Content uploaded by Jean Louis Van Belle
Author content
All content in this area was uploaded by Jean Louis Van Belle on Nov 25, 2021
Content may be subject to copyright.
Pair production and annihilation as a nuclear process
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
19 November 2020
Abstract
The phenomenon of matterantimatter pair creation and annihilation is usually taken as confirmation
that, somehow, fields can condense into matterparticles or, conversely, that matterparticles can
somehow turn into lightlike particles (photons and/or neutrinos) – which are nothing but traveling fields
(electromagnetic or, in the case of the neutrino, some strong field, perhaps). However, pair creation
always requires the presence of a nucleus. We, therefore, wonder whether pair creation and
annihilation cannot be analyzed as part of some nuclear process.
We argue the usual nuclear reactions involving protons and neutrons can effectively account for the
processes of pair creation and annihilation. We therefore argue that the need to invoke some quantum
field theory (QFT) to explain these highenergy processes would need to be justified much better than it
currently is.
Contents
Introduction .................................................................................................................................................. 1
Pair production as a nuclear process ............................................................................................................ 4
Pair annihilation as a nuclear process ........................................................................................................... 6
Stable and unstable particles: equilibrium and nonequilibrium states ....................................................... 7
Conclusions ................................................................................................................................................. 10
Post scriptum (15 May 2021): SLAC’s E144 experiment as a refutation? ................................................... 11
Post scriptum to the post scriptum (25 November 2021): Nope ! ............................................................. 11
Annex I : Preliminary thoughts on a deuteron model ................................................................................ 13
The electron cloud versus the proton core ............................................................................................. 13
The proton as a strong charge assembly? .............................................................................................. 14
Annex II : Recap of the zbw interpretation of spin and matterantimatter ............................................... 16
Modelling spin and antimatter ............................................................................................................... 16
Matter/antimatter and force/antiforce .................................................................................................. 18
1
Pair production and annihilation as a nuclear process
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
19 November 2020
Introduction
The phenomenon of matterantimatter pair creation and annihilation is usually taken as confirmation
that, somehow, fields can condense into matterparticles or, conversely, that matterparticles can
somehow turn into lightlike particles (photons and/or neutrinos) – which are nothing but traveling fields
(electromagnetic or, in the case of the neutrino, some strong field, perhaps). However, pair creation
always requires the presence of a nucleus
1
and one may, therefore, legitimately wonder whether the
electron and positron were not already present somewhere, somehow. Of course, we need to be
scientific here and show where and how exactly, so that is what we will try to do here.
Carl Anderson’s original discovery of the positron involved cosmic rays hitting atmospheric molecules, a
process which involves the creation of unstable particles including pions.
2
Cosmic rays themselves are,
unlike what the name suggests, no rays – not like electromagnetic gamma rays, at least – but highly
energetic protons and atomic nuclei. Hence, they consist of matterparticles, not of photons. The
creation of electronpositron pairs from cosmic rays involves these pions as intermediate particles:
1. The π+ and π− particles have net positive and negative charge of 1 e+ and 1 e− respectively.
According to mainstream theory, this is because they combine a u and d quark but – abandoning
the quark hypothesis
3
– we may want to think their charge could be explained, perhaps, by the
1
The usual reason that is quoted here has to do with excess energy and momentum that, somehow, needs to be
absorbed. The Wikipedia article on pair creation, which quotes or summarizes from J.H. Hubbell’s 2006 overview
article on electronpositron pair production by photons, says this: “The photon must be near a nucleus in order to
satisfy conservation of momentum, as an electron–positron pair produced in free space cannot both satisfy
conservation of energy and momentum.” We think this explanation does not quite cut it.
2
The discovery of the positron is, without any doubt, to be credited to the tireless efforts of Carl Anderson in the
early 1930s. In contrast, the discovery of the pion – both experimentally as well as theoretically – is a more
complicated matter. Nobel Prizes in Physics were awarded to Yukawa in 1949 for his theoretical prediction of the
existence of mesons, and to Cecil Powell in 1950 for developing and applying the technique of particle detection
using photographic emulsions, which were effectively used to confirm the existence of what was then referred to
as charged πmesons in an international effort led by Cecil Powell. However, some credit a young Indian scientist
at the Bose Institute in Calcutta (now Kolkata), Bibha Chowdhury, with the actual discovery. She effectively
discovered traces of the heavy ionized particles using photographic plates and apparently published on these
discoveries in not less than three articles for the Nature journal in 1941 and 1942. As for its theoretical
foundations, we think Yukawa’s concept of a strong force makes sense, but we never quite understood the idea of
it having to be mediated by a 100 MeV virtual quantum. See our paper on the nature of Yukawa’s force and charge.
3
You may be so familiar with quarks that you do not want to question this hypothesis anymore. If so, let me ask
you: where do the quarks go when a charged pion disintegrates into a muonelectron (or positron), or into highly
energetic photons? We think the invention of the concept of strangeness by Murray GellMan and Kazuhiko
Nishijima in the 1950s may or may not have been useful as a mathematical concept. However, we feel this concept
started a rather strange life of its own as it would effectively serve – much later – as the basis for the quark
2
presence of a positron (or an electron in the case of a π−)! They effectively disintegrate into a
muonelectron () which, in turn, will emit a neutrino
4
and morph into an electron or its
positively charged antimatter counterpart (e).
2. The neutral pion is a very different animal: it (usually) disintegrates into two photons
5
which, in
turn, somehow both morph into an electron and a positron – so we get two electrons and two
positrons, and so that is the process which we want to think about in this paper.
The illustration below shows the (1) ingredients (the highly energetic proton and an atmospheric
molecule) and (2) final products (one muonelectron/positron pair, two electron/positron pairs
6
, and a
neutron) of this remarkable process.
Figure 1: Pion production from cosmic rays (source: Wikipedia)
Before we get into the nittygritty of it all, we should make some few preliminary remarks:
1. Note that the illustration above might suggest that the whole process – from start to end – does not
respect the charge conservation principle: the charge of the incoming proton is e+, while the charges of
the intermediate products (π+, π−, π− and a neutron) add up to zero. However, while the lifetime of a
(free) neutron is close to eternity (about 15 minutes), we argue one should think of it as combining a
proton and an electron.
7
Hence, the proton balance before and after is OK, but we are missing an
hypothesis which – for a reason we find even stranger than the concept of strangeness itself – was officially
elevated to the status of a scientific dogma by the Nobel Prize Committee for Physics.
