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The concepts of charge, elementary ring currents,

potential, potential energy, and field oscillations

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

30 April 2021 (revised on 11 September 2022

i

)

Abstract

This paper explores some more advanced questions in what we refer to as our realist interpretation of

quantum mechanics – which is based on a ring current model of elementary particles. More specifically,

we wonder how elementary ring currents – charged oscillations – can create the spherically symmetric

(electro)static potential we associate with electrons and protons: why are there no small variations

because of the motion of the pointlike charge inside? We argue the question does not arise because of

the Planck-Einstein relation. Part of the argument offers some thoughts on the quantum-mechanical

application of the least action principle.

Finally, we explore the possible relation between asymmetric (static) potentials and the idea of curved

spacetime (gravity). While the latter topic is surely not directly related to the question we started out,

some of the same thinking triggered a reflection on it, and that is why we discuss it in one of the

annexes to this paper.

Contents

Introduction .................................................................................................................................................. 1

The magnetic field of a ring current ............................................................................................................. 2

The nature of the electrostatic potential ...................................................................................................... 5

The least action principle in quantum mechanics ........................................................................................ 8

Conclusions ................................................................................................................................................. 10

Annex I: The Complementarity and Uncertainty Principles ........................................................................ 14

Annex II: Relativistic kinetic energy and momentum ................................................................................. 18

Annex III: Static potentials and spacetime curvature ................................................................................. 25

i

We acknowledge the 21st anniversary of the 9/11 events.

1

Introduction

In previous papers, we related a lot of classical concepts to the weird world of quantum physics. We

explained the quantization of matter- and field/light(like)-particles in terms of two- and three-

dimensional ring currents and electromagnetic field oscillations, respectively. Let us write out the

Planck-Einstein relation for the electron (e), proton (p), and photon (), respectively

ii

:

We also explained previously that we think of the electron oscillation as a planar oscillation (an

oscillation in two dimensions only). In contrast, in order to explain the measurable properties of a

proton – notably its size – the math suggests a spherical oscillation (an oscillation in all of the three

dimensions of physical space). We will not come back to that here. What we should note here, is that

the electron and proton (rest) mass – or, noting the mass-energy equivalence (which, by the way, is also

explained by this electron and proton model

iii

) their energy – appear as fundamental constants of

Nature⎯as fundamental as the elementary charge (qe), lightspeed (c) or Planck’s quantum of action (h

or, in reduced form, ħ

iv

). To put it simply: we have electrons and protons in Nature, and they come in

one version only.

In contrast, photons – which we think of as field-particles (light

v

) – can have any energy, as long as this

energy is some difference between two energy states of the electron (or between the atomic or

molecular orbitals, to be precise). However, these (stable) energy states also respect the Planck-Einstein

relation. We may think of atomic orbitals, for example, as being separated not only by a certain amount

of energy but also by an amount of (phyical) action that is equal to h. Hence, when an electron

transitions from one level to the next – say from the second to the first – then it will not only lose energy

ii

We use the γ symbol as a symbol for a photon but, later in the text, also for the Lorentz factor. The context makes clear what

is what.

iii

The neutron is not a stable particle outside of the nucleus, even if they decay rather slowly as compared to other unstable

particles: their mean lifetime is a bit less than 15 minutes (for the latest measurements, see this recent overview in the Nature

journal). We, therefore, developed a neutron model which combines the idea of a proton and an electron oscillation.

iv

We interpret the de Broglie frequencies in the context of matter-particles as orbital rather than linear frequencies. The

concepts of angular momentum and physical action merge effectively into one and the same in the context of orbital motion.

The reduced form of Planck’s constant is, therefore, the one that is most relevant. We remind the reader that linear motion

involves the concepts of linear momentum and (linear) wavelengths. Linear wavelengths and loop circumferences are related

by the λ = 2r formula, with r the (average or effective) radius of the loop. Needless to say, linear wavelengths (and, therefore,

the non-reduced form of Planck’s constant) are relevant in the context of lightlike particles, which travel linearly at the speed of

light.

v

Photons are lightlike particles that are associated with electromagnetic oscillations. In previous papers, we argued that

neutrinos must be the photon-like field-particles associated with the proton oscillation(s), but we cannot delve into that here.

2

but also one unit of ħ. The photon that is emitted packs both. The energy is given by the Rydberg

formula

vi

:

We have the fine-structure constant here

vii

, and the principal quantum numbers n1 and n2. Consider the

transition from the second to the first level, for example, for which the 1/n12 – 1/n22 = 1/12 – 1/22 factor

is equal to 0.75. Hence, the photon energy should be equal to (0.75)·ER ≈ 10.2 eV. This corresponds to a

wavelength λ = c/f = hc/E 122 nm. To give you a better idea of what that is, we may note that it is a

wavelength in the UV-C spectrum, and that its length corresponds to the size of a large molecule.

viii

What we wrote above just illustrates the photon model: it is consistent and requires no further

explanation. We, therefore, think field oscillations present no mystery.

ix

What we are, therefore, left

with, is the question as to how a ring current – a charged oscillation – can create a spherically symmetric

electrostatic potential. Indeed, if we think of an electron as a ring current, then Maxwell’s laws give us

the magnetic dipole moment, and the magnetic field which keeps the current going, but what about the

electric potential? What generates it? And why would it be spherically symmetric?

