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This paper offers a few reflections on the Universe and cosmological theories, including a possible explanation of dark matter (which we think of antimatter), plus an explanation of SRT/GRT based on an analysis of the argument of the quantum-mechanical wavefunction.
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The Finite Universe
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
30 March 2021
Abstract
This paper offers a few reflections on the Universe and cosmological theories, including a possible
explanation of dark matter (which we think of antimatter), plus an explanation of SRT/GRT based on an
analysis of the argument of the quantum-mechanical wavefunction.
Contents
Introduction: forces and scales ..................................................................................................................... 1
The geometric approach to gravity ............................................................................................................... 3
The closed Universe ...................................................................................................................................... 4
The mass of the Universe .............................................................................................................................. 5
Dark matter is antimatter ............................................................................................................................. 5
Beginnings ..................................................................................................................................................... 7
An oscillating Universe? ................................................................................................................................ 8
Life on Earth and elsewhere… .................................................................................................................... 9
The cosmological constant .......................................................................................................................... 12
Annex I: The wavefunction and special relativity ....................................................................................... 13
Annex II: The wavefunction and general relativity ..................................................................................... 19
1
Introduction: forces and scales
All of the energy of the Universe adds up to a constant. Singularities points of infinite potential are
mathematical abstractions: they are not real. Particles are finite quanta: their energy/mass is finite, and
they pack a finite amount of physical action. Stable particles pack one or multiple units of ħ (angular
momentum): E = ħ = hf = h/T. For unstable particles, the Planck-Einstein relation is not valid. The
wavefunction of unstable particles involves an additional decay factor :


The sign of the coefficient A models matter/antimatter, while the sign of the complex exponent
(iEt/ħ) captures the spin direction of matter/antimatter particles.
1
Light-particles differ from matter-
particles because they carry no charge. Their oscillation (if photons are electromagnetic oscillations,
then neutrinos must be nuclear oscillations) is, therefore, not local: they effectively travel at the speed
of light.
The relevant distance scale for nuclear interactions (e.g. the motion of charges within the deuteron
nucleus) is about 1 to 100 fm (1015 m). The size of atomic electron orbitals whose energy is
electromagnetic is of the order of the Bohr radius, whose order of magnitude (1011 m) differs with a
factor of 10,000 from the picometer scale. We may, therefore, say that, roughly speaking, the fine-
structure constant () separates nuclear from electromagnetic and atomic distance scales.
The fine-structure constant also separates the scales of the classical electron radius (re = rC = 2rB), its
Compton radius (rC = ħ/mec = rB), and the electron’s atomic orbital radius (rn = n2rC/),with n the
principal quantum number):

Mass is, quite simply, inertia to a change in the state of motion. The equations below model mass-
without-mass particles: all mass is electromagnetic (mC) or nuclear mass (mN).

The order of magnitude of the electromagnetic and nuclear force may be compared using the standard
electromagnetic and nuclear parameters in the orbital energy equations above.
2
However, to compare
forces, we must use the same numerical values for mC = mN. Evaluating the functions above at r = a, we
get:
1
See Annex IV, V and VI of our paper on ontology and physics.
2
The plus or minus sign is there because the nuclear charge keeps like charges together, while the Coulomb force
repels them. The sign also depends on the reference point for the potential energy: the U = 0 point may be chosen
at r = or at r = 0. We often make abstraction of the sign when comparing the magnitudes of forces, energy, or
distance scales.
2

This should not surprise us: we define the range parameter here as the distance r = a for which the
magnitude of the two forces (whose direction is opposite) is the same. The form factor and, hence, the
nature of the two forces, is very different, though.
Figure 1 shows the electromagnetic and nuclear potential functions: the nuclear potentials yields an
inverse-cube law for the nuclear force, which explains why it is apparent at very short range only: for
larger r, the combined nuclear and electromagnetic potential function approaches the electromagnetic
potential.
Figure 1: The nuclear potential well arising from 1/r and a/r2 functions
3
The geometric approach to gravity
The energy equation for gravitational orbitals follow from Kepler’s laws for the motion of the planets
3
:


The kinetic and potential energy (per unit mass) add up to zero instead of c2 (nuclear and
electromagnetic orbitals), which is why a geometric approach to gravity makes eminently sense: massive
objects simply follow a geodesic in space, and there is no gravitational force in such geometric approach.
We may also compare the standard parameters by equating m to mC (in practice, this means using the
mass and charge of the electron in the equation below) and r to a
4
:




