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Ontology and physics

Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil

17 February 2021

1

Abstract

This papers concludes our excursions into the epistemology/ontology of physics. We provide a basic

overview of the basic concepts as used in the science of physics, with practical models based on orbital

energy equations. We hope to make a difference by offering an alternative particle classification based

on measurable form factors.

Contents

Prolegomena ................................................................................................................................................. 1

The ring current model of elementary particles ........................................................................................... 4

Time and relativity ........................................................................................................................................ 5

The wavefunction and its (relativistically invariant) argument .................................................................... 9

Rutherford, Bohr, Dirac, Schrödinger, and electron orbitals ...................................................................... 10

The two fundamental forces (Coulomb and nuclear/strong) ..................................................................... 12

The nuclear range parameter and the fine-structure constant .................................................................. 15

Conclusions ................................................................................................................................................. 17

Annex I: Dirac’s energy and Schrödinger’s wave equation ......................................................................... 18

Annex II: The quark hypothesis ................................................................................................................... 21

Strange kaons .......................................................................................................................................... 21

Transient oscillations: what is real? ........................................................................................................ 22

An analysis of non-equilibrium states ..................................................................................................... 23

The math of transients ............................................................................................................................ 24

Hermiticity and reversibility .................................................................................................................... 25

Annex III: A complete description of the Universe ..................................................................................... 26

Annex IV: A complete description of an event ............................................................................................ 28

Annex V: The nature of anti-matter ............................................................................................................ 31

Annex VI: A new matrix algebra? ................................................................................................................ 34

1

We found an error in our rendering of Dirac’s energy equation in Annex I. We corrected this on 16 September

2022 and added a note there, accordingly. No other changes were made.

1

Prolegomena

Why is it that we want to understand quarks and wave equations, or delve into complicated math

(perturbation theory

2

, for example)? We believe it is driven by the same human curiosity that drives

philosophy. Physics stands apart from other sciences because it examines the smallest of smallest⎯the

essence of things, so to speak.

Unlike other sciences (the human sciences in particular, perhaps), physicists also seek to reduce the

number of concepts, rather than multiply them⎯even if, sadly, enough, they do not always a good job

at that. The goal is to arrive at a minimal description or representation reality. Physics and math may,

therefore, be considered to be the King and Queen of Science, respectively.

The Queen is an eternal beauty, of course, because Her Language may mean anything. Physics, in

contrast, talks specifics: physical dimensions (force, distance, energy, etcetera), as opposed to

mathematical dimensions⎯which are mere quantities (scalars and vectors).

Science differs from religion in that it seeks to experimentally verify its propositions. It measures rather

than believes. These measurements are cross-checked by a global community and, thereby, establish a

non-subjective reality. The question of whether reality exists outside of us, is irrelevant⎯a category

mistake (Ryle, 1949). All is in the fundamental equations. We are part of reality.

An equation relates a measurement to Nature’s constants. Measurements – such as the energy/mass of

particles, or their velocities – are relative but that does not mean they do not represent anything real.

On the contrary.

Nature’s constants do not depend on the frame of reference of the observer and we may, therefore,

label them as being absolute. The difference between relative and absolute concepts corresponds to the

difference between variables and parameters in equations. The speed of light (c) and Planck’s quantum

of action (h) are parameters in the E/m = c2 and E = hf, respectively. In contrast, energy (E), mass (m),

frequency (f) are measured quantities.

Feynman (II-25-6) is right that the Great Law of Nature may be summarized as U = 0 but that “this simple

notation just hides the complexity in the definitions of symbols is just a trick.” It is like talking of “the

night in which all cows are equally black” (Hegel, Phänomenologie des Geistes, Vorrede, 1807). Hence,

the U = 0 equation needs to be separated out. We would separate it out as:

2

Analyzing phenomena in terms of first-, second-,… nth-order effects is useful as a rough approximation of reality

(especially when analyzing experimental data) but, as Dirac famously said, ”neglecting infinities […] is not sensible.

Sensible mathematics involves neglecting a quantity when it is small – not neglecting it just because it is infinitely

great and you do not want it!" (Dirac, 1975) Perturbative theory often relies on a series expansion, such as the

series expansion of relativistic energy/mass::

We do not immediately see the relevance (need) of this formula when solving practical problems.

2

Energy is measured as a force over a distance: we do work with or against the force.

3

Forces are forces between charges. If there is an essence in Nature, it corresponds to the concept of

charge. We think there is only one type of charge: the electric charge q. Charge is absolute: an electron

in motion or at rest has the same charge. That is why Einstein did not think much of the concept of

mass: the mass of a particle measures its inertia to a change in its state of motion, and gravitation is

likely to reflect the geometry of the Universe: a closed Universe, which very closely resembles Cartesian

spacetime but not quite.

We imagine things in 3D space and one-directional time (Lorentz, 1927, and Kant, 1781). The imaginary

unit operator (i) represents a rotation in space. A rotation takes time and involves distance: we rotate a

charge from point a to point b. A radian, therefore, measures an angle () as well as a distance and a

time. We usually think of angular velocity as a derivative of the phase with respect to time, though:

The Lorentz force on a charge is equal to:

F = qE + q(vB)

If we know the (electric field) E, 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

To make the dimensions come out alright

4

, we need to associate the s/m dimension with the imaginary

unit i. This reflects Minkowski’s metric signature and counter-clockwise evolution of the argument of

3

Potential energy is defined with respect to a reference point. The reference point may be taken at an infinite

distance () of the charge at the center of the potential field, or at the charge itself (r = 0). Sign conventions

depend on the choice of the reference point.

4

E is measured in newton per coulomb (N/C). B is measured in newton per coulomb divided by m/s, so that’s

(N/C)(s/m). Note 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 a·ei and a·e−i functions with left and right-handed

polarization, respectively.

3

complex numbers, which represent the (elementary) wavefunction = aei.

5

The nature of the nuclear

force is different, but its structure should incorporate relativity as well.

6

The illustration below provides the simplest of simple visualizations of what an elementary particle

might be⎯an oscillating pointlike charge:

Figure 1: The ring current model

7

Erwin Schrödinger referred to it as a Zitterbewegung

8

, and Dirac highlighted its significance at the

occasion of his Nobel Prize lecture:

“It is found that an electron which seems to us to be moving slowly, must actually have a very high

frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us.

As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light.

This is a prediction which cannot be directly verified by experiment, since the frequency of the oscillatory

motion is so high, and its amplitude is so small. But one must believe in this consequence of the theory,

since other consequences of the theory which are inseparably bound up with this one, such as the law of

scattering of light by an electron, are confirmed by experiment.” (Paul A.M. Dirac, Theory of Electrons and

Positrons, Nobel Lecture, December 12, 1933)

The actual motion of the pointlike charge might be chaotic but this cannot be verified: we measure

averages (cycles) only. The regularity (periodicity) of motion makes it deterministic. High velocities

introduce probability: quantum physics adheres to probabilistic determinism. H.A. Lorentz told us there

is no need to elevate indeterminism to a philosophical principle:

“Je pense que cette notion de probabilité [in the new theories] serait à mettre à la fin, et comme

conclusion, des considérations théoriques, et non pas comme axiome a priori, quoique je veuille bien

admettre que cette indétermination correspond aux possibilités expérimentales. Je pourrais toujours

garder ma foi déterministe pour les phénomènes fondamentaux, dont je n’ai pas parlé. Est-ce qu’un esprit

plus profond ne pourrait pas se rendre compte des mouvements de ces électrons ? Ne pourrait-on pas

5

720-degree symmetries and the boson/fermion dichotomy are based on a misunderstanding of the imaginary

unit representing a 90-degree rotation in this or that direction.

6

For an analysis of the relativity of magnetic and electric fields, see Feynman, II-13-6.

7

The British chemist and physicist Alfred Lauck Parson (1915) proposed the ring current or magneton model of an

electron, which combines the idea of a charge and its motion to represent the reality of an electron. The combined

idea effectively accounts for both the particle- as well as the wave-like character of matter-particles. It also

explains the magnetic moment of the electron.

8

Zitter (German used to be a more prominent language in science) refers to a rapid trembling or shaking motion.

4

garder le déterminisme en en faisant l’objet d’une croyance? Faut-il nécessairement ériger l’

indéterminisme en principe?" (H.A. Lorentz, Solvay Conference, 1927)

Velocities can be linear or tangential (orbital), giving rise to the concepts of linear versus angular

momentum. Angular momentum and Planck’s quantum of action have the same physical dimension. It is

that of a Wirkung: force (N) times distance (m) times time (s). Orbitals imply a centripetal force, and the

distance and time variables becomes the length of the loop and the cycle time, respectively. When

motion is linear, the length of the loop is a (linear) wavelength, which is 2π times the radius: we

distinguish h and its reduced version ħ = h/2π.