Before we try to answer that question, let us briefly discuss the magnetic field and analyze its

properties. This may look like a digression, and, to a large extent, it is. However, we recommend the

reader not to skip it: it is intended to understand the nature and detail of the questions above better.

The magnetic field of a ring current

An intuitive understanding of magnetic fields generated by ring currents may be gained by analyzing the

magnetic field of a superconducting ring. The perpetual current(s) in a superconductor behave just like

electrons in an electron orbital in an atom:

1. No field energy is radiated out, nor absorbed. The system, therefore, behaves like a perpetuum

mobile, in which kinetic and potential energy always add up to a constant over the cycle.

vi

To not overcomplicate matters, we only consider the shells (orbitals) here, as given by the principal quantum number (n). Spin

or angular momentum (up or down) and spin coupling (between electron spin and nuclear spin) give us a fine and hyperfine

structure within the principal energy levels, but let us abstract away from that as for now.

vii

The fine-structure constant is an electromagnetic constant which, in the current system of SI units, is co-defined with the

electric and magnetic constants. Their CODATA values are related as follows:

The fine-structure constant has several interesting uses (e.g. scaling constant, coupling constant), which we discussed in an

introductory paper (Layered motions and the meaning of the fine-structure constant) and is, therefore, a remarkably versatile

and very interesting constant, but it is not a mystery.

viii

This short-wave ultraviolet light (UV-C) is the light that is used to purify water, food or even air. It kills or inactivate

microorganisms by destroying nucleic acids and disrupting their DNA. It is, therefore, harmful. Fortunately, the ozone layer of

our atmosphere blocks most of it: otherwise, you and I would not be here writing and reading.

ix

As mentioned above, we also think it makes sense to think of neutrinos as the photons of the nuclear force⎯which we

associate with a three-dimensional oscillation whose energy is given by the proton mass. However, we will not go into that here

because that is rather speculative. We must also further develop the idea of proton states – it is hard to immediately see how

they could be like the electron states we discussed above – and we have not worked on that, yet.

3

2. There is also no heat or any other frictional effect: no thermal motion of electrons, nuclei,

atoms, or molecules as a whole and, therefore, no heat (thermal) radiation or absorption.

x

3. Superconducting currents also involve electron pairs (Cooper pairs), and the Pauli exclusion

principle applies: the spin angular momentum (as opposed to the orbital angular momentum) of

the two electrons will be opposite. The pair will, therefore, be in its lowest (spin) energy state.

That is why superconductivity is said to be a quantum-mechanical phenomenon which we can

effectively observe at the macroscopic level. Since 1961 (the experiments by Deaver and Fairbank in the

US and, independently, by Doll and Nabauer in Germany

xi

), we know this field is quantized. To be

precise, the product of the charge (q) and the magnetic flux (Φ), which is the product of the magnetic

field B and the area of the loop S, – will always be an integer (n) times h

xii

:

q·Φ = q·BS = n·h

This quantization does not imply that we should assume that the magnetic field itself must, somehow,

consist of (discrete) field quanta. Not at all, really. The magnetic field is just what it is: a finite quantized

magnetic field. There is absolutely no need whatsoever to think of virtual particles here.

The equation above makes it clear the field cannot be separated from the circulating charge, which –

because electrons form Cooper pairs in superconductors – is twice the electron charge. This explains

why the basic flux unit is defined as:

The current loop will have a magnetic moment (μ) equal to the product of the current and the surface

area of the loop (μ = I·π·a2 = I·S) and, because the current in the loop is equal to the charge times the

frequency of the orbit (I = q·f), we can write the magnetic moment as μ = q·f·S. Now, we also know the

(potential) magnetic energy is calculated as the product of the magnetic moment and the magnetic

field: Umag = μ·B. We can, therefore, show that the Planck-Einstein relation is valid here again. Indeed,

the (magnetic) energy is an integer multiple of Planck’s constant times the frequency of the current

xiii

:

x

Heat is, of course, electromagnetic radiation as well, so it gets radiated or absorbed in discrete units (photons) as well. The

question as to why some superconducting materials freeze into a state that permits superconductivity to happen at higher

temperatures than the usual very-close-to-zero Kelvin temperature is an interesting one, into which we have not delved.

xi

It may be noted that the theoretical prediction (quantization of the flux trapped by a superconducting ring) had been made by

F. London in 1950 already, so the mentioned physicists knew what they were looking for.

xii

Φ = BS is a vector (dot) product but – because of the set-up – reduces to an ordinary scalar product: Φ = BS = BScosθ = BS.

As usual, it is always instructive to check the physical dimensions: the magnetic field is expressed in N/C times s/m, while the

surface area is expressed in m2. Hence, [q·BS] = C·(N/C)·(s/m)·m2 = N·m·s, which is effectively the physical dimension of

Planck’s quantum of action.

xiii

We should take the directions of the magnetic moment and the magnetic field into account and, hence, write the energy a

vector dot product (Umag = −μ·B) but here we are interested in the magnitudes. Of course, the reader may object that the

magnetic moment and the magnetic field should align: we, therefore, might have orthogonal vectors and a vector product

which is equal to zero. Also, the Umag = −μ·B formula is normally used in the context of an external magnetic field, so that is a

quite different magnetic field than our B here. In the presence of an external field, we should also introduce precessional

motion of the charged pair, which involves an oscillation and whose energy we should also be able to analyze in terms of kinetic

and potential energy. How this works out geometrically – not approximately but exactly – is an interesting question, but we will

not dig into it here.