4
To be precise, the process involves the emission of two neutrinos: a neutrino and its socalled antimatter
counterpart. We think of neutrinos as lightlike particles, so there is no opposite charge here: we think the two
neutrinos differ only in their spin.
5
There are other decay modes, of course, but this is the principal one, and so we will look at this mainly.
6
Because the will disintegrate into an e (the lifetime of a muonelectron is 2.210−6 s), you may think of the
final products as three electronpositron pairs (and some neutrinos, of course) and – lest we forget – the neutron.
Note that a lifetime of 2.210−6 s is considered to be (almost) an eternity in particle physics. The Wikipedia entry
on the microsecond has an animated gif which gives you an idea of what such time interval actually means.
7
We know this sounds outrageous but we think it is justified because of the neutron decay reaction. A neutron
does not decay into quarks or some other exotic thing. It decays into a proton and an electron: n0 → p+ + e− + 0.
Simple. We do not understand why some academics find it so difficult to accept what is written here or, worse,
simply refuse to consider it as an alternative for the quark hypothesis.
3
electron. It should be added somewhere.
8
Where, exactly? We do not know, yet. It must be something
with the atmospheric molecule.
9
2. It is plain weird – or artificial, we should say, perhaps – that neutral pions are, somehow, being
thought of as being similar to (charged) π particles. The casual lumping together of π particles and
neutral pions under one and the same banner (pions) is like saying protons and neutrons are nucleons,
both. That is an obvious truth, of course, but we do not learn much by it: we need to get into the nitty
gritty of neutron decay and other nuclear processes to understand how different they actually are,
right? And the difference between neutral and charged pions is even starker.
For starters, neutral pions have a much shorter lifetime – in the order of 10−18 s only – than π+ and π−
particles, whose lifetime is a much more respectable 2.610−8 s. Something you can effectively measure,
in other words.
10
And then, charged pions carry charge. Neutral pions do not. Huge difference! In short,
despite similar energies, neutral pions do not seem to have a lot in common with π+ and π− particles.
Historically, charged pions were discovered in the late 1930s (and further confirmed in the 1940s), while
the neutral pion was discovered in very different experiments in the 1950s only. We, therefore, wonder
why neutral pions and π+ and π− particles are to be thought of as, somehow, being similar particles.
11
8
All kinds of weird things may happen to the number of charged particles – especially if they are only intermediate
particles – but the matterantimatter pair creation of annihilation does respect the overarching charge
conservation law. Charge, momentum (linear and angular), and energy are always conserved, somehow. The
invention of a zillion weird quantum numbers does not fundamentally challenge this.
9
There are many possibilities here. The most obvious is an ionization of the atmospheric molecule: there is a very
good reason why the upper layer of the atmosphere is referred to as the ionosphere, indeed! However, such
ionization may not be the direct result of an electron being ripped out of a shell. The highly energetic proton might,
perhaps, knock out one of the neutrons in the nucleus! It could then morph into a neutron by capturing an
electron: p+ + e− → n0 + 0. Is this what happens? We do not know. The point is this: in highenergy physics, we
should forget about particles being conserved – obviously – but we should not forget total charge must be
conserved, somehow.
10
The point estimate of the lifetime of a neutral pion of the Particle Data Group (PDG) is about 8.510−17 s. Such
short lifetimes cannot be measured in a classical sense: such particles are usually referred to as resonances (rather
than particles) and the lifetime is calculated from a socalled resonance width. We may discuss (and criticize) this
approach in a future version of this paper. Just note that, even at the speed of light, these particles would only
travel (8.510−17 s)·(3108 m/s) = 25.5108 m−9. That length is about 500 times the radius of a hydrogen atom, and
a particle with a rest mass of 135 MeV can surely not aspire to travel anything near lightlike. So, yes, thinking of it
as some kind of local unstable resonance – something which happens at the scale of an atom itself – is quite
appropriate.
11
Quark theorists say they have this in common: they all consist of a quark and an antiquark. We wonder what
they mean by that – not approximately, but exactly? What explains the very different lifetimes and the very
different decay modes? Aitchison and Hey answer this question in two volumes (Gauge Theories in Particle Physics,
2013) but, frankly, we find such long answer rather complicated and, therefore, unconvincing. The short
explanation is that the neutral pion decays via the electromagnetic force, while the charged pions decay because
of the weak force. We read this as follows: the neutral pion consists of opposite (electric) charges (which do not
necessarily need to be quarks for us) while the charged pions (also) involve something else, which is not
necessarily some weak force (we think of a force as holding something together, rather than as something pulling
something apart) but, perhaps, some strong force. Such strong force must have a different geometry than the
electromagnetic force or – who knows? – might act on a different charge, or both perhaps. However, we do not
necessarily think of the concept of color charge here.
4
Even the energy difference is quite substantial (when measured in terms of the electron mass, that is):
the neutral pion has an energy of about 135 MeV, while π+ and π− particles have an energy of almost
140 MeV. To be precise, the difference is about 4.6 MeV. That is quite a lot: the electron rest energy is
0.511 MeV only.
12
So it is not stupid to think that π+ and π− particles might carry an extra positron or
electron, somehow. In our notsohumble view, this is as legitimate as thinking – like Rutherford did –
that a neutron should, somehow, combine a proton and an electron.
13
The whole analysis – both in the QED as well as in the QCD sector of quantum physics – would radically
alter when thinking of neutral particles – such as neutrons and π0 particles – not as consisting of quarks
but of protons/antiprotons and/or electrons/positrons cancelling each other’s charges out. We have not
seen much – if anything – which convinces us such thinking cannot possibly be correct. We, therefore,
believe a more realist interpretation of quantum physics should be possible for highenergy phenomena
as well. With a more realist theory, we mean one that does not involve quantum field and/or
renormalization theory. Such new theory would not be contradictory to the principle that, in Nature, the
number of (charged or neutral) particles is no longer conserved, but that total (net) charge is actually
being conserved, always. Hence, charged particles could appear and disappear, but they would be part
of neutral particles. All particles in such processes are very shortlived anyway, so what is a particle
here? We should probably think of as an unstable combination of various bits and bobs, isn’t it?
However, we readily admit this was probably the longest introduction to a paper – ever – and that,
nevertheless, some of the reasoning above may be considered to be rather sloppy and general. Let us,
therefore, be much more precise.
