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Lectures on Physics Chapter VII :

The nuclear force and quaternion math

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

13 April 2021 (revised on 30 September 2022)

The special problem we try to get at with these lectures is to maintain the interest of the very enthusiastic and

rather smart people trying to understand physics. They have heard a lot about how interesting and exciting physics

is—the theory of relativity, quantum mechanics, and other modern ideas—and spend many years studying

textbooks or following online courses. Many are discouraged because there are really very few grand, new,

modern ideas presented to them. Also, when they ask too many questions, they are usually told no one really

understands or, worse, to just shut up and calculate. Hence, we were wondering whether or not we can make a

course which would save them by maintaining their enthusiasm. This paper is the seventh chapter of such (draft)

course.

i

Abstract

This paper is a basic introduction to quaternion math. In fact, we do not talk much about it at all. We

just show how our electron and proton models – and their wavefunction representations – can be

usefully phrased using Hamilton’s set of imaginary units (i, j and k), which we interpret as rotation

operators (one for each possible plane of rotation: the xy-, yz-, and xz-plane). In addition, we added

some remarks on interpretation which, perhaps, were not so clear in our other papers or on which we –

quite simply – have some more subtle or nuanced ideas now.

i

We paraphrase Feynman’s introduction to his Lectures on Physics here. Quoting one of Feynman’s other lines, we

would say: all is intended for fun, of course! No one should feel offended (we are thinking of someone in particular

here now).

For those who track the history and versions of this paper, we did not fundamentally revise the body of this paper

but added more detailed explanations and remarks (reflecting the new nuance on some of our earlier ideas, such

as writing Planck’s quantum of action as a vector rather than a scalar quantity, or the possibility that charge might

have a fractal structure). We also recategorized it as one of our more basic papers in what we refer to as the K-12

level physics project, which we effectively think of as basic lectures on physics for amateur physicists who do not

have any formal qualifications in the field, but who do master some of the more advanced math one needs to

appreciate the beauty of both classical as well as quantum physics. We also added an annex with some

philosophical thoughts, because we were dragged into them in the course of rewriting this paper and so we

thought it would not be a bad idea to take some older (admittedly, very simple) material and put it here.

1

The nuclear force and quaternion math

Contents

The ring current model ................................................................................................................................. 2

Basic algebra ............................................................................................................................................. 2

The elementary ring current (electron) in motion.................................................................................... 5

The wave function and Schrödinger’s wave equation .............................................................................. 6

Force and field strengths .............................................................................................................................. 6

Fields, orbital motion, and imaginary units .................................................................................................. 8

The nuclear oscillation .................................................................................................................................. 9

General principles ..................................................................................................................................... 9

Calculations ............................................................................................................................................. 10

The asymmetry between positive and negative charge ......................................................................... 11

Particle representations using wavefunctions and rotational operators ................................................... 13

Recap ....................................................................................................................................................... 13

2D versus 3D oscillations ........................................................................................................................ 13

Hamilton’s rotational operators and scattering matrices ...................................................................... 14

Limitations of Hamiltonian’s quaternion math ....................................................................................... 16

Potential wells and tunneling ..................................................................................................................... 18

Philosophical annex .................................................................................................................................... 19

The essence of time ................................................................................................................................ 19

Theories of Everything: U = 0 .................................................................................................................. 21

The meaning of the fine-structure constant ........................................................................................... 23

Why the (absolute) velocity of light makes sense .................................................................................. 25

References .................................................................................................................................................. 29

2

The nuclear force and quaternion math

The ring current model

Basic algebra

The ring current or Zitterbewegung model of an electron (Parson, 1905; Breit, 1928, Schrödinger, 1930;

Dirac, 1933; Hestenes, 2008, 2019) interprets the elementary wavefunction as a radius

or position vector:

ψ = r = a·e±iθ = a·[cos(±θ) + i · sin(±θ)]

We use boldface to denote the sine and cosine vectors: the 90-degree phase shift (sin(θ + π/2) = cosθ)

amounts to a rotation in 3D space which can be represented by using the imaginary unit (i) as a vector

(rotational) operator: sinθ = i·cosθ. Taking the derivative of the wavefunction with respect to time,

yields the (orbital or tangential) velocity (dr/dt = v), and the second-order derivative gives us the

(centripetal) acceleration vector (d2r/dt2 = a). The plus/minus sign of the argument of the wavefunction

gives us the direction of spin, and we may, perhaps, add a plus/minus sign to the wavefunction as a

whole to model matter and antimatter, respectively (the latter assertion remains very speculative

though).

ii

The point to note is this: the sign of the imaginary unit () indicates the direction of spin and,

interpreting 1 and −1 as complex numbers (cf. the boldface notation), we do not treat as a common

phase factor: e+iπ equals e−iπ in algebra (because both equal −1) but not in geometry! To put it simply,

when you go from here to there (+1 to −1, in this particular case), it may not matter how you get there

from a mathematical point of view but in physics, it surely does!

iii

ii

The force is nothing but the Lorentz force and can, therefore, always be broken down in an electric and a

magnetic field vector (or, for the nuclear force, their equivalents). The direction of the magnetic field vector is

given by a vector cross-product and is, therefore, given by a right-hand rule. We think antimatter is governed by

left-hand rules. When using a negative coefficient, the wavefunction will incorporate such left-hand rules. See our

(speculative) paper on cosmological hypotheses., which presents our hypothesis on the nature of anti-matter

(which we advanced in quite a few papers already) quite concisely. As for our remark on the nuclear force having

an equivalent structure, see our paper on the nuclear force hypothesis for a rather complete exploration of it.

iii

See our paper on Euler’s wavefunction and the double life of

−

1, October 2018. This paper is one of our very early

papers – a time during which we developed early intuitions – and we were not publishing on RG then. We also

develop much of the same ideas in a more recent RG paper on the geometry of the wavefunction. We basically

take Feynman’s argument on base transformations apart.

3

One orbital cycle packs Planck’s quantum of action, which we can write either as the product of the

energy (E) and the cycle time (T), or the momentum (p) of the charge times the distance travelled, which

is the circumference of the loop λ in the inertial frame of reference

iv

:

h = E·T = p·λ ħ = E = pa

Notes: The reader may want us to elaborate some more on the uncertainty principle. We hesitate to do because we think this

(in)famous principle obscures rather than clarifies matters. The indeterminacy is nothing ontological. Two factors are

simultaneously at play:

(1) We do not know the initial condition of the system and we can, therefore, not determine the exact state of the system at

a specific point in time.

(2) Also, the pointlike charge in elementary particles is photon-like: its rest mass is zero, and any mass it acquires is relativistic

only. Its orbital velocity, therefore, equals lightspeed, which makes it impossible to determine its position at any point in

time.

Of course, the latter remark only applies to the wavefunction of elementary oscillations. Having said that, electrons in atomic

orbitals also achieve relativistic velocities, even if they are only a fraction of lightspeed. In any case, the point is this: the

uncertainty in the position is because we cannot measure its precise position. Hence, we are talking statistical uncertainty:

nothing ontological. We would, therefore, rather talk of a complementarity principle, instead.

v

However, now that we are here,

let us make the following brief remarks about it:

⎯ Heisenberg’s original formulation of the principle was Δx·Δp ≥ h. It was Kennard’s rather straightforward derivation one

year later which established the modern σx·σp ≥ ħ/2 inequality. Kennard’s formulation comes with a rather straightforward

general mathematical proof

vi

and is, therefore, technically more correct.

⎯ However, it should not lead us to think that physical action (Wirkung) comes in packets of h/2 or ħ/2 rather than h or ħ.

That is nonsensical in our view. To be precise, photon, electron, proton, and other elementary oscillations in Nature pack

one full unit of h = 6.62607015×10-34 N·m·s

vii

, with the use of h or ħ in the h = E·T = p·λ ħ = E = pa depending on the

use of T or λ or their reduced forms ( = T/2π and a = λ).

We add one more reflection. It should be noted that E and T (or = T/2π) are scalars. Hence, the h in the h = E·T equation is,

obviously, a scalar too. In contrast, one may be tempted to think we should write p and, possibly, the radius distance a, as

vectors. One may, therefore, be tempted to write Planck’s quantum of action as a vector too: should we write h or ħ (boldface)

rather h or ħ? We will leave it to the reader to explore that question. Our opinion on it is rather straightforward:

⎯ The direction of the momentum rotates with position and, hence, we could effectively rewrite h = p·λ as λ = h/p. (we do so

in a diagram that we used here and there). However, as the momentum rotates with position, the vectorial variant of

Planck’s quantum of action would do the same and would have to have the same direction so as to make the equation

work. Hence, no vector notation is needed. Also, no direction can be associated with an orbital wavelength (which is what

the λ in these equations usually stands for: it is nothing but a circumference). When everything is said and done, it appears

like a simple choice. In any case, if one would like to write p as a vector whose direction varies in time, then one would

iv

We use the symbol here to denote the cycle time measured in radians. It is not to be confused with the mean

lifetime of a particle (as used in the wavefunction below). Unstable particles can be modeled adding a transient

(real) exponential:

.

v

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

vi

See, for example, the concise proof of Kennard’s inequality in the Wikipedia article on the uncertainty principle.

vii

This is an exact value. Planck’s quantum of action is a precisely defined constant of Nature. As such, it is part of

the new set of physical constants since the 2019 revision of the system of SI units. We feel this revision is complete

now (we think of Occam’s razor principle when define completeness here) and defines all of physics: all of physics

(classical and quantum mechanics) is encapsulated in Maxwell’s equations and the Planck-Einstein relation!

