us it packs one wavelengthor one cycle. The integrity of
that cycle is associated with its energy (E) and its cycle time (T) or alternatively with its momentum
(p) and its wavelength (λ), as evidenced in the E·T = p·λ = h relation:




So, yes, the equations above sort of vaguely tell us that if we think of measuring some distance or some
time as we do when we probe an electron with photons that we will have to content ourselves with
92
See, for example, Feynman’s discussion of using photons to try to detect electrons as part of the two-slit
experiment.
93
See, for example, our paper on Relativity, Light, and Photons.
94
Just so you can imagine what we are talking about, this is short-wave ultraviolet light (UV-C). It is the light that is
used to purify water, food or even air. It kills or inactivate microorganisms by destroying nucleic acids and
disrupting their DNA. It is, therefore, harmful. Fortunately, the ozone layer of our atmosphere blocks most of it.
30
measuring it in wavelength units (λ) or, equivalently, in cycle time units (T = 1/f = λ/c).
However, when probing electrons as part of Compton scattering processes, we are beating the usual
game here: we are actually measuring an effective radius of interference not in terms of the wavelength
of the light that we are using but in terms of the difference in the wavelength of the photon that goes in
and comes out of the scattering process. That’s what the Compton scattering formula tells us:


We already mentioned that the 1 cosθ factor on the right-hand side of this equation goes from 0 to 2
as θ goes from 0 to π. Hence, the maximum possible change in the wavelength is equal to 2λC, which we
get from a head-on collision with the photon being scattered backwards at 180 degrees.
95
However, that
doesn’t answer the question in regard to the 2π factor: the h/mc factor is still the Compton wavelength,
so that is 2π times the radius: λC = 2π·a = 2πħ/mc. In fact, mentioning the C value for the maximum
difference in wavelength introduces an additional factor 2.
Who ordered that? Let us advance a possible explanation: we are not saying it is the explanation, but it
may be an explanation. It goes like follows.
The electron-photon excitation: a temporary spin-2 particle?
We think the energy of the incident photon as an electromagnetic oscillation, that is is temporarily
absorbed by the electron and, hence, the electron is, therefore, in an excited state, which is a state of
non-equilibrium. As it returns to equilibrium, the electron emits some of the excess energy as a new
photon, and the remainder gives the electron as a whole some additional momentum.
How should we model this? One intriguing possibility is that the electron radius becomes larger because
it must now incorporate two units of h or, when talking angular momentum, two units of ħ. So we
should, perhaps, re-do our calculation of the Compton radius of our electron as follows:







This may actually workas long as we remember the energy and mass factors here are the combined
energies and masses of the electron and the photon. Of course, this triggers the next question: what’s
the typical energy or mass of the incoming photon? To demonstrate the Compton shift, Compton used
X-ray photons with an energy of about 17 keV, so that’s about 3.3% of the energy of the electron, which
is equal to 511 keV. For practical purposes, we may say the photon doesn’t change the energy of the
electron very much but, of course, it is significant enough to cause a significant change in the state of
95
The calculation of the angle of the outgoing photon involves a different formula, which the reader can also look
up from any standard course. See, for example, Prof. Dr. Patrick R. LeClair’s lecture on Compton scattering, which
we referenced already. The reader should note that the 1 cosθ is equal to 1 for π, and that there is no change
in wavelength for θ = 0, which is when the photon goes straight through, in which case there is no scattering.
31
motion of the electron.
96
Hence, the excited state of an electron may involve a larger radiustwice as large, approximately. Do
we think it explains the above-mentioned factor 2? We will let the reader think for himself here as we
haven’t made our mind up on it yet.
Planck’s quantum of action as a vector
What is uncertainty in quantum mechanics? How does it relate to the philosophical concepts of freedom
and determinism? We already briefly discussed this, but let us walk over the basics once more. Richard
Feynman notes that, “from a practical point of view”, we have “indeterminability in classical mechanics
as well.”
97
He gives the example of water splashing over a dam:
“If water falls over a dam, it splashes. If we stand nearby, every now and then a drop will land on
our nose. This appears to be completely random, yet such a behavior would be predicted by
purely classical laws. The exact position of all the drops depends upon the precise wigglings of
the water before it goes over the dam. How? The tiniest irregularities are magnified in falling, so
that we get complete randomness. Obviously, we cannot really predict the position of the drops
unless we know the motion of the water absolutely exactly.
While it is, effectively, not practically possible to know the motion of the water “absolutely exactly”, the
deeper question is: is it theoretically possible? In classical mechanics, we assume it is. In quantum
mechanics, we assume it is not. Quoting Feynman once again, we may say that “the Uncertainty
Principle protects quantum mechanics.” Why should this be so?
We think it’s plain nonsense: there is no reason whatsoever to assume the uncertainty in classical
mechanics in any different from the uncertainty in quantum mechanics. The uncertainty in quantum
mechanics is not anything more than the Ungenauigkeitimprecision, which is the term Heisenberg
first used to describe the apparent randomness in our mathematical description of Nature
98
we
encounter in classical mechanics as well. It is just an imprecision, indeedas opposed to the weird
metaphysical quality which Heisenberg would later claim it to be
99
and which, without any precise
96
Arthur Compton actually did not fire photons into free electrons but into electron shells bound into atoms: it is
only because the binding energy between the nucleus and the orbital electron is much lower than the energy of
the X-ray photons that one could think of the electrons as being free. In fact, the experiment knocked them out of
their orbitals!
97
See: Feynman’s Lectures, III-2-6.
98
We gratefully acknowledge the Wikipedia article on Heisenberg for this information. It states that it was
effectively in Copenhagen in 1927, to be precise that Heisenberg developed his uncertainty principle but that
in the letter he wrote to Wolfgang Pauli to describe it he used the term Ungenauigkeit rather than Ungewissheit
or Unbestimmtheit (indefiniteness). The reader should note that German was probably more important than
English or French as a language of science back then.
99
We think the term ‘indefiniteness’ (Ungewissheit or Unbestimmtheit in German) might better capture this
metaphysical quality. When saying this, we hastily add we do not believe such metaphysical quality applies to
quantum mechanics: probing electrons with photons is a complicated process but we do not believe it is
fundamentally different from, say, a blind man probing some object by taking it in his hands and trying to feel it or
if you prefer a more accurate metaphor carefully scanning some object by firing spherical elastic balls at it and
carefully keeping track of how they bounce back.