Hence, the force of gravity if considered a force is about 1042 weaker than the two forces we know
(electromagnetic and nuclear).
5
Gravity acts on matter-particles (matter-particles carry charge) as well
as light-particles (photons
6
, neutrinos) no matter their energy which is why we think efforts to model
gravity as a residual force must fail: a force acts on a chargebut photons/neutrinos carry no charge!
General relativity establishes an equivalence between the description of gravity as a force and its
description as curved space. A (large) mass results in a curvature of space, but the (relativistic) mass of
photons must curve space as well and we must, therefore, rely on the superposition principle to arrive at
a complete description. The description in terms of curved space and geodesics might feel more natural
but is difficult in terms of the math: it is easier to just superpose forces to arrive at the equations of
motion.
7
3
See, for example, the MIT OCW reference course on orbital motion. We believe the law is relativistically correct
because the velocity is an orbital/tangential velocity (the concept of escape velocity is often used as a synonym
but, for elliptical orbitals, acquires a slightly different meaning) and because in accordance with relativity theory
we use the relativistic mass concept m = m0.
4
The physical dimension of the gravitational constant is usually written as m3kg1s2. The dimensional equation
below shows this is equivalent to the Nm2kg2 dimension:




 
5
Feynman (Lectures, I-2, Table 2-3) gives a factor of the order 1040. Perhaps he uses photon masses and energies:
the fine-structure constant effectively also appears as a photon-electron coupling constant (see our paper on the
meaning of the fine-structure constant, section V).
6
See the Eddington observation (29 May 2019), which unequivocally proves light is bent by large masses.
7
The analysis becomes complicated when considered not two but three bodies in motion (cf. the three-body
problem).
4
The closed Universe
The Universe is finite and all matter/energy in the Universe revolves around a (mathematical) center-of-
mass. Its (global) curvature must, therefore, be positive. Photons and neutrinos traveling outwards
must, therefore, also follow elliptical orbitals: what goes around, must come around.
These far-out light-particles may or may not have been created because of large-scale matter-antimatter
annihilation events when the Universe came into existence. We think there is a lot of antimatter in the
Universe because we think of dark matter/energy as antimatter. We are not very used to thinking of
antimatter. However, in our matter-world, matter-antimatter pairs are created routinely
8
: it is only
because of the predominance of matter in our world, that antimatter is not very present.
The closed Universe expands and may, at some point in time, collapse back into itself. Its expansion
and its possible collapse means the finite energy in the Universe gets distributed over an ever-
increasing volume. Hence, the Universe’s energy density E = E/V (measured in J/m3 or N/m2) goes down.
The relation between the concepts of energy, temperature and entropy are complicated and we will not
dig into them here. As for now, the decreasing energy density of the Universe may be equated with a
decreasing density of matter (including antimatter
9
), and temperature may be defined as the (average)
kinetic energy of the (stable) particles inside of a volume: two volumes with different mass densities
think of a different number of particles here are at the same temperature if the (average) kinetic
energy of the particles is the same. The current average temperature in the Universe is measured from
cosmic microwave background radiation and is about 2.73 K (−270.42 °C).
The age of the Universe is currently estimated to be about 13.8 billion (109) years. The Universe
expands, and the bits and pieces that sent us radiation from the limits of the observable Universe moved
further out in the meanwhile, and at pretty impressive speed. Astronomers think the end-to-end
distance across the Universe is of the order of 46 billion lightyears. A diameter of 46 billion (lightyears) is
about 1.666… times the diameter of the observable Universe (27.6 lightyears), and the estimated speed
of expansion (the radial velocity vr) of the Universe must, therefore, be equal to:


Two thirds of the speed of light? How can we know such things? Relativistic redshift: photons emitted by
stars who are moving away from us still travel at the speed of light but their wavelength shifts to a
lower-energy spectrum.
If we think of radial velocity, we should think of a tangential velocity too: the Universe might have
angular momentum too, right? Yes, but there is no agreement on the question of whether or not the
Universe has an actual center. More generally speaking, measurements depend on the choice of the
8
Carl Anderson (1932) discovered the positron while analyzing the ion trail of cosmic rays, using a cloud chamber
and ordinary photographs.
9
Antimatter is not to be equated with negative matter. Negative matter does not exist (negative energy does not
exist either: negative potential energy is only measured as negative because of the choice of the U = 0 reference
point at infinity).
5
reference frame: if we choose the reference frame to be that of the Universe itself, all of the angular
momentum of the bits and pieces inside of it should add up to zero.
The mass of the Universe
The Wikipedia article on the observable Universe states that the total mass of ordinary matter in the
universe can be calculated using the critical density and the diameter of the observable universe (whose
radius should correspond to the above-mentioned 13.8 billion lightyears) to be about 1.5 × 1053 kg.
The article distinguishes ordinary matter (about 74% of which (in mass) is hydrogen) from dark matter
and dark energy. In fact, it states that ordinary matter which is understood to be matter-particles
might constitute only 4.9% of all matter, with the rest being either dark matter (26.8%) or dark energy
(68.3%).
10
We, therefore, need to clarify what dark matter/energy might actually be.
Dark matter and dark energy is matter/energy which is assumed to be out there but which cannot be
detected. Black holes? No! Even black holes emit radiation as they evaporate (Hawking radiation).
11
We
also have all of the light-particles at the edge of our Universe orbiting out and which who knows
when? might come back in again.
12
We think dark energy/dark matter is antimatter, so let us explain that.
Dark matter is antimatter
The electromagnetic force has an asymmetry: the magnetic field lags the electric field. The phase shift is
90 degrees. We can use complex notation to write the E and B vectors as functions of each other.
Indeed, the Lorentz force on a charge is equal to: F = qE + q(v×B). Hence, if we know the (electric field) E,
then we know the (magnetic field) B: B is perpendicular to E, and its magnitude is 1/c times the
magnitude of E. We may, therefore, write:
B = iE/c
The minus sign in the B = iE/c expression is there because we need to combine several conventions
here. Of course, there is the classical (physical) right-hand rule for E and B, but we also need to combine
the right-hand rule for the coordinate system with the convention that multiplication with the imaginary
unit amounts to a counterclockwise rotation by 90 degrees. Hence, the minus sign is necessary for the
consistency of the description. It ensures that we can associate the aeiEt/ħ and aeiEt/ħ functions with left
and right-handed spin (angular momentum), respectively.
10
The Wikipedia article on dark matter distinguishes between dark matter and dark energy based on the
matter/energy density in space.
11
Of course, black holes also swallow matter/energy and, if there is enough of such matter around, they keep
growing.
12
As a child, I thought a lot of the radiation that comes from far away galaxies and star systems might be absorbed
by galaxies and star systems that are closer to us, and that this might explain why we cannot see (detect) dark
matter. That idea is not valid, however, because energy in must, over the long run, equal energy out. Another
objection is that, while the idea that galaxies and star systems may hide or eclipse (other) far-away galaxies and
star systems from our line of sight is valid, such eclipses should, of course, be temporary only (different rotational
velocities).
6
Now, we can easily imagine an antiforce: an electromagnetic antiforce would have a magnetic field
which precedes the electric field by 90 degrees, and we can do the same for the nuclear force (EM and
nuclear oscillations are 2D and 3D oscillations respectively). It is just an application of Occam’s Razor
principle: the mathematical possibilities in the description (notations and equations) must correspond to
physical realities, and vice versa (one-on-one). Hence, to describe antimatter, all we have to do is to put
a minus sign in front of the wavefunction. [Of course, we should also take the opposite of the charge(s)
of its antimatter counterpart, and please note we have a possible plural here (charges) because we think
of neutral particles (e.g. neutrons, or neutral mesons) as consisting of opposite charges.] This is just the
principle which we already applied when working out the equation for the neutral antikaon
13
:
 
 
 
 
   
   
The point is this: matter and antimatter are each other opposite, literally: the wavefunctions aeiEt/ħ and
aeiEt/ħ add up to zero, and they correspond to opposite forces too! Of course, we also have lightparticles,
so we have antiphotons and antineutrinos too.
We think this explains the rather enormous amount of so-called dark matter and dark energy in the