4

have to write the radius vector as a vector which varies in time too so as to be consistent. The two are always

perpendicular and, hence, their product would be equal to a scalar anyway.

⎯ The more fundamental objection we now have against writing Planck’s constant as a vector quantity, is its physical

dimension: it is, effectively, a product of three factors: newton × meter × second. This suggests it would be difficult to

associate a specific direction with it: time has a direction (it goes one way only), but it is not spatial. We add a diagram

here which we find useful to appreciate and relate all of the relevant physical dimensions.

The basic conclusion which the reader should note is this: uncertainty in quantum physics is a case of statistical uncertainty. It is

nothing ontological. As such, it is pretty much the same as classical indeterminacy, even if the system is fully determined.

The Lorentz transformation formulas in the diagram above remind us of the relativity of all of the fundamental concepts we

use: force, energy, momentum, and space and time itself. Only Nature’s fundamental constants are absolute. The concepts and

constants then combine in what is – or should be – a representation of what we vaguely refer to as reality.

We must get on with the matter at hand now: let us, therefore, move to the next section of this paper.

5

The elementary ring current (electron) in motion

When considering an electron in motion, a linear component combines with the orbital velocity to yield

a radius which varies relativistically (Figure 1, Vassallo & Di Tommaso, 2019). The argument of the

wavefunction incorporates both special as well as general relativity theory.

viii

Figure 1: Relativistic decrease of the electron radius

The ring current model distinguishes the idea of an electron and a (pointlike) charge: the electron is

charge in motion. The small anomaly in the magnetic moment of an electron can easily be explained by

assuming the photon-like charge inside of the electron has some small (but not infinitesimally small)

spatial dimension itself.

ix

In fact, its structure might be fractal and, because this is our seventh lecture,

we do not hesitate to lay out the difficulties in interpreting these things and explain to you why we think

that might be the case. The reasoning is as follows.

The fine-structure constant (α ≈ 0.0073 ≈ 1/137) is the scaling factor which distinguishes the classical,

Compton and Bohr radii of the electron: 2.818 fm ≈ α·0.386 pm ≈ α·52.917 pm. This series suggests we

might be talking some fractal structure, indeed! Such structure may or may not be finite: we associate

the classical electron radius with the radius of the poinlike charge, but perhaps we can drill down

further. You may wonder why would we do that? We have all the answers now, right?

No. We do not. We entertain the possibility of another layer in what we should probably refer to as the

structure of charge because of the rather large size of the classical electron radius (which we think of as

the size of the charge inside of the electron in this model) as compared to the size of a proton. Indeed,

the proton radius is about 0.83-0.84 fm

x

, so that is less than 1/3 of 2.818 fm. That begs a rather plain

viii

We demonstrated this ad nauseam because we think it is an important point: it confirms the intuition that the

wavefunction represents something real (something that does not depend on our reference frame). See, for

example, our paper on de Broglie’s matter-wave.

ix

See our explanation in our very first lecture (quantum behavior).

x

For the detail of the calculations, and also how they match to the most recent measurements of the proton

radius, see our paper on nuclear oscillations and neutrinos. The key calculation is this:

This value is well within the results of JLAB’s PRad measurements, which measured the proton radius to be equal

to rp = 0.831 ± 0.007stat ± 0.012syst fm. It is also within the 2018 CODATA value for the proton radius, which is equal

to rp = 0.8414 ± 0.0019 fm. We, therefore, feel this calculation is spot on. We will come back to it in this paper.

6

common-sense question: how could a positive equivalent of the pointlike charge inside of an electron

possibly fit into a proton? The assumption of a fractal charge structure might provide the answer here.

The wave function and Schrödinger’s wave equation

As part of this introduction to the ring current model, we must, perhaps, also say a few words about

wave equations – and, more in particular, Schrödinger’s wave equation. We will be brief because it is

not very relevant to this paper. However, for the sake of completeness, let us remind ourselves of the

basics here. Schrödinger’s wave equation for a single electron in free space (no other charges and,

therefore, no potential energy terms) can be written as:

The effective mass (which is half the total mass of the electron) is the relativistic mass of the pointlike

charge as it whizzes around at lightspeed. Dirac (1933) confused it with some motion of the electron

itself, further compounding de Broglie’s mistaken interpretation of the matter-wave as

a linear oscillation: the ring current model assumes orbital rather than linear motion. We have twice the

electron mass in Schrödinger’s wave equation for electron orbitals: he made not one but two mistakes

(not accounting for spin angular momentum, and adding the effective mass of two electrons to get a

mass factor), but they make the wave equation come out all right.

The point to note here is that the imaginary unit in Schrödinger’s equation also acts as a rotation

operator. Schrödinger’s wave equation, therefore, models (planar) orbital energy states (including

subshells). The physical dimension of ħ/m is m2/s which, combined with the 1/m2 dimension of the

2 operator, yields the required 1/s dimension on both sides of Schrödinger’s equation.

xi

Force and field strengths

Newton’s force law tells us the (centripetal) force which keeps the pointlike charge in its orbit equals the

relativistic mass of the pointlike charge times its (centripetal) acceleration: F = ma. The relativistic

correct expression of Newton’s force law is, of course, F = dp/dt = d(mv)/dt. However, we assume the

pointlike charge has zero rest mass: all of its mass is relativistic, and it is what Feynman referred to as

the effective mass of the charge. Furthermore, the energy equipartition theorem tells us half of the mass

of the electron must be kinetic, while the other half is (potential) field energy. We can, therefore,

calculate the force in two ways:

xi

For a detailed exploration of the physical dimensions in all of the hocus-pocus associated with Schrödinger’s wave

equation, see our paper on the language of (quantum) math. Such dimensional is quite a headache but, once you

get it right, you can play with it, which is what we do when calculating things like the velocity and acceleration

vectors in the next section of this paper. In some of our earlier writings, we thought the wavefunction itself must

have some physical dimension: if it is a position vector, then it would be the meter, right? However, if that

dimension (m) would be there, then it would come with its coefficient (the radius or distance from the center), and

not in Euler’s function (e±iθ). Also, we entertained the idea that the imaginary unit itself might have an s/m

dimension because it folds one axis (say, the distance axis) into another (say, the time axis). Those are early

mistakes. We should probably scrutinize our earlier papers (especially those that we put on viXra.org) and correct

all these things if we have time for this, which we do not: we should explore the future rather than dwell on the

past.

7

1. We can calculate the velocity and acceleration vectors

xii

:

The magnitude of the force must, therefore, be equal to:

2. Energy is force times distance. Assuming the centripetal force is constant, we do not need to integrate

and may simple write

xiii

:

A force at the sub-atomic scale which gives a mass of about 106 gram an acceleration of 1 m/s per

second is rather humongous at the sub-atomic scale. The field strength (force per unit charge) is equally

tremendous:

xii

We recommend the reader to do a dimensional analysis on these derivations as well. Dimensional analysis is one

of the best ways to make sense of complicated equations.

xiii

The reader may be puzzled by the distance over which the force acts being equated to 2a: the circumference of

the loop is equal to = 2a. Why do we equate this to 2a? Is our calculation, perhaps, somewhat off, by a factor 2

or ? One might think, like Richard Feynman (III-2-4), that “this is a kind of dimensional analysis […] and that we

need not trust our answer to within factors like 2, π, etc.” However, we think the expression is correct. The energy

conservation equation for a relativistic oscillator can be written as (see p. 9 of one of our papers on the oscillator

model):

The kx2/2 term is the potential energy (potential energy depends on position, which is the position x along the line

of oscillation in this case) and γmc2 = γ2m0c2 is the kinetic energy here (so it is not the usual classical mv2/2 = mc2/2

factor). The force is expended over the diameter d = 2r (twice the radius). The reader should note we think the

poinlike charge has zero rest mass: all of its mass is relativistic and, hence, the γmc2 expression is easier to

interpret than the γ2m0c2 expression, but the reader can do some graphs and easily interpret this limit of the latter

expression. That is all we will say about the 2, ½ or other factors such as π or 2π factors in equations like this.

Alternatively, the reader may want to consider the factor as a correction to the curvature of the path or – quite

simply – follow Feynman and brush everything off by considering such calculations to possibly be off by such

factors. One quickly gets into philosophical discussions about the true nature of the elementary charge oscillation

(see, for example, our remarks about thinking about the elasticity of spacetime itself in our revised paper on the

nuclear force hypothesis) but that is, most probably, not the way to go.

8

To help you to put this value into perspective, we note the most powerful man-made accelerators reach

field strengths of the order of 109 N/C (1 GV/m) only. Finally, this is a ring current model, and we can,

therefore also calculate the electric current:

This is a standard household-level current and, therefore, an equally enormous value considering the

scales involved. We, therefore, leave it to the reader to consider the credibility of these models. All that

we can say here is that rather tremendous energies are needed to smash protons, neutrons, or small

nuclei into each other and break them up into some unstable temporary configuration which academics

refer to as hypothetical quarks or gluons or other temporary constructs, which then quickly reconfigure

back to into stability.

xiv

Hence, the forces must be equally mindboggling.

xv

We, therefore, feel these

values might make sense.