The ring current model of elementary particles

The ring current model is a mass-without-mass model of elementary particles. It analyzes them as

harmonic oscillations whose total energy – at any moment (KE + PE) or over the cycle – is given by E =

ma22. One can then calculate the radius or amplitude of the oscillation directly from the mass-energy

equivalence and Planck-Einstein relations, as well as the tangential velocity formula⎯interpreting c as a

tangential or orbital (escape

9

) velocity.

Such models assume a centripetal force whose nature, in the absence of a charge at the center, can only

be explained with a reference to the quantized energy levels we associate with atomic or molecular

electron orbitals

10

, and the physical dimension of the oscillation in space and time may effectively be

understood as a quantization of spacetime.

Tangential velocities imply orbitals: circular and elliptical orbitals are closed. Particles are pointlike

charges in closed orbitals. We do not think non-closed orbitals correspond to some reality: linear

oscillations are field particles, but we do not think of lines as non-closed orbitals: the curvature of real

space (i.e. the Universe we happen to live in) suggest we should⎯but we are not sure such thinking is

productive (efforts to model gravity as a residual force have failed so far).

Space and time are innate or a priori categories (Kant, 1781). Elementary particles can be modeled as

pointlike charges oscillating in space and in time. The concept of charge could be dispensed with if there

were not lightlike particles: photons and neutrinos, which carry energy but no charge.

The pointlike charge which is oscillating is pointlike but may have a finite (non-zero) physical dimension,

which explains the anomalous magnetic moment of the free (Compton) electron. However, it only

appears to have a non-zero dimension when the electromagnetic force is involved (the proton has no

9

The concepts of orbital, tangential and escape velocity are not always used as synonyms. For a basic but

complete introduction, see the MIT OCW reference course on orbital motion.

10

See, for example, Feynman’s analysis of quantized energy levels or his explanation of the size of an atom. As for

the question why such elementary currents do not radiate their energy out, the answer is the same: persistent

currents in a superconductor do not radiate their energy out either. The general idea is that of a perpetuum mobile

(no external driving force or frictional/damping terms). For an easy mathematical introduction, see Feynman,

Chapter 21 (the harmonic oscillator) and Chapter 23 (resonance).

5

anomalous magnetic moment and is about 3.35 times smaller than the calculated radius of the pointlike

charge inside of an electron). What explains ratios like this? There is no answer to this: we just find

these particles are there: their rest mass/energy behave like Nature’s constants: they are simply there.

We have two forces acting on the same (electric) charges: electromagnetic and nuclear. One of the most

remarkable things is that the E/m = c2 holds for both electromagnetic and nuclear oscillations, or

combinations thereof (superposition theorem). Combined with the oscillator model (E = ma22 = mc2

c = a), this makes one think of c2 as an elasticity or plasticity of space.

Why two oscillatory modes only? In 3D space, we can only imagine oscillations in one, two and three

dimensions (line, plane, and sphere).

Photons and neutrinos are linear oscillations and, because they carry no charge, travel at the speed of

light. Electrons and muon-electrons (and their antimatter counterparts) are 2D oscillations packing

electromagnetic and nuclear energy, respectively. The proton (and antiproton) pack a 3D nuclear

oscillation. Neutrons combine positive and negative charge and are, therefore, neutral. Neutrons may or

may not combine the electromagnetic and nuclear force: their size (more or less the same as that of the

proton) suggests the oscillation is nuclear.

2D oscillation

3D oscillation

electromagnetic force

e (electron/positron)

orbital electron (e.g.: 1H)

nuclear force

(muon-electron/antimuon)

p (proton/antiproton); n0 (neutron)

Composite (stable or transient)

?

D+ (deuteron)? pions (π/ π0)?

corresponding field particle

(photon)

(neutrino)

The theory is complete: each theoretical/mathematical/logical possibility corresponds to a physical

reality, with spin distinguishing matter from antimatter for particles with the same form factor.

Time and relativity

Panta rhei (Heraclitus, fl. 500 BC). Motion relates the ideas of space (position) and time. Spacetime

trajectories need to be described by well-defined function: for every value of t, we should have one, and

only one, value of x. The reverse is not true, of course: a particle can travel back to where it was. That is

what it is doing in the graph on the right. The force that makes it do what it does is some wild oscillation

but it is possible: not only theoretically but also practically.

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Figure 2: A well- and a not-well behaved trajectory in spacetime

Time has one direction only because we describe motion (trajectories) by well-behaved functions. In

short, the idea of motion is what gives space and time their meaning. The alternative idea is spaghetti

(first graph).

The idea of an infinite velocity makes no sense: our particle would be everywhere and we would,

therefore, not be able to localize it. Likewise, the idea of an infinitesimally small distance is a

mathematical idea only: it underlies differential calculus (the logic of integrals and derivatives) but

Achilles does overtake the tortoise: motion is real, and the arrow reaches its goal (Zeno of Elea).

Light-particles (photons and neutrinos, perhaps

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) have zero rest mass and, therefore, travel at the

speed of light (c): the slightest acceleration accelerates them to lightspeed. Light-particles, therefore,

acquire relativistic mass or momentum (F = dp/dt).

The p = mc = γm0c function behaves in a rather weird way (Figure 3): the Lorentz factor () goes to

infinity as the velocity goes to c, and m0 is equal to zero. Hence, we are multiplying zero by infinity.

Figure 3: p = mvv = γm0v for m → 0

The function reminds one of the Dirac function (x): the sum of probabilities must always add up to one.

If we measure the position of a particle at x = x at time t = t, then the probability function collapses at

P(x, t) = 1.

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We think of neutrinos as 3D oscillations and they may, therefore, have some non-zero rest mass or, to be

precise, some inertia to a change in their state of motion along all possible directions of motion. In contrast, the

two-dimensional oscillation of the electromagnetic field vector (photon) is perpendicular to the direction of

motion and we therefore have no inertia in the direction of propagation.

7

Figure 4: The Dirac function (x) as the limit of a probability distribution (Feynman, III-16-4)

We may imagine a wavefunction which comes with constant probabilities: |ψ|2 = |aei|2 = a2. The

wavefunction ψ is zero outside of the space interval (x1, x2). We have an oscillation in a spatial box

(Figure 1), which packs a finite amount of energy. All probabilities have to add up to one, and so we

must normalize the distribution.

Figure 5: Elementary particle-in-a-box model

The energy (and equivalent mass) of a harmonic oscillation is given by E = ma22 = m2f2. We can,

therefore, write:

This gives us a physical normalization condition based on the total energy of the particle and the

physical constants c and ħ. The wavefunction itself represents energy densities⎯energy per unit volume

(V) unit, or force per area unit (A):

E = E/V, and [E] = [E/V] = Nm/m3 = N/m2 = [F/A]

8

The volume V and the energy E are the volume and energy of the particle, respectively⎯and the area A

and force F are the orbital area and the centripetal force, respectively. The physical dimension of the

components of the wavefunction is, therefore, equal to [] = N/m2: force per unit area. All other things

being equal (same mass/energy), stronger forces make for smaller particles.

12

The illustration below (Figure 6) imagines how the Zitterbewegung radius of an elementary particle

decreases as one adds a lateral (linear) velocity component to the motion of the pointlike charge: it

decreases as it gains linear momentum. Why is that so? Because the speed of light is the speed of light:

the pointlike charge cannot travel any faster if we are adding a linear component to its motion. Hence,

some of its lightlike velocity is now linear instead of circular and it can, therefore, no longer do the

original orbit in the same cycle time.

Figure 6: The Compton radius must decrease with increasing velocity

13

Needless to say, the plane of oscillation of the pointlike charge is not necessarily perpendicular to the

direction of motion. In fact, it is most likely not perpendicular to the line of motion, which explains why

we may write the de Broglie relation as a vector equation: λL = h/p. Such vector notation implies h and p

can have different directions: h may not even have any fixed direction! It might wobble around in some

regular or irregular motion itself!

12

The time dependency is in the phase (angle) of the wavefunction = t = Et/ħ. We may say that Planck’s

quantum of action scales the energy as per the Planck-Einstein relation E = ħ = hf = h/T, with T the cycle time.

We may say Planck’s quantum of action expresses itself as some energy over some time (h = ET) or as some

momentum over a distance (h = p). If the pointlike charge spends more time in a volume element (or passes

through more often), the energy density in this volume element will, accordingly, be larger.

13

We borrow this illustrations from G. Vassallo and A. Di Tommaso (2019).