32
definition, physicists now refer to as ‘uncertainty’. The concept of ‘uncertainty’ effectively mixes two
different connotations:
(3) an imprecision in our measurements that is due to the wave-particle nature of electrons and other
elementary particlesa duality which the ring current model, combining the idea of charge and its
motion, captures from the outset (as opposed to trying to cope with the duality in some wave
equation or other dual mathematical descriptions); and
(4) a rapid magnification of what Feynman refers to as ‘the tiniest irregularities’ but which are, in
practice, purely statistical.
This is reflected in what Feynman refers to as the ‘great equation of quantum mechanics’the wave
Needless to say, we do acknowledge the force is not a simple contact force here: the structure of the
electromagnetic force is complicated, as evidenced from the fact that Volume II of Feynman’s Lectures is the most
voluminous of all three, even if all it does is go through the consequences of Maxwell’s set of equations. In fact,
Feynman shows that, when using four-vector notation, Maxwell’s equations reduce to two only:
The A and j are four-vectors A combines the classical electric potential and the (magnetic) vector potential A,
so A = (, A) and so I should probably use boldface rather than straight face (yes, I just invented that term right
here and now). If you are reading this footnote, it means you are damn serious in trying to figure this outso, yes,
please do email me if you have some ideas here. I promise I will reply. The point that I want to make here is this:
the first relation above models the dynamics of Maxwell’s equations. The second just states the principle of charge
conservation. Combine that with the idea of various modes of an oscillation in space and in time, and… Well…
That’s as close as I got to modelling the Great Mystery in Physics.
The implications of this ‘re-write’ of Maxwell’s equations are indeed quite deep. From the first equation, we get
this:
The relation is, obviously, extremely beautiful: the ratio of two physical (but four-dimensional) realities equals the
product of a mathematical operator and a (one-dimensional) physical (scalar) constant. I honestly don’t have a clue
about what it might mean, but it must be a very importantly relation. Why? Because the electric constant 0 is
related to lightspeed and the fine-structure constant. We will let you google those relationships. Again, if you feel
you make sense of them, please do not hesitate to email us: I think about them pretty much all of the time.
Feynman had to get his Nobel Prize and, therefore, probably deceives us here and there. However, in Vol. II, he is a
genius at work and, for me, I think the depth of his equations is quite obvious here:
33
equation for an electron in free space
100
101
:




We replaced the m2c2/h2 by the Compton wavelength λC = h/mc in the formula above because we
effectively do think as the distance scale within which we can localize the electron in a particle-like
sense. The dissipation which this wave equation models is, therefore, not the dissipation of the electron
itself but, effectively, our rapidly increasing uncertainty about where it might be at a later point in time.
This, of course, triggers the question: why would it increase so rapidly? The answer is simple. It’s just
scale: Planck’s quantum of action can (also) be written as the product of momentum over some distance
(h = p·λ)
102
and, in light of the very tiny distance scale (λ), the momentum (p) is, effectively, relatively
largewhich is why any imprecision in its magnitude or direction quickly translates into huge
uncertainties concerning the actual position of the elementary particle that we are trying to describe.
This is clear enough
103
, but the smart reader will, of course, immediately ask for more clarity: do we
think the uncertainty is in the direction or in the magnitude of the momentum? This is an essential
question, indeed! If the uncertainty is purely statistical, then what’s the hidden variable? We think the
hidden variable may be in the motion of the plane of rotation of the Zitterbewegung charge. Hence, we
think the statistical uncertainty has to do with the direction rather than the magnitude of the
momentum. We may, perhaps, focus the idea of the reader with the illustration below. It shows the
precessional motion of the plane of rotation of a spinning top. It reminds us of the vector quality of the
angular momentum. We should, therefore, probably write Planck’s quantum as a vector: h (or, to use
the right unit here, ħ) instead of h (or ħ)just like we would use L (the boldface to denote a vector)
rather than L for angular momentum. Look at the wobbling angular momentum of the spinning top
below, and then think of the plane of rotation of our pointlike charge.
100
This is just one of the many expressions of it. We took it from Feynman’s Lectures, section I-48-6. Feynman did
not substitute m2c2/h2 for 1/λC2 in his presentation. We must, therefore, assume he did not think of any possible
association or relation with the Compton wavelength and/or radius of an electron.
101
Needless to say, free space means we don’t think of any electromagnetic field or any other force field here.
Schrödinger adapted this equation to include a central force field: a hydrogen nucleus (a proton) attracting the
electrons around it. It is good to note that Dirac refers to such wave equations as the equations of motion of an
electron, because that’s what they are.
102
The h = E·T and h = p· λ are complementary but equivalent descriptions of the electron in the ring current
model.
103
The reader will immediately recognize this is just the same mechanism as the (statistical) mechanism which
makes chaos theory comprehensible and, therefore, rational and logically true.
34
Figure 9: A wobbling plane of rotation
104
Of course, we should remind ourselves we only have regularity in this motion because of the force field,
which is gravitational in the example of the spinning top (it’s the Fg vector). In quantum mechanics, the
force field would be some magnetic field: think of the Stern-Gerlach experiment here.
105
The question
then becomes: what’s the motion in the absence of a magnetic field?
We do not have the answer to that question but we do not believe that motion would be entirely
random: one would expect that interactions with other charged and non-charged particles (think of the
interaction with photons which we just described) would not only result in some change of the linear
state of motion of an electron but in changes in the direction of its angular momentum as well. These
are, of course, assumptions we may not be able to test because the measurement itself the probing
with photons, or the measurement of the angular momentum by applying some magnetic field would
cause such changes themselves! This is, in fact, what Heisenberg’s Ungenauigkeit is all about!
106
104
The illustration is taken from Wikimedia Commons under the CC-BY-SA 2.5 license so I need to acknowledge the
author: Xavier Snelgrove.