We might add yet more calculations of the magnetic moment (including an explanation of the small

anomaly, which we attribute to the pointlike charge having a small but not infinitesimally small physical

dimension), but we refer the interested reader to other papers here.

xvi

Also, we will insert some of these

calculations when we talk about the nuclear oscillation in the next section(s) of this paper.

Fields, orbital motion, and imaginary units

The ring current model assumes the existence of an electromagnetic field which keeps the pointlike

charge in its orbit. The centripetal force may, therefore, be modelled as the Lorentz force (F):

F = q·(E + v×B) = q·(E + c×E/c) = q·(E + 1×E) = q·(E + j·E) = (1+ j)·q·E

We use a different imaginary unit here (j instead of i) because the plane in which the magnetic field

vector B is rotating differs from the E- plane. These two planes may be referred to as the xy– and xz-

planes in 3D Cartesian space.

xvii

Animations of circularly polarized electromagnetic waves usually show

the electric field vector only, and animations which show both E and B are usually linearly polarized

waves. We, therefore, reproduce the simplest of images below (Figure 2). The E-plane is the plane of the

paper, and the B-plane is only visible as a line going from 1 to -1 (or from i to -i).

xiv

We are probably a bit aggressive in terms of language here. Let us nuance our statement: we think all of these

hypothetical short-lived matter- or lightlike particles may be form factors in some kind of new scattering matrix

algebra. See our paper on the Zitterbewegung hypothesis and the scattering matrix.

xv

We will soon show the forces inside of a proton or a muon-electron are even more mindboggling because they

add another four or five orders of magnitude in such calculations.

xvi

Unsurprisingly, the order of magnitude of the dimension of the pointlike charge is given by the fine-structure

constant, which relates the classical electron radius to its Compton radius, which we think of as the effective radius

of interference of an electron with (pointlike) photons. See, for example, our first (alternative) lecture on physics.

xvii

The magnetic field vector lags the electric field vector. Hence, the plane is the xz-plane⎯not the yz-plane. We

should note the right-hand rules combines with the conventional definition of a current (flow of positive charge).

Note that we write 1 as a vector (boldface) or a complex number: 1 = 1 + i·0. Depending on mathematical

convention, we might also have written as 1 = 1 + i·0 or 1 = 1 + i·0.

9

Figure 2: Plane of E vector (plane of B vector orthogonal to E plane)

The E and B vectors – taken together – also form a plane. This plane is not static: it changes all the time

as a result of the rotation of both the E and B vectors. Hamilton’s quaternion math lends itself naturally

to represent rotations and rotating planes. We could refer to the plane formed by the E and B vectors as

the E–B plane but, in line with Hamilton’s quaternion algebra, we will refer to it as the k-plane. We,

therefore, have three imaginary units (basic quaternions), which are related one to another as follows:

i2 = –1; j2 = –1; k2 = –1; i·j = k; j·i = –k

The first three rules are the usual ones in complex number math: two successive rotations by 90 degrees

bring one from 1 to –1. The order of multiplication in the other two rules (i·j = k and j·i = –k) gives us not

only the k-plane but also the spin direction. All other rules in regard to quaternions (e.g., i·j·k = –1) can

be derived from the above-mentioned basic quaternion rules.

The nuclear oscillation

General principles

Do we need three imaginary units? Two imaginary units (or rotational directions) will do for modeling

electromagnetic oscillations (E and B field vectors), but not when trying to model nuclear oscillations. To

explain the proton radius, for example, we think of a nuclear oscillation as an orbital oscillation

in three rather than just two dimensions. We, therefore, have two (perpendicular) orbital oscillations,

with the frequency of each of the oscillators given by ω = E/2ħ = mc2/2ħ (energy equipartition theorem).

Each of the two perpendicular oscillations packs one half-unit of ħ only (this also explains the results of

Mach-Zehnder one-photon interference experiments). Such spherical view of a proton fits with packing

models for nucleons and yields the experimentally measured radius of a proton:

The 4 factor here is the one distinguishing the formula for the surface of a sphere (A = 4πr2) from the

surface of a disc (A = πr2). We may now write the proton wavefunction as a combination of two

elementary wavefunctions:

10

The solutions to the wave equation will combine various combinations (products) of i- and j-terms, and

simplifying them will, therefore, require the use of quaternion algebra.

Calculations

We said we would offer some more calculations, which is what we will do here. However, we want to be

brief and we, therefore, will limit our discussion here to a mere summary of the detailed calculations we

presented in our paper on the proton radius. Let us start by presenting the key result of the latest

precise measurements of the proton radius, which were done by the PRad (proton radius) team at

Jefferson Lab (JLab), and which measured the root mean square (rms) charge radius of the proton as:

rp = 0.831 ± 0.007stat ± 0.012syst fm.

xviii

Using the CODATA definition of ħ and the CODATA value for the proton mass, we can do a more precise

calculation for the ap = 4ħ/mpc ≈ 0.84 fm equation above:

This value is not within the 0.831 0.007 fm interval, but it is well within the wider

rp = 0.831 ± 0.007stat ± 0.012syst fm interval. It is also within the 2018 CODATA value for the proton

radius, which is equal to rp = 0.8414 ± 0.0019 fm.

xix

We, therefore, think our theoretical proton radius is,

essentially, correct. We must make the following remarks here:

⎯ We arrived at this calculation, the detail of which is contained in the above-referenced paper on the

proton radius, by a thorough reflection on the (measured and theoretical) magnetic moment of the

proton and – more importantly – on the precession which a magnetic particle such as the proton

then must have in a magnetic field.

xx

⎯ Our reflections on precession lead us to insert 2 factor. To be precise, we use a μL = 2qeħ/mp

2.02… J/T value for the magnetic proton which, we argue, differs from the CODATA value with a 2

xviii

See: https://www.jlab.org/prad/collaboration.html. We shared our theoretical model of the proton with Prof.

Dr. Randolf Pohl (whom we highly regard as one of the geniuses behind the 2019 revision of SI units) and the PRad

team. However, we did not receive any substantial comments so far, except for the PRad spokesperson (Prof. Dr.

Ashot Gasparan) confirming the Standard Model does not have any explanation for the proton radius from first

principles, which encouraged us to continue our research as an amateur physicist. In contrast, Prof. Dr. Randolf

Pohl was more skeptical, and suggested the concise calculations come across as numerological only. We hope our

rising RI score might help to make him change his mind.

xix

We quickly tried to verify the history behind the new CODATA value but could not find anything in terms of

methodology. We will do some more search in the NIST bibliographies.

xx

We understand the PRad experiment was, essentially, an e-p scattering experiment and, hence, our reflections

on the proton’s magnetic moment and the precession of the proton ring current do not apply to the experiment

itself. In fact, we think that explains the discrepancy on the rough 0.83 and 0.84 fm measurements. In addition, we

must assume that, just like the electron, that the proton’s magnetic moment must have some anomaly too: if the

photon-like (negative) charge in an electron has some small but non-zero dimension, then it is only logical to

assume the oscillating (positive) charge inside of a proton will have some incredibly small but non-zero dimension

too. Whether or not it is also of the order of the fine-structure constant, and how that would affect our

calculations, is a question we did not examine any further.

11

factor because of precession. This probably comes across as an ad hoc shortcut to the result, but we

think the numerical result is too precise. Hence, we do not think it is too good to be true: if a

number is spot on in a calculation, it must be correct, right?

xxi

⎯ The problem, which is one that we readily admit, is that the formula for the magnetic moment of an

electron (μe = qeħ/2me 9.274 J/T) gives us the CODATA value (apart from the anomaly, of course)

without the need for any correction factor because of precession. Of course, if an electron is some

ring current as well, then it must precess as well.

⎯ The only reason we can think of is that NIST researchers never really thought about any correction

for precession when reporting on proton radius, magnetic moment, and other proton

measurements and – as they do – distill their best point estimate (including confidence interval) out

of all of these experiments. We looked on the NIST site but, frankly, we could not find much in terms

of methodology. We sent an email to the NIST Public Affairs section with a request to guide us to the

necessary materials in this regard. Annex I to our paper with the proton radius calculations offers a

full discussion of the perceived issue and, hence, we must refer the reader there.

[…]

The added detail and the remarks above may not improve on the readability of this paper: the first

version of it was much shorter. However, we must soldier on and, therefore, we will elaborate a bit on

what, at first, comes across as some kind of fundamental asymmetry in the structure of an electron and

a proton. We will then go to the most important part of this paper: why quaternion math (three

rotational operators rather than just one or two) might or might not be useful.