9

Figure 6 also shows that the Compton wavelength (the circumference of the circular motion becomes

a linear wavelength as the classical velocity of the electron goes to c. It is now easy to derive the

following formula for the de Broglie wavelength

14

:

The graph below shows how the 1/γβ factor behaves: it is the green curve, which comes down from

infinity (∞) to zero (0) as v goes from 0 to c (or, what amounts to the same, if β goes from 0 to

1). Illogical? We do not think so: the classical momentum p in the λL = h/p is equal to zero when v = 0, so

we have a division by zero. We may also note that the de Broglie wavelength approaches the Compton

wavelength of the electron only if v approaches c.

Figure 7: The 1/γ, 1/β and 1/γβ graphs

15

The combination of circular and linear motion explains the argument of the wavefunction, which we will

now turn to.

The wavefunction and its (relativistically invariant) argument

We will talk a lot about wavefunctions and probability amplitudes in the next section, so we will be brief

here. When looking at Figure 6, it is obvious that we can use the elementary wavefunction (Euler’s

formula) to represents the motion of the pointlike charge by interpreting r = a·eiθ = a·ei·(E·t − k·x)/ħ as its

position vector. The coefficient a is then, equally obviously, nothing but the Compton radius a = ħ/mc.

16

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

14

You should do some calculations here. They are fairly easy. If you do not find what you are looking for, you can

always have a look at Chapter VI of our manuscript.

15

We used the free desmos.com graphing tool for these and other graphs.

16

When discussing the concept of probability amplitudes, we will talk about the need to normalize them because

the sum of all probabilities – as per our conventions – has to add up to 1. However, the reader may already

appreciate we will want to talk about normalization based on physical realities⎯as opposed to unexplained

mathematical conventions or quantum-mechanical rules.

10

frame of reference by using Lorentz’s equations

17

:

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

18

:

E0 is, obviously, the rest energy and, because p’ = 0 in the reference frame of the electron, the

argument of the wavefunction effectively reduces to E0t’/ħ in the reference frame of the electron itself.

Besides proving that the argument of the wavefunction is relativistically invariant, this calculation also

demonstrates the relativistic invariance of the Planck-Einstein relation when modelling elementary

particles.

19

This is why we feel that the argument of the wavefunction (and the wavefunction itself) is

more real – in a physical sense – than the various wave equations (Schrödinger, Dirac, or Klein-Gordon)

for which it is some solution.

In any case, a wave equation usually models the properties of the medium in which a wave propagates.

We do not think the medium in which the matter-wave propagates is any different from the medium in

which electromagnetic waves propagate. That medium is generally referred to as the vacuum and,

whether or not you think of it as true nothingness or some medium, we think Maxwell’s equations –

which establishes the speed of light as an absolute constant – model the properties of it sufficiently

well! We, therefore, think superluminal phase velocities are not possible, which is why we think de

Broglie’s conceptualization of a matter particle as a wavepacket – rather than one single wave – is

erroneous.

20

Rutherford, Bohr, Dirac, Schrödinger, and electron orbitals

A particle will always be somewhere but, when in motion, its position in space and time should be

thought of as a mathematical points only. The solution to the quantum-mechanical wave equation are

17

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.

18

One can use either the general E = mc2 or – if we would want to make it look somewhat fancier – the pc = Ev/c

relation. The reader can verify they amount to the same.

19

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.

20

See our paper on matter-waves, amplitudes, and signals.

11

equations of motion (Dirac, 1930). The electron in an atomic or molecular orbital moves at an (average)

velocity which is a fraction of lightspeed only. This fraction is given by the fine-structure constant and

the principal quantum number n:

The velocities go down, all the way to zero for n → , and the corresponding cycle times increases as

the cube of n. Using totally non-scientific language, we might say the numbers suggest the electron

starts to lose interest in the nucleus so as to get ready to just wander about as a free electron.

Table 1: Functional behavior of radius, velocity, and frequency of the Bohr-Rutherford orbitals

n

1

2

3

4

5

6

7

8

9

rn n2

1

4

9

16

25

36

49

64

81

vn 1/n

1

0.500

0.333

0.250

0.200

0.167

0.143

0.125

0.111

ωn 1/n3

1

0.125

0.037

0.016

0.008

0.005

0.003

0.002

0.001

Tn n3

1

8

27

64

125

216

343

512

729

The important thing is the energy formula, of course, because it should explain the Rydberg formula,

and it does:

The calculations are based on the assumption that, besides energy, electron orbitals also pack a discrete

amount of physical action⎯a multiple of Planck’s quantum of action, to be precise:

The orbital energies do not include the rest mass/energy of the Zitterbewegung (zbw) electron itself

(0.511 MeV). In fact, they are tiny as compared to the electron’s rest mass: 13.6 eV for n = 1 orbital of

the hydrogen atom 1H. This is the Rydberg energy (ER) in the formula above. It is the combined kinetic

and potential energy of the electron in the (first) Bohr orbital. Using the definition of the fine-structure

constant (as per the 2019 revision of SI units) and the rest energy (E0 = m0c2) of the electron, we can

write it as:

Schrödinger’s model of the hydrogen atom does not fundamentally differ from the Bohr-Rutherford

model

21

but includes non-elliptical/non-symmetrical orbitals, which obey the vis-viva (literally: ‘living

force’) equation. For the gravitational force, this equation is written as:

21

Around 1911, Rutherford had concluded that the nucleus had to be very small. Hence, Thomson’s model – which

assumed that electrons were held in place because they were, somehow, embedded in a uniform sphere of

positive charge – was summarily dismissed. Bohr immediately used the Rutherford hypothesis to explain the

emission spectrum of hydrogen atoms, which further confirmed Rutherford’s conjecture, and Niels and Rutherford

12

The parameter a is the length of the semi-major axis: a > 0 for ellipses but infinite () or negative (a < 0)

for non-closed loops (parabolas and hyperbolas, respectively). The Universe is closed and all lightlike

particles (photons and neutrinos) must, therefore, return. Einstein’s view that the nature of the

gravitation may not reside in a force but in the mere geometry of the Universe (our Universe, which we

live in), therefore, makes sense. In any case, efforts to model the gravitational force as a residual force

have failed⎯so far, at least.

The two fundamental forces (Coulomb and nuclear/strong)

The idea of a particle assumes its integrity in space and in time. Non-stable particles may be labeled as

transients (e.g. charged pions

22

) or, when very short-lived, mere resonances (e.g. neutral pion or tau-

particle

23

). Hence, the Planck-Einstein relation does not apply: we cannot model them as equilibrium

states. We think the conceptualization of both the muon- as well as the tau-electron in terms of particle

generations is unproductive.

The muon’s lifetime – about 2.2 microseconds (10−6 s) – is, however, quite substantial and we may,

therefore, consider it to be a semi-stable particle. This explains why we get a sensible result when using

the Planck-Einstein relation to calculate its frequency and/or radius. Inserting the 105.66 MeV (about

207 times the electron energy) for its rest mass into the formula for the zbw radius

24

, we get:

The mean lifetime of a neutron in the open (outside of the nucleus) is almost 15 minutes, and the

Planck-Einstein relation should, therefore, apply (almost) perfectly, and it does:

jointly presented the model in 1913. As Rydberg had published his formula in 1888, we have a gap of about 25

years between experiment and theory here. It should be noted that Schrödinger’s model accounts for subshells

but still models orbital electrons as spin-zero electrons (zero spin angular momentum). It, therefore, models

electron pairs, which explains the ½ factor Schrödinger’s wave equation, which – we think – is relativistically

correct.

22

The mean lifetime of charged pions is about 26 nanoseconds (10−9 s), which is about 1/85 times the lifetime of

the muon-electron. We have no idea why charged pions are lumped together with neutral pions, whose lifetime is

of the order of 8.410−17 s only. An accident of history? If anything, it shows the inconsistency of an analysis in

terms of quarks.

23

The (mean) lifetime of the tau-electron is 2.910−13 s only.

24

See the derivation earlier in the text:

13

The 1/4 factor is the 1/4 factor between the surface area of a sphere (A = 4πr2) and the surface area of a

circle (A = πr2).

25

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 oscillators would be equal to = E/2ħ = mc2/2ħ: each of the two perpendicular

oscillations would, therefore, pack one half-unit of only.

26

According to the equipartition theorem, each

of the two oscillations should each pack 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.

However, the calculation of the radius above is quick-and-dirty only. It applies perfectly well for the

(stable) proton, but we cannot immediately reconcile it with the idea of a neutron consisting of

consisting of a ‘proton’ and an ‘electron’, which are the final decay products of a (free) neutron. We

should immediately qualify the ‘proton’ and ‘electron’ idea here: the reader should effectively think in

terms of pointlike charges here⎯rather than in terms of a massive proton and a much less massive

electron!