4

This shows that the energy in a magnetic field is quantized as well, just like the magnetic field. Of course,

you may argue think that the frequency (f) can take on any value, but that is not the case: the current,

surface area, and frequency must have values that are consistent with the field strength which, as we

showed above, can only take on the discrete values that are given by the q·Φ = q·BS = n·h equation.

Hence, we have an analogy here between a superconductor and the ring current that we associate with

electrons, and now we must wonder about where the analogy does not work. What are the differences

between the magnetic field created by a perpetual current in a superconductor and the field that we

might associate with the electron ring current? We can readily think of two differences:

1. An electron ring current – or the 3D ring current in a proton – has one pointlike charge only: we

are not talking electron pairs here.

2. A far more important difference, however, is this: the charges in perpetual current(s) in a

superconductor are held into place by the superconducting material. In contrast, the charge in

electrons, protons, or charges in atomic/molecular orbitals is supposed to be held in place by

the magnetic field only.

The latter remark raises a fine-tuning question: the smallest disturbance should cause the equilibrium

state to collapse.

xiv

This is an important question, which we should solve: if quantum mechanics is

mysterious, then this is the mystery! In fact, it is the only mystery in our view, and we will briefly re-

reflect on it in a few moments.

Let us first return to the question we started out with: what is the energy in the electric field? And, if

such electrostatic energy is real – as real as magnetic energy – then how is it quantized⎯not

approximately, but exactly? Indeed, we must draw the reader’s attention that the analysis of the

magnetic field in the context of superconduction does not involve an electrostatic potential. We have a

magnetic field only, and we know the strength of magnetic fields is only 1/c that of the electric field. We

may think of the speed of light (c) here as a force scaling constant here and, while the electric and

magnetic fields obey relativity theory, this constant (the ratio between E and B) is absolute.

xv

Another useful remark – to which we shall return later – is that the analysis above is based solely on the

idea of unit charges: we are not thinking of currents in terms of ampere (C/s) of charge densities (C/m3

or C/m2) but in terms of one unit charge (qe) rotating at a certain frequency. This establishes a bit of an

ambiguous (not straightforward) relation with an analysis in terms of Maxwell’s equations.

xvi

We will

soon talk about that, but we must first make a few more notes, which may also look like a bit of a

digression but, just like what we wrote above, the reader should probably go through it to appreciate

the argument that follows.

xiv

This may also be related to the fact that superconducting currents occur only in extremely cold materials: thermal motion

may be thought of as a disturbance that would cause the superconducting current to collapse.

xv

Feynman offers an interesting analysis of what he refers to as the relativity of electric and magnetic fields but that should not

confuse the reader: thinking of the magnetic field as something relative and, therefore, something that might not be real, is a

mistake: Feynman’s analysis only tells us we should look at electromagnetic fields as a whole, and how it gets ‘cut up’ – so to

speak – between E and B vectors does, of course, depend on the reference frame.

xvi

This remark will become very relevant in a few moments.

5

The nature of the electrostatic potential

For the clarity of the argument, we must first make the elementary distinctions between:

1. The electrostatic potential at some point r, which we write as (r) and we usually think of as being

caused by some charge q1 at the zero point (0) of the reference frame:

2. The electrostatic potential energy U(r), which is only there if we put another charge q2 at point r

0

xvii

:

3. The electrostatic force between two charges is equal to:

F1 is the force on charge q1, e12 is the unit vector from q1 to q2, and F2 is the force on charge, which is

equal and opposite to F1. The minus sign in the force equation is there because like charges (++ or −−)

will repel each other, while opposite charges (+− or −+) will attract each other. Hence, we have a

directional sign here.

In contrast, the minus sign in the potential energy formula is conventional because we happen to agree

on measuring potential energy by doing work against the force: we could just as well agree on

measuring by doing work (force over a distance) with the force. In addition, we also need to choose the

zero point for energy⎯which is usually taken at r = but one might well choose it at r = 0. Combining

these two choices, we have not less than four potential energy curves depending on what we happen to

agree on in terms of conventions. The nature of potential energy is, clearly, very different of that of

kinetic energy. Indeed, kinetic energy is always positive: it does, therefore, not depend on any

convention. This may suggest that kinetic energy is more real in a physical sense. In fact, our model of

matter-particles (charge in motion) would effectively suggest that, and we do like to think of reality in

terms of charge in motion. However, such philosophical statement is of no relevance here: we must

move on to the next and final definition: the field.

4. The electrostatic field (E) is defined in terms of a force per unit charge (qe). The question is: do we

divide by q1 or by q2 here? The potential is associated with q1, so we must divide by q2, of course!

However, we will usually calculate the field in a very different way. Indeed, while fields must – inevitably

– be defined in terms of forces and charges, we can calculate the (electric) field by taking the derivative

xvii

The reader may find this repetition of standard definitions rather tedious, but I note that some respected authors are not

always as clear when writing about potential and potential energy. The concepts should not be mixed or confused. Potential

energy is measured as work which, in turn, is (an integral of) force over a distance. For a force to be present, we must have two

charges attracting or repelling each other, in line with Newton’s simple but deep third law of motion in classical mechanics: all

forces occur in pairs such that if one object exerts a force on another object, then the second object exerts an equal and

opposite reaction force on the first. That is what is explained above too (point 4).