Pair production as a nuclear process
The overview below (Figure 2) lists of all of the decay modes of a proton.
Figure 2: The decay modes of the π0 resonance
The table shows what we know already: a neutral pion (π0) usually – this means 98.8% of the time here –
decays into two photons. Occasionally (almost 1.2% of the time), it decays into a photon and an
electronpositron pair but, according to Wikipedia, this is actually also a twophoton decay with one of
12
Of course, it is much smaller when compared to the proton (rest) energy, which it is about 938 MeV.
13
See our short history of quantummechanical ideas or our paper on protons and neutrons.
5
the photons decaying into an electronpositron pair. Once in a million (see the 10−6 fractions), or once in
a billion (see the 10−9 fractions), it decays into something else. We may or may not come back to those
other modes in a later version of this paper. Let us first think about the main decay mode: two highly
energetic photons. How energetic, exactly? And what happens with these photons, then?
Gamma rays from radioactive decay (nuclear gamma rays) carry energies up to 8 MeV, but so here we
must be talking 67 MeV photons (half of the 134 MeV energy of the pion).
14
That is huge. When
interacting with the electromagnetic fields inside of an atom and, presumably, within a nucleus itself,
such photon must rip all apart – and it does! This is probably why the naturally occurring process of pion
decay in the upper layers of our atmosphere usually shows the two photons creating electronpositron
pairs when interacting with other nearby matterparticles (as shown in Figure 1, indeed). So how does
that happen – not approximately but exactly? We must, of course, think of the four principal nuclear
processes here:
1. Neutron decay
15
: n0 → p+ + e− + 0
2. Electron capture by a proton: p+ + e− → n0 + 0
3. Positron emission by a proton (i): 0 + p+ → n0 + e+
4. Positron emission by a proton (ii): γ + p+ → n0 + e+ + 0
The latter two processes are very different
16
but yield the same: a proton emits a positron and becomes
a neutron. However, the process we are interested in here is, of course, the positron emission which
involves the photon absorption. So we think of a sequence like this:
1. The nucleus absorbs the gammaray photon by a protonneutron Verwandlung
17
: p+ → n0 + e+.
We have a proton less, but an extra neutron and a positron now.
2. The nucleus returns to its original state when the extra neutron decays back into a proton, while
emitting an electron.
18
Hence, the equation is this:
14
See the values for the momentum in the final column of the PDG table.
15
All of these processes involve neutrinos. We were first tempted to not distinguish between neutrinos and
antineutrinos. We effective think of neutrinos as lightlike particles, so that is like photons, but involving a different
force and, therefore, a different energy. Because they do not carry any charge (no electric and also no strong
charge – or whatever else you might invent as a charge), the difference between neutrinos and antineutrinos
must, therefore, be related to their spin only, which we interpret as being physical somehow (all of our theories
are geometric and, therefore, physical). Spin is, therefore, always in one of two possible geometric directions, and
the prefix (anti) may, therefore, not be useful when talking neutrinos. We, therefore, opted to denote
antineutrinos with an underscore (0) instead of the usual overline (). It makes for easier typing too!
16
The first is the 1951 CowanReines experiment (bombarding protons with neutrinos). The second describes +
decay. We refer to one of our papers on this for a more detailed description.
17
We prefer this German word to the English: transformation. We admit it is not scientific. We note the proton
neutron transformation involves a neutrino. Where does that come from? We do not know: it may be energy from
outside, but we think it should come from some internal strong field. We admit this is speculative. We put the
neutrino in the final equation: the reader can verify we have it in both sides of the equation, which lends credibility
to the hypothesis of using internal energy only here.
18
Our accounting of neutrinos here is somewhat sloppy. We are not so worried about that. If the neutrinos are
antineutrinos of each other, they should annihilate and provide some extra (strong) energy – whatever that might
be. If not, then we would effectively need to keep track of them much more carefully than we do here. It should be
noted that some of the other decay modes of neutral pions involve neutrinos. Hence, we do not feel our rather
6
+ p+ + 0 → n0 + e+ → p+ + e− + e+ + 0
The net result is the → e− + e+ equation that we needed. You will have to admit this is a much more
elegant way to explain matterantimatter pair production out of photons than the usual hocuspocus,
isn’t it? However, science is not necessarily about elegance.
19
Science is about what makes sense and
who does not. Hence, if this makes sense (which remains to be seen), we should also explain matter
antimatter annihilation in a way that shows electric charge does not get magically lost somehow! Let us
see if we can do this.
Pair annihilation as a nuclear process
Let us think practically here too: the positron will meet an electron and there will be mutual annihilation
but where, exactly? The positron is likely to meet an electron that is part of some atom. Will it engage
with one of the electrons in the electrons shells? Maybe. However, if we would think of a neutron as
consisting of a proton with an electron
20
, we may imagine the positron to, perhaps, interact with the
nucleus. Our positron is probably highly energetic and so it will, effectively, tear through the electron
shells without any (meaningful) interaction with them or – more likely, perhaps – it may shear them all
off without losing much energy at all.
21
Hence, we might imagine a process that is the reverse of the
positron emission by a proton. Instead of 0 + p+ → n0 + e+, we get this
22
:
n0 + e+ → p+ + 0
We might refer to this as positron capture by a neutron, and some scientific articles actually do explore
this, although we are not sure whether or not there is some experimental evidence for this.
23
The
question, of course, is this: how would a neutron do this?
We think of the neutron consisting of a proton and a neutron, in which case the incoming positron must
annihilate the nuclear electron. Can we prove this? No. Can we rule out this is not possible? No. But we
do think it makes a lot of sense.
relaxed approach here (which basically amounts to saying that we might examine this more in detail later) as a
serious issue. The reader has the right to disagree, of course. He may also want to think it through for himself by
adding more detail to the analysis.
19
As Dirac famously remarked, quantum field theory and perturbation approaches are surely not about elegance
and beauty!
20
The reader will, in the meanwhile, have understood that we love Rutherford’s original hypothesis of the neutron
combining a proton with a nuclear electron: we think it remains relevant and extremely productive. For a short
introduction to Rutherford’s ideas here, see our short history on quantummechanical ideas, in which we analyze
some of Rutherford’s remarks in this regard in his paper on ‘The Structure of the Electron’ at the 1921 Solvay
Conference.
21
If we take the example of atmospheric molecules, the reader should remember those molecules are mostly
ionized already, so there are no electron shells to start with even!