27

Both the ‘proton’ and the ‘electron’ carry the elementary (electric) charge but we think both

must be simultaneously bound in a nuclear as well as in an electromagnetic oscillation. In order to

interpret v as an orbital or tangential velocity, we must, of course, choose a reference frame. Let us first

jot down the orbital energy equation for the nuclear field, however

28

:

25

Cf. the 4π factor in the electric constant, which incorporates Gauss’ Law (expressed in integral versus differential

form).

26

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).

27

We do not have a hydrogen-like model here!

28

A dimensional check of the equation yields:

We recommend the reader to regularly check our formulas: we do make mistakes sometimes!

14

Figure 8: Two opposite charges in elliptical orbitals around the center of mass

29

The mass factor mN is the equivalent mass of the energy in the oscillation

30

, which is the sum of the

kinetic energy and the potential energy between the two charges. The velocity v is the velocity of the

two charges (qe+ and qe−) as measured in the center-of-mass (barycenter) reference frame and may be

written as a vector v = v(r) = v(x, y, z) = v(r, , ), using either Cartesian or spherical coordinates.

We have a plus sign for the potential energy term (PE = akeqe2/mr2) because we assume the two charges

are being kept separate by the nuclear force.

31

The electromagnetic force which keeps them together is

the Coulomb force:

The total energy in the oscillation is given by the sum of nuclear and Coulomb energies and we may,

therefore, write:

29

Illustration taken from Wikipedia. For the orbital equations, see the MIT OCW reference course on orbital

motion.

30

We will use the subscripts xN and xC to distinguish nuclear from electromagnetic mass/energy/force. There is

only one velocity, however⎯which should be the velocity of one charge vis-á-vis the other. We hope we made no

logical mistakes here!

31

We have a minus sign in the same formula in our paper on the nuclear force because the context considered two

like charges (e.g. two protons). As for the plus (+) sign for the potential energy in the electromagnetic orbital

energy, we take the reference point for zero potential energy to be the center-of-mass and we, therefore, have

positive potential energy here as well.

15

The latter substitution uses the definition of the fine-structure constant once more.

32

Dividing both sides

of the equation by c2, and substituting mN and mC for m/2 using the energy equipartition theorem,

yields:

It is a beautiful formula

33

, and we could/should probably play with it some more by, for example,

evaluating potential and kinetic energy at the periapsis, where the distance between the charge and the

center of the radial field is closest. However, the limit values vπ = c (for rπ → 0) and rπ = 0 (for vπ → c) are

never reached and should, therefore, not be used.

One might hope to find a way to relate the orbital energy equations to the formula for the zbw radius to

get a specific value not only for the neutron radius a – which should, hopefully, be very near to 0.84 fm

(the proton/neutron diameter

34

) – but also for the range parameter of the nuclear force.

35

However, as

we will show below, things are probably not that easy.

The nuclear range parameter and the fine-structure constant

At the very least, we have an order of magnitude for this range parameter now. This order of magnitude

may be calculated by equating r to a in the formula above

36

:

32

One easily obtains the keqe2 = ħc identity from the

formula. We think the 2019 revision of SI units

consecrates all we know about physics.

33

The a in the formula(s) above is the range parameter of the nuclear force, which is not to be confused with the

Zitterbewegung (zbw) radius!

34

The neutron radius should, in fact, be slightly larger than the proton radius because of the energy difference

between a proton and a neutron, which is of the order of about 1.3 MeV (about 2.5 times the energy of a free

electron). We note there is no CODATA value for the neutron radius. This may or may not be related to the

difficulty of measuring the radius of a decaying neutral particle or, more likely, because the neutron mass/energy is

not considered to be fundamental. However, one must get the range parameter a out of the formulas, somehow,

and we, therefore, think experimental measurements of the (free) neutron radius are crucially important. As for

quarks, we are happy to see NIST does not dabble too much into the quark hypothesis. At best, they are purely

mathematical quantities (combining various physical dimensions) to help analyze and structure decay reactions of

unstable particles, but that is being taken care of by the Particle Data Group.

35

The reader should note that our neutron model implies a neutral () dipole, which relates to our previous efforts

to develop an electromagnetic model of the deuteron nucleus. See our paper on the electromagnetic deuteron

model.

36

The range parameter is usually defined as the distance at which the nuclear and Coulomb potential (or the

forces) equal each other. See: Ian J.R. Aitchison and Anthony J.G. Hey, Gauge Theories in Particle Physics (2013),

section 1.3.2 (the Yukawa theory of force as virtual quantum exchange).

16

The ħ/mc constant is, obviously, equal to the classical electron radius re 2.818 fm (10−15 m)⎯which is

of the order of the deuteron radius (about 2.128 fm) and which is the usual assumed value for the range

parameter of the nuclear force.

37

We think it is a significant result that the lower limit for the range parameter for the nuclear force must

be at least twice at large. An upper limit for this range parameter must be based on the experimentally

measured value for the radius of atomic nuclei. The scale for these measurements is the picometer

(10−12 m). The nucleus of the very stable iron (26Fe), for example, is about 50 pm.

38

The radius of the

large (unstable) uranium (92U) is about 175 pm.

The fine-structure constant may be involved again: 5.536 fm times 1/ yields a value of about 77 pm.

We think this is a sensible value for the (range of the upper) limit for the (nuclear) range parameter,

which will, of course, depend on the shape (eccentricity) of the actual orbitals.

Of course, the stability of the nucleus of an atom is determined by other factors, most notably the

magnetic coupling between the nucleons and the electrons in the atomic (sub)shells. This should,

somehow, explain the ‘magic numbers’ explaining the (empirical) stability of nuclei, but the exact

science behind this seems to be beyond us.

39

The meaning of the fine-structure constant becomes somewhat clearer now:

• The fine-structure constants relates the classical electron radius, Compton radius and the Bohr

radius of an electron: re = ħ/mc = rC = 2rB

• The Bohr radius is the distance where the combined electromagnetic and nuclear potential (1/r

– a/r) approaches the electromagnetic potential (1/r). Hence, we might say that the Compton

radius separates nuclear from electromagnetic scale.

We may remind the reader here of the (average) radius of electron orbitals:

Nuclear orbitals – or combined nuclear-electromagnetic, we should say – orbitals are of the order of

ħ/mc. We think this is not a coincidence but, as with magic numbers, it will take a while for the more

numerology-oriented physicists to figure out a more exact explanation.

We should, of course, raise the obvious question here: this model – combining electromagnetic and

nuclear force – yields a distance scale which is not compatible with the neutron radius (0.84 fm). That is

why we think the neutron must be a genuine nuclear oscillation. We are a bit at a loss, however, as to

37

See footnote 36.

38

This is Feynman’s calculated radius of a hydrogen atom, but the measured radius of the hydrogen nucleus is

about half of it. To be precise, the empirical value is about 25 pm according to the Wikipedia data article on atomic

radii. We leave it to the reader to think about the 1/2 factor and the fine-structure constant as a scaling parameter.

39

See the Wikipedia article on magic numbers (nuclei).

17

how to model that exactly? Perhaps we should return to modelling the neutron as a massive proton

being enveloped by the pointlike charge.

40

Conclusions

When reading this, my kids might call me and ask whether I have gone mad. Their doubts and worry are

not random: the laws of the Universe are deterministic (our macro-time scale introduces probabilistic

determinism only). Free will is real, however: we analyze and, based on our analysis, we determine the

best course to take when taking care of business. Each course of action is associated with an anticipated

cost and return. We do not always choose the best course of action because of past experience, habit,

laziness or – in my case – an inexplicable desire to experiment and explore new territory. Is that free will?

We are not sure. Ontology is the logic of being. The separation between consciousness and its object is

no more real than consciousness' inadequate knowledge of that object. The knowledge is inadequate only

because of that separation.

41

Hegel completed the work of philosophy. Physics took over as the science

of that what is. It should seek to further reduce rather than multiply concepts.

Brussels, 17 February 2021

40

For suggestions in this regard, see our paper on the mass-without-model for protons and neutrons.

41

Quoted from the Wikipedia article on Hegel’s Phänomenologie des Geistes (1807).

18

Annex I: Dirac’s energy and Schrödinger’s wave equation

Preliminary note (added on 16 September 2022): We found an error in our rendering of Dirac’s energy equation.