6

(with regard to the position) of yet another concept that we must introduce: the scalar potential. The

scalar potential is written as φ and, using the (vector) differential operator (gradient) rather than a

one-dimensional derivative (both the position and the electric field are three-dimensional vectors), we

write:

If moving charges are involved and if we, therefore, would have dynamic or magnetic effects, then we

must also bring in the magnetic potential, which is usually written as A. That is not a scalar but a vector

(potential). Let us remind the reader of the formula, which we must assume he has seen before

xviii

:

As a philosophical intermezzo – and to keep the mind sharply focused – we recommend the reader to

try to think of the question: what is real here, and what is the added mathematical/logical

representation of such reality? We think A and A/t might be the real thing, while E and are just

mathematical or logical concepts we use to describe the real thing, but we will let the reader think that

through. Indeed, while the above may all look exceedingly simple and logical for the academic scientist,

we do recommend the reader who does not use these formulas daily to rederive these formulas and

deeply reflect about them to get some feel for all that we are writing about here.

Let us now go back to the problem that we are trying to solve here. Indeed, the reader may still wonder:

what is the problem that we are trying to solve – not approximately but exactly? It is this: our electron

and proton models do not think of charge as something that is infinitesimally small and static. Not at

all: we think of an electron as a (charged) ring current. Hence, then we should, perhaps, have a dynamic

electrostatic field? A field that varies with time because of the lightlike orbital velocity of the pointlike

charge?

Let us push the question further: should we, perhaps, think in terms of a uniform charge density ()

here? Can we assume that, because of the lightlike velocity of the pointlike charge, charge is distributed

uniformly? But, if so, is it a uniform ring of charge, or a uniform disk or – ideally – a uniformly charged

sphere? Let us think this through.

If there would be a charge distribution, and it would be static, we could calculate the potential at point r

as the following integral:

The dVi are infinitesimally small volume elements with charge densities i, and the distances ri are, of

course, the distance between r to these volume elements. However, to get the kind of spherically

symmetric potential that we associate with the potential caused by an electron, we need to either

assume (i) that the ring current model is, somehow, compatible with the idea of a uniformly charged

sphere or, else, (ii) that the distances ri are much larger than the electron scale so that small variations in

charge densities and distances do not matter and/or cancel each other out.

xviii

See Feynman’s overview of all electromagnetic theory in his Lectures (II-15, table 15-1). Note that the equation reduces to

the more usual E = − equation if A/t = 0, i.e., when we are analyzing (electro)statics only.

7

However, it is easy to argue this cannot be the case. Indeed, we do not see how assumption (i) can be

motivated. Calculations of the electrostatic field around a ring current usually focus on the x- or y-

component of the electric field only

xix

but the formulas make it clear that the field is not spherically

symmetric because the formulas do not have an easy 1/r2 dependence. The formula for the z-

component of the electric field of a ring current in the xy-plane, for example, is this:

So, we must, perhaps, resort to – and closely examine – the latter assumption (ii), i.e., the idea of

variations cancelling each other out⎯for the time being, at least. Let us not rush to conclusions,

however, and think some more about it.

What if the charge densities are not only non-uniform but also non-static? We must add the time

variable and, because relativity tells us a field oscillation travels at lightspeed, also introduce the concept

of the retarded time t’ = t – r/c

xx

:

Again, such time-varying potential is not compatible with the spherically symmetric static potential that

we associate with an elementary particle, and it does not look like these small variations in charge

densities and distances do not matter and/or cancel each other out. Of course, we might, perhaps,

assume the spherical potential is a far-field potential only but the question then becomes: could we

design an experiment showing near-field variations at the electron or proton scale?

The answer is: yes⎯of course! Why? Because we can measure such small quantities as Planck’s

quantum of action and the magnetic (dipole) moment of elementary particles with great precision, and

we do so at the femto- and picometer scale.

We must also note, once again, that the E = B/c equation, which we get from Maxwell’s equations, tells

us that the magnitude of the electric field is c times that of magnetic fields! Hence, we may confidently

state that, if there would be small-scale variations in the electrostatic potential around an electron or a

proton, we would or should have discovered these already.

xxi

xix

See, for example, the Hyperphysics discussion on ring currents, from which we copied the formulas in the text. The formulas

are all equivalent but use different concepts: the total charge Q of the ring current or, alternatively, the concept of linear

charge density (), which we can then multiply with 2R (R is the radius of the ring) to obtain the same charge.

xx

For the concept of retarded time, see, for example, Feynman’s Lectures, I-28-2. Feynman also offers various interesting

calculations of potentials based on dynamically changing charge distributions (Volume II, various chapters), including interesting

expansions yielding power series which help distinguishing between near, inbetween and far-field potentials. Of course, none

of these answer the even more fundamental question: what is the nature of the electrostatic force? From an ontological

perspective, all we can say is that it is just there, and the best way to describe it is probably in terms of charge/mass ratios (cf.

the electron and proton mass as fundamental constants of Nature). We will write some more about this in the annexes to this

paper.

xxi

The reader can easily verify that there is no dearth of research in this specific area by, for example, searching on

ResearchGate combining the electron, oscillation, ring current and/or Zitterbewegung keywords. The current precision of

measurement of physics may also be illustrated by the (relative) standard error of the CODATA values for the fine-structure

constant and electric and magnetic constants, which are all the same since the 2019 redefinition of SI units (see footnote vii),

which is equal to 1.510−15. If there is any true metaphysical or ontological ‘uncertainty’ in physics (which we, personally, do

8

This is a rather unpleasant conclusion because all our past research was focused on showing why the

ring current model of elementary particles makes so much sense, so are we now opening an entirely

new can of worms?