105
The reader will know that in order to separate particles based on their magnetic moment being up or down
the magnetic field cannot be homogenous: a field gradient is necessary so as to produce an actual force on the
particle. We can, therefore, not prove that an electron or whatever other elementary particle with spin will
effectively line up in a homogenous magnetic field as well. In fact, we should note something which may surprise
the reader: an actual Stern-Gerlach experiment has actually never been done with electronsor with other
charged particles. Stern-Gerlach experiments are always done with electrically neutral particles, such as potassium
atoms (see, for example, this rather typical lab experiment for students) or, in the original experiment, silver
atoms. There is a very practical reason for this, of course: it’s because any electric charge in the magnetic field in
the Stern-Gerlach apparatus would be subject to a Lorentz force which would be much larger than the force
resulting from the magnetic moment. At the same time, I find it a rather weird state of affairs: most, if not all, of
mainstream quantum theory hinges on the assumption that an electron has two spin states only (up versus down)
but, at the same time, no experiment has actually verified this. It is not that no one ever suggested it. H. Batelaan,
T. J. Gay, and J. J. Schwendiman, for example, wrote a rather intriguing letter to the Physical Review journal in
1997, explaining in very much detail how the Stern-Gerlach experiment could be modified to also split an electron
beam based on the magnetic moment being up or down. Their proposal is based, in fact, on a proposal of the
French physicist Léon Brillouin. Brillouin was an active participant in the 1927 Solvay Conference and his proposal,
which he published in 1928 (L. Brillouin, Proc. Natl. Acad. Sci. U.S.A. 14, 755, 1928), can be traced straight back to
the discussions there. In light of the venerable history of this proposal, it is very strange no one has followed up on
it. One would think this should be a higher priority than the new US$600m accelerator project which Dr. Consa
talks about in his scathing review of the current state of affairs in the field of experimental research.
106
We may usefully remind the reader Heisenberg’s very first arguments were entirely based on this kind of
physical reasoning. Think of Heisenberg’s thought experiment involving a microscope using gamma rays instead of
35
Hence, in the absence of a magnetic field (read: for a particle in free space), we would not be able to
ascertain the direction of Planck’s quantum written as a vector (h or, in its reduced form, ħ) but, while
we would think of it as being fairly random, we would not doubt its direction is the result of the motion
of the plane of rotation of the Zitterbewegung charge which, itself, is the result of previous interactions
and events. Such randomness would, therefore, be random but it would not be mysterious!
So how would we write the Uncertainty Principle then? Simple, we just write h and p as vectors. We get
the p·λ = h expression from the Planck-Einstein relation and we just apply it to the ring current geometry
(with λ the de Broglie wavelength
107
). We must then introduce the idea of a non-precise direction of h so
we can write something like this:
Δh = Δ(p·λ)
Plain vector algebra and remembering Δh denotes an uncertainty in the direction of the h and p
vectors, not of its magnitude then gives us the expression you are used toHeisenberg’s (in)famous
Uncertainty Principle:
h = ΔΔλ
That’s all what it is. By way of conclusion, we may, perhaps, add one more remark: why are we so sure
the uncertainty must be in the direction of the angular momentum of the electron, rather than in its
magnitude?
The honest answer is: we cannot be sure. We are basically just siding with Einstein’s instinct here: the
velocity of light is the velocity of light, Planck’s quantum of action is Planck’s quantum action, and the
classical velocity of the electron is the classical velocity of the electron. Hence, if we assume the
Compton radius of the electron and the motion of the charge inside is defined by h and c, and if we
assume the linear motion of the same charge is defined by the classical (linear) velocity of the electron
(v), then there is no room for uncertainty! The uncertainty must, therefore, be in the wobbling of h.
Nothing more. Nothing less.
108
optical light: it is very obvious Heisenberg does not think of uncertainty as something metaphysical here! Instead,
he plainly relates the uncertainty to the interaction between the observer the act of probing or measuring and
what is being observed or measured.
107
For a geometric interpretation of the de Broglie wavelength, we refer to our discussion of the Compton
wavelength and what happens to it when the Zitterbewegung charge inside of the electron is also subjected to
some linear motion.
108
We are very much aware of our rather arrogant tone here. I am happy to get other opinions here. To be sure, I
would treat them as being philosophical rather than scientificas philosophical as Einstein’s or our own intuition
here. However, that should not prevent one from having a healthy logical discussion on scientific matters.
36
Conclusions
The professional physicist will point out that (1) the lines above were definitely unscientific and (2)
despite our claim we would give you a real explanation of Compton scattering, we failed to produce all
of the details of the process. Indeed, how does it work, exactly?
We effectively did not work out all of the details, but we do have some very strong clues here on how a
definitive explanation including all relevant variables, including the mentioned plane formed by the
outgoing photon and the recoiling electron may look like. The plane of the Zitterbewegung oscillation,
for example, must without any doubt play a crucial role in determining the angles of the incoming as
well as the outgoing photon.
Also, it is quite obvious that the circular motion of the zbw charge must explain the π or 2π factor. It is
not a size factor: the size of the electron is of the order of the Compton radius (a) but, because of the
circular motion, it is actually the circumference of the motion (λ = 2πa) that must enter the Compton
scattering formula. The image of a hand sling throwing a stone comes to mind, of coursebut that’s
probably too simplistic: throwing some mass out by converting circular to linear motion is easy enough
to imagine, but here we’re not talking mass.
At the same time, our models are all ‘mass without mass’ models and so they show that energy is,
ultimately, motion: an oscillating force on a charge in case of the electron or, in case of the photon, an
oscillating electromagnetic field. The motion that’s associated with an electron is circular, while the
motion of the oscillating electromagnetic field is linear. As such, we may figuratively speaking say
that ‘circular’ motion is being converted into ‘linear’ motion, and vice versa.
In terms of the ‘mechanics’ of what might actually be going on when a photon is absorbed or emitted by
an electron, that’s all what we can give you for the time being ! You’ll have to admit we can’t quite show
you what’s going on inside of the box, but we did open it at least, didn’t we?
Just in case you think we didn’t, we invite you – once again to google the papers explaining the S-
matrix of Compton scattering processes.
109
We are sure you will prefer to re-read our paper and who
knows? start doing some pretty calculations yourself!
There is, perhaps, one question we still need to answer. Why and when did things go wrong in the
history of scientific thought here?
Now, I have not had the time to do much historical research here, but I do have a few clues, which I will
sum up in some separate concluding remarks.