22
Note that the reverse of the reaction involves an antineutrino. Indeed, from a mathematical point of view,
opposite spin and time reversal both amount to the same: the same wavefunction but with an opposite sign for
the imaginary unit.
23
A casual google effort yielded a list with an arXiv.org article: Mikhail Khankhasayev and Carol Scarlett, Positron
on Neutron capture reaction, radiative corrections and neutron electric dipole moment, May 2013. However, this
article is a theoretical exploration only. Theoretical as it is, it confirms the neutrino in the reaction must be an
antineutrino, so that confirms our hypothesis.
7
So we lost a neutron and we gained a proton, but the state of the nucleus before and after must be the
same – sort of at least, right? That is taken care of by the electron: we must assume this electron is
highly energetic too and will, therefore, also be able to tear through the electron shells without any
interaction.
24
This electron should be captured by a proton so as to restore the original nucleus state:
p+ + e− → n0 + 0
So, yes, the two processes together yield the e+e− annihilation process we wanted to see:
n0 + e+ → p+ + 0
p+ + e− → n0 + 0
⎯⎯⎯⎯⎯⎯⎯
p+ + n0 + e+ + e− → p+ + n0 + 0 + 0 → p+ + n0 + 2
Note that the assumption here is that the neutrino and antineutrino will decay into two photons with
opposite spin. How does this happen, exactly? That is a question we cannot answer for the time being.
However, we feel it is more reasonable to argue that strong field energy inside of the nucleus could,
somehow, be converted into electromagnetic field energy
25
– much more reasonable than the
pointblank creation of matterparticles out of field energy, in any case!
The key point is this: this process explains matterantimatter annihilation as a nuclear process too. There
is, therefore, no need for quantum field theory !
Stable and unstable particles: equilibrium and nonequilibrium states
We have been talking about protons, neutrons, electrons, and their antimatter counterparts – real
matterparticles. And about photons and – to a very limited extent – neutrinos. Things that we know to
exist in any meaningful way: they last for a while, or even permanently (except when they happen to be
part of a highenergy event, of course). So what are those pions, then?
You tell me. I do not worry about them too much. They are some kind of unstable state – a
disequilibrium state, in other words: some transient electromagnetic oscillation
26
with one or more
elementary charges whirling around in it. When a car gets destroyed in some accident, we are usually
interested in the victims – not in the exact details of what debris flies where exactly. We are not, in any
case. We summed up our vision of what makes sense in several ironic rewrites of Feynman’s Lectures.
We rewrote his introduction to quantum physics, for example, as follows
27
:
24
Note that the positron is going through a potential well, while the electron is going through a potential barrier.
25
And vice versa, of course. The fourth reaction (photon absorption by a proton) which is so crucial in our
reasoning here is actually a process which is not well understood. Instinctively, we feel it should, probably, also
involve a photonneutrino conversion, somehow. Because spin (angular momentum) is conserved, this probably
involves pair production of photons and/or neutrinos. For a more profound analysis of what might or might not be
going on in a nucleus, we may refer to our paper on protons and neutrons.
26
You might wonder: perhaps some strong oscillation too? If you find it useful to think like that, I do not mind. Not
at all, really. But I would appreciate if you could elaborate what you could possibly mean with that. Something
neutrinolike, perhaps?
27
See our Lectures on Physics, Chapter I: Quantum Behavior.
8
Newton thought that light was made up of particles, but then it was discovered that it behaves like a
wave. Later, however (in the beginning of the twentieth century), it was found that light did indeed
sometimes behave like a particle. Historically, the electron, for example, was thought to behave like a
particle, and then it was found that in many respects it behaved like a wave. Hence, the challenge is to
find a description that takes account both of the wave as well as of the particlelike character of both
matter as well as lightparticles. We may refer to both as wavicles but – for historical reasons – this term
did not become household language.
Lightparticles are known as photons. Photons carry electromagnetic energy, but they do not carry charge.
In contrast, matterparticles always carry charge. If they are neutral – think of a neutron or an atom – they
will carry both positive and negative charges. We should, therefore, think of them as composite particles.
Elementary particles are stable. Composite particles consist of elementary particles and may be stable or
unstable. An atom is an example of a stable composite particle. A neutron is stable inside of the nucleus
but unstable as a free particle: it spontaneously disintegrates into a proton and an electron. This process
involves the emission of a neutrino, which ensures energy is conserved. We think of a neutrino as a
lightlike particle: it also carries energy but no charge.
28
Electrons and protons are elementary matterparticles. They are stable because they are wavicles in an
equilibrium state – in the sense that their fundamental cycle is given by the PlanckEinstein relation: T =
1/f = h/E.
29
They are stable but not indestructible. Highenergy collisions between protons – or between
protons and antiprotons – yield unstable particles which disintegrate back into stable particles. Because
they are unstable, such particles should not be referred to as particles but as transients or, when very
shortlived, as resonances.
The Higgs particle is an example of an extremely shortlived resonance: its lifetime is of the order of 10−22
seconds. Even at the speed of light – which an object with an estimated rest mass of 125 GeV/c2 can never
aspire to attain – it cannot travel any further than 0.3 femtometer (0.310−15 m) before it disintegrates.
Such distance is smaller than the radius of a proton, which is in the range of 0.83 to 0.84 fm. Labelling it as
a particle is, therefore, hugely misleading. Likewise, quarks have also never been directly observed or
isolated. Their existence is and remains, therefore, a mere hypothesis, which we will not entertain in
these lectures because we have no need for it: highenergy physics studies disintegration processes,
which involve nonequilibrium states⎯and we will not study these in our lectures.
30
These highenergy
collisions are interesting though because they show that protons must have some internal structure. We
think of such structure not in terms of quarks or gluons
31
, but in terms of the motion of the elementary
charge. Paul Dirac wrote the following on that:
28
The nature of this energy is not electromagnetic, however. Electromagnetic energy is related to electromagnetic
forces. We may, therefore, think of the energy of a neutrino as being related to the strong(er) force inside of a
proton or a neutron.
29
For a broad overview of our assumptions, which amount to a fullblown realist interpretation of particle or
quantum physics, see our Principles of Quantum Physics.
30
If you want to know what we think of the quark hypothesis, we think this hypothesis results from an
unproductive approach to analyzing disintegration processes: GellMan and Kazuhiko Nishijima studied
disintegration processes of Kmesons back in the 1950s, and invented new quantities that are supposedly being
conserved in these processes. One of these quantities was referred to as strangeness (see the analysis of Kmesons
in Feynman’s Lectures). These strange new concepts then started to lead an even stranger life of their own.