In any case, we do not think highly of Dirac’s assumptions and development of his wave equation, as will be

obvious from what is written below. We only corrected the formula and then expand on our electron model and

what we think of as the correct energy concept to be used for a free electron. For a more complete treatment of

Dirac’s wave equation, we refer to a new annex (Annex II) of our most popular paper (de Broglie’s matter-wave:

concept and issues). In light of its popularity (its RI score is higher than 96% of research items published in 2020),

we should probably do a more substantial revision or re-edit of that paper, but time in a man’s life is limited –

especially if such man has a totally different day job.

In his Nobel Prize Lecture, Dirac starts by writing the classical (relativistic) energy equation for a particle

(an electron) as:

This equation raises obvious questions and appears to be based on a misunderstanding of the

fundamental nature of an elementary particle⎯which, in the context of Dirac’s lecture

42

, is a free

electron. We do not agree with his concepts and definitions. According to the Zitterbewegung

hypothesis (which Dirac mentions prominently) and applying the energy equipartition theorem, half of

the energy of the electron will be kinetic, while the other half is the energy of the field which keeps the

pointlike (zbw) charge localized. The pointlike charge is photon-like

43

and, therefore, has zero rest mass:

it acquires a relativistic or effective mass m = me/2. Its kinetic energy is, therefore, equal to

44

:

Dirac refers to the pr in the equation as momentum, but this must represent potential energy in the

reference frame of the particle itself. If the oscillation’s nature is electromagnetic, then this potential

energy is given by

45

:

It is useful to write the orbital energy equation as energy per unit mass:

42

https://www.nobelprize.org/uploads/2018/06/dirac-lecture.pdf

43

We avoid this term, however, because photons do not carry charge: this distinguishes light-particles (photons

and neutrinos) from matter-particles.

44

This equation is relativistically correct because (i) the velocity v is an orbital/tangential velocity and (ii) we use

the relativistic mass concept. The velocity v is equal to the speed of light (c) but, in a more general treatment (e.g.

elliptical orbitals), v should be distinguished from c.

45

U(r) = V(r)·qe = (ke·qe/r)·qe = ke·qe2/r with ke 9109 N·m2/C2. Potential energy (U) is, therefore, expressed in joule

(1 J = 1 N·m), while potential (V) is expressed in joule/Coulomb (J/C).

19

We may also write this in terms of the relative velocity = v/c and the fine-structure constant

46

:

When adding a linear component to the orbital motion of the pointlike charge, the electron oscillation

will move linearly in space and we can, therefore, associate a classical velocity ve and a classical

momentum pe with the Zitterbewegung oscillation. We discussed and illustrated this sufficiently in the

body of our paper. We must now distinguish the rest energy of the electron (E0) and its kinetic energy,

which, referring to the classical momentum, we will denote by Ep = E − E0. Writing E as E = mc2 again, we

can use the binomial theorem, to expand the energy into the following power series

47

:

This formula separates the rest energy E0 = m0c2 from the kinetic energy Ep, which may, therefore, be

written as:

Schrödinger’s wave equation models electron orbitals whose energy excludes the rest energy of the

electron. We are not sure whether Dirac’s wave equation correctly integrates this rest energy again: are

Dirac’s pr (r = 1, 2, 3,…) references to the 2, (2)2,.. terms in the power series? We think of this series

expansion as a mathematical exercise only: we are not able to relate them to anything real⎯we think of

forces and/or potentials here!

We offer further comments on the use of wave equations to model motion in the Annex to our paper on

the matter-wave.

48

We think it is rather telling that Richard Feynman does not bother to present Dirac’s

46

Since the 2019 revision of the SI units, the electric, magnetic, and fine-structure constants have been co-defined

as ε0 = 1/μ0c2 = qe2/2αhc. The CODATA/NIST value for the standard error on the value ε0, μ0, and α is currently set

at 1.51010 F/m, 1.51010 H/m, and 1.51010 (no physical dimension here), respectively. We use the me = m/2

once more. To quickly check the accuracy and, more importantly, their meaning, we recommend the reader to do

a dimensional check. We have a purely numerical equation here (all physical dimensions cancel):

47

See Feynman’s Lectures, I-15-8, and I-15-9 (relativistic dynamics). The expansion is based on an expansion of m =

m0:

This is multiplied with c2 again to obtain the series in the text.

48

Jean Louis Van Belle, De Broglie’s matter-wave : concepts and issues, May 2020.

20

wave equation in his Lectures on Physics (1963). We think it is because he cannot make sense of it

either. Feynman’s wave equation for a free particle is the following

49

:

This equation incorporates the integrity of Planck’s quantum of action as the unit of the angular

momentum of the oscillation (cf. the iħ factor). The (scalar) potential φ can be electromagnetic, nuclear

or a combination thereof, acting on the (electric) charge q. Assuming the scalar potential varies with

time, the vector potential A is probably to be derived from the Lorenz gauge condition in

electromagnetic theory:

For a time-independent scalar potential, which is what we have been modeling so far, the Lorentz gauge

is zero (·A = 0) because the time derivative is zero: φ/t = 0 ·A = 0.

50

The magnetic field,

therefore, vanishes. The time-dependent magnetic field – or its nuclear equivalent – should absorb half

the energy in accordance with relativity theory

51

and it should then be easy to develop the equivalent of

Maxwell’s equations for the nuclear force field using the theorems of Gauss and Stokes.

49

See: Feynman, III-21, Schrödinger’s equation in a magnetic field and his equation of continuity for probabilities.

We took the liberty of writing 1/i as − i. We also multiplied the right-hand side of Feynman’s equation with

(−1)(−1) = +1, and substituted the dot product of the −iħ − qA operators for the square of the same operator.

50

The Lorenz gauge does not refer to the Dutch physicist H.A. Lorentz but to the Danish physicist Ludvig Valentin

Lorenz. It is often suggested one can choose other gauges. We do not think so. We think the gauge is given by

relativity theory, and that is the same for time-dependent and time-independent fields. It does vanish, however,

time-independent fields (cf. electromagnetostatics). See our remarks on the vector potential and the Lorentz

gauge in our paper on the electromagnetic deuteron model.

51

When using natural units (c = 1), the relativity of electric and magnetic fields becomes more obvious.

21

Annex II: The quark hypothesis

Strange kaons

Kaons (aka K-mesons

52

) are supposed to consist of a strange quark (or its antimatter counterpart) and

some other quark (the up or down quark, or its antimatter counterpart). Like pions (-mesons), the

charged kaons and the neutral kaon have very little in common, except a somewhat similar mass: a bit

less than 500 MeV/c2, so that is about half of the proton/neutron mass. However, charged kaons have a

(mean) lifetime of 12.4 nanoseconds (10−9 s) – quite comparable to the mean lifetime of charged pions

(about 2610−9 s)

53

– while the mean lifetime of a neutral kaon is… Well… We have two neutral kaon-

particles⎯one with a shorter and one with a longer lifetime: KL0 (about 5210−9 s) and KS0 (8910−12 s).

54

Like pions, kaons were first seen in decays of cosmic rays in bubble chambers or on photographic plates.

In fact, pions and kaons are closely related, as shown in Feynman’s drawings (III-11-14) of the decay

reaction of a − and a 0.

Many different reactions are possible. The Particle Data Group lists all of them. Feynman focuses very

much on the hypothetical reactions that do not happen, such as this one:

K0 + p → 0 + +

We have an intermediate (neutral) lambda baryon

55

(0) here: it is very massive – about 1115.6 MeV/c2

– but also short-lived: as shown in figure (b) above, it decays into a − and a proton (p). The charged pion

decays into a muon (or antimuon) and, therefore, ultimately into an electron (or positron), so we should

not be concerned with it, either. The question here is: why do we observe K0 + p → 0 + + reactions

56

52

Mesons are defined as subatomic particles composed of an equal number of quarks and antiquarks, usually one

of each, bound together by strong interactions (read: the strong force).

53

The mean lifetime of a neutral pion (0) is 8.410−17 s. If the charged pion can be thought of as a transient, then

the neutral pion is just an extremely short-lived resonance.

54

We would rather think of a KS0 particle as a very short-lived resonance, as opposed to a somewhat more robust

transient particle, but let us go along with the argument.

55

A baryon is supposed to consist of an odd number of quarks, usually three.

56

We use an underbar (K) instead of an overbar to denote the antimatter counterparts of a particle out of laziness

(we do not want to use the equation editor all of the time).

22

but not the K0 + p → 0 + + reaction? We think antimatter differs from matter only because of opposite

spin – or, to be precise, because of its opposite spacetime signature

57

– but, surely, the 0 + + come

with two possible directions of spin as well, don’t they?