Universe. Dark matter is called dark because it does not appear to interact with the electromagnetic
field: it does not seem to absorb, reflect, or emit electromagnetic radiation, and is, therefore, difficult to
detect. That should not be a surprise: antiphotons would not be absorbed or emitted by ordinary
matter. Only anti-atoms (i.e. think of a antihydrogen atom as a antiproton and a positron here) would do
so.
14
13
See Annex IV, V and VI of our paper on ontology and physics.
14
The opposite spacetime signature of antimatter is, obviously, equivalent to a swap of the real and
imaginary axes. This begs the question: can we, perhaps, dispense with the concept of charge
altogether? Is geometry enough to understand everything? We are not quite sure how to answer this
question but we do not think so: a positron is a positron, and an electron is an electronthe sign of the
charge (positive and negative, respectively) is what distinguishes them! We also think charge is
conserved, at the level of the charges themselves (see our paper on matter/antimatter pair production
and annihilation). We, therefore, think of charge as the essence of the Universe. But, yes, everything
else is sheer geometry.
7
Beginnings
It is tempting to think all began with matter/antimatter creation but, as far as we know, protons and
electrons are the only stable elementary particles. We also have neutrons, but they are unstable outside
of the nucleus. We also think our model of the neutron as a composite proton-electron equilibrium state
inside of a nucleus makes a lot of sense.
15
In short, we are left with an enormous amount of protons and electrons an overall electrically neutral
ensemble which combined to form hydrogen in one of its two stable isotopes (1H and 2H
16
). Protons
are nuclear oscillations of the positive (pointlike) Zitterbewegung (zbw) charge, while electrons are
electromagnetic oscillations of the negative zbw charge. Two forces, two particles. We see no need to
complicate history by invoking asymmetries or other ad hoc hypotheses.
We may, therefore, think of the conditions that were prevalent at the time of the Big Bang as being the
same as those that are prevalent in a star: thermonuclear fusion reactions generated heat and
lightparticles (photons as well as neutrinos
17
). The thermonuclear reactions first burn hydrogen and,
because of gravity, the more massive nuclei are drawn to the center, where nuclear fusion produces
even more massive elements but with the balance going from producing to consuming heat, as a result
of which the core eventually collapses. The implosion is then followed by a huge explosion, with non-
homogeneous fragments going everywhere.
Does this answer the question of where protons and electrons or the positive and negative charges at
their core came from? No. Should we be interested in the original distribution of protons and
neutrons? Probably, but as Feynman (III-2-6) notes:
“The tiniest irregularities are magnified […], so that we get complete randomness. […] Given an
arbitrary accuracy, no matter how precise, one can find a time long enough that we cannot
make predictions valid for that long a time. […] This length of time is not very large. It is not that
the time is millions of years if the accuracy is one part in a billion. The time goes, in fact, only
logarithmically with the error, and it turns out that in only a very, very tiny time we lose all our
information.”
One may also think of Einstein’s statistical analysis of the random walk.
18
Let us quickly illustrate this.
15
See our paper on the mass-without-mass model of electrons, protons, and neutrons. Also see our paper on
(matter/antimatter) pair production and annihilation as a nuclear process, in which we show charge, energy/mass,
and (linear and angular) momentum are being preserved, always.
16
The deuterium nucleus consists of a neutron and a proton and, therefore (n = p + e), of two protons and two
electrons.
17
We think of neutrinos as the equivalent of photons for the nuclear force: photons carry electromagnetic energy,
while neutrinos carry nuclear energy. Neither carry charge: only matter-particles do.
18
Einstein’s analysis of the random walk is part of one of his 1905 (Annus Mirabilis) papers. It has a rather
impossible title: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden
Flüssigkeiten suspendierten Teilchen (On the Motion of Small Particles Suspended in a Stationary Liquid, as
Required by the Molecular Kinetic Theory of Heat).
8
Figure 2: The random walk (illustration taken from Feynman I-6-3)
The idea here is that we start at point x = 0 and that we move by either take a step forward (i.e. away
from our starting point x = 0) or backward (toward x = 0) but that the direction of each step is random.
19
The resulting motion is described by (i) the horizontal coordinate N, which is the total number of steps
taken and (ii) the vertical coordinate DN, which is the net distance moved from the starting position. The
illustration (Figure 2) shows three possible outcomes after 30 steps, two of which involve the same final
position but different paths. Now, it is impossible to reconstruct what happened which path was
actually taken from analyzing the final position only because the information about the individual
steps got lost.
It is fairly to show that, after N steps, we will have covered an expected distance from the x = 0 point
that is equal to
20
:

As you can see, we might expect to be at x = 2 after four steps, but progress quickly slows: after 100
steps, the expected distance of travel is only x = 10, and for N = 1000, it is equal to 1000 31.6 only.
The good news here is that, if ever you get lost, you are usually not too far from the point where you got
lost. 
An oscillating Universe?
Was there one Big Bang, or were there several at different distances in space and in time? We do not
know. It might help to try to think about a contracting Universe: hydrogen but also more massive
elements now contracting into local clusters of mass and energyforming stars, that first expand and
then collapse/explode again.
We think the coming-into-being of the Universe was probably not a linear process and, therefore, we
should not exclude there were several Big Bangs at different places and times. The ultimate re-collapse
19
We should not be killing Schrödinger’s cats to determine where we go, but the logic and the math are very
similar.
20
These are notations the reader should familiarize him- or herself with. We also recommend the reader reviews
first-order approximations of functions, including complex functions. We refer to one of our blog posts on Euler’s
formula here. Einstein’s approach to the random walk is a good way to familiarize oneself not only with the basic
notions of statistics but also to get a more intuitive feeling of physical scales and ratios.
9
of the Universe will not be a linear process either! But both the Universe’s expansion as well as its
collapse are governed by the same laws and should, therefore, not be too different.
Life on Earth and elsewhere…
It is quite remarkable how life evolved: atoms formed more stable or less energetic combinations by
sharing or exchanging valence electrons, and molecules then arranged into organisms: vulnerable to
high-energy radiation (but protected from it by atmospheric layers on planetscool matter) but
adaptable to changing circumstances. How crawling insects and conscious Homo Sapiens came into
being is weird but explainable: insects are more robust to changing circumstances, while man adapted
its environment or searched other environments to ensure better chances of survival and successful
reproduction.
We too obey the laws of the Universe: we burn a lot of energy to get our cars rolling, or produce mobile
phones. Data and information that is not immediately useful is stored for future purposes, or discarded
as useless.
Is there life on Venus, or on Mars? Or in other planetary systems? Maybe. Probably. Is it useful for us?
Most probably noton the contrary: other life forms are probably likely to harm us We should take
Stephen Hawking’s warnings in that regard very seriously, I think!
21
We should also add that the COVID
crisis showed we are not very good at dealing with new viruses, even if amazingly pharmaceutical
companies were able to develop a vaccine against it in less than a year.
22
Of course, we should add that communication could be difficult. Indeed, the nearest star system to that
of our Sun is the Alpha Centauri system, and it is about 4.3 lightyears away. Hence, messages from here
to there or vice versa would take 4.3 lightyears to get there. Perhaps they are trying to reach us, but
the intensity of the signals they send might be too weak to detect. However, after reading the Wikipedia
article on communication with extraterrestrial intelligence, my sense is that, if there would be intelligent
life out there within reasonable communication distances (say, 5 to 50 lightyears from here), we
should have conclusively detected signals already.
Just to add to your sense of wonder, we may add that NASA’s Voyager 2 is currently only 18.7 billion km
away, so that is only about 0.002 lightyears away.
23
:-/
21
If aliens visit us, the outcome would be much as when Columbus landed in America, which didn't turn out well
for the Native Americans.” (Hawking, 2010, quoted in the Wikipedia article on extraterrestrial life, which also
mentions other thinkers such (such Jared Diamond) expressing similar concerns)
22
We are writing mid-March 2021 in Belgium here, almost one year after the government-imposed lock-down
measures.
23
NASA launched the Voyager 2 space probe was back in 1977 (almost 44 years ago!), and it has left the Solar
system now. It yielded valuable data on the outer planets Jupiter, Saturn, Uranus, and Neptune when it passed
these planets in 1979, 1981, 1986 and 1989, respectively. Another spacecraft designed to leave the Solar system
NASA’s New Horizons probe was launched in 2006, and explored the Kuiper belt, including the dwarf planets
Pluto and Charon (one of Pluto’s ‘moons’, but its diameter is about half that of Pluto). It will take a while before
any of these interstellar probes will reach a nearby star: Voyager 1 is scheduled to enter the Sirius system 40,000
years from now, while Voyager 2 is headed for star AC +79 3888. We will probably have lost contact with these
probes by then. 
10
11
So what is left? Wonder! In one of the introductory chapters of his Lectures on Physics (I-3-7), Feynman
writes this:
“A poet once said, “The whole universe is in a glass of wine.” We will probably never know in
what sense he meant that, for poets do not write to be understood. But it is true that if we look
at a glass of wine closely enough we see the entire universe. There are the things of physics: the
twisting liquid which evaporates depending on the wind and weather, the reflections in the
glass, and our imagination adds the atoms. The glass is a distillation of the earth’s rocks, and in
its composition we see the secrets of the universe’s age, and the evolution of stars. What
strange array of chemicals are in the wine? How did they come to be? There are the ferments,
the enzymes, the substrates, and the products. There in wine is found the great generalization:
all life is fermentation. Nobody can discover the chemistry of wine without discovering, as did
Louis Pasteur, the cause of much disease. How vivid is the claret, pressing its existence into the
consciousness that watches it! If our small minds, for some convenience, divide this glass of
wine, this universe, into partsphysics, biology, geology, astronomy, psychology, and so on
remember that nature does not know it! So let us put it all back together, not forgetting
ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all!”
He wrote that in the early 1960s. In the meanwhile, we have the worldwide web enhancing
intelligence
24
and, effectively, we are currently witnessing the introduction of artificial intelligence itself.
We have also entered the areas of nanotechnology and biotechnology. I summarized these advances as
follows in one of my blog posts
25
:
I just wrapped up my writings on physics (quantum physics) with a few annexes on the
(complex) math of it. And then a friend of mine sent me this image of the insides of a cell. There
is more of it on where it came from. Just admit it: it is truly amazing, isn't? I suddenly felt a huge
sense of wonder - probably because of the gap between the simple logic of quantum physics
and this incredible complex molecular machinery.
24
I would not have been able to develop all this knowledge without it!
25
See: Reading Feynman, All of Physics, 23 February 2021.
12
I quote: "Seen are Golgi apparatus, mitochondria, endoplasmic reticulum, cell wall, and
hundreds of protein structures and membrane-bound organelles. The cell structure is of a
Eukaryote cell i.e. a multicellular organism which means it can correspond to the cell structure
of humans, dogs, or even fungi and plants." These images were apparently put together from
"X-ray, nuclear magnetic resonance (NMR) and cryoelectronic microscopy datasets."
I think it is one of those moments where it feels great to be human.
We have nothing to add to this, except for two annexes on special and general relativity, respectively.