We do not think so, and we will show – in a few moments – why. However, we will first inject some

more material to make sure that our conclusions – the answer to all the questions we raised above – are

as solid as can be.

The least action principle in quantum mechanics

Let us think some more about one of the obvious conclusions that comes out of most of the discussions

above: the energy is quantized – that we know – but the question is: how, exactly? Hence, this must also

apply to electrostatic energy but, yes, how should we think of that?

When thinking about the quantization of energy, it is always useful to carefully consider the quantum-

mechanical variant of the least action principle. If we let a charge in some electromagnetic field move

freely, it will follow a trajectory which respects both the least energy as well as the least action principle.

The minimum energy principle is, effectively, usually not sufficient to determine the trajectory which our

charge should follow⎯at least not when conservative forces are involved which is, obviously, the case

here.

xxii

So, we also need the minimum or least action principle to determine the path or trajectory of

our charge: it tells us the charge will lower its total energy (kinetic and potential) by moving along a path

which minimizes (physical) action

xxiii

:

Now, the Planck-Einstein relation tells us (physical) action comes in units of h, and the least action

principle must, therefore, be modified to incorporate this.

xxiv

It is rather sad that the concept of physical

action – and the principle of least action – is remarkably poorly explained in most physics textbooks⎯or,

worse, not at all! Hence, let me quickly give you an intuitive explanation and interpretation of it – with a

warning, however: the explanation will be precise or, let us say, scientifically correct, but the

interpretation of it, however, may come across as philosophical or, worse, speculative!

Consider this: to move from one point to another, some energy is needed over some distance or – what

amounts to the same – some momentum during some time. We think this energy and/or momentum is

extracted from the field and effectively comes in units of h:

not think is the case), then this very tiny value might reflect it. However, we think future technology and experiments will

further reduce its value and we, therefore, do not think of it as some metaphysical or ontological ‘uncertainty’.

xxii

See our paper on the nuclear force hypothesis, which considers non-conservative potentials, which we end up dismissing.

xxiii

For a full development of the least action principle – both from a classical as well as a quantum-mechanical perspective – we

refer to Feynman’s Lectures, Volume II, Chapter 19 (The Least Action Principle). We think its central place – right in the middle

of the middle Volume – is no coincidence. The reader should note that the minus sign in the KE − PE expression depends on the

convention we choose for the sign of the potential energy. For the reader who will go through Feynman’s development, we

must note that we have the impression that Feynman is not always consistent in this regard (choice of sign for (potential)

energies).

xxiv

I love the German term for physical action: Wirkung. It just captures the concept so much better. There are good reasons

why scientists such as Einstein preferred German over English, although we must, of course, note that German was then –

much more than now – an oft-used language in the sciences.

9

h = 6.62607015×10−34 Nms = ΔE·Δt = Δp·Δx

It is here that, for the first time in this paper, we write out what is referred to as the Uncertainty

Principle which – in our view – is not about some ontological certainty. However, we wrote about that

elsewhere, and so we will not dwell on it.

xxv

The question here is this: does this h = ΔE·Δt = Δp·Δx relation – combined with the least action

principle – imply that motion itself comes in discrete bits, and that the energy and momentum in the

fields must come in discrete amounts too?

We think it does. We think that it is very hard to escape the conclusion that the motion of a charge is

not continuous: it must come in very small – but not infinitesimally small – space-time bits. Discrete

amounts that are given by the h = 6.62607015×10−34 Nms = ΔE·Δt = Δp·Δx equation. Of course, as we

explained above, that does not imply that the fields themselves cannot be continuous: the flux comes in

discrete values, but the fields can be – and probably are – continuous. Indeed, we repeat, once again,

that we do not believe in virtual field-particles.

xxvi

On a philosophical note, one might say this finally solves Zeno’s paradox: the mathematical description

of motion is a continuous function, of course, but the Uncertainty Principle (or, let us refer to it, as

complementary rather than uncertainty) strongly suggests that the underlying reality of motion is

discrete and that, then, should explain why we cannot actually keep dividing distances into ever smaller

bits to prove that Achilles can never overtake the tortoise.

xxvii

However, we keep digressing from the matter at hand. What is that we can say now about it? It is this:

while the

and

formulas above suggest that small

variations in charge densities and distances should result in small spatial asymmetries in the

electrostatic potential, we argue these variations are smaller than the ΔE·Δt = h threshold for an

actual field variation.