The asymmetry between positive and negative charge

The electron also comes in a more massive but unstable version: the muon-electron.

xxii

The muon energy

is about 105.66 MeV, so that is about 207 times the electron energy. Its lifetime is much shorter than

that of a free neutron

xxiii

but longer than that of other unstable particles: about 2.2 microseconds (10−6

s). Now that is fairly long as compared to other non-stable particles – all is relative! – and that may

explain why we also get a sensible result when using the Planck-Einstein relation to calculate its

frequency and radius.

xxiv

xxi

As the reader will note from our remark on Dr. Pohl dismissing our results as ‘numerological only’ (see footnote

xviii), we are a bit upset about it. We, therefore, invite our readers here to examine the question a bit more in

depth and, if any substantial error in our logic would be found, to tell us where it is – not approximately but

precisely.

xxii

You may also have heard about the tau-electron but that is just a resonance with an extremely short lifetime, so

the Planck-Einstein relation does not apply: it is not an equilibrium state. To be precise, the energy of the tau

electron (or tau-particle as it is more commonly referred to

xxii

) is about 1776 MeV, so that is almost 3,500 times the

electron mass. Its lifetime, in contrast, is extremely short: 2.910−13 s only. We think the conceptualization of both

the muon- as well as the tau-electron in terms of particle generations is unproductive: stable and unstable

particles are, generally speaking, very different animals!

xxiii

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

xxiv

This presumed longevity of the muon-electron should not be exaggerated, however: the mean lifetime of

charged pions, for example, is about 26 nanoseconds (10−9 s), so that is only 85 times less.

12

Indeed, the CODATA value for the Compton wavelength of the muon is 1.17344411010−14 m

0.00000002610−14 m, and if you divide this by 2 to get a radius instead of a wavelength, you get the

same value: about 1.8710−15 m. Hence, our oscillator model seems to work for a muon as well! Why,

then, is it not stable? We think it is because the oscillation is almost on, but not quite.

xxv

In contrast, the

exercise for the tau-electron does not yield such sensible result: the theoretical a = ħ/mc radius does not

match the CODATA value.

xxvi

We think this confirms our interpretation of the Planck-Einstein relation as

modelling stable particles. We can, finally, also use our model to calculate the centripetal force which

must keep the charge in its orbit for the muon-electron and the ratio of this force for the electron and the

muon:

If a force of 0.106 N is pretty humongous, then a force that is 42,753 times as strong, may surely be called

a strong force, right? What about the force inside of a proton? The proton mass is about 8.88 times that

of the muon, and it is about 2.22 times smaller. Once again, we get this strange 1/4 factor. The Planck-

Einstein relation gives us a frequency, but it also gives us the angular momentum of a (stable) particle.

Hence, a proton is also definitely not some more massive or stable antimatter version of a muon-electron.

A force calculation using the same logic yields the following force magnitude:

Now, these force values (4,532 N for a muon and 89,349 N for a proton) may or may not make sense to

you

xxvii

, but the point that we want to make here is this: the proton is the proton, and there is no muon-

like (more) massive (semi-stable) version of the proton! This leads us to think that the negative and

positive charge are not just each other’s opposite. Matter-antimatter are each other’s opposite –

literally – but positive and negative charge are not: the negative charge appears to be fractal (cf. the

fine-structure constant as a scaling constant, relating classical, Compton and Bohr radius of electrons),

but the proton does not. Of course, in the antimatter world, roles are reversed, and it is governed by

xxv

The reader can verify this by re-calculating the Compton wavelength from the radius we obtain and the exact

CODATA values (with or without the last digits for the uncertainties) for the constants and variables. The reader

will see that the value thus obtained falls within the uncertainty interval of the CODATA value for the Compton

wavelength. We will let the reader think about this result and its meaning as part of the exercise.

xxvi

CODATA/NIST values for the properties of the tau-electron can be found here: https://physics.nist.gov/cgi-

bin/cuu/Results?search_for=tau. To go from wavelength to radius and vice versa, one should divide or multiply by

2π.

xxvii

A few back-of-the-envelope calculations reveal we should not be too worried we are modelling a black hole

here. The more informed reader may, for example, calculate the Schwarzschild radius of a proton to verify we are,

effectively, talking about entirely different orders of magnitude here.

13

left-hand rules rather than right-hand rules (think, for example, of the phase of the magnetic field vector

leading rather than lagging the phase of the electric vector).

Particle representations using wavefunctions and rotational operators

Recap

Let us recap the basics of what we presented so far. We have a ring current or mass-without-mass

model of elementary particles which considers particles as harmonic oscillations whose total energy – at

any moment (KE + PE) or integrated 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

xxviii

)

velocity, as we did above.

The model gives a sensible explanation of the quantization of spacetime. Particles are effectively

interpreted as 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 = nħ = nhf = nh/T. We do

not model particles as wavepackets but as definite orbital oscillations of the (electric) charge.

xxix

We

may, therefore, write the wavefunction of stable particles as:

Applying Occam’s Razor Principle (i.e., matching all mathematical possibilities in the expression with

experimentally verified (physical) realities), we think of the sign of the coefficient A as modeling

matter/antimatter, while the sign of the complex exponent (iEt/ħ) captures the spin direction of

matter/antimatter particles.

xxx

The wavefunction of unstable particles (transients) involves an additional

decay factor :

Light-particles differ from (stable) 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.

2D versus 3D oscillations

The proton model introduced a 3D oscillation, as opposed to the 2D models of the electron (and the

muon-electron) that we briefly presented above. Indeed, to explain measurable properties such as

radius and magnetic moment of a proton, we must introduce the idea of two (perpendicular) orbital

oscillations, with the frequency of each of the oscillators given by ω = E/2ħ = mc2/2ħ (energy

equipartition theorem). Each of the two perpendicular oscillations packs one half-unit of ħ only (this also

xxviii

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.

xxix

Both electromagnetic and nuclear forces act on the electric charge and we, therefore, will simply refer to the

electric charge as charge.

xxx

See Annex IV, V and VI of our paper on ontology and physics.

14

explains the results of Mach-Zehnder one-photon interference experiments). Such spherical view of a

proton fits with packing models for nucleons and yields the experimentally measured radius of a proton:

The 4 factor here is the one distinguishing the formula for the surface of a sphere (A = 4πr2) from the

surface of a disc (A = πr2). We can, therefore, effectively write the proton wavefunction as

a combination of two elementary wavefunctions:

Two imaginary units (i and j) rather than just one (i). Now that we have made this step (a rotational

operator for two planes in 3D space (say, the xy- and yz-plane in your particular choice of a Cartesian

space), we must introduce the third one: k (a rotation in the xz-plane if we stick to the definition of the i

and j operators here). Why would we need it?

The answer is this: when combining transients with stable oscillations and analyzing interactions, the

i·j = k; j·i = –k products must, inevitably, pop up in the calculations. Hence, one had better use Hamilton’s

space to make sense of it all.

Hamilton’s rotational operators and scattering matrices

We will make our life easy and just copy some basic thoughts from our paper on what Bombardelli

(2016) refers to as a new scattering matrix approach. Let us just throw it on paper. Here we go. We

apologize for reemphasizing points we made before.

1. Applying Occam’s Razor Principle (i.e., matching all mathematical possibilities in the expression with

experimentally verified (physical) realities), we think of the sign of the coefficient A as modeling

matter/antimatter, while the sign of the complex exponent (iEt/ħ) captures the spin direction of

matter/antimatter particles. The wavefunction of unstable particles (transients) involves an additional

decay factor :

2. Based on the considerations above, one can, for example, analyze the famous K0 + p → 0 + + decay

reaction and write it as follows

xxxi

:

xxxi

Of course, there are further decay reactions, first and foremost the 0 + + → − + p + + reaction. We chose the

example of the K0 + p reaction because Feynman uses it prominently in his discussion of high-energy reactions

(Feynman, III-11-5). In fact, we may well say it is this reaction which triggered all of the rubbish on strange quarks:

a decent analysis of the various properties of processes was substituted for ontological nonsense. The key mistake

is always this: an apple has no essence beyond its properties of taste, color, form, etcetera. John Locke was right.

Immanuel Kant acknowledged to have been inspired by David Hume (see, for example, the usually well-written

summary in the Stanford Encyclopedia of Philosophy article on this), but Hume’s inspiration is that of John Locke.

We feel the diagrams we presented on page 4 of this paper wrap up the philosophical discussion on these things.

15

The minus sign of the coefficient of the antikaon wavefunction reflects the point we made above:

matter and antimatter are each other opposite, and quite literally so: the wavefunctions –AeiEt/ħ and

+AeiEt/ħ add up to zero, and they correspond to opposite forces and different energies too!

xxxii

To be

precise, the magnetic field vector is perpendicular to the electric field vector but – instead of lagging the

electric field vector by 90 degrees (matter) – it will precede it (also by 90 degrees) for antimatter, and

the nuclear equivalent of the electric and magnetic field vectors should do the same (we have no reason

to assume something else).

xxxiii

Indeed, the minus sign of the wavefunction coefficient (A) reverses both

the real as well as the imaginary part of the wavefunction.

However, it is immediately obvious that the equations above can only be a rather symbolic rendering of

what – paraphrasing Wittgenstein (die Welt ist was der Fall ist) – might be the case. First, we cannot

model the proton by an AeiEt/ħ wavefunction because we think of it as a 3D oscillation. We must,

therefore, use two rather than just one imaginary unit to model two oscillations. To put it simply: we

need to think of two planes of rotation here. This may be solved by distinguishing i from j and thinking of

them as representing rotations in mutually perpendicular planes. Hence, we write the proton as

xxxiv

:

In addition, the antikaon may combine an electromagnetic (2D) and a nuclear (3D) oscillation and we

may, therefore, have to distinguish more than two planes of oscillation.