31
See our remarks on the quark hypothesis in footnote 30. As for gluons, these are supposed to carry the strong
force. We see no need to invent new particles to carry forces: the concept of fields – electromagnetic or other –
should do. The idea of forcecarrying particles resembles 19th century aether theory: there is no need for it, so why
should we entertain it?
9
“Quantum mechanics may be defined as the application of equations of motion to particles. […]
The domain of applicability of the theory is mainly the treatment of electrons and other charged
particles interacting with the electromagnetic field⎯a domain which includes most of low
energy physics and chemistry.
Now there are other kinds of interactions, which are revealed in highenergy physics and are
important for the description of atomic nuclei. These interactions are not at present sufficiently
well understood to be incorporated into a system of equations of motion. Theories of them have
been set up and much developed and useful results obtained from them. But in the absence of
equations of motion these theories cannot be presented as a logical development of the
principles set up in this book.
We are effectively in the preBohr era with regard to these other interactions. It is to be hoped
that with increasing knowledge a way will eventually be found for adapting the highenergy
theories into a scheme based on equations of motion, and so unifying them with those of low
energy physics.”
32
These words were written in 1958 but still ring true today. What about quantum field and perturbation
theory? Dirac thought they could not be true.
33
We think the situation is a lot worse: no one seems to be
able to clearly state why they were invented or what problem they are supposed to solve. It is probably
a question to be left to the history of science: no one uses quantum mechanics in practical theory
anyway. The study of semiconductors, for example, just takes the main results out of quantum physics
and then develops more realist workable models based on these results.
34
In a pragmatic interpretation
of what physics should and should not be, we think that is maybe not great (because untrue), but good
enough (because practically workable). In addition, as Dr. Consa usefully notes
35
, it sustains large
research institutions and consumes budgets that would otherwise would probably be spent on R&D in
the defense sector anyway. We may, perhaps, add a final remark on Dirac. In the Preface to the fourth
and last edition (1958) of his Principles of Quantum Mechanics (1930) – from which we quote above –
Dirac also writes this:
“In presentday highenergy physics the creation and annihilation of charged particles is a frequent
occurrence. A quantum electrodynamics which demands conservation of the number of charged particles
is therefore out of touch with physical reality. So I have replaced it by a quantum electrodynamics which
includes creation and annihilation of electronpositron pairs.
This involves abandoning any close analogy with classical electron theory, but provides a closer
description of nature. It seems that the classical concept of an electron is no longer a useful model in
physics, except possibly for elementary theories that are restricted to lowenergy phenomena.”
32
Paul A.M. Dirac, The Principles of Quantum Mechanics, 4th edition (1958), p. 312.
33
In regard to Dirac’s skepticism, the Wikipedia article on Paul Dirac, quotes this from his last paper (The
inadequacies of quantum field theory, 1984): "It effectively contains his last and final judgment on quantum field
theory: "These rules of renormalization give surprisingly, excessively good agreement with experiments. Most
physicists say that these working rules are, therefore, correct. I feel that is not an adequate reason. Just because
the results happen to be in agreement with observation does not prove that one's theory is correct." The other
quotes refer to the lack of a good theory, with a ‘good theory’ being defined as mentioned above: “a scheme
based on equations of motion.” See our paper on the meaning of uncertainty and the geometry of the
wavefunction.
34
At least, that is what my son – who is currently finishing a master’s degree in engineering – tells me.
35
See: The Rotten State of QED.
10
From what we wrote in this paper, the reader will understand we could not agree more with the former
part of this statement. However, we do not agree with the latter part. We find Rutherford’s concept of
nuclear electrons (or neutrons combining a proton and an electron, somehow) amazingly productive and
rich, and we think all that is needed to save the ‘old quantum mechanics’ is to think of pair creation and
annihilation as nuclear processes – involving interactions with protons and neutrons, and also involving
neutrinos. We may, therefore, qualify these interactions as strong rather than electromagnetic
interactions, but such qualification is – for us, at least – not a license to multiply concepts by invoking
color charges, quarks, gluons, or whatever other virtual particles one might come up with. We also think
the classical concept of a field will do. The quantization of a field is a useful concept, but we think it has
got nothing to do with fields condensing, somehow, in real or virtual (stable or not) particles.
36
We
welcome good arguments on why we should think otherwise.
Conclusions
What new physics are we looking at, then? Probably some form of neoclassical physics: the old
quantum physics augmented by a thorough exploration of what might be going on inside of a nucleus by
further exploring Rutherford’s idea of the nuclear electron, which joins one or more protons and
provides the necessary attractive strong force to overcome the enormous electromagnetic forces which
should push protons apart. Can we hope to uncover its structure? The challenge is to model the nucleus
of deuterium: deuteron. We think of the neutron and the proton as two protons with an electron which
– not unlike the valence electrons in some molecule – keep the cell together. Of course, trying to model
how two positive and one negative elementary charge might be dancing together – combining not one
but two very different forces (the electromagnetic force – which we know very well from what happens
outside of the nucleus – and a strong(er) force, which we do not know at all) sounds endlessly much
more complicated than the threebody problems you are used to.
37
Can it be done at all? We are not sure. All we know is that not many have been trying since Bohr’s young
wolves hijacked scientific discourse after the 1927 Solvay Conference and elevated a mathematical
technique – perturbation theory – to the scientific dogma which is now referred to as quantum field
theory. In the meanwhile, we offer a few remarks in the annex which may or may not help to enlighten
and/or focus the discussion some more. We also warmly recommend the work of Andrew Meulenberg
on deep relativistic electrons (and/or electron deep orbits) and the related modeling of a neutron as an
excited hydrogen state.
38
Jean Louis Van Belle, 19 November 2020
36
See, for example, our analysis of quantized magnetic fields in the context of a ring current in a superconductor in
our paper on the concept of a field. The quantization does not imply that we should assume that the magnetic field
itself must, somehow, consist of (discrete) field quanta. Not at all. The magnetic field is just what it is: a finite
quantized magnetic field.
37
Note that, as far as we know, no general closedform solution exists for the general threebody problem either!
Is this a new form of quantummechanical uncertainty? To not add to the usual mysterymongering, we would
surely prefer to not use terms like this!