109
As mentioned in the introduction, we rather randomly googled the following two papers:
https://arxiv.org/abs/1512.06681 and https://arxiv.org/abs/1810.11455. You may get others. This input-output
calculus approach allows for endless refinements and, hence, endless researchwhich is always nice from an
academic point of view, of course.
37
A brief history of bad ideas
Dr. Oliver Consa traces all of the nonsense in modern physics back to the Shelter Island (1947), Pocono
(1948) and Oldstone (1949) Conferences, about which he writes the following:
“After the end of World War II, American physicists organized a series of three transcendent
conferences for the development of modern physics: Shelter Island (1947), Pocono (1948) and
Oldstone (1949). These conferences were intended to be a continuation of the mythical Solvay
conferences. But, after World War II, the world had changed. The launch of the atomic bombs in
Hiroshima and Nagasaki (1945), followed by the immediate surrender of Japan, made the
Manhattan Project scientists true war heroes. Physicists were no longer a group of harmless
intellectuals; they had become the powerful holders of the secrets of the atomic bomb.”
110
Secrets that could not be kept, of course. The gatekeepers did their best, however. Julius Robert
Oppenheimer father of the atomic bomb and prominent pacifist at the same time was, effectively,
one of them.
The Solvay Conferences actually continued after World War II, and Niels Bohr and Robert Oppenheimer
pretty much dominated the very first post-WW II Solvay Conference, which was held in 1948. Bohr does
so by providing the introductory lecture ‘On the Notions of Causality and Complementarity
111
, while
Oppenheimer’s ‘Electron Theory’ sets the tone for subsequent Solvay Conferences—most notably the
one that would consecrate quantum field theory (QFT), which was held 13 years later (1961).
112
Significantly, Paul Dirac is pretty much the only one asking Oppenheimer critical questions. As for Albert
Einstein, I find it rather strange that despite him being a member of the scientific committee
113
he
actually hardly interferes in discussions. It makes me think he had actually lost interest in the
development of quantum theory.
Let us go back to the discussions between Dirac and Oppenheimer on the ‘Electron Theory’ paper in
1948. It may effectively mark one of the watershed moment: the point where things went wrong. In
fact, both Oppenheimer and Dirac make rather historical mistakes here:
1. Robert Oppenheimer uses perturbation theory to arrive at some kind of ‘new’ model of an electron,
based on Schwinger’s new quantum field theory (QFT)which, as we now know, do not lead anywhere.
2. Paul Dirac then challenges the use of these theories making the following sensible comment:
All the infinities that are continually bothering us arise when we use a perturbation method,
110
Oliver Consa, Something is rotten in the state of QED, February 2020.
111
See the 1948 Solvay Conference report on the ULB’s digital archives.
112
Institut international de physique Solvay (1962), La théorie quantique des champs: douzième Conseil
de physique, held at the Free University of Brussels, 9 to 14 October 1961.
113
Einstein was a member of the Solvay scientific committee from the very first conference (1911) representing,
in typical style, a country (Austria, not Germany) rather than an institution or just being a member in some
personal capacity till 1948. He was not a member of the 1951 scientific committee. The reason might well be age
or a lack of interest, of course: Einstein was 72 years in 1951, and would die four years later (1955).
38
when we try to expand the solution of the wave equation as a power series in the electron
charge. Suppose we look at the equations without using a perturbation method, then there is no
reason to believe that infinities would occur. The problem, to solve the equations without using
perturbation methods, is of course very difficult mathematically, but it can be done in some
simple cases. For example, for a single electron by itself one can work out very easily the
solutions without using perturbation methods and one gets solutions without infinities.
114
At the same time, he is too stubbornly foolish himself
115
: he simply keeps defending his un-defendable
electron equation which, of course, does not lead anywhere either:
We would not get infinities if we solve the wave equations without using a perturbation
method [but] if we look at the solutions which we obtain in this way, we meet another difficulty:
namely we have the run-away electrons appearing. Most of the terms in our wave functions will
correspond to electrons which are running away
116
, in the sense we have discussed yesterday
and cannot correspond to anything physical. Thus nearly all the terms in the wave functions
have to be discarded, according to present ideas. Only a small part of the wave function has a
physical meaning.
It is, however, very weird that Dirac does not follow through on his own conclusion:
Only a small part of the wave function has a physical meaning. We now have the problem of
picking out that very small physical part of the exact solution of the wave equation.
It is the ring current or Zitterbewegung electron, of course! It is that trivial solution that Schrödinger had
found, and which Dirac mentioned very prominently in his 1933 Nobel Prize lecture
117
. The other part of
the solution(s) is (are), effectively, bizarre oscillations which Dirac here refers to as ‘run-away electrons’.
Dirac just loved his own wavefunction too much to jettison it. “Kill your darlings” is a common piece of
advice given by unexperienced writers. Dirac should have thought about it, but so he did not. Here
again, the Nobel Prize Committee for Physics may actually have consecrated bad theory that, as a result
of the award, became difficult if not impossible to backtrack on.
That is my take on it but we will, of course, let you think about this for yourself. 
Jean Louis Van Belle, 25 April 2020
114
See pp. 282-283 of the report of the 1948 Solvay Conference, Discussion du rapport de Mr. Oppenheimer.
115
I think it is significant that Paul A.M. Dirac was no longer invited nor even mentioned in the 1951 Solvay
Conference. Niels Bohr and Wolfgang Pauli made subsequent remarks that clearly show Paul Dirac was the only
one who actually dared to challenge what had, by then, become mainstream theory.
116
See our own remarks in the beginning of this paper on this major problem of the wave equation: actual matter-
particles do not dissipate.
117
See p. 17 of this paper.
39
Annex: Light, photons and the Mach-Zehnder interference experiment
We did not say all that much about photons in this paper. We only gave you the basic model: a photon
as a one-cycle oscillation. The photon model may look simpler than the electron model but statements
like that are probably nothing but a matter of taste. Electromagnetic oscillations are rather complicated.
Let us, therefore, add a few remarks to this paper so as to illustrate this complexity.
Polarization states based on the electric field vector only
You have probably seen an illustration like the one below a zillion times already: it shows linear,
elliptical, and circular polarization states for lightand for a photon, because we think of a photon as an
electromagnetic oscillation too.
Figure 10: Linear, elliptical, and circular polarization states (electric field vector only)
So what is wrong with this image, which we borrowed from Feynman’s Lectures?