Last but not least, we should note that the math becomes even more complicated because the planes of

oscillation of the antikaon and the proton are likely to not coincide. We, therefore, think some modified

version of Hamilton’s quaternion approach may be applicable, in which case we have i, j and k rotations.

xxxii

See our previous remarks on the lag or precession of the phase factor of the components of the wavefunction.

Needless to say, masses and, therefore, energies are positive, always, but the nature of matter and antimatter is

quite different.

xxxiii

We think this explains dark matter/energy as antimatter: the lightlike particles they emit, must be

antiphotons/antineutrinos too, and it is, therefore, hard to detect any radiation from antimatter. See our paper on

cosmology.

xxxiv

We use an ordinary plus sign, but the two complex exponentials are not additive in an obvious way (i j). Note

that t is the proper time of the particle. This point should be appreciated by noting that the argument of the

(elementary) wavefunction a·ei, as a whole, is invariant.

16

Furthermore, each of these rotations will be specific to each of the particles that go in and come out of

the reactions, so we must distinguish, say, the iK, jK, kK, from the i, j, k rotations.

xxxv

The j and k rotations may be reserved for the two perpendicular (nuclear) rotations, while Euler’s

imaginary unit (i) would model the electromagnetic oscillation (not necessarily perpendicular to any of

the two components of the nuclear oscillation). In addition, we must note these planes of rotations are

likely to rotate in space themselves: the angular frequency of the orbital rotations has a magnitude and

a direction. If an external field or potential is present, then the planes of oscillation will follow the

regular motion of precession. In the absence thereof, the angular rotation will be given by the initial

orbital angular momentum (as opposed to the spin angular momentum).

[…]

The average reader – which is nobody except us: amateur physicists with some sense for math – must

be thinking we are talking Chinese now. It is not. Read all of the above again and, hopefully, you will

experience the same Aha-Erlebnis as we did.

Limitations of Hamiltonian’s quaternion math

Our ambition was and remains to develop a credible neutron model. However, we must admit we are

stuck. When rewriting our key paper on it, we concluded this:

“We cannot use the electron or proton model for the neutron – at least not directly – because

we consider the neutron to combine positive and negative charge. We consider the non-zero

magnetic moment of the neutron to justify this hypothesis.

xxxvi

Because the formalism of the

wavefunction, which we introduced above, applies to the motion of one charge only, we cannot

relate it to our neutron hypothesis.”

Hamilton’s formalism gets into trouble when there is no real (i.e., ontological) center to the analysis

xxxvii

or, what amounts to the same, when we are analyzing multiple charges rotating (or, let us use language

xxxv

The K and subscripts denote the (neutral) antikaon and lambda-particle, respectively. We use an underbar

instead of an overbar to denote antimatter in standard script (i.e., when not using the formula editor).

xxxvi

Again, the neutron is not stable outside of the nucleus. We, therefore, think of it as a composite particle. Of

course, the reader who is well versed in the history of nuclear research will probably be aware of Schrödinger’s

Platzwechsel model for nuclei, and so we must say a few words about it (we will also do that in the text itself later

on). The Platzwechsel theory is an early hypothesis of Schrödinger suggesting that a neutron and a proton swap

places or, to be more precise, assemble and disassemble into each other. This may or may not happen, but we see

no research supporting this. The Wikipedia article on the Yukawa potential says a few words about it in its history

section, but we will let the reader google more about it. Indeed, we do not dig into it because modeling the

neutron as a combination of a proton and a (nuclear) electron – its decay products – is not in line with our world

view here. Indeed, we repeat – once more – that charge is the more fundamental concept when discussing forces

and fields. Indeed, a force acts on a charge, and we think of mass as the inertia to a change in the state of motion

of the charge (or, plural if applicable, charges) inside of a (neutral or charged) particle.

xxxvii

Just to be clear, such ontological center is the naked charge for us. What is it? For the time being, it is the

pointlike charge in our electron or proton model. Its dimension is very small, but not infinitesimally small (that’s

just the gist of our explanation of the anomalous magnetic moment). Our papers present a consistent mass-

without-mass model of reality. Charge is the essential concept. Charge-without-charge models make no sense to

us because we firmly believe the 2019 revision of SI units is complete in Occam’s definition of completeness (OK,

that is a very simplified rendering of our thoughts on physical dimensions, but it should do for the time being).

17

that captures multiple dimensions, zittering around each other). We must, then, fall back to classical

approaches, and the only classical approach which comes to mind is the superposition principle.

xxxviii

In practical terms, that amounts to adding to scalar and dynamic potentials resulting from the presence

of multiple pointlike charges being added. The statics are easy. The dynamics are not: they reflect the

age-old two- or three-body problem.

xxxix

Where can we find some inspiration to modeling multiple charges in motion? We do not know. It is on

our list of research topics. All that we can offer on that right now are Feynman’s thoughts on it, which –

despite the rather advanced nature of some of his lectures – are extremely rudimentary. The screenshot

below is all he has to offer on it:

[…]

Is it useful? Maybe. Maybe not. Feynman is great, but a number of his derivations (including his heuristic

derivation of Schrödinger’s equation) are based on the classical idea of a particle which, despite our love

of the classical equations (we mean those of the late 19th century and early 20th century), we think of as

an outdated concept.

The mass of particles – which are only real when energies and momenta are well above the limit set by

Planck’s quantum of action – makes us think the only real thing – in an ontological sense – is the concept

of charge. Elementary particles are nothing but charge in motion. Hence, it is the concept of charge –

not the particle idea as such – which must be used consistently as some kind of ontological anchor when

analyzing the equations of physics. As this and other papers of us make clear, we must think in terms of

(elementary) oscillations of pointlike (but not infinitesimally small) charges rather than in terms of

particles when moving from classical mechanics to quantum mechanics. Hence, Feynman’s potential

math will not do the trick. Some kind of mathematical treatment of the dynamics involved in Planck or

sub-Planck interactions between a charge and the (dynamic) fields that surround such zittering or

oscillating pointlike charge (we repeat: such charge is very very small (pointlike, yes –sure, the ontology

xxxviii

The Wikipedia article on the superposition principle is not bad, if only because it incorporates Feynman’s

views on it.

xxxix

We do not think of Wikipedia too highly because, from our own experience trying to join a Wiki discussion, we

noticed that the topic of Zitterbewegung is pretty much cornered by its moderators. However, elementary topics

such as the two-body problem are covered rather well. We also think of Wikipedia’s article on the three-body

problem as rather excellent: the animation which juxtaposes a gravitational versus an elastic three-body problem

is outright inspiring. We, therefore, would like to explore it further – again, if we have time and energy left in our

rather limited life.

18

or epistemology beyond is and remains very complicated

xl

) but not infinitesimally small) view has and

must continue to be injected. However, how we deal with multiple charges zittering or rotating each

other – in full respect of both Maxwell’s equations as well as the fundamental Planck-Einstein-de Broglie

equations – remains unsolved.

Potential wells and tunneling

We should, perhaps, add a few words on tunneling and potential wells.

xli

These concepts – which are

closely related to Heisenberg’s uncertainty principle – should not be thought of as being mysterious

and/or incomprehensible. One must 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, a particle breaking through a ‘potential wall’ or

coming out of a potential ‘well’ is likely to just use a temporary opening corresponding to a very classical

trajectory in space and in time. It is not unlike shooting bullets through the blades of a fast-rotating

airplane propeller: if the bullets are ‘fast enough’, they will get through, and higher-speed bullets (speed

relative to the angular velocity of the propeller) are more likely to get through.

In our paper(s) on electron propagation

xlii

, we offer some thoughts on how electrons ‘jump’ from one

element in a lattice to another element: it is a ‘stop-go’ motion which involves tunneling. Following quote

out of it might help you to think things through for yourself: “[The electron] will tend to stay where it is.

From time to time, it does tunnel through. The question now becomes: when and how does it do that?

That is a bit of a mystery, but one should think of it in terms of dynamics. We modeled particles as charges

in motion. Hence, we think of an atom as a dynamic system consisting of a bunch of elementary (electric)

charges. These atoms, therefore, generate an equally dynamic electromagnetic field structure. We,

therefore, have some lattice structure that does not arise from the mere presence of charges inside but

also from their pattern of motion.”

In short, if one has a dynamic view of potentials, it is rather obvious charged particles can tunnel through

following a quite classical trajectory – respecting the least action principle combined with the Planck-

Einstein relation (applied to both energy and momentum).

xl

I had a burst of creativity two years ago. It brought the RG score to what it is now: from 0 to about 180 now.

Nothing great, but nothing small either: I am, apparently, top 1% of reads of authors who came online in 2020.

Nothing more. Nothing less. I have decided to use the little credibility which such stats bring to me to go all out on

my project, which is scientific and philosophical at the same time (it is my firm conviction that the two disciplines

must quickly merge so as to remain societally acceptable as ‘truth’). As I state in my profile, my objective is to dig

into “the theoretical foundations and epistemology (interpreted as the genealogy or archaeology of ideas) of

modern quantum physics (both QED as well as QCD).” I am well underway, and I am grateful to the ResearchGate

owners and editors for allowing me to express my opinions to a larger public.

xli

As mentioned in the text body of this paper, this section may seem out of place in this paper. However, when

talking rotations or rotations within rotations, we feel one should say a few words about dynamics in space or in

lattices (real-life matter) tout court.

xlii

See our (alternative) lectures on physics, in particular lecture III (the field concept) IV (electron propagation in a

lattice) and V (moving charges, radiation, and near- and far-fields). The quote above was taken out of our paper on

Feynman’s Time Machine.