38
Andrew Meulenberg and JeanLuc Paillet, Highly relativistic deep electrons and the Dirac equation, Conference
paper for the 22nd International Conference on Condensed Matter Science, Assisi (Italy), Sept 813 2019 (to be
published in JCMNS).
11
Post scriptum (15 May 2021): SLAC’s E144 experiment as a refutation?
Since this was published, I received various comments. I also substantially finetuned my ideas on the
notion of charge and the (possible) nature of antimatter. This should probably lead me to substantially
revise this paper. However, because of a lack of time, I will just copy the most significant exchange here,
which resulted from a ResearchGate discussion:
Dear JeanLouis – You wrote in the abstract of your paper: "We argue the usual nuclear reactions
involving protons and neutrons can effectively account for the processes of pair creation and annihilation.
We therefore argue that the need to invoke some quantum field theory (QFT) to explain these high
energy processes would need to be justified much better than it currently is.”
Note that pair production was also proven with experiment #E144 to be possible by converging photons
to a single spot in space without any heavy nucleus being present: McDonald K et al. (1997) Positron
Production in Multiphoton LightbyLight Scattering, Phys. Rev. Lett. 79, 1626.
(http://www.slac.stanford.edu/exp/e144/).
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.79.1626
Best Regards, André Michaud
I have not had the time or energy yet to study the references above in detail but, if correct, they would,
of course, substantially modify the analysis.
Post scriptum to the post scriptum (25 November 2021): Nope !
I recently did have a bit of time to review the E144 experiment. The SLAC website has a lot of references
to papers for which no subscription is needed, and we briefly look at a 42page review paper (Studies of
nonlinear QED in collisions of 46.6 GeV electrons with intense laser pulses, February 1999) written by
about 20 coauthors/experimenters/scientists who have closely associated with the experiment.
We are everything but convinced. The article accounts for the result of the experiment in terms of
mainstream QED analysis, and effectively thinks of the pair production being the result of the theoretical
‘BreitWheeler’ pair production process from photons only. However, the experiment fails to properly
account for the incoming beam of electrons. It is a rather audacious but we think one cannot exclude
that the powerful laser photons  coming from the opposite direction, into the incoming electrons  sort
of turn the 'electromagnetic' force in some of the incoming electrons (the force that keeps the zbw
charge in the electron in its oscillation) inside out. Hence, charge would not get 'lost' or be 'created', but
simple switch its 'spacetime' or 'force' signature. We have added a annex to this paper (Annex II) which
describes what we are thinking both more eloquently as well as more precisely.
In other words, the 'mistake' that is made in the ‘interpretation’ of this experiment is, in my notso
humble view, the same mistake that was made by both de Broglie and Dirac when they first came out
with the wavefunction and wave equation that set quantum physics on an entirely different historical
path: 'spin' is not being taken into account straight from the start. This paper too thinks of both the
electrons as well as the highenergy photons as theoretical spinzero particles, which do not exist.
We feel that our geometric interpretation of the elementary wavefunction – which models electrons,
protons, their antimatter counterparts as well as photons and, possibly, neutrinos as ‘nuclear’ lightlike
12
particles, and which further builds on Schrödinger's original Zitterbewegung model for an electron
39

should be able to account for the results of the experiment in a way that does not involve the usual
‘hocuspocus’ (a term Richard Feynman used in his treatment of GellMann’s explanation of nuclear
processes invoking ‘strange’ properties of particles such as… Well… Strangeness) which is so typical of
the field nowadays. We once again refer to Annex II for a short two/threepage summary of this more
‘realist’ interpretation of quantum mechanics and, for further detail, we refer the reader to our other
papers on ResearchGate.
39
We credit David Hestenes once again for picking up the work that Schrödinger had left in the early 1990s, even
his work did not result in the kind of ‘scientific revolution’ that is clearly needed in the field of QED. We gratefully
acknowledge Oliver Consa for producing a much more specific account of where it all went wrong after World War
II (Something is wrong in the state of QED, Oct 2021). In fact, we preferred the earlier title of a first version of this
paper: Something is rotten in the state of QED (Feb 2020), but the newer version is more polite as well as more
precise on what exactly.
13
Annex I : Preliminary thoughts on a deuteron model
The electron cloud versus the proton core
The electron and the proton both incorporate the elementary charge. However, the form factor(s) in any
model of an electron and a proton must be very different:
1. The electron appears as a pointlike charge whizzing around some center. The mass of this cloudlike
object as a whole
40
is relatively small () and the size of this Zitterbewegung (zbw)
electron is given by its Compton radius
41
(
).
Such zbw or ring current model
42
of an electron gives rise to a distinction between the electron itself and
the zbw charge at its core, which has zero rest mass but – as it whizzes about at lightspeed – acquires a
relativistic mass m = me/2.
43
The orbital motion of the zbw charge inside generates the magnetic
moment of the electron, and the small anomaly in the magnetic moment (of the order of Schwinger’s
α/2π factor) can be explained when assuming the zbw charge itself cannot be infinitesimally small: the
calculations effectively yield the classical electron radius (). This, then,
gives rise to the idea of an electron cloud, as opposed to the idea of a pointlike particle.
44
2. A proton, in contrast, is (much) more massive and much smaller: its mass is about 1,836 times that of
the electron () and its radius was recently remeasured to be about
, which is about 3.35 times smaller than the classical electron radius, and about 460 times
smaller than its Compton radius, which we think of as the effective interaction radius of a free
electron.
45
40
We interpret the mass of the electron as the inertia to a change in its state of motion in what is, essentially, an
application of Wheeler’s ‘mass without mass’ idea: the mass is the equivalent mass (m = E/c2) of the energy in the
motion of the pointlike charge. This mass combines (i) the energy in the oscillatory motion of the pointlike charge –
which can be described using the elementary wavefunction (r = a·eiθ, with the sign representing the basic spin
direction), and (ii) its relativistic mass as a result of its velocity (c). For our electron model, see section 14 in our
paper on quantum behavior.
41
The Compton radius is the reduced Compton wavelength (aC = C/2π) which, paraphrasing Prof. Dr. Patrick
LeClair, we effectively interpret as “the scale above which the electron can be localized in a particlelike sense.”
42
We use the two terms interchangeably, although Alfred Lauck Parson’s original ring current model (1915) has
very different incarnations, and Zitterbewegung theorists themselves – the most prominent of which is probably
Dr. Hestenes – still struggle with various interpretations.