118
Nothing much,
except that Feynman makes the same mistake here as many professors explaining polarization states of
photons: he only thinks of the electric field vector E. There is a magnetic field vector B too, whose
118
We, once again, gratefully acknowledge Caltech (and Michael Gottlieb and Rudolf Pfeiffer, of course) for making
Feynman’s Lectures available online as a common reference for amateur physicists. This illustration was taken
from Feynman’s Lecture on the polarization of light. Note that, interestingly, Feynman expresses the Ex and Ey
components in both real as well as complex notation. I am not sure, but it suggests Feynman knew he was
presenting one aspect of the electromagnetic oscillation only.
40
direction in space will always be perpendicular to E as well as to the velocity vector c.
119
In fact, we may
also write B as B = i·E/c or i·E/c
120
.
We are not sure why physicists make that mistake. There may be two reasons:
1. The strength of the magnetic field is 1/c times that of the electric field, so they may think that is
negligibly small.
2. There may be a magnetic field but Lorentz force law tells us there is no force if there is no
charge. Hence, as long as the force does ‘t have a charge to play with (think of an electron here),
we should perhaps not consider the magnetic field.
Let us tackle the second consideration first. It is nonsensical because the magnetic field has a reality of
its own and the magnetic effect should, therefore, always be considered. To be precise, because of the
absence of a charge and, therefore, some velocity v we can use in the F = qE + qvB formula for the
Lorentz force, we should actually switch to a relativistically correct mathematical framework (read: four-
vector notation) and, as we do so, talk about the magnetic vector potential instead of the magnetic field.
Because we just want to make some basic remarks here, we will not dwell on this and just refer the
reader to more advanced reading material here.
121
The point is: the second reason is not a good reason
because it is false.
The first reason is nonsensical as well but the argument has to be somewhat more subtle here: it has got
to do with scales and natural units. Let us say a few words about that in a separate sub-section so as to
provide for some structure here.
The relative strength of the electric and magnetic field
We can, effectively, only talk about the relative strength of the electric and magnetic field in terms of
natural time and distance units. Why? Think of this: if we change our time and distance units so as to
ensure we measure c as being equal to unity (c = 1), then the B = E/c formula for the magnitudes of the
field vectors E and B reduces to B = E.
We should, of course, immediately add that simplifications like this are always quite tricky because the
physical m/s dimension of c continues to matter even if the numerical value of c is 1. Indeed, the electric
field strength is measured in newton per coulomb (N/C), but the magnetic field strength is, effectively,
expressed in units of (N/C)/(m/s) = (N·s)/(C·m) = T (tesla). Having said that, you immediately see what
119
You may not have seen a vector notation for c but just think of it as a velocity the velocity of the photon, of
course and then you will appreciate both the magnitude as well as the direction must be taken into account in a
physical analysis of what might or might not be happening.
120
The + or sign depends on our orientation, which determines whether we should rotate E/c clockwise or
counter-clockwise to obtain B. Needless to say, we use boldface letters to represent geometric vectors the
electric (E) and magnetic field vectors (B), in this case. There is a risk of confusion between the energy E and the
electric field E because we use the same symbols but the context (and our italics (E instead of E), if we do not
forget them) should make clear what is what.
121
We did a few blog posts on this, but these stuck rather closely to Feynman’s treatment of the matter, so we
may as well direct the reader to Feynman’s Lectures here too.
41
happens: we can no longer say one field is more important than the other. Both are equally relevant
and, therefore, need to be part of the analysis. So what are we saying here?
We are actually saying a linear polarization state is not really linear. We need to think of the combined
electromagnetic field of a photon making its way through space. Of course, you may say natural time
and distance units are just what they are natural units and you may wonder why we would want to
work with them. I have no definite answer to that question, but think of this. We may not only want to
think of natural time and distance units but also of some kind of natural unit for the force, the energy or
the (linear or angular) momentum. Force, energy, linear and angular momentum are currently
expressed in N (newton), N·m, N·s, and N·m·s, respectively. This shows that, if we already have a new
unit for distance and time (let us write those units as mγ and sγ respectively), we will just need to
calculate a new unit for the force too. We will write that unit as Nγ for the time being and we will
calculate it based on…
Well... Based on what?
It should be based on some physical law, obviouslya law such as the one we used to define natural
units for time and distance, which is this: the speed of light is always equal to c, regardless of the frame
of reference. It is obvious we can I’d say, should use the Planck-Einstein relation now. If we have the
wavelength and the cycle time for a photon any photon, really (a highly energetic gamma ray photon,
a visible light photon, a low-frequency radiowave photon or whatever) then we know the following
relations will always be trueand also regardless of the frame of reference:

You will say: what is that Fγ force in that h = Fγ·λγ·Tγ relation? However, you should not worry too much
about that now: think of it as the force that comes with the idea of an oscillating electromagnetic field
122
and just note the relation(s) above become(s) self-evident when dividing h by h and doing some
substitution
123
:
The point is this: if you are defining natural time and distance units based on the absolute velocity of
light, then you should also define a natural force unit based on the Planck-Einstein relation. If you do so
122
The electric and/or magnetic field are usually defined in terms of a force per unit charge. Indeed, we will say the
field strength is so many newton per coulomb, with an added complication for the magnetic field because of the
mentioned second per meter (s/m) factor: the physical dimension of the magnetic field, which we write as [B], is
equal to [B] = (N/C)/(m/s) = (N/C)·(m/s).
123
You may want to note the p = E/c relation becomes self-evident when using the mass-equivalence relation E/m
= c2 and writing p as p = mc = (E/c2c = E/c. Also, because light and, therefore, photons travel at the speed of light,
the λ/T = c relation is equally self-evident. The reader may sometimes wonder why we would be writing simple
things like this, but we hope these small digressions help in understanding what we are trying to understand. 
42
i.e. if you do re-define time, distance, and force units so as to make sure the numerical value of c and h
are equal to unity
124
then you also get natural units for energy and (linear or angular) momentum.