19

Philosophical annex

We felt some of the smaller remarks in this paper were quite philosophical. Such is the nature of our

research: on our RG profile page, we write that we are mainly interested in the theoretical foundations

and epistemology (interpreted as the genealogy or archaeology of ideas) of modern quantum physics

(both QED as well as QCD). That is and remains the case. So, we thought it might not be a bad idea to take

some older (admittedly, very simple) material

xliii

and put it here.

The essence of time

Immanuel Kant’s ideas on the nature of space and time were interesting. They go back to the great

Enlightenment thinker David Hume whose ideas can, in turn, be traced further back in time to earlier

thinkers—until the trace disappears in the mists of history. Studying the genealogy of concepts and

ideas – think about the evolution of the concept of energy, for example – is an interesting exercise but

that is not what we are going to do here. So, what is that we want to do then? We want to offer a small

reflection on the concept of time.

You know time has one direction only: it is not like a (spatial) coordinate axis along which we can

measure position in both directions: this direction or its opposite. You also know there are some general

principles in physics that reflect that idea of time going in one direction only—time ticking away, lost

forever. The idea is very real. In fact, the title of what I think of as Prof. Dr. David Hestenes’ most

significant article (he must be credited with starting the modern thinking about the interpretation of

quantum mechanics that we, obviously, like so much

xliv

) is quite significant: Quantum Mechanics of the

Electron Particle-Clock.

xlv

So, what is the essence of time? In classical mechanics, physicists will talk about thermodynamics and

entropy and tell you that entropy (energy dispersal) will always increase, and so that is why physical

processes are not reversible—in practice, not in theory: we cannot actually build a perpetuum mobile. It

will not work because of friction or heat. That is because the modern idea of energy says that energy has

no direction. However, that idea is not very absolute. Kinetic energy is, obviously, associated with (linear

or orbital) momentum and, hence, the idea of directional energy is not something which should be

discarded. One example is this: if some charged particle is going to tunnel through what is referred to as

xliii

We once wrote a rather lengthy manuscripts, for which there was initial interest from scientific publishers

(Morgan & Claypool (IOP Concise Physics series), World Scientific Publishing (WSP), and Springer Verlag). However,

when we look at the manuscript now, we understand why this publishing project went nowhere: too many words,

not enough academic references. Our 30-odd papers on ResearchGate do the trick of explaining some basic ideas

much better. We are grateful for the service offered, and hold the RG forum in high esteem.

xliv

See: David Hestenes, The Zitterbewegung Interpretation of Quantum Mechanics, Found. Physics., Vol. 20, No.

10, 1990, pp. 1213–1232.

xlv

In case you prefer a more classical or more eminent reference, Richard Feynman also advances the idea of the

particle clock in a short series of lectures (QED: The Strange Theory of Light and Matter) for laymen. These lectures

were transcribed and published by Ralph Leighton, just before Feynman died, from cancer, at the relatively young

age of 69. It is a delightfully simple introduction to quantum math, but towards the end there are some

inaccuracies, so we wonder if Feynman has thoroughly reviewed it. Ralph Leighton definitely goes off track when

writing about the fine-structure constant as some ‘God-given number’. We do not like such guru-like talk. Feynman

probably did not either but – as mentioned – it makes us think Feynman may not have reviewed that otherwise

fine little booklet very carefully.

20

a potential barrier, it will need some direction: if you want to ram a door, you’d better get the direction

right. In any case, that is classical physics.

xlvi

Quantum physicists will tell you a more complicated story involving CP-symmetry breaking and its

corollary: CPT invariance. Yes. CPT symmetry and all the reflections that surround it are truly great stuff.

I have written about these things a lot.

xlvii

However, when someone would ask me to explain why time has one direction only, I would not refer to

any principle in physics. I will tell him or her it is just because our mind works that way. If the person I

am talking to has no background in physics or math, I will refer to philosophers such as Hume or Kant

and repeat what they told us: time is just a category of our mind but – unlike spatial categories (up or

down, left or right, front or back) – it has one direction only. Our world would not make sense if we

would allow this or that object to go back in time. That is why we find all of those time travel movies so

interesting: they are intriguing because they do not make sense.

xlviii

But, then, why would that be so? What is the ontological basis here? If the person I would be talking to

has only very basic knowledge of math and physics, I would give him or her a very simple answer. No

convoluted concepts. The answer is this: time has one direction only because – if it would not – we

would not be able to describe trajectories in spacetime by a well-behaved function. There is really no

need to think of entropy or of other explanations. The diagrams below illustrate the point. The

spacetime trajectory in the diagram on the left is not kosher, because our object travels back in time a

couple of times. 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 very wild oscillation – obviously – but, while it is wild, it is possible: not only theoretically

but also practically. In short, all possible trajectories in space and time imply that time has one direction

only.

Hence, it is easy to see that our concept of time going in one direction, and in one direction only, implies

that we get well-behaved functions for trajectories—for motion in other words. The idea of motion, in

fact, is what gives space and time their meaning. The alternative idea is spaghetti: that is the first graph.

xlvi

That is why we talked about potential wells and tunneling in this paper, even if it looks like a rather random

section.

xlvii

If you are interested in these things, you can check my blog, which has various posts on these fascinating topics.

See, for example, https://readingfeynman.org/2014/05/11/time-reversal-and-cpt-symmetry-iii/, or just use the

search function with key words such as time reversal or CPT symmetry.

xlviii

Do the test: you will always find something that just is not right about the movie. However, Jason Hise – a

specialist in computer animation, whom I was in touch with because I love his gif-files with rotations within

rotation – told me I should see Predestination because he doesn’t find any logical mistake there. He refers to his

animations as 4D reality. We think that is nonsense: (relativistic) 3D space and 1D time will do.

21

Theories of Everything: U = 0

My papers regularly get comments or reactions of people who urge me to look into some kind of scalar

theory of everything (STOE). Look at the exchange below. I will not reveal the identity of the person, but

his RG score is much higher than mine.

X: “The strong and weak nuclear forces are ad hoc additions because the electric field is postulated to be

a characteristic of particles. If one builds a model that has the magnetic field as the characteristic of the

most elementary particle (that is, it is a magnet with N and S poles), [one can] apply Maxwell's equations

using the B as the cause of E. Experiment has shown the Biot-Savart Law is slightly wrong and suggest

magnets (the most elementary particles) are emitted from the current flow. Scalar Theory of Everything

(STOE) unites the big, the small, and the four forces (GUT) by extending Newton's model.” [my italics]

Me. “I agree with the qualification of weak and strong force as 'ad hoc additions' to a pre-war model

which was pretty solid but, for historical reasons, was apparently abandoned. I am not so sure about an

STOE: mass-without-mass models are easy (inertial mass then comes from the energy of an oscillatory

charge - neutral particles like neutrinos combining positive and negative charge), but charge-without-

charge? However, I will look at it when I have time.”

I do not believe in scalar theories. Of course, space and time are concepts that enable reality to meet

our mind, but we should not confuse them with reality. A very minimal distinction – which, I am

surprised, many physicists do not seem to make – is the distinction between mathematical space (think

of the Cartesian coordinate space) and physical space (think of the idea of curved space around massive

objects, for example). We will not dwell on this. We only want to give you a few pointers to think for

yourself, and we will do so by presenting what I personally think of as Feynman’s best jokes in his

Lectures series. It answers that question I get from the very few friends who sort of vaguely appreciate

what I am trying to do: what is that fundamental equation or that Theory of Everything that Stephen

Hawking was looking for?

xlix

When I get that question, I will usually say something like along the lines of

what follows.

Everything is motion.

l

At the macro-level, a force is that what changes the state of motion of an object.

At the micro-level, a force is that what changes the motion of a pointlike (but not infinitesimally small)

charge. Mass is a measure of inertia—resistance to a change of the state of motion. When we choose a

natural unit to measure force, we choose a natural unit to measure mass. A force has to grab onto

something. At the macro-level, that something is mass. At the micro-level, that something is charge.

So, what about that strong or weak force that is supposed to describe what goes on inside of the

nucleus? That is, unfortunately, an area where scientists lost their base. They should have talked to good

xlix

I enjoyed the movie on Hawking’s life (The Theory of Everything, 2014). At the same time, some of Hawking’s

one-liners in that movie are obviously taken out of context or might, quite simply, be factually incorrect.

l

You may want to interpret this as Heraclitus’ panta rhei principle. That is fine. However, this idea is much older

than that – think of the thoughts that are attributed to Simplicius and Plato himself, for example – and, in any case,

physicists cannot do much with an idea without an equation. In other words, do not engage in too much

philosophy here.