Let us look at the PDG listings of kaon reaction. […] Surprise, surprise! These do not list the reactions

involving K0 particles! Why not? We are not sure. It is very confusing: Feynman’s account does not

match the PDG picture. Neutral particles are supposed to be their own antiparticles, no? Yes. We think

so, at least.

So what can we say? Nothing much. Let us focus on the instability part.

Transient oscillations: what is real?

Feynman argues one needs the concept of strangeness to explain why this or that reaction does not take

place, but the argument does not convince us⎯especially because strangeness is not always conserved.

When that happens, the decays are supposed to be weak decays (as opposed to strong, nuclear decays),

which, according to Feynman (and the inventors of these strong and weak interactions) also need not

respect this new strangeness conservation law. The Particle Data Group effectively invokes CP or T

violation regularly: the ubiquitous symmetry-breaking which explains everything that cannot be

explained.

58

Anything goes, it seems.

The thing that grabs my attention much more is the shape of the wavefunctions, which we copied from

the same source (Feynman III-11-6):

The coefficients – the C− coefficient above – that we get out of the Hamiltonian system of equations for

the K0 and/or K0 system is not a stable wavefunction: we get a transient or – the boundary between

transients and resonances is not clear-cut – a resonance: an unstable energy state, to which we cannot

apply the Planck-Einstein relation (E = ħ).

We admire this business of trying to reduce the complexity of the situation on hand through the

introduction of the quark hypothesis but, paraphrasing H.A. Lorentz, we do not immediately see the

57

See p. 34 to 36 of our paper on quantum behavior (modeling spin and antimatter).

58

Combined CPT-symmetry must hold, however. See the discussion on our blog.

23

need to elevate quarks (and the related form factors) to ontological status.

59

Our criticism is, therefore,

not as scathing as our criticism on the ‘discovery’ of the Higgs field/particle

60

, but the nature of our

criticism remains the same.

It is, of course, quite OK to resort to mathematical techniques – we are dealing with some kind of factor

analysis to find the S-matrix (scattering or Spur-matrix

61

) here – when we cannot explain some reaction

which happens or does not happen on the basis of the classical conservation laws (conservation of

charge, energy, physical action, and linear and angular momentum), but it is not OK to recognize this is

just some kind of engineering approach to find a numerical approximation to a problem that, basically,

amounts to a (much more complicated) three-body problem.

Was the 1969 Nobel Prize for Murray Gell-Mann justified? First, it should have been shared with others

who were working on similar analyses of non-equilibrium states (Yuval Ne'eman, George Zweig and

(many) others) and, second, the Nobel Prize in Physics is usually not awarded for a significant

breakthrough in numerical or mathematical analysis. Gell-Mann did not discover some new physical law

or physical reality. Englert and Higgs did not either.

The End of Science – all that is left is engineering, right? – is not easy to digest. :-/ In any case, we should

not get too philosophical here. Let us look at those coefficients and try to find out what they might

mean.

An analysis of non-equilibrium states

We will closely follow Feynman’s treatment here but simplify and add our own remarks. It starts off with

a rather typical set of Hamiltonian equations for what Feynman refers to as the K0K0 system but we think

of it as simply modelling two opposite spin states of the same neutral particle:

Feynman then gives you the usual Spiel – transformation to another set of base states and the

associated trial solutions – but with a notable exception: the frequency in the ae−it wavefunction is

not a real number which we get from the Planck-Einstein relation ( = E0/ħ or = A/ħ or some linear

combination hereof). No! This time it is a complex number = + i. Of course, we know what that

means: a friction term, or a driven oscillation⎯a transient, in short (as opposed to a pure harmonic

oscillation). That is what is depicted above (Feynman’s Fig. 11-6). As for the values for and , Feynman

writes this:

“Since nobody knows anything about the inner machinery, that is as far as Gell-Mann and Pais

could go. They could not give any theoretical values for α and β. And nobody has been able to

do so to this date. They were able to give a value of β obtained from the experimentally

59

The comments of H.A. Lorentz in regard of the ‘new’ quantum-mechanical theories at the occasion of the last

Solvay Conference (1927) he had been in charge of, were this: ‘Ne pourrait-on pas garder le déterminisme en

faisant l’objet d’une croyance? Faut-il nécessairement ériger l’indéterminisme en principe?’

60

See our Smoking Gun Physics paper, July 2019.

61

The terms are certainly not synonymous: related, at best!

24

observed rate of decay into two π's (2β = 1010 s−1), but they could say nothing about α. […] There

are some rough results which indicate that the α is not zero, and that the effect really occurs—

they indicate that α is between 2β and 4β.

62

That is all there is, experimentally.”

Feynman’s conclusion is this:

“The analysis we have just described is very characteristic of the way quantum mechanics is

being used today in the search for an understanding of the strange particles. All the complicated

theories that you may hear about are no more and no less than this kind of elementary hocus-

pocus using the principles of superposition and other principles of quantum mechanics of that

level.”

We agree⎯but we are even less sure now about the question of whether or not Murray Gell-Mann

deserved a Nobel Prize for Physics. Perhaps the next Prize should go to the Wolfram Physics project.

The math of transients

The math of transients is not so difficult: it suffices to multiply the wavefunction (let us refer to our

unstable particle as U, so we can denote something stable as S) with a real-valued negative exponential:

The illustration below shows how this works: both the real and imaginary part of the wavefunction –

think of the electric and magnetic field vector here, for example – lose amplitude and, therefore, energy.

Where does the energy go? It cannot get lost, so we must assume it goes into the field, where it

contributes to progressively building up another oscillation. The combined particle-field combination

will, therefore, be something stable (S) that conserves energy (and, therefore, mass):

We may apply the usual interpretation to the and factors:

62

We definitely do not agree with Feynman here! See our remarks on the very different order of magnitude

between the and parameters!

25

1. The in the e−t decay function gives us the mean lifetime of the unstable particle ( = 1/) and,

as Feynman points out, such mean lifetime will be of the order of 10−9 to 10−12 seconds.

2. The in the Ae−t decay function is equal to E/ħ and will generally be a frequency (its dimension

is s−1) that is much larger than . The frequency of an electron, for example, can be calculated

as:

As we can see, we have a difference of 10 orders of magnitude (1010) between and here, and an

electron is not very massive as compared to a proton! Of course, this explains that transient or resonant

particles do not last very long, but still pack like 1010 cycles during their short lifetime!

We should wrap up and let us, therefore, make one final remark in regard to asymmetries in Nature.

Spin may be left or right-handed, but the imaginary part – think of the magnetic field vector in an

electromagnetic oscillation – will always lag 90 degrees behind the real part or – if you like – it will lead

the real part by 270 degrees. This does not only define an absolute direction of rotation in space, but it

also introduces an asymmetry, which – in our view – should also help to explain why certain reactions

do not take place. We think this must be the cause of the CP- and T-breaking that is observed in such

reactions. However, always remember we still have (combined) CPT-symmetry!

Hermiticity and reversibility

Physicists try to model these reactions and processes using the following rather general matrix equation,

which has an S- or A-matrix at its center: it operates on some state |ψ to produce some other state

|ϕ: ϕ|A|ψ

We can now take the complex conjugate:

ϕ|A|ψ* = ψ|A†|ϕ

A† is, of course, the conjugate transpose of A – we write: A†ij=(Aji)* – and we will call the operator (and

the matrix) Hermitian if the conjugate transpose of this operator (or the matrix) gives us the same

operator matrix, so that is if A† = A. Many quantum-mechanical operators are Hermitian, and we will

also often impose that condition on the S-matrix.

63

Why? Because you should think of an operator or an

S-matrix as a symmetric apparatus or a reversible process. It is as simple as that. We, therefore, think

that the Hermiticity condition amounts to a simple reversibility condition

64

and, as mentioned above, we

think certain processes may not be reversible because of the asymmetry in the wavefunction itself!

So do physicists really resemble econometrists modeling input-output relations?

65

We think the answer

is yes, and no! The main difference is the complexity: those ϕ| and |ψ states should probably be

rewritten as multidimensional arrays, and there are a lot of constraints on that matrix S (or A)!

63

See Feynman, Vol. III, Chapter 8 (the Hamiltonian matrix) for a rather pleasant explanation of the game.

64

See our paper on the difference between a theory, a calculation, and an explanation. Also see our blog post on

the end (?) of physics.

65

See our (somewhat disrespectful) blog post on the end (?) of physics.

26

We will qualify this statement in the next section (Annex III).