In other words, we think the formulas above – which calculate the electric or electrostatic potential

based on (static or dynamic) charge densities – are only applicable to larger-scale densities and

distances: the oscillation of the charge inside an electron or a proton is, in each and every way,

elementary, and we should, therefore, not worry about the theoretical question we raised in this paper:

the Planck-Einstein relation answers it.

xxviii

xxv

We prefer the concept of complementarity over the concept of uncertainty because we feel uncertainty is a rather vague

concept (see our paper on Uncertainty and the geometry of the wavefunction). In contrast, complementarity does refer to

clear-cut mathematical concepts such as complementary variables or functions. It is these clear-cut mathematical concepts –

rather than any metaphysical or ill-defined uncertainty – which are used in, for example, the proof of the Kennard inequality

(this proof is based on the preconceived notion of a composite wave (a wavepacket rather than a precise wave) and then just

relates the two distributions through the Fourier transform).

xxvi

Our previous papers make it abundantly clear that we do not think highly of modern quantum field theory.

xxvii

For an overview of the various expressions of Zeno’s paradox (or, we should use the plural, Zeno’s paradoxes), see, for

example, the Wikipedia article on it. The reader may want to google other references as well.

xxviii

This argument is remarkably similar to Feynman’s calculation of the size of a hydrogen atom based on the Uncertainty

Principle, which he himself refers to as a heuristic argument. We do not think of as being merely heuristic: we think it is

theoretically solid, but interpretations of it may differ. In any case, Einstein’s 1905 article on special relativity is often referred to

as being heuristic too, so we should not look down on so-called heuristic arguments.

10

Let us add some more remarks now that we are here. In our papers, we mentioned several times that

we do not exclude that the actual motion of the hypothetical pointlike charge inside an electron or a

proton might be chaotic or irregular. However, it must be regular enough because phenomena such as

Compton scattering and quantum-mechanical interference are real and are associated with very precise

concepts such as frequencies, wavelengths, energy levels and inertial mass⎯even if we think of these

concepts as being statistically meaningful. Hence, the idea that small amounts of energy (ΔE) and

momentum (Δp) might be borrowed and returned in very small time and space intervals (Δt, Δx), in

full respect of the ΔE·Δt = Δp·Δx ħ

xxix

complementarity principle, makes a lot of sense to us.

xxx

[…]

So, that is it, then? Yes. Problem solved. End of paper.

Let us sum it all up in a summary section, adding some more formalism here and there, and highlighting

the most salient conclusions. For the reader who would not be satisfied after that, we may point to the

two annexes to this paper, which provide more reading material to the reader who likes to dig in even

deeper.

Conclusions

The two most basic equations in the realm of quantum mechanics are, most probably, (1) the Planck-

Einstein relation and (2) Einstein’s mass-energy equivalence relation which, using the obvious

mathematical = 2f relation

xxxi

, we can combine as:

xxix

We also consider non-linear motion here, which is why we use ħ = h/2. An additional ½ factor is often added because of the

distinction between kinetic and potential energy but, within a cycle, these will vary between 0 and the total energy of the

system and we, therefore, think the ½ factor should not be used here.

xxx

The reader may wonder what our opinion is about this intra-particle motion of charge: is it regular or irregular,

chaotic/indeterministic or deterministic? We believe it is regular, and deterministic. However, because of the high frequency

and speed (the charge moves at lightspeed, we argue), we cannot determine the exact position of the charge at any point of

time, and we also do not know the initial condition of the system in any practical problem. Hence, it appears as random or –

using scientific language that we, personally, do not find all that scientific – as quantum-mechanical uncertainty. We would

prefer a term such as ‘quantum-mechanical immeasurability’ or something. We repeat this remark elsewhere in this paper, but

it is an important point to reflect on, so we do not mind repetition of it.

xxxi

When expressing a frequency as an angular frequency, we might say we are measuring time in radians rather than seconds.

Think of it like this: based on the frequency, we can use the cycle time T = 1/f as a natural time unit. Of course, this natural time

unit will still be expressed or measured in seconds, but we may now think of τ = T/2π as the cycle time expressed or measured

in radians rather than seconds. Of course, the radian is a length unit so we should also define a natural length unit. Now, we will

want to think of elementary particles in terms of pointlike charges whizzing around at lightspeed (such charges have zero rest

mass but acquire relativistic mass because of their motion) so we will want to associate an orbital radius a = /c with these

particles and consider that to be the natural distance unit. The radius will then correspond to the length of a radian (1 rad), and

the length λ = 2πa (circumference of the circle defined by a) is then the distance over the loop. It is then easy to see that the

cycle time T will be associated with λ = 2π rad, while will be associated with a = 1 rad. The orbital velocity c can then be

written as c = a = λf. The radius a defines the effective radius of interference of the charge (think of an electron here) with,

say, photons (think neutrinos in case of a proton). We refer to it as the Compton radius which, of course, is just the reduced

Compton wavelength: a = /c = E/ħc = mc/ħ.

11

The c2/h ratio is equal to 1.356392489652131050 (not approximately but exactly), which is a rather

humongous number, as you can see.

xxxii

It also has the somewhat weird (kgs)−1 dimension, which may

not say much when you do not think too much about it, but so we are talking about a frequency per unit

mass here. Yes, it is one of those cases in physics where a simple verbal expression (frequency per mass

unit) says more than the mere expression in SI units. So, what can we say about it?

Nothing much, perhaps, except that it probably shows the true nature of mass, quite apart from it being

a measure of inertia (appearing in Newton’s first law of motion which – in its relativistically correct

form

xxxiii

– is written as F = dp/dt = d(m·v)/dt): matter is nothing but charge in motion. Such

interpretation is not mainstream but, when everything is said and done, it is consistent with Wheeler’s

‘mass without mass’ ideas and – more importantly, probably – with the 2019 revision of the system of SI

units, in which mass also appears as a derived unit from more fundamental constants now, most notably

Planck’s constant.