22

philosophers. They refer to quantum chromodynamics (QCD) like one would talk of another planet. It is

not. That one equation exists. It is just Feynman’s equation of the Theory of Everything

li

:

U = 0

That is a full-blown Theory of Everything. In fact, it is the Theory of Everything. Without any doubt

whatsoever. Why? All of the laws and equations in physics – Standard Model or not (it works for any

theory—even nonsensical ones) – can be written like this. We wrote U and 0 in boldface, so it can be a

vector equation, or a matrix equation. Yes. We just take all of our laws and equations and we re-write

them as a mn matrix equation: Ui = 0. What are m and n? Think of m and n as the dimensions of the

most complicated equation on your list. To give a very simple example, note that we can always re-write

the simple x = v·t equation of motion as the following multi-dimensional matrix equation

lii

:

So we can write every single law or equation in physics as U1 = 0, U2 = 0,… Ui = 0,…

liii

and we can then

add them all to get the Theory of Everything:

U1 + U2 + … + Ui +… + Uæ = U = 0

What’s that æ subscript? It is the dimension of the Universe. Why? Because it is the number of

equations we need to describe it. Or… Well… No. Maybe æmn is the dimension of the Universe.

[…]

Surely, you’re joking, Mr. Feynman! You are right. He was joking when he wrote this. But think about it.

It triggers some fundamental questions. What is a dimension? Is it just a number or some mn product?

How are they related to the degrees of freedom in a physical system? Is a dimension physical or is it a

li

Feynman develops the U = 0 equation for the Theory of Everything in what I think is one of his best tongue-in-

cheek arguments in his seminal Lectures on Physics (http://www.feynmanlectures.caltech.edu/II_25.html#Ch25-

S6). Its title sounds serious enough: The Invariance of the Equations of Electrodynamics. There is a lot of criticism

on Feynman’s seminal Lectures on Physics—especially the volume on quantum mechanics. For example, Prof. Dr.

Ralston (How To Understand Quantum Mechanics, 2017, p. 10-40) writes that “Feynman’s magic fails when he gets

tangled up trying to derive quantum mechanics from Stern-Gerlach and ad-hoc notation.” We do not agree with

such comments. The great value of Feynman’s Lectures is that they make you think for yourself. It took me two

decades to get through them, and I am still discovering subtleties I had not seen before. There are serious flaws in

them, for sure, but these are just the flaws of the Copenhagen interpretation of quantum mechanics, which

Feynman did not dare to challenge—although he was tantalizing close to it in some of his Lectures. His chapter on

identical particles (http://www.feynmanlectures.caltech.edu/III_04.html) is – in my not so humble view – the one

that is most misleading.

lii

We think of the weirdness of proposals suggesting that 3D space may have multiple dimensions folded into it.

Nonsense: one must stick to one representation when developing geometric ideas. You cannot suddenly swap

from Cartesian to polar coordinates without making the necessary transformations of your variables.

liii

You may wonder how we can get the x = v·t equation, with x, v, and t as matrix quantities, in the Ui = 0 format

but you should be able to figure that out for yourself. It involves the idea of an inverse of a matrix. You can google

that.

23

purely mathematical concept? There is no answer to this because the concept of a mathematical

dimension is quite ambiguous. It is as ambiguous as the concept of a mathematical space.

liv

In physics, it is somewhat less ambiguous – we think we know what a kg or a joule represents, and we

know how the physical space that we are living in looks like – but it is not as straightforward as teachers

or university professors want us to believe. In fact, that is another thing we want you to think about as

you are going through this book: what is a dimension? What is it in math? What is in physics? What is it

when we are trying to make sense of the world?

OK. What’s the point here? The point is: the Theory of Everything is already there! It is the Standard

Model. It just has a structure we do not really like because it is overly complicated. The equations do not

look good: everyone feels they can be simplified, somehow.

lv

All Theories of Everything are about

simplifying – and I am talking about a reduction in the number of symbols that we use to represent our

knowledge here – what we already know. The knowledge, itself, is complete. That is why few brilliant

young people enter the field of physics. I recommend brilliant young people to go for engineering

studies: apply the knowledge that is there already, because that knowledge is pretty much complete.

The meaning of the fine-structure constant

As we are – clearly – rambling here, I will now insert some very common-sense remarks on Plato’s world

view, which is based on a dichotomy that penetrates all of our thinking:

(1) The world of math, which is based on ultimate ideas, such as continuity, infinity, and

infinitesimally small (read: zero-dimension) quantities, which allow us to cut up space and time

intervals into things that do not really exist, and;

(2) The world of physics, which combines the continuity of fields with the ultimate discrete unit of

the world, which is Planck’s quantum of action, which – in conjunction with mankind’s

definitions of c and other fundamental constants as part of the 2019 revision of the system of SI

units – mankind defines to be equal to 6.62607015×10-34 N·m·s.

Before I move to the easy bit of my Plato-like remarks, I want to talk about the complicated things:

definitions and equations. The reader may or may not appreciate these, but they are very necessary:

statements that cannot be reflected by a precise mathematical relationships are entirely useless. So let

us make us some very precise remarks before we make easy philosophical remarks.

Six out of the sixteen chapters of my early manuscript are devoted to explaining the fine-structure

constant as:

⎯ A scaling constant of what is, clearly, some kind of (finite or infinite) fractal structure of reality;

liv

If you think that concept is not ambiguous, read this early blog post of mine:

https://readingfeynman.org/2013/10/27/many-spaces/.

lv

This must sound sacrilegious to Gell-Mann (he died in 2019, so I should not be worried: let him RIP – rest in

peace) and his colleagues, as they spent all of their life simplifying the wild zoo of particles through the

introduction of the concepts of quarks and gluons. We applaud this effort. However, we feel the so-called QED

sector of the so-called Standard Model of physics can be simplified substantially by injecting plain sense geometry

and classical physics. We must go back to the intuitions of dear old Einstein and dear old H.A. Lorentz. See our

paper on the history of quantum-mechanical ideas. Scientists must get back on track. We strayed away from the

Iron Path for far too long now. Period.

24

⎯ A coupling constant, relating energy exchanges between fields and the (linear or orbital)

momentum of the charged oscillations which we, rather vaguely, refer to as matter-particles;

and, finally;

⎯ The bridge between all properties of the so-called vacuum: electric permittivity or magnetic

permeability.

The beauty of the 2019 revision of SI units remains unexplored, as far as I can see, but my personal

understanding of the fine-structure constant is, most probably, best summarized in its properties as a

scaling constant. Why? Because the equations incorporate Planck’s (reduced) constant most directly.

Think of the following. The formula for the classical electron radius is this:

It basically defines the fine-structure constant as a scaling constant because, yes, it is nothing but the

ratio between the electron’s Compton radius and the size of the pointlike charge inside which generates

the anomaly in its magnetic moment

lvi

:

This equation relates the fine-structure constant not only to Planck’s quantum of action but also to

Maxwell’s equations: it co-defines both the electric permittivity factor (ε0) as well as the magnetic

permeability factor (c2 = 1/ ε0μ0) through the rather obvious relation between both that comes out of,

yes, Maxwell’s equation once again.

lvii

We can ramble on, but we prefer the reader to think through these relations all alone, preferably with a

good glass of wine or other good company.

lviii

lvi

We offered easy explanations of the anomaly in the electron’s magnetic moment in our very first lecture on

quantum behavior (read: the first of a series of easy papers). The reader should not think we are married to such

prima facie explanations! We are not. We keep every option open. We talked about fractal charge structure before

but – to be very honest – we feel such fractal structures must be finite. Why? Because reality is finite: concepts

such as infinite fractal structures are mathematical idealizations – in our not-so-humble view of reality, which the

reader must take not just with one but many grains of salt.

lvii

The reader who has read other papers of ours will note we are generally appreciative of Wikipedia articles.

However, we engaged into a tentative re-write of the article on Schrödinger’s or Hestenes’ concept of

Zitterbewegung and were appalled by the way early editors tend to ‘corner’ a topic. We offered some

contributions, but they were rejected. That does not matter: I have no ego. However, I am rather appalled that the

Wikipedia articles on electric permittivity and magnetic permeability do not reflect the most basic relation

between them, which is the c2 = 1/ ε0μ0. We have no real opinion on things, except for noting what approaches

make more sense that others. We recommend the reader to adopt the same critical attitude when trying to make

sense of reality. We surely do not claim to offer the truth here: we only share what makes sense to us! Nothing

more. Nothing less.

lviii

We will let the reader google the context of Feynman’s famous quote: “A poet once said: the whole universe is

in a glass of wine.” The truth of this quote is undisputable, even if you do not drink wine. Reality is reality, and it

confronts us rather forcefully.

25

Why the (absolute) velocity of light makes sense

The cartoon below – which was sent to me by Dr. Giorgio Vassallo, who is just one of the many kind

academics who encouraged me to follow my own path in my rather weird philosophical search for truth

in physics – explains the nature of this paper.

So, let us tell you a very basic and, hopefully, pleasant story about lightspeed. Relativistic speeds –

velocities that are a substantial fraction of c, that is – are not uncommon when we are discussing

elementary particles – as opposed to the large-scale objects we are used to. So, you know: we can no

longer treat mass as some constant. Mass increases with velocity. Mass has to increase because of

the absolute speed of light. If it would not increase, a force would be able to accelerate an object – any

object, really – to an infinite speed: it would take some time – especially if it is a big mass and a small

force – but, in the end, it would accelerate to an infinite speed.