Annex III: A complete description of the Universe

We have a ϕ| state in, a |ψ state out and we should relate both through the S-matrix

66

:

ϕ|S|ψ

The deterministic worldview implies reversibility and we should, therefore, also be able to go back from

the final state |ψ (the ket in Dirac’s bra-ket notation) to the initial state ϕ| by taking the complex

conjugate: ϕ|A|ψ* = ψ|A†|ϕ

The complex conjugate implies a reversal of spin, not of actual time, although both look the same from a

formal (mathematical) point of view: time goes in one direction only and C, P and T-symmetry may be

broken, but the combined CPT-symmetry should hold, always.

Both the initial as well as the final state vectors consist of a bunch of matter- and light-particles. Matter-

particles carry charge and may be stable or not. The -factor will be non-zero for unstable particles,

which implies they have a finite decay time = 1/:

In contrast, the lifetime of stable particles is infinite and is, therefore, equal to zero:

The -factor may be positive or negative – representing up or down spin respectively – but is always

positive. The -factor will also be usually much larger than and we may, therefore, write: || >> .

is the (angular) frequency of the particle and is given by the energy (for unstable particles, this energy

is the initial energy) and the Planck-Einstein relation: = = E/ħ. We assume the coefficients Ak (the

amplitude of the oscillation) are normalized: the mass/energy of all particles (Ek = mkc2) must add to the

total energy of the system (1 < k < n), with n the total number of particles in the initial or final state. The

ϕ|S|ψ statement, therefore, takes care of the mass/energy conservation principle.

The initial and final state may consist of a different number of particles and we take n to the largest

number of the two. However, in general, particle interactions are rather simple – like two particles

interacting to yield three or four other particles, or one particle decaying into a limited number of other

(stable or unstable) matter-particles, with light-particles (photons) taking care of the excess energy and

(linear/angular) momentum. The physicist should, therefore, consider the ϕ|S|ψ statement to

describe a single event.

Events change the potential fields surrounding the (charged) particles and, therefore, one event is to be

associated with static (stable) potentials. Particle-field interactions must also obey the mass-energy

66

It is interesting, historically speaking, that John Archibald Wheeler (whom we know from the mass-without-mass

models of elementary particles) and Erwin Schrödinger independently developed the idea of the S-matrix (the s

stands for scattering, not for Spur) in the late 1930s/early 1940s.

27

equivalence relation and the Planck-Einstein relation, which is why we refer to light-particles as field

particles.

The choice of a unique reference frame also takes care of the conservation of linear momentum because

it splits the E0t’/ħ factor over its Evt/ħ and px components (see the section on the relativistic invariance

of the wavefunction in this paper).

What about the conservation of the total amount of angular momentum (spin)?

67

This data (or

information) is implicit in the particle wavefunctions as well

68

and so we are left with the charge

conservation law only:

• Neutral matter-particles (e.g. neutrons) consist of an equal number of opposite charges (the

neutron, for example, consists of a positive and negative charge).

• Light-particles carry no charge but they do carry energy and (linear and angular) momentum.

Photons carry electromagnetic energy/momentum, and neutrinos carry nuclear (strong)

energy/momentum.

This, then, is the only constraint on the ϕ|S|ψ particle reaction: the total charge of the matter-

particles going in must equal the total charge of the matter-particles going out.

69

There is no need for

baryon, lepton, or strangeness conservation laws.

In short, we may complement Feynman’s Unworldliness equation with Dirac’s statement or definition of

events as described above:

ϕj|Sj|ψj for all events j (j = 1, 2,… )

Feynman’s U = 0 equation describes the laws of the Universe, which govern the events, happening

simultaneously and/or in succession. All events that are possible, are real, and have been listed (in

rather excruciating detail) by the Particle Data Group.

The statements above are not a formula for happiness. Happiness is a state of mind which emerges from

regularly taking the best course of action when taking care of personal (human) business, which

contributes to good habits. Taking care of others first (i.e. developing a sense of duty) is the best way to

take care of oneself.

Any irreversibility of actions or processes must be rooted in the asymmetry in the wavefunction: the

imagery part lags the real part by a right angle. Mankind should, perhaps, reverse its course of action.

67

We obviously should not be measuring spin here in terms of up or down but quantify the exact amount of spin as

well as keep track of all directions in space.

68

We can apply the quantum-mechanical angular momentum to the wavefunction. More in general, the quantum-

mechanical operators gives us all of the physical characteristics that are implicit in the wavefunction that describes

the particle.

69

For matter-antimatter pair creation/annihilation, see our paper on this topic.

28

Annex IV: A complete description of an event

The illustrations below (taken from Feynman, III-11-5) show the basic decay reactions of the neutral

kaon once again:

⎯ A K0 will decay into a + and a −, which decay into a muon (+ or −) and, therefore (ultimately),

into a positron or an electron (e+/e−).

70

⎯ A K0 particle

71

will interact with a hydrogen nucleus (proton) and produce a 0 and +. The rather

massive (neutral) lambda baryon

72

(about 1115.6 MeV/c2) will decay into a − and a proton (p).

K0 + p → 0 + + → − + p + +

Feynman focuses very much on reactions that do not take place, such as this one:

K0 + p → 0 + +

We wonder why: if you take the antimatter-counterpart of the K0 particle, you should take the

antimatter counterpart of the proton too, and the K0 + p → 0 + + should be possible. Antimatter has

an opposite spacetime signature and is, therefore, not just matter with opposite charge.

73

The

wavefunction models spin and physicists do not doubt charge conservation, so that is all covered.

70

Because we live in a matter-world, positrons annihilate with a nearby electron.

71

We use an underbar (K) instead of an overbar to denote the antimatter counterparts of a particle out of laziness

(we do not want to use the equation editor all of the time).

72

A baryon is supposed to consist of an odd number of quarks, usually three.

73

We think of the kaon as consisting of two opposite charges. This is not unlike the (much more stable) neutron,

whose magnetic moment can only be explained by a asymmetric combination of a proton-like charge and a

(nuclear) electron-like charge. The antimatter counterpart of a neutron, therefore, consists of the antimatter-

counterparts of the proton and the electron (antiproton and positron). This explains why an antineutron and an

antikaon are not just neutrons/kaons with reversed spin. There is no need to invoke ‘strangeness’ here.

29

Let us focus on the K0 + p → 0 + + event. We have a ϕ| state in, a |ψ state out and we should relate

both through the scattering-matrix

74

: ϕ|S|ψ

This statement raises the question: ϕ|S|ψ is equal to what? It is equal to a probability or – to be more

precise – to a complex number = Aei(+i)t = ϕ|S|ψ that corresponds to the probability P = ||2. This

probability is constant if it is equal to 1, i.e. when we are not considering other possibilities.

75

We will take ϕ| and |ψ to be an 1n vector and 1m vector, respectively. The dimension of the

scattering matrix is, therefore, nm.

76

We write:

Let us write it all out by substituting K0, p, 0 and + for their complex representation

77

:

The and are empirical (i.e. measured) values: is the (initial) energy (per cycle) of the particle going

in (and coming out of) the reaction, and is the inverse of the mean lifetime of the (un)stable particle (

= 1/).

In short, we have the laws of the Universe and we have events, which are happening (and, therefore, are

real (ϕj|Sj|ψj = 1).

ϕj|Sj|ψj = 1 for all events j (j = 1, 2,… )

The hf vector dot product reduces to the scalar hf because force structures are in accordance with

special relativity theory (STR): the magnetic field vector is perpendicular to the electric field vector, and

74

It is interesting, historically speaking, that John Archibald Wheeler (whom we know from the mass-without-mass

models of elementary particles) and Erwin Schrödinger independently developed the idea of the S-matrix (the s

stands for scattering, not for Spur) in the late 1930s/early 1940s.

75

We may always consider the antimatter counterpart of the reaction, of course!

76

Note that we do not respect the usual convention of reading bra-kets from right to left: we see no use for it, and

Feynman also uses the usual left-to-right notation in his introduction to the matrix algebra of quantum mechanics

(Feynman, III-5). Note that we take issue with the supposed 720-degree symmetries which Feynman supposedly

proves for spin-1/2 particles. For a more general interpretation of the wavefunction and state vectors, see our

papers on Feynman’s ‘time-machine’ and the geometry of the wavefunction.

77

The minus sign for the antikaon wavefunction has to do with its antimatter nature. We will discuss that in the

next section (Annex V).

30

its magnitude lags it by 90 degrees, and the nuclear equivalent should do the same (we have no reason

to assume something else).

78

All that is left is modeling interdependencies between events so as to arrive at a complete

representation of how the Universe evolves (we cannot simply superpose events if they are

interdependent

79

).

78

See our paper on ontology and physics. The lagging aspect of the imaginary aspect is why we write the

elementary wavefunction as Aeit rather than as Ae−it.

79

An event has a start and an end time and is, therefore, associated with a time interval. All events with

overlapping time intervals are, in principle, interdependent. See Feynman’s treatment of dependent/independent

particles.