The f/m ratio is, of course, valid for all matter or – let us be precise – for all (stable) elementary particles

(note that the electron and proton (and their anti-matter counterparts) are stable, but the neutron is

not). However, it is important to note that, while the f/m ratio is the same for both the electron as well

as the proton mass, the q/me and q/mp ratios are, obviously, very different. We, therefore, do associate

two very different charge oscillations with them: we think of the electron and proton as a two- and

three-dimensional ring current, respectively. Hence, while these specific oscillator equations are,

theoretically and mathematically, compatible with any mass number, we do not think of the electron

and proton energies as variables but as constants of Nature themselves.

Now, this description of elementary particles in terms of a ring current or a two- or three-dimensional

oscillation is precise but, of course, we cannot say anything about what might really be happening: all

that we know is that the motion of the pointlike charge is regular enough to give meaning to our

concepts of frequency

xxxiv

, orbital motion, magnetic moment, and the concepts of mass and energy

itself. We also know that any variations or irregularities do not result in any loss of energy through

radiation or other oscillations of the electromagnetic field outside of the elementary particle. We do not

observe any electric dipole moment, for example.

We think this can only be explained because the very small variations in energy (E) and/or (linear or

angular) momentum (p) are associated with space and time intervals (x or t) that are also extremely

small. In other words, we think these variations are smaller than the ΔE·Δt = h threshold that would be

required to result in an actual field variation.

xxxii

I checked on Wikipedia, and a number with 50 zeros would be referred to as one hundred quindecillion (using the Anglo-

Saxon short scale) or one hundred octillions (using the non-English long scale of naming such astronomic numbers).

xxxiii

We write it out here because we will use this in the other annex, where we will talk about the relativistic oscillator. The

formula is relativistically correct because both m and v are not constant, but functions varying in time as well. That is why we

cannot take them easily out of the d()/dt brackets as for now: we will do so in the mentioned annex on the oscillator.

xxxiv

As for our interpretation of this, we do think the frequencies and the motion are determined – not in a way that we can

measure them, nor can we determine the position of the charge at any point of time. Why not? Because of the high frequencies

and, yes, the incredibly velocities (lightspeed, in fact). We can also not determine the initial conditions of the system for the

same reason. We believe this explains quantum-mechanical ‘uncertainty’. In short, we do not think of matter-waves as

wavepackets with an undefined frequency: see our paper on de Broglie’s matter-wave for an explanation of what we think of as

an erroneous interpretation of what matter-waves might actually be. Also, we would prefer terms like ‘quantum-mechanical

immeasurability’ rather than philosophical or ontological terms such as (quantum-mechanical) uncertainty or indeterminism.

12

The Planck-Einstein relation can thus be associated with the (in)famous uncertainty principle, which we

should probably always write as two complementary relations

xxxv

:

ΔE·Δt h

Δp·Δx h

h = 6.62607015×10−34 Nms

xxxvi

Let us now go straight to the crux of the matter at hand. The equations above show where classical

electromagnetic theory – Maxwell’s equations – hits a logical roadblock. Indeed, Maxwell’s equations

give us the electric and magnetic field vectors (E and B) based on the concept of charge densities and

(electric) currents j:

xxxvii

:

Maxwell’s equations

E = /0

Gauss’ law

E = −B/t

Faraday’s law

B = 0

no magnetic charges (no flux of B)

c2B = j/0 + E/t

We could, for example, calculate the electrostatic or scalar potential from the

and

formulas – which we get from Maxwell’s equations – but these should result in

small spatial asymmetries in the electrostatic potential⎯which we do not observe.

xxxviii

The conclusion,

therefore, must be that these variations – be they variations in energy (E), momentum (p), fields (E

or B), currents (j) or densities () – are effectively too small in an ontological sense: we can only

think of them as something virtual

xxxix

: they do not combine to produce a real current, a real field

oscillation, some real momentum, some real energy⎯nothing that we can think of as a being something

physical: whatever these variations are, they lack (ontological) wholeness, coherence, or unity.

xl

Of course, what remains to be done is to relate the notions above to the usual quantum-mechanical

mathematical formalism, such as canonical commutation relations. The gist of the matter is simple

enough, however. The ΔE·Δt h and Δp·Δx h relations establish pairs of related variables, and the

xxxv

This is in line with other remarks we made: it would probably be better to replace the term ‘Uncertainty Principle’ by

‘Complementary Principle’ or some other term that better reflects the mathematical reality of complementarity between both

classical and quantum-mechanical variables, which is what we have here.

xxxvi

This is, once again, an exact value. We think the 2019 reform of the system of SI units might complete mankind’s

understanding of Nature.

xxxvii

As mentioned, if the reader is familiar with the same equations expressed in terms of the scalar and vector potential, they

would be equally enlightening.

xxxviii

The reader might think we would not be able to measure such variations because of the very same Planck-Einstein or

uncertainty principle, but we think such reasoning is erroneous: we are able to measure quantities as small as Planck’s quantum

of action itself, or as small as the magnetic (dipole) moment of elementary particle, and with great precision⎯at the femto-

and picometer scale, to be precise! We must also note that the E = B/c equation, which we get from Maxwell’s equations, tells

us that the magnitude of the electric field is c times that of magnetic fields! Hence, we may probably confidently state that, if

there would be small-scale variations in the electrostatic potential around an electron or a proton, these would have been

measured already.

xxxix

We do not like to use this term, because the reader may think we support the concept of virtual particles as used quantum

field theory, which we do not. We made this remark several times already, but it is an important one, so – here also – we do not

mind repetition.

xl

We are using entirely non-scientific language here, but we think the language might meet the definition of pragmatism⎯in

the sense of our ideas being practical, or practical enough.