This should suffice to show that the idea of the relative strength of the electric and magnetic field does
not make much sense. However, we want to add an interesting argument here. You may or may not
know that we can define the electric and magnetic constants 0 and 0 in terms of the fine-structure
constant α, the lightspeed c and Planck’s quantum of action h:




It is now easy to see that the electric and magnetic constants become each other’s reciprocal if, and only
if, we use a natural unit for h as well:


We may have lost the reader by now so what’s the point? The point is this: you should think of a linearly
polarized photon as a two-dimensional oscillation as well. The oscillating electric field vector is just one
of the two aspects of the electromagnetic oscillation and, hence, you should always think of the
magnetic field vector too! When you do that, the hocus-pocus of quantum-mechanical interference and
diffraction experiments should disappear.
125
Note: The h2 = 1 condition effectively corresponds to two different possibilities: h = 1 or h = 1. This is in
line with our interpretation of h as a vector quantity (h instead of h). The h = 1 possibility corresponds to
matter, while the h = 1 corresponds to antimatter. It is nothing but a reversal of spatial
directionsboth linear as well as rotational (or circular) directions.
The geometry of a polarized photon
To describe the exact geometry of a linearly, circularly, or elliptically polarized photon taking into
account both the electric as well as the magnetic field one needs some more sophisticated geometric
formulas. We did not work this out fully, but the following remarks may encourage the reader to explore
the exact geometry of a photon somewhat further.
The geometry of the ellipse (or ellipsoid) relates the geometry of the circle (or the sphere) to the
geometry of a (wave)length.
126
We will want to think of an electromagnetic field vector rotating around
the direction of wave propagation here so we will limit ourselves to a two-dimensional analysis here.
124
We would like to mention once again you should not forget about the physical dimension of the c and h
constants. Writing c = h = 1 is, basically, meaningless because of the physical dimension of these constants: c is a
velocity (expressed in m/s or mγ/sγ or whatever time and distance unit you’d want to choose), while h is expressed
in units of physical action: force times distance times time.
125
See, for example, our classical explanation of one-photon Mach-Zehnder interference.
126
An ellipse is a shape in two dimensions only, while an ellipsoid is a shape in three dimensions. Think of the
difference between a circle and a sphere here.
43
Have a look at the a, b, and c lengths in the illustration below. If b goes to zero, a goes to c. Also, it is
easy to see that the two focal points (F2 and F1) will become the start and end point of a line.
Figure 11: The latus rectum formula: a·p = b2
Conversely, if we let the two focal points approach each other, we will get a perfect circle when they
coincide. The length p then becomes the radius of the circle, and the latus rectum formula becomes a
a·p = b2 = a·b = b2 = a2. We can now relate the circumference of a circle to a linear distance. Just think of
flattening a circle by stretching the ellipse, as illustrated below
127
.
Figure 12: Flattening the circle by stretching the ellipse
If we denote the radius of our circle by r, then it is easy to see the circumference of the ‘linear ellipse’
will be equal to 2πr. It is equally obvious that the length of the line will then be equal to half of the
circumference of the circle (πr). In other words, when the eccentricity of the ellipse (e = c/a
128
) goes
127
These illustrations were provided under the Creative Commons CC BY-SA 4.0 license so we acknowledge the
source and/or the author here.
128
The symbol e obviously does not refer to Euler’s number here, and the c here is just the distance from the
center to the focal point(s).
44
from 0 to 1, the length of the circular arc πr think of it as a circular length
129
becomes a simple linear
length.
130
What is the point here? As mentioned, we did not work this out fully. We just wanted to draw attention
to the fact that most analyses of the polarization state of a photon are rather simplistic because they
forget about the magnetic field vector. Nothing more, nothing lessfor the time being, at least!
131
Let us conclude this Annex by a short reflection on a question which few people seem to care about: if
light or photons may interfere constructively or destructively, what happens with the energy? The
question is quite obvious for the case of destructive interference. Indeed, what happens there? Does the
energy just disappear as two photons annihilate each other? Maybe. Maybe not. Let us think about that
in a separate section.
What happens to the energy when there is destructive interference?
It is an interesting question: if two light beams are exactly 180 degrees out of phase, we get destructive
interference and the resulting beam should have zero energy. Hence, the question is: where is the
energy of the incoming beams now? The energy conservation principle tells us it should be somewhere,
right? Right.
We did not spend too much time on this but, when googling, we found one good discussion on the
Physics StackExchange site, one very good video on MIT’s OpenCourseWare, and one very good paper
written by a former Princeton faculty member on it. These, taken together, give us a pretty reasonable
answer to the question: photons or the two halves of a photon in a one-photon Mach-Zehnder
interference experiment
132
do not destroy each other when coming together at the detection
129
However, it seems that this is a non-existing term. Everyone seems to go through the trouble of writing such
length as ‘the length of the circular arc.’
130
We may mention a rather weird mathematical fact here: there is no easy formula for the circumference (or
perimeter as it is often referred to) of an ellipse. See the Wikipedia entry on it or this very interesting paper of Prof.
Dr. Paul Abbott.
131
The next version of this paper may hold some more clues! 
132
Note the tricky question at the end of the video: “How is it that we get constructive interference in one beam
and destructive interference in another beam, when the two paths are the same in the two cases?” Fortunately,
we get a quick answer from one of the comments on the video: reflecting off of the back of a beam splitter does
not introduce a phase shift, while reflecting off of mirrors, and the front side of a beam splitter, will introduce a
180 degree (i.e. a half wavelength) phase shift. It took me quite a while to think this through, and I am actually still
not finished with it. For me, it just confirms the tricky physics of these mirrors and beam splitters in these so-called
one-photon Mach-Zehnder experiments is what explains the equally tricky results and conclusions. We think an
entirely classical explanation of the interference of a photon with itself should be possible, and that it is likely
based on the fact that beam splitters also result in linear polarization. However, we admit we did not have the
time, energy, and courage so far to fully work out the details of such explanation. We invite the interested reader
to further engage and if possible work out such fully-fledged solution him- or herself.
45
pointwhether it be a screen or a photon detector.
133
They just bounce back
134
or if there is no screen
(continue to) travel along happily.
The other side of the coin (or the question, I should say) is also interesting: if there is constructive
interference, we should get twice the energy when the beams are in phase. Why? We illustrated this
before
135
but let us, for the convenience of the reader, quickly review the basics here. They are this:
1. These interference experiments one photon at a time or more likely because easier to do in a
student lab environment always involve a beam splitter. Such beam splitter does not only split the
beam into two: it also changes its shapewe get two linearly polarized beam.