43
The other half of the electron mass is in the field which keeps it going. Note that this is just one of the possible
interpretations of the electron Zitterbewegung, which Schrödinger stumbled upon as a trivial solution to Dirac’s
wave equation for the free electron. For more background, see the paper mentioned in footnote .
44
The term is usually reserved for electron in atomic or molecular orbitals, whose radius is of the order of the Bohr
radius, which is related to the Compton radius by the same finestructure constant (rB = aC/α). However, which our
model makes clear the term would be valid for a free electron as well. The ae = αaC = α2rB relation gives rise to the
interpretation of the finestructure constant as a scaling constant. Some think of a fractal structure here (e.g.
Oliver Consa, 2018), but such fractal structure is not consistent with the idea of a zbw charge with zero rest mass.
45
The results of the 2019 PRad experiment yielded a point estimate of about 0.831 fm. While this value differs
only slightly from the 0.841 value that was measured by Pohl (2010) and Antognini (2013), we think the PRad value
is credible because it is very consistent with the anomalies (radius as well as magnetic moment) one can calculate
using a ring current model. See our paper on anomalies, the finestructure constant and the proton radius. As for
14
Hence, if we think of a neutron as, somehow, consisting of a proton and an electron, the picture which
emerges is that of a proton in an electron cloud.
To the various radii of the electron – its radius as a pointlike charge (the hard core of a free electron), as
a free electron, and as an electron in an atomic or molecular orbital – we should now, somehow, add
the radius of an electron as a deep relativistic electron in the nucleus. That, then, is, briefly, the
challenge that we are trying to highlight here.
Needless, to say, this new radius must be calculated from a force equation that, somehow, combines
the electromagnetic as well as the strong force inside a nucleus and – so we believe – in the neutron
itself. What can we say about the strong force? Apart from some very preliminary exploratory thoughts,
nothing much.
For those preliminary thoughts, we refer to previous speculative thoughts
46
. Here, we will limit ourselves
to a brief discussion of the idea that the proton might, perhaps, consist of charge that is being held
together by the strong force. If so, a decent proton model might, perhaps, reveal the geometry or
structure of the strong force.
The proton as a strong charge assembly?
One way of calculating electromagnetic mass is to assemble a uniform sphere of charge. This approach
uses the Coulomb law only (electrostatics) and one wonders: might we think of using a different force
formula – based on a model for the strong(er) force which, presumably, holds the proton together
47
– to
determine some kind of ‘strong’ mass (as opposed to electromagnetic mass) for the proton?
We refer to Feynman’s calculations of the necessary integral
48
for the electrostatic force, which yields
the following formula for a sphere of radius a with total charge Q:
Substituting Q for the elementary charge, and U for the total energy of the electron, we get:
We get a rather annoying 3/5 factor here. In a later lecture devoted to electromagnetic mass only
49
,
Feynman tries one or two different definitions of electromagnetic mass in an attempt to explain this, but
admits he fails and concludes the electromagnetic mass model does not work for an electron. We are
not so sure: a 3/5 factor is not nice, but it is pretty good, isn’t it? At least we get the correct order of
the difference between a charge and an interaction radius, this should speak for itself but the reader may want to
review their agreed definitions.
46
See, for example, our papers on the nature of Yukawa’s charge and force, which also offer some speculative
thoughts on a possible wave equation for the strong force.
47
When applying a simple ring current model for the proton, one can calculate the centripetal force that would be
needed to hold the pointlike charge in orbit: the calculations yield a rather astonishing value (about 89,349 N, to
be precise) in comparison to the calculations of the centripetal force inside of the electron (0.106 N). See footnote
43 for the reference.
48
See : Feynman’s Lectures, Vol. II, Chapter 8, section 1.
49
See: Feynman’s Lectures, Vol. II, Chapter 28, Electromagnetic mass.
15
magnitude and – who knows? – perhaps some formula allowing for a varying charge density might solve
the problem!
However, we should keep our wits about us here! It is actually not the 3/5 factor but the finestructure
constant which is annoying here! That is a much more annoying factor in this context: we are wrong by a
factor which is approximately equal to (3/5)·(1/137) now! In fact, this charge assembly model of an
elementary particle would fit a proton much better than an electron because our cloud model of an
electron assumes a pointlike charge with zero rest mass. In contrast, our proton is not cloudlike and,
hence, a charge assembly model based on some ‘strong’ force would make sense, right?
50
[…]
Maybe. Maybe not. Probably not. We should not forget we want to explain why things stay
together⎯not why they fall apart! Hence, a deuteron model should probably more think in terms of
modeling a strong force between opposite charges: the two protons being held together by a nuclear
electron. This is, in fact, the basic idea behind the new deuteron model which we hope will be found one
day: it would confirm that one should effectively think of a neutron as consisting of a proton with a
nuclear electron: no quarks, no gluons!
50
We developed a ring current model for the proton in previous papers, but we now feel it raises as many
questions as it answers. See, for example, our paper on the mass, radius and magnetic moment of electrons and
protons.
16
Annex II : Recap of the zbw interpretation of spin and matterantimatter
Modelling spin and antimatter
A good theory should respect Occam’s Razor⎯the lex parsimoniae: one should not multiply concepts
without necessity. The need for new concepts or new principles – such as the conservation of
strangeness, or postulating the existence of a new force or a new potential
51
– should, therefore, be
continuously questioned. Conversely, when postulating the existence of the positron in 1928 – which
directed experimental research to a search for it and which, about five years later, was effectively found
to exist – Paul Dirac unknowingly added another condition for a good theory: all of the degrees of
freedom in the mathematical description should map to a physical reality.
It is, therefore, surprising that the mainstream interpretation of quantum mechanics does not integrate
the concept of particle spin from the outset because the + or − sign in front of the imaginary unit (i) in
the elementary wavefunction (a·e−i· or a·e+i·) is thought as a mathematical convention only. This non
used degree of freedom in the mathematical description then leads to the false argument that the
wavefunction of spin½ particles has a 720degree symmetry. Indeed, physicists treat −1 as a common
phase factor in the argument of the wavefunction.
52
However, we should think of −1 as a complex
number itself: the phase factor may be +π or, alternatively, −π: when going from +1 to −1 (or vice versa),
it matters how you get there⎯as illustrated below.