31

Annex V: The nature of anti-matter

The sign of the imaginary unit in the wavefunction distinguishes the spin direction (up or down). We

should, effectively, not treat −1 as a common phase factor in the argument of the wavefunction.

80

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.

81

Combining the + and − sign for the imaginary unit with the direction of travel, we get four mutually

exclusive particle wavefunctions (think of an electron here):

Spin and direction of travel

Spin up (J = +ħ/2)

Spin down (J = −ħ/2)

Positive x-direction

= Aexp[i(kx−t)]

* = Aexp[−i(kx−t)] = Aexp[i(t−kx)]

Negative x-direction

χ = Aexp[−i(kx+t)] = Aexp[i(t−kx)]

χ* = Aexp[i(kx+t)]

We may now combine these four possibilities with the properties of anti-matter. If the imaginary part of

the wavefunction of matter-particles lags the real part with 90 degrees

82

, then the imaginary part must

precede it by 90 degrees for antimatter-particles.

Not only does this explain the different nature of matter and antimatter, but it also explains matter-

antimatter pair production and/or annihilation. If matter is right-handed (travels counterclockwise),

then antimatter is left-handed (travels clockwise).

83

To model this mathematically, we must distinguish the algebraic identity cos(−) = −cos() from the

geometric identity r = −(−r). We may also treat the argument or the angle as a vector, i.e. a quantity

with a direction. Hence, we write cos() = −cos(−), which implies cos() cos(−): they are each other’s

80

Mainstream physicists effectively 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.

81

The quantum-mechanical argument is technical, and I did not reproduce it in this book. I encourage the reader

to glance through it, though. See: Euler’s Wavefunction: The Double Life of – 1. Note that the e+iπ e−iπ expression

may look like horror to a mathematician! However, if he or she has a bit of a sense for geometry and the difference

between identity and equivalence relations, there should be no surprise. If you are an amateur physicist, you

should be excited: it is, effectively, the secret key to unlocking the so-called mystery of quantum mechanics.

Remember Aquinas’ warning: quia parvus error in principio magnus est in fine. A small error in the beginning can

lead to great errors in the conclusions, and we think of this as a rather serious error in the beginning!

82

To visualize this, one can think of the electric and magnetic field vectors, but remember the nuclear force has an

equivalent structure.

83

This definition depend, of course, on your perspective on the line of travel, i.e. on the observer’s line of sight. For

an introduction to symmetries and asymmetries in Nature, see Feynman’s Lectures, Vol. III, Chapter 17 and Vol. I,

Chapter 52.

32

opposite! Likewise, we will take the opposite of the imaginary part, i.e. sin() = −sin(−). Note that these

equations only make sense when thinking of and − as vector quantities, as shown below.

We, therefore, get the following wavefunctions for matter (S) and antimatter (S), respectively

84

:

The question is now: how does this apply to transients or resonances (unstable particles)? The

antimatter particle must decay, so the e−t factor must be there too! It turns out that this works out

alright:

We, therefore, wrote the equations for the interaction between the neutral antikaon (K0) and the

proton (p) as we did (see Annex IV):

So, this is all very nice. Matter and antimatter are each other opposite: the wavefunctions Aei and −Aei

add up to zero, and they correspond to opposite forces too! Of course, we also have lightparticles: there

should be antiphotons and antineutrinos too, right? Right. All is consistent: Occam is happy.

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?

84

We use underbars instead of overbars in the text out of laziness: we do not always want to use the formula

editor.

33

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 electron⎯the 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.

85

We, therefore, think of charge as the essence of the Universe. Everything else is sheer geometry.

The Greek philosophers were right all along! Panta rhei: everything flows!

Note on the meaning of i2 = −1 (two rotations by i):

Another way of thinking about complex numbers as vectors is to think of their real and imaginary parts,

and apply simple trigonometry. We can then write:

• Matter: ei = cos + isin = cos + cos( - /2) : imaginary part lags the real part by 90 degrees.

• Antimatter: −ei = −cos − isin = −cos + cos( + /2) : imaginary part precedes by 90 degrees

That is why we refer to antimatter as having an opposite spacetime signature: we think of

positive/negative spin (the sign of the imaginary unit (i) in the Aeit wavefunction) as being measured

from the positive real axis while, for anti-matter, we measure it with reference to the negative real axis.

This, of course, only makes sense if we think of time going in one direction only, which it does!

We mentioned the matter and antimatter wavefunctions add up to zero: Aeit − Aeit = 0, but the

associated energies or energy densities

86

do not, of course:

|Aeit|2 + |−Aeit|2 = |A|2|eit|2 + |−A|2|−eit|2 = 2A2

85

See our paper on matter/antimatter pair production and annihilation.

86

See our paper on the geometry and meaning of the wavefunction.

34

Annex VI: A new matrix algebra?

Antimatter has an opposite spacetime signature: we think of positive/negative spin (the sign of the

imaginary unit (i) in the Aeit wavefunction) as being measured from the positive real axis while, for

anti-matter, we measure it with reference to the negative real axis. This, of course, only makes sense if

we think of time going in one direction only, which it does!

87

The matter and antimatter wavefunctions add up to zero if the coefficients A are the same: Aeit − Aeit

= 0. However, the associated energies or energy densities

88

do not, of course, which explains the

conservation of energy in matter/antimatter pair creation/annihilation processes:

|Aeit|2 + |−Aeit|2 = |A|2|eit|2 + |−A|2|−eit|2 = 2A2

We can write the wavefunctions of stable/unstable matter/antimatter particles as U/U and S/S

respectively:

1.

2.

3.

4.

Let us analyze a typical kaon decay event:

K0 + p → 0 + + → − + p + +

We can represent this event using the scattering matrix (S):

Writing it all out, yields the following two equations:

87

See our paper on ontology and physics.

88

See our paper on the geometry and meaning of the wavefunction.

35

We have four coefficients and only two equations: the system is, therefore, not fully determined. Let us

think this through.

We have a ϕ| state in, a |ψ state out and we can relate both through the S-matrix

89

:

ϕ|S|ψ

The deterministic worldview implies reversibility and we should, therefore, also be able to go back from

the final state |ψ (the ket in Dirac’s bra-ket notation) to the initial state ϕ|.

The inverse of a (square) matrix

90

is given by:

[Ajk] is the matrix of cofactors Ajk, and [Ajk]T is its transpose. The assumption, of course, is that A is

invertible and that it is, therefore, not singular (or degenerate).

So we now have four equations and four coefficients. Hence, the system should be solvable.

However, we know not all reactions are reversible: C- and P-symmetry (charge and parity) is not always

conserved, and even the combined CP-symmetry is not always conserved. However, we do know the

combined CPT-symmetry is conserved, always.

We are not sure, but we think the problem is solved by our interpretation of antimatter, and our use of

a negative wavefunction to represent them. Among other things, it shows why neutral particles such as

neutrons or kaons (which we think of composite particles consisting of opposite charges) are not

identical to their antimatter counterpart. It also explains dark matter/energy: the antiphotons and

antineutrons that come out of low- and/or high-energy reactions would normally not interact with

89

It is interesting, historically speaking, that John Archibald Wheeler (whom we know from the mass-without-mass

models of elementary particles) and Erwin Schrödinger independently developed the idea of the S-matrix (the s

stands for scattering, not for Spur) in the late 1930s/early 1940s. Note that we do not respect Dirac’s right-to-left

reading of the ϕ|S|ψ expression. We see no good reason whatsoever to do so. Note that we use an underbar

(instead of an overbar) for antimatter in ordinary writing (i.e. when not using the formula editor).

90

We are not sure what to do with non-square matrices (m-by-n matrices for which m n). However, we note such

matrices may have a left or right inverse, and we should be able to work with that. If A is m-by-n and the rank of A

is equal to n, then A has a left inverse: an n-by-m matrix B such that BA = I. If A has rank m, then it has a right

inverse: an n-by-m matrix B such that AB = I.

36

matter. Hence, even if the K0 + p → 0 + + → − + p + + would not be reversible, its antimatter

counterpart should be real:

K0 + p → 0 + + → − + p + +

Assuming all else is equal (spin, momentum, and energy), this yields a similar matrix equation:

Writing it all out, we get two additional equations:

We think the considerations above should fully determine the system (four coefficients, four

equations), but we admit we need to study it all some more.

One thing is for sure: Dirac’s Principles of Quantum Mechanics need a thorough rewrite. We think he got

lost in incomplete and/or misinterpreted math: in light of his rather courageous use of Occam’s Razor

principle (prediction of the proton), he should have been more consistent.