The cosmological constant
All of the above was written before I checked the Wikipedia article on the cosmological constant,
according to which the expansion of the universe is actually accelerating! I quote:
In 1998 two teams of astrophysicists, one led by Saul Perlmutter, the other led by Brian
Schmidt and Adam Riess, carried out measurements on distant supernovae and show that the
speed of galaxies recession in relation to the Milky Way increases over time. The universe is in
accelerated expansion. [] The universe would contain a mysterious dark energy producing a
repulsive force that counterbalances the gravitational braking produced by the matter
contained in the universe (see standard cosmological model). For this work, Perlmutter
(American), Schmidt (American-Australian), and Riess (American) jointly received the Nobel
Prize in physics in 2011.
I am sure there must be a rational explanation for this. Perhaps the Universe is just a blob in a larger
cluster of Universes, most of which would also consist of antimatter (our explanation of dark matter)?
Brussels, 30 March 2021
13
Annex I: The wavefunction and special relativity
Particles are finite quanta: their energy/mass is finite, and they pack a finite amount of physical action.
Stable particles pack one or multiple units of ħ (angular momentum): E0 = ħ = hf = h/T. For unstable
particles, the Planck-Einstein relation is not valid. The wavefunction of unstable particles involves an
additional decay factor :


The sign of the coefficient A captures the difference between matter and antimatter, while the sign
of the complex exponent (iEt/ħ) captures the direction of spin (angular momentum).
26
Light-particles
differ from matter-particles because they carry no charge. Their oscillation (if photons are
electromagnetic oscillations, then neutrinos must be nuclear oscillations) is, therefore, not local: they
effectively travel at the speed of light.
The energy in the wavefunction is the rest energy of the particle, which we think of as a wavicle: its
essence is an oscillating pointlike charge. We, therefore, think of the elementary wavefunction to
represents the motion of the pointlike charge by interpreting r = A·eiθ = A·ei·(E·t k·x)/ħ as its position
vector. The coefficient A is then, equally obviously, nothing but the Compton radius A = rC = ħ/mc. The r
= A·eiθ = A·ei·(E·t k·x)/ħ expression shows how classical motion adds a linear component to the argument of
the wavefunction (see Figure 3).
Figure 3: The Compton radius must decrease with increasing velocity
27
The relativistic invariance of the argument of the wavefunction is then easily demonstrated by noting
that the position of the pointlike particle in its own reference frame will be equal to x’(t’) = 0 for all t’.
We can then relate the position and time variables in the reference frame of the particle and in our
frame of reference by using Lorentz’s equations
28
:
26
See Annex IV, V and VI of our paper on ontology and physics.
27
We borrow this illustrations from G. Vassallo and A. Di Tommaso (2019).
28
We can use these simplified Lorentz equations if we choose our reference frame such that the (classical) linear
motion of the electron corresponds to our x-axis. See Feynman’s Lectures, I-15-2.
14


When denoting the energy and the momentum of the electron in our reference frame as Ev and p =
m0v, the argument of the (elementary) wavefunction a·ei can be re-written as follows
29
:





Besides proving that the argument of the wavefunction is relativistically invariant, this calculation also
demonstrates the relativistic invariance of the Planck-Einstein relation when modelling elementary
particles.
30
Needless to say, the plane of the local oscillation is not necessarily perpendicular to the direction of
(linear) motion, nor must we assume the local oscillation is necessarily planar. For a proton, one must
apply an extra factor (4) to calculate its Compton radius:
 

The 4 factor is the 4 factor which distinguishes the formula for the surface area of a sphere (A = 4πr2)
from the surface area of a circle (A = πr2).
31
We effectively think of an oscillation in three rather than just
two dimensions only here: the oscillation is, therefore, driven by two (perpendicular) forces rather than
just one, and the frequency of each of the two oscillations is equal to = E/2ħ = mc2/2ħ: each of the two
perpendicular oscillations would, therefore, pack one half-unit of ħ only
32
, and applying the
equipartition theorem each of the two oscillations packs half of the total energy of the proton. This
spherical view of neutrons (and protons) as opposed to the planar picture of an electron fits nicely
with packing models for nucleons.
33
Let us analyze the argument of the wavefunction more in detail. We wrote it as:


29
We use the relativistically correct p = mv equation, and substitute m for m = E/c2.
30
The relativistic invariance of the Planck-Einstein relation emerges from other problems, of course. However, we
see the added value of the model here in providing a geometric interpretation: the Planck-Einstein relation
effectively models the integrity of a particle here.
31
Cf. the 4π factor in the electric constant, which incorporates Gauss’ Law (expressed in integral versus differential
form).
32
This explanation is similar to our explanation of one-photon Mach-Zehnder interference, in which we assume a
photon is the superposition of two orthogonal linearly polarized oscillations (see p. 32 of our paper on basic
quantum physics, which summarizes an earlier paper on the same topic).
33
We think a neutron consists of a positive and a negative charge, and combines an electromagnetic as well as a
nuclear oscillation. See the above-mentioned paper on ontology and physics.
15
The momentum of a photon (and, we must assume, a neutrino
34
) is equal to p = mc = mc/c2 = E/c, with
E = Ev = Ec. The equation above is, then, equal to:



We can, therefore, see that the argument of the wavefunction for a particle traveling at the speed of
light vanishes! This is not easy to interpret. It is not like time has no meaning anymore but relativistic
time dilation becomes absolute: in our frame of reference, we think of the clock as the photon as
standing still. To put it differently, all of its energy is in its motion, and it derives all of its energy from its
momentum.
For particles that are not traveling at the speed of light, we still have the two terms:
The dimensional analysis of the Ev/ħ and the p/ ħ is rather instructive and shows the argument (of
phase) of the wavefunction has no physical dimension:
This makes sense because the phase of the wavefunction is measured in radians which can be used both
as distance as well as time units. One can appreciate this idea when re-writing the phase as:

The p = mv = Ev/c2 relation allows us to rewrite the argument of the wavefunction also as:


This relation, too, can be easily verified
35
:


The point is this: an elementary particle packs one unit of physical action (ħ) per oscillation cycle, that
is and, when in motion, we think of this as expressing itself as a combination of (i) angular momentum
(and, therefore, rotational energy) and (ii) linear momentum.
Now, the functional behavior of the t’ = (t vx/c2) function may not be immediately obvious: goes
from 1 to infinity () as v goes from 0 to c, and time dilation may, therefore, not be immediately
understood. Hence, a graph may be useful. To produce one, we write x as a function of t: x(t) = vt. The t’
function can, therefore, be rewritten as:
34
We think of the neutrino as the light-particle of the nuclear force: just like a photon, it does not carry charge, but
it carries nuclear energy.
35
We use the 



 equation here.
16


The 1 factor is the inverse Lorentz factor, and its function (for positive v) is the arc of the first quadrant
of the unit circle, as illustrated below. It is, therefore, easy to see that, for any velocity v (0 < v < c), t’ will
be smaller than t, which illustrates the point.
Figure 4: The inverse Lorentz factor (1) as a function of
Likewise, the behavior of the = (Evt px)/ħ function may also not be immediately obvious, but
rewriting it as = (E0t)/ħ and taking what we wrote about the t’ = 1t function shows that the
phase of the wavefunction shows the same time dilation.
Note: The reader should not think we established a non-heuristic logical proof of special relativity based
on the reality of the wavefunction. If anything, we only showed that quantum mechanics is fully
consistent with special relativity (and, as we will show in the following annex, with general relativity).
We do think, however, that we did show what the relativistic invariance of the argument of the
wavefunction actually means, and that quantum mechanics and relativity theory mutually confirm each
other.
That does not amount to an intuitive understanding of special relativity, of course. Understanding
(special) relativity theory intuitively may not be possible, but the following considerations may or may
not help the reader to play some more with it.
When observing a object which is moving sideways with velocity v, we may think of its velocity v as a
tangential velocity.
Figure 5: Tangential velocity
Of course, you will say that most objects are not moving sideways only, but also towards or away from
us. However, such motion along the line of sight (which we will refer to as the radial velocity) can be
determined from the red- or blueshift of the light we use to determine the position of the object (in
order for us to able to track the position of an object in what we refer to as the inertial reference
17
frame it has to emit or reflect light). Hence, if we can determine both the tangential as well as the
radial velocity, we can add the two velocity components to get the combined velocity vector.
It is good to specify what is relative and what is not here: the distance between us, the observer, and the
object is not relative: there is no length contraction along the line of sight. Also, in the reference frame
of the object (which we will refer to as the moving reference frame), the (tangential) velocity of our
reference frame will be measured just the same: v. Finally, the speed of light does not depend on the
reference frame, either. Clock speeds, however, will depend on the reference frame, which gives rise to
the distinction between t and t’.
Because there is no length contraction along the line of sight, its length will be measured the same in
the inertial and moving reference frame. Lightspeed is used as the yardstick in both reference frames
and we must, therefore, conclude this distance must be measured using non-moving clocks. In other
words, we must assume the same clock is used here.
36
In contrast, the relative velocity of the reference
frames is measured using moving clocks:

 

When combining this with the t = 1t relation (which establishes time dilation
37
), we get the relativistic
length contraction equation:

 
 
We get the same graph (Figure 4): for any velocity v (0 < v < c), ds’ will be smaller than ds, and s’ will,
therefore, be smaller than s
38
, which illustrates the point.
There is little to add, except for a few remarks on geometry perhaps:
1. If the distance between the origin of the inertial reference frame and the s = s’ = 0 point is equal to a
(the same in both reference frames, remember!), then we may measure that distance in equivalent time
units by dividing it by the speed of light. This amounts to measuring the distance a as a time distance. Of
course, we can always go back to measuring a as a distance by multiplying the time distance by c again:
we then get the distance expressed in light-seconds, i.e. as a fraction or multiple of 299792458 m.
36
This is not a matter of synchronization: we must assume the clock that is used to measure the distance from A to
B does not move relative to the clock that is used to measure the distance from B to A. It is one of these logical
facts which makes it difficult to understand relativity theory intuitively: clocks that are moving relative to each
other cannot be made to tick the same. An observer in the inertial reference frame can only agree to a t = t’ = 0
point (or, as we are talking time, a t = t’ = 0 instant, we should say). From an ontological perspective, this entails
both observers can agree on the notion of an infinitesimally small point in space and an infinitesimally small instant
of time. Indeed, both observers also have to agree on the s = s = 0 point!
37
We get the time dilation equation from writing s as a function of t: s(t) = vt and substituting in the Lorentz
transformation:



38
See footnote 36: observers need to agree both on the t = t’ = 0 as well as on the s = s’ = 0 point!
18
In fact, we think a good understanding of the absolute nature of the speed of light, and a deeper
understanding of the equivalence of using time and spatial distances may be all what can be provided in
terms of a more intuitive understanding of relativity theory. Indeed, when everything is said and done,
we are always measuring things in one specific reference frame: swapping back and forth between
reference frames is a rather academic exercise which does not clarify all that much: the laws of physics
(mass-energy equivalence, Planck-Einstein relation, force law, etcetera) are the same in every reference
frame and, hence, students should probably consistently focus on understanding these rather than
relativity, as relativity is just a logical consequence of these laws!
In any case, let us agree on writing a which is, of course, the length of the base of the triangle in Figure
5 as a spatial distance but assume all spatial distances are measured in light-seconds. This also implies
that we can write the velocities v, vt, and vr as relative velocities , t, and r, respectively.
Let us, indeed, introduce the radial velocity again now. We can then write the velocity vector as = t +
r, with t = ds/dt = ds/dt. The length of the hypotenuse will, therefore, be equal to a + rt.
Pythagoras’s Theorem then gives us the following equation:
(a + rt)2 = a2 + (tt)2
a2 + r2t2 + 2art = a2 + t2t2
(t2 r2)t = 2ar
Multiplying both sides with c2, yields an equation in terms of the usual velocities measured in m/s:
(vt2 vr2)t = 2acvr
It is a nice equation, but there is probably not all that much we can do with it.
39
2. Figure 5 introduces the concept of the phase (), which we measure in radians, and the angular
frequency , whose dimension is s1. The two are related through the = t equation and, also using
the v = a equation, it will be easy for the reader to verify the following relation:

 



 



 


We leave it to the reader to establish the relations for the variables in the moving reference frame.
39
The reader will probably know Pythagoras’s Theorem does not apply to curved spacetime, but here we are
talking about special relativity only. Note that the ac factor gives us a radial distance expressed in meter again (not
in light-seconds). We are a little bit puzzled to what this expression might mean geometrically, so any suggestion
and/or correction of our readers is most welcome!
19
Annex II: The wavefunction and general relativity
We know a clock goes slower when placed in a gravitational field. To be precise, the closer the clock is to
the source of gravitation, the slower time passes. This effect is known as gravitational time dilation.
40
This cannot be explained by writing the argument of the wavefunction as a function of its energy Ev and
its momentum p. We will, therefore, distinguish (i) the rest energy of the particle outside of the
(gravitational) field (E0) and (ii) the potential energy it acquires in the field (Eg). The total energy as
measured in the equivalent of the inertial frame of reference (which is the reference frame without
gravitational field, i.e. empty space), and the argument of the wavefunction, can therefore be written as:
E = E0 + Eg E0 = E Eg


This effectively shows the frequency of the oscillation is lower in a gravitational field. At first, the
analysis looks somewhat counterintuitive because the convention is to measure potential energy (PE) as
negative (the reference point for PE = 0 is usually taken at infinity, i.e. outside of the gravitational field).
However, when noting extra energy must be positive (i.e. when taking the reference point for PE = 0 at
the center of the gravitational field, or as close to the source as possible
41
), all makes sense.
We hopes this provides a more intuitive understanding of gravitational time dilation based on the
elementary wavefunction.
The reader should note this analysis is also valid for an electromagnetic or nuclear potential, or for any
potential (which may combine two or all three of the forces
42
). We may refer the reader here to
Feynman’s rather excellent analysis of potential energy in the context of quantum physics in his
Lectures, in which he also explains the nature of quantum tunneling.
43
However, we think Feynman’s
analysis suffers from a static view of the potentials involved.
We think one should have a dynamic view of the fields surrounding charged particles. Potential barriers
or their corollary: potential wells should, therefore, not be thought of as static fields: they vary in
time. They result from two or more charges moving around and creating some joint or superposed field
which varies in time. Hence, we think a particle breaking through a ‘potential wall’ or coming out of a
potential ‘well’ is just using a temporary opening corresponding to a very classical trajectory in space
and in time. We, therefore, think there is no need to invoke an Uncertainty Principle.
40
See, for example, the Wikipedia article on gravitational time dilation.
41
A gravitational field comes with a massive object which is usually taken to have a (finite) radius.
42
We are not aware of any successful attempt proving the gravitational force may be analyzed as some residual
force resulting from asymmetries or other characteristics of the two forces which we consider to be fundamental
(electromagnetic and nuclear). The jury is, therefore, still out on the question of whether or not we should think of
the gravitational force as a pseudoforce. We, therefore, still think of Einstein’s geometric approach to gravity
(curved spacetime) as an equivalent analysis. The question may be entirely philosophical: it should be possible to
also come up with a geometric interpretation of the electromagnetic and nuclear forces but, because of their
multidimensional character (2D/3D, respectively), this may not be easy.
43
See: Feynman’s Lectures, Potential energy and energy conservation (III-7-3).
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