That is not OK. Infinity is a nice mathematical concept, but you will agree that, in reality, it is kind of a

weird thing. In fact, we may want to think that, if objects are around long enough, all kinds of forces

might cause all of these objects to reach an infinite speed and that would be very inconvenient. So, we

should effectively impose some absolute speed cap on the Universe. Now, if you would be God, and you

would have to regulate the Universe by putting a cap on speed, how would you do that?

First, you would probably want to benchmark speed against the fastest thing in the Universe, which are

those photons. Why should they be the fastest thing in the Universe? Because they have no rest mass

and so they can effectively travel at the speed of light: c. So that is the fastest thing in the Universe now:

it is the speed of a signal, really.

lix

So now you want to put a speed limiter on everything else, so it can

only travel at some fraction of the speed of light. That fraction (v/c) is just a ratio between 0 and 1, of

course.

lix

If you know anything about quantum mechanics, you will know that the phase velocity of a composite wave

packet may be superluminal. In fact, it usually is. However, this phase velocity is just a mathematical concept. It is

not something real that is traveling through space. In other words, it cannot carry any information. Only the shape

of the wave can carry information (I mean a proper signal here). And the shape of the wave travels with the group

velocity of the wave packet, which is always smaller than c. See our paper on de Broglie’s matter-wave once again.

26

Now, because you are God, you do not want to police around. So, you want something mechanical: you

want to burden everything with an intricate friction device, so as to make sure the friction goes up

progressively as v/c goes to 0 to 1. You do not want something linear because you want the friction to

become infinite as v/c goes to 1, so that’s when v approaches c. So, that is one thing you have figured

out in your design.

Of course, you will also want a device that can cope with everything: electrons, bicycles, spaceships,

solar systems—whatever you can think of. The speed limit applies to all. But then you do not need too

much force to accelerate a proton as compared to, say, that new spaceship that was just built on planet

X.

So, you think about brakes and engines and all that but, after a while, you realize it is probably better to

just ask your best engineers to finalize your design. You all sit together, and you explain your problem

and the design requirements. One of them, Newton, will tell you that, when applying a force to an

object, its acceleration will be proportional to its mass. So, he goes to the blackboard and writes it down

like this: F = m·a. Of course, you tell him you know that already, and that this is exactly your problem:

even the smallest force can accelerate the heaviest object to crazy speeds—to infinite speeds! You just

need to apply the force long enough. Newton shrugs his shoulders and sits down again.

Now Lorentz gets up and points to the mass factor in the formula: m should go up with speed, he says.

And it should go up progressively—as per God’s design, he says. Lorentz is always well prepared, so he

has a print-out with some formulas and graphs and sticks it on the blackboard. Here is an easy formula

that does the trick, he says.

Look here, he says. That is how we can put a speed cap on bicycles, spaceships, and galaxies. The red

graph is for m = 1/2, the blue one for m = 1, and the green one for m = 3. In the beginning, nothing much

happens: the thing picks up speed, but its mass does not increase all that much. Why not? Because you

do want to allow everyone to move their stuff around, right? But when it gets a bit crazy, then the

friction kicks in, and very progressively so as the speed gets closer to the speed of light.

Now you stare at this for a few seconds, but you tell Lorentz you do not want to discriminate: it looks

like we are putting more aggressive brakes on the green thing than on the blue or the red thing, right?

However, Lorentz says that is not the case. There is no discrimination here: his factor is the

same per unit mass. The graphs show the product of the mass and his Lorentz factor, which is actually

represented by the blue line—because that is the one for m = 1. So, yes, the green thing will actually

27

have better brakes, but that is just commensurate with its mass. You want the lorry to come with better

brakes, right? And bicycle brakes will not do for a car either, right?

You look again, and you think that makes sense. But then you hesitate, of course. You do not want to

change the Laws of the Universe, as that would be messy. It would surely upset Newton because he is

pretty fussy about you tampering with stuff. So, you look at both and you say: what are the implications

for the force law? Newton nods in agreement: yes, what about it? We do not want to change it, he says,

because there are a zillion devices that work on it now. We cannot do a total recall, can we?

Lorentz is still at the blackboard, but he tells Newton: it is not a problem. We are going to use the same

force law. We are just going to distinguish two mass concepts: the mass at rest, and the mass at some

velocity v. Just put a subscript – mv – and then you use this. He jots this on the blackboard:

Now Newton stares at that, and he takes a few minutes. You think he is going to turn it down because

his formula is… Well… Newton’s formula, right? But… No. Something weird happens: Newton nods and

agrees! He gets up, shakes hands with Lorentz and says: excellent job! Perfect fix!

So, you are delighted, and you tell Lorentz he can pick and choose his men and build it.

lx

Newton walks out, and Lorentz stays behind. Suddenly you see some worry on this face, and so you ask:

what’s up? You are not happy with your own thing? He sighs and says: my formula is the only thing that

can do the trick because, yes, you want it to be progressive. It needs to be something based on the idea

of the mass unit. But this mechanical thing has some weird implications. You ask: what implications?

Now Lorentz starts a discussion on a guy you have never heard about – Albert Einstein – and he starts

mumbling about time dilation and length contraction. He says Newton’s formula came with Galilean

relativity, and that we will need a new concept of relativity. But you want to move on by now, and so

you tell Lorentz to hire that Einstein and just get on with it.

[…]

So… Well… That is what we will do also. We will just get on with it. That is what we did by looking at all

that wavefunction math.

We also wrote – rather jokingly – that infinity is a nice mathematical concept but that it is weird to think

of what it could possibly mean in reality. This is actually a rather deep philosophical statement. You

should think through Zeno’s paradoxes. Differential calculus shows that the idea that we can keep

lx

The Lorentz formula seems to be the only one that can do the trick. Of course, there is no proof that other

formulas would not work but, in any case, our Universe is what it is, and so the Lorentz factor is what it is.

However, it is an interesting exercise to try some other formulas. The factor makes us think of the

formula for a circle: , and so you might think some similar formula might also do the trick. Try it. It

does not.

28

splitting some interval in time or in space in smaller and smaller bits – going on forever (so that’s, funny

enough

lxi

, the idea of a limit in math) – is not incompatible with Achilles overtaking the tortoise, or the

idea of an arrow being somewhere while flying through space, but it is good to think through those

paradoxes. We need math to describe reality – whatever idea we have about it – but Planck’s quantum

of action, and the finite speed of light, seems to tell us our mathematical ideas are what they are:

idealized notions to describe something finite.

lxii

Play with a graphing tool: see what happens if the m0 term in the mv = γm0 goes to zero. The velocity has

to be c. Reality is not ambiguous: reality, and our understanding of it, is one.

lxi

Think about what I would call that funny: the mathematical definition of a limit involves the idea of infinity. So

that is a pretty clear example of a contradictio in terminis, no?

lxii

The rather philosophical discussion on the mathematical consistency of Dirac’s delta function is a nice example

of a paradox in quantum mechanics. We will not entertain such discussions here (we did so, just a little bit, here

and there on our blog and in some RG papers we just cannot find right now). Not because we do not like them – on

the contrary – but because they have little practical value in trying to move towards some understanding of it all.

However, we do encourage the reader to look into this. It is fun. For starters, the reader may want to think of how

a link function can map the infinite [0, +] set of real numbers to the finite [0, c] interval.

29

References

The reference list below is limited to the classics we actively used, and publications of researchers whom

we have been personally in touch with:

⎯ Richard Feynman, Robert Leighton, Matthew Sands, The Feynman Lectures on Physics, 1963

⎯ Albert Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik, 1905

⎯ Paul Dirac, Principles of Quantum Mechanics, 1958 (4th edition)

⎯ Conseils Internationaux de Physique Solvay, 1911, 1913, 1921, 1924, 1927, 1930, 1933, 1948

(Digithèque des Bibliothèques de l'ULB)

⎯ Jon Mathews and R.L. Walker, Mathematical Methods of Physics, 1970 (2nd edition)

⎯ Patrick R. LeClair, Compton Scattering (PH253), February 2019

⎯ Herman Batelaan, Controlled double-slit electron diffraction, 2012

⎯ Ian J.R. Aitchison, Anthony J.G. Hey, Gauge Theories in Particle Physics, 2013 (4th edition)

⎯ Timo A. Lähde and Ulf-G. Meissner, Nuclear Lattice Effective Field Theory, 2019

⎯ Giorgio Vassallo and Antonino Oscar Di Tommaso, various papers (ResearchGate)

⎯ Diego Bombardelli, Lectures on S-matrices and integrability, 2016

⎯ Andrew Meulenberg and Jean-Luc Paillet, Highly relativistic deep electrons, and the Dirac equation,

2020

⎯ Ashot Gasparian, Jefferson Lab, PRad Collaboration (proton radius measurement)

⎯ Randolf Pohl, Max Planck Institute of Quantum Optics, member of the CODATA Task Group on

Fundamental Physical Constants

⎯ David Hestenes, Zitterbewegung interpretation of quantum mechanics and spacetime algebra (STA),

various papers

⎯ Alexander Burinskii, Kerr-Newman geometries (electron model), various papers

⎯ Ludwig Wittgenstein, Tractatus Logico-Philosophicus (1922) and Philosophical Investigations

(posthumous)

⎯ Immanuel Kant, Kritik der reinen Vernunft, 1781