53
51
We think of the invention of the concept of strangeness by Murray GellMan and Kazuhiko Nishijima in the 1950s
here. This concept started a rather strange life of its own and would later serve as the basis for the quark
hypothesis which – for a reason we find even stranger than the concept of strangeness itself – was officially
elevated to the status of a scientific dogma by the Nobel Prize Committee for Physics.
As for the invention of a new force or a new potential, we are, obviously, referring to the Yukawa potential. This
hypothesis – which goes back to 1935 – might actually have been productive if it would have led to a genuine
exploration of a stronger shortrange force on an electric charge⎯or, if necessary, the invention of a new charge.
Indeed, if the electromagnetic force acts on an electric charge, it would be more consistent to postulate some new
charge – or some new wave equation, perhaps – matching the new force. Unfortunately, theorists took a whole
different route. They invented a new aether theory instead: it is based on the medieval idea of messenger or
virtual particles mediating forces. The latter led to the invention of gluons which – as a concept – we find at least
as weird as quarks.
52
Mainstream physicists therefore think one can just multiply a set of amplitudes – let us say two amplitudes, to
focus our mind (think of a beam splitter or alternative paths here) – with −1 and get the same physical states.
53
The quantummechanical argument is technical, and I did not reproduce it in this book. I encourage the reader
to glance through it, though. See: Euler’s Wavefunction: The Double Life of – 1. Note that the e+iπ e−iπ expression
may look like horror to a mathematician! However, if he or she has a bit of a sense for geometry and the difference
between identity and equivalence relations, there should be no surprise. If you are an amateur physicist, you
should be excited: it is, effectively, the secret key to unlocking the socalled mystery of quantum mechanics.
Remember Aquinas’ warning: quia parvus error in principio magnus est in fine. A small error in the beginning can
lead to great errors in the conclusions, and we think of this as a rather serious error in the beginning!
17
Figure 3: e+iπ e−iπ
Combining the + and − sign for the imaginary unit with the direction of travel, we get four mutually
exclusive structures for the electron wavefunction:
Spin and direction of travel
Spin up (J = +ħ/2)
Spin down (J = −ħ/2)
Positive xdirection
= exp[i(kx−t)]
* = exp[−i(kx−t)] = exp[i(t−kx)]
Negative xdirection
χ = exp[−i(kx+t)] = exp[i(t−kx)]
χ* = exp[i(kx+t)]
Table 1: Occam’s Razor: mathematical possibilities versus physical realities (1)
We may now combine these four possibilities with the properties of antimatter. Indeed, we think
antimatter is different from matter because of its opposite spacetime signature. The logic here is the
following. Consider a particular direction of the elementary current generating the magnetic moment
(we effectively define spin as an (elementary) current
54
). It is then quite easy to see that the magnetic
moment of an electron (μ = −qeħ/2m) and that of a positron (μ = +qeħ/2m) would be opposite. We may,
therefore, associate a particular direction of rotation with an angular frequency vector ω which −
depending on the direction of the current − will be up or down with regard to the plane of rotation.
55
We can, therefore, associate this with the spin property, which is also up or down.
We, therefore, have another table with four mutually exclusive possibilities, which we should combine
with the possible directions of travel in Table 1
56
:
Matterantimatter
Spin up
Spin down
Electron
μ−e = −qeħ/2m
μ−e = +qeħ/2m
Positron
μ+e = +qeħ/2m
μ+e = −qeħ/2m
Table 2: Occam’s Razor: mathematical possibilities versus physical realities (2)
Table 2 shows that (1) the ring current model also applies to antimatter but that (2) antimatter has a
different spacetime signature. Abusing Minkowski’s notation, we may say the spacetime signature of an
54
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.
55
To determine what is up or down, one has to apply the ubiquitous righthand rule.
56
The use of the subscripts in the magnetic moment may be confusing, but should not be: we use −e for an
electron and +e for a positron. We do so to preserve the logic of denoting the (positive) elementary charge as qe
(without a + or a − in the subscript here).
18
electron would be + + + + while that of a positron would be + − − −.
57
Table 1 and Table 2, therefore,
complement each other. However, instead of this opposite ‘spacetime signature’ interpretation, it is
probably more productive to just think of matter and antimatter as having opposite
wavefunctions⎯literally, as we argue above and below.
Matter/antimatter and force/antiforce
From the above, it follows that matter and antimatter are each other opposite, and quite literally so: the
wavefunctions –AeiEt/ħ and +AeiEt/ħ add up to zero, and they correspond to opposite forces and different
energies too!
58
The opposite spacetime signature may also be analyzed as an antiforce in what we referred to as the
Zitterbewegung model of electrons and protons.
59
We To be precise, the magnetic field vector is
perpendicular to the electric field vector but – instead of lagging the electric field vector by 90 degrees
(matter) – it will precede it (also by 90 degrees) for antimatter, and the nuclear equivalent of the electric
and magnetic field vectors should do the same (we have no reason to assume something else).
60
Indeed,
the minus sign of the wavefunction coefficient (A) reverses both the real as well as the imaginary part of
the wavefunction.
However, it is immediately obvious that the equations above can only be a rather symbolic rendering of
what might be the case. First, we cannot model the proton by an AeiEt/ħ wavefunction because we think
of it as a 3D oscillation. We must, therefore, use two rather than just one imaginary unit to model two
oscillations. This may be solved by distinguishing i from j and thinking of them as representing rotations
in mutually perpendicular planes. Hence, we should probably write the proton as
61
:
57
In case the reader wonders why we associate the + +++ signature with the positron rather than with the
electron, the answer is: convention: 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. However, we leave it to
the reader to further think about this, as this is quite deep as an ‘intuition’ and, therefore, we feel we have not
quite fathomed it out ourselves !
58
See our previous remarks on the lag or precession of the phase factor of the components of the wavefunction.
Needless to say, masses and, therefore, energies are positive, always, but the nature of matter and antimatter is
quite different.
59
See, for example, our paper on basic quantum math or our introductions to the Zitterbewegung model.
60
We think this explains dark matter/energy as antimatter: the lightlike particles they emit, must be
antiphotons/antineutrinos too, and it is, therefore, hard to detect any radiation from antimatter. See our paper on
cosmology.
61
We use an ordinary plus sign, but the two complex exponentials are not additive in an obvious way (i j). Note
that t is the proper time of the particle. The argument of the (elementary) wavefunction a·ei is invariant. We refer
to Annexes II and III of this paper for an analysis of the wavefunction in the context of SRT and GRT.