136
Hence, you should
think of two linearly polarized beams, which we will refer to as beam a and b, respectively. These waves
then arrive at the detector the screen or the single-photon detector
137
with a phase difference
unless the apparatus has been set up to ensure the distances along both paths are exactly the same.
Hence, the general case is that we would describe a by cos(ω·t k·x) = cos() and b by cos( + Δ)
respectively.
In the classical analysis, the difference in phase (Δ) will be there because of a difference of the path
lengths
138
and the recombined wavefunction will be equal to the same cosine function, but with
argument + Δ/2, multiplied by an envelope equal to 2·cos(Δ/2). We know this sounds rather monstrous
but we are just applying basic calculus here.
139
Indeed, to make a long story short, we can just write this:
cos() + cos( + Δ) = 2·cos( + Δ/2)·cos(Δ/2)
133
If it is a screen, we require a photon beam so as to provide sufficient energy to excite the electrons of the atoms
that make up the screen, so we can literally see the effect as these excitation states go back to normal and re-emit
photons of the same frequency as the incoming photons in all directions, including the line of sight formed by our
eye and the screen. One-photon Mach-Zehnder interference experiments obviously involve a much more delicate
experimental set-up involving one-photon detectors!
134
Again, if it is a screen and we cannot see the effect then we just have a dark spot on the screen: there is no
interaction here. In other words, the photon(s) must effectively bounce back.
135
See the above-mentioned reference to our paper on one-photon Mach-Zehnder interference. Alternatively, we
may also refer to Chapter XV of our 2019 manuscript on quantum mechanics.
136
You can check various sources here but the Wikipedia article on beam splitters is quite easy to read and quite
complete: most polarizing beam splitters including those in student labs use birefringent materials to split light
into two beams of orthogonal polarization states.
137
If we have one photon only, then our wave reduces to one cycle of the electromagnetic oscillation only. It is,
therefore, hard to talk about a beam. However, the principle of interference is the same: the principle of
superposition tells us all we have to do is to add the fields. If the field direction is the same, we get constructive
interference. If it is opposite, we get destructive interference.
138
Feynman’s path integral approach to quantum mechanics allows photons (or probability amplitudes, we should
say) to travel somewhat slower or faster than c, but that should not bother us here.
139
We are just applying the formula for the sum of two cosines here. If we would add sines, we would get sin() +
sin( + Δ) = 2·sin( + Δ/2)·cos(Δ/2). Hence, we get the same envelope: 2·cos(Δ/2).
46
2. What this tells us, is that we always get a recombined beam with the same frequency. However,
when the phase difference between the two incoming beams is small, its amplitude is going to be much
larger. To be precise, it is going to be twice the amplitude of the incoming beams for Δ = 0.
In contrast, if the two beams are out of phase, the amplitude is going to be much smaller. To be equally
precise as above, it is going to be zero if the two waves are 180 degrees out of phase (Δ = π), as shown
below.
Figure 13: Constructive and destructive interference for linearly polarized beams
3. That may make prima facie sense but, in fact, it does not. Why not? Because we know the energy of
an oscillation is proportional to the square of the amplitude. Hence, twice the amplitude means four
times the energy, and zero amplitude means zero energy. The energy conservation law is being violated:
photons are being multiplied or, conversely, are being destroyed.
Let us be very explicit about the energy calculation here. In order to do so, let us show you the set-up of
a typical Mach-Zehnder interference experiment:
Figure 14: The Mach-Zehnder interferometer
So what is going on here? Before the photon enters the beam splitter, we have one wavefunction: the
photon. When it goes through, we have two probability amplitudes
140
that somehow recombine and
140
We already referred to our paper(s) or, alternatively, chapter VIII of our manuscript for a geometric
interpretation of the wavefunction of a photon. When analyzing interference in this context, the wavefunction
47
interfere with each other. What we want to do here is to explain this classically. So let us look at the
apparatus. We have two beam splitters (BS1 and BS2) and two perfect mirrors (M1 and M2). An incident
beam coming from the left is split at BS1 and recombines at BS2, which sends two outgoing beams to
the photon detectors D0 and D1. More importantly, by playing with the distances between the beam
splitters and the mirrors, the interferometer can be set up to produce a precise interference effect
which ensures all the light goes into D0. Alternatively, the setup may be altered to ensure all the light
goes into D1.
Let us now discuss the energy calculations.
4. The energy conservation principle tells us that, when the incoming beam splits up at BS1, the energy
of the a and b beam will be split in half too.
141
We know the energy is given by (or, to be precise,
proportional to) the square of the amplitude (let us denote this amplitude by A).
142
Hence, if we
normalize or standardize the energy of the incoming beam by equating it to one, then we will want the
energy of the two individual beams to add up to A2 = 12 = 1, then the (maximum) amplitude of the a and
b beams must be 1/√2 of the amplitude of the original beam, and our formula becomes:
(1/√2)·cos() + (1/√2)·cos( + Δ) = (2/√2)·cos( + Δ/2)·cos(Δ/2)
This reduces to (2/√2)·cos() for Δ = 0. Hence, we still get twice the energy (2/√2)2 equals 2 when the
beams are in phase and zero energy when the two beams are 180 degrees out of phase.
This does not make sense. We must have made some mistake in our modeling here but what mistake,
exactly?
5. The only mistake we might have made is in the assumption that the direction of the (linear)
polarization of the two beams is the same. They are not: the two linear waves must be orthogonal to
each other, and that is why we cannot just add the amplitudes of the a and b beamsbecause they
have different directions.
All makes sense now. If the a and b beams after being split from the original beam are linearly
polarized, then the angle between the axes of polarization should be equal to 90 degrees to ensure that
the two oscillations are independent. We can then add them like we would add the two parts of a
complex number. Remembering the geometric interpretation of the imaginary unit as a
counterclockwise rotation, we can then write the sum of our a and b beams as:
(1/√2)·cos() + i·(1/√2)·cos( + Δ) = (1/√2)·[cos() + i·cos( + Δ)]
What can we do with this? Not all that much, except noting that we can write the cos( + Δ) as a sine for
Δ = ± π/2. To be precise, we get:
concept gives way to the concept of a probability amplitude which we associate with a possible path rather than a
particle.
141
Again, the term ‘beam’ may not seem to be appropriate in the context of one photon only, but think of it as a
one-cycle beam. 
142
If we would reason in terms of average energies, we would have to apply a 1/2 factor because the average of
the sin2 and cos2 over a cycle is equal to 1/2.
48
(1/√2)·cos() + i·(1/√2)·cos( + π/2) = (1/√2)·(cos i·sin) = (1/√2)·ei·
(1/√2)·cos() + i·(1/√2)·cos( π/2) = (1/√2)·(cos + i·cos) = (1/√2)·ei·
This gives us the classical explanation we were looking for:
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.
6. What about the energy equation now? In other words, what about that 1/√2 factor now? If the ei·
and ei· wavefunctions can, effectively, be interpreted geometrically as a physical oscillation in two
dimensions which is, effectively, our interpretation of the wavefunction
143
then then each of the
two (independent) oscillations will pack one half of the energy of the wave. Hence, if such circularly
polarized wave splits into two linearly polarized waves, then the two linearly polarized waves will
effectively, pack half of the energy without any need for us to think their (maximum) amplitude should
be adjusted.
Let us be precise here. If we think of the x-direction as the direction of the incident beam in the Mach-
Zehnder experiment, and we would want to also think of rotations in the xz-plane, then we need to
need to introduce some new convention here. Let us introduce another imaginary unit, which we will
denote by j, and which will represent a 90-degree counterclockwise rotation in the xz-plane.
144
We then
get the following classical explanation for the results of the one-photon Mach-Zehnder experiment:
Photon
polarization
At BS1
At mirror
At BS2
Final result
RHC
Photon (ei· = cos +
i·sin) is split into
two linearly polarized
beams:
Upper beam (vertical
oscillation) = j·sin
Lower beam
(horizontal oscillation)
= cos
The vertical oscillation
gets rotated clockwise
and becomes j·j·sin
= j2·sin = sin
The horizontal
oscillation is not
affected and is still
represented by cos
Photon is
recombined. The
upper beam gets
rotated counter-
clockwise and
becomes j·sin. The
lower beam is still
represented by cos
The photon
wavefunction is given
by cos + j·sin = e+j·.
This is an RHC photon
travelling in the xz-
plane but rotated
over 90 degrees.
143
We can assign the physical dimension of the electric field (force per unit charge, N/C) to the two perpendicular
oscillations.
144
This convention may make the reader think of the quaternion theory but we are thinking more of simple Euler
angles here: i is a (counterclockwise) rotation around the x-axis, and j is a rotation around the y-axis.
49
LHC
Photon (ei· = cos
i·sin) is split into
two linearly polarized
beams:
Upper beam (vertical
oscillation) = j·sin
Lower beam
(horizontal oscillation)
= cos
The vertical oscillation
gets rotated clockwise
and becomes
(j)·(j)·sin = =
j2·sin = sin
The horizontal
oscillation is not
affected and is still
represented by cos
Photon is
recombined. The
upper beam gets
rotated counter-
clockwise and
becomes j·sin. The
lower beam is still
represented by cos
The photon
wavefunction is given
by cos j·sin = ej·.
This is an LHC photon
travelling in the xz-
plane but rotated
over 90 degrees.
Of course, we may also set up the apparatus with different path lengths, in which case the two linearly
polarized beams will be out of phase when arriving at BS1. Let us assume the phase shift is equal to Δ =
180° = π. This amounts to putting a minus sign in front of either the sine or the cosine function. Why?
Because of the cos( ± π) = cos and sin( ± π) = sin identities. Let us assume the distance along the
upper path is longer and, hence, that the phase shift affects the sine function.
145
In that case, the
sequence of events might be like this:
Photon
polarization
At BS1
At mirror
At BS2
Final result
RHC
Photon (ei· = cos +
i·sin) is split into
two linearly polarized
beams:
Upper beam (vertical
oscillation) = j·sin
Lower beam
(horizontal oscillation)
= cos
The vertical oscillation
gets rotated clockwise
and becomes j·j·sin
= j2·sin = sin
The horizontal
oscillation is not
affected and is still
represented by cos
Photon is
recombined. The
upper beam gets
rotated counter-
clockwise and
because of the longer
distance becomes
j·sin( + π) = j·sin.
The lower beam is still
represented by cos
The photon
wavefunction is given
by cos j·sin = ej·.
This is an LHC photon
travelling in the xz-
plane but rotated
over 90 degrees.
LHC
Photon (ei· = cos
i·sin) is split into
two linearly polarized
beams:
Upper beam (vertical
oscillation) = j·sin
Lower beam
(horizontal oscillation)
= cos
The vertical oscillation
gets rotated clockwise
and becomes
(j)·(j)·sin = =
j2·sin = sin
The horizontal
oscillation is not
affected and is still
represented by cos
Photon is
recombined. The
upper beam gets
rotated counter-
clockwise and
because of the longer
distance becomes
j·sin( + π) = +j·sin.
The lower beam is still
represented by cos
The photon
wavefunction is given
by cos + j·sin = e+j·.
This is an RHC photon
travelling in the xz-
plane but rotated
over 90 degrees.
145
The reader can easily work out the math for the opposite case (longer length of the lower path).
50
What happens when the difference between the phases of the two beams is not equal to 0 or 180
degrees? What if it is some random value in-between? Do we get an elliptically polarized wave or some
other nice result? Denoting the phase shift as Δ, we can write:
cos + j·sin( + Δ) = cos + j·(sin·cosΔ + cos·sinΔ)
However, this is also just a circularly polarized wave, but with a random phase shift between the
horizontal and vertical component of the wave, as shown below. Of course, for the special values Δ = 0
and Δ = π, we get cos + j·sin and cos j·sin once more.
Figure 15: Random phase shift between two waves
Mystery solved? We think so, but we invite you to re-do the geometry and the related math. If anything,
we hope the argument shows you Bell’s No-Go Theorem should not prevent you from trying to go
everywhere. 
PS: The above argument suggests two energy states are, effectively, separated by Planck’s quantum of
actionfor an electron as well as for a photon. That is what the En = n·h·f relation suggests. If so, we
have a rather natural explanation for the half-photon explaining one-photon Mach-Zehnder interference
here: the half-photon is something real. 
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