13

relation makes it clear we cannot simultaneously reduce energy and cycle time differentials (or

momentum and distance differentials) and think of them as (possibly) infinitesimally small quantities.

The conclusion, hence, is this: Planck’s quantum of action tells us infinitesimally small differentials are

a mathematical concept only, just like infinite velocity or infinite distances are mathematical concepts

only.

xli

We may rephrase this as follows:

Maxwell’s equations only make sense as soon as the concepts of charge densities (expressed in

coulomb per volume or area unit: C/m3 or C/m2) and currents (expressed in C/s) start making sense,

which is only above the threshold of Planck’s quantum of action and within the quantization limits set

by the Planck-Einstein relation. This leads us to a simple and philosophical summary of all of physics:

Quantum Mechanics = All of Physics = Maxwell’s equations + the Planck-Einstein relation

xlii

xli

We hesitate to equate mathematical concepts to logical concepts because – just one example – the idea of infinite velocity is

logically not consistent: an object traveling at infinite speed is everywhere along its trajectory at any point in time⎯it is,

therefore, everywhere and, therefore, nowhere (we cannot localize it). Also, an infinitesimally small object has no (physical)

dimension whatsoever and, therefore, cannot exist. Mathematical concepts are, therefore, not always logical⎯not in a

common-sense (or pragmatic) sense, at least.

xlii

What about special and general relativity theory? SRT (the absoluteness of lightspeed) comes out of Maxwell’s equations: all

that Einstein did was to provide a rather heuristic argument to make it more palatable, we might say. [As an added benefit, we

got Einstein’s mass-energy equivalence relation out of it, of course!] And then you may think of GRT as yet another heuristic

argument combining SRT and Mach’s principle. So, yes, we do have it all here!

14

Annex I: The Complementarity and Uncertainty Principles

We did quite a few more ontological or epistemological papers on physics already

xliii

, but we never paid

much attention to the complementarity principle. In fact, we also did not pay much attention to the

uncertainty principle either, so let us try to relate both here – because we do think of both principles as

being related, if not the same altogether! [The advanced reader should also relate these notions to the

geometry of complex or quaternion algebra (imaginary units (i, j and k) as rotation operators

xliv

),

quantum-mechanical operators, wavefunctions, statements on CP- and T-symmetry, the constraints on

S-matrices or Hamiltonians, etcetera. We did that in previous papers, and so we refer the reader there.]

Let us run through the key model, or the mother of all examples, so to speak. We may, effectively,

elucidate the canonical commutation relation for momentum and position a little bit, using the by now

familiar ring current model for an electron, which is illustrated below (Figure 1).

Figure 1: The Zitterbewegung model of a charged elementary particle

We think of the (elementary) wavefunction as representing the position (r = ae−iθ = acosθ + iasinθ)

xlv

)

of the pointlike charge on its orbit in terms of its coordinates x = (x, 0) = (acosθ, 0) on the real axis and y

= (0, y) = (0, asinθ) on the imaginary axis (y = ix). Now, we also have a momentum vector p = mv = mc

here which, considering the geometry of the situation and writing the angular frequency as a polar

vector = v/a = c/a, we can write as:

p = jmc = jma

Now, this combination of a new imaginary unit

xlvi

(j) and a (vector) product of a radius and angular

frequency becomes a bit complicated in terms of notation, so we will just write the equation in terms of

magnitudes:

Now, we must, of course, now add the concepts of (linear) momentum in quantum physics: we talk

linear motion and, therefore, linear momentum, then. We will also be talking classical linear velocity of

xliii

See, for example, Ontology and Physics, February 2021.

xliv

On the usefulness of quaternion math in physics, see our paper on the nuclear force and quaternion math.

xlv

We use boldface for the sine and cosine because we think of them as vectors: they have a magnitude and a direction here.

The imaginary unit (i) then functions as a rotation operator. This is readily understandable when thinking of sine and cosine as

essentially the same (cyclical) function⎯except of a phase difference of 90 degrees (/2).

xlvi

We use a different imaginary unit as rotation operator here (j), so as to distinguish it from i. Needless to say, i and j are

orthogonal to each other so, thinking of them as vectors and following Hamilton’s notation for basic quaternions, we may write:

ij = 0 = 0.

15

the electron as a whole⎯as opposed to the (hypothetical) motion of this pointlike charge which we

introduced to explain mass, radius, magnetic moment and other so-called intrinsic properties of an

electron.

xlvii

We, therefore, we need to understand its relativistic nature (it is an invariant under Lorentz

transformations of the reference frame) and understand how the introduction of classical linear motion

amounts to adding a linear component – think of it as drift velocity – to the velocity of the pointlike

charge (see Figure 2).

Figure 2: The (ring current) radius of an electron decreases with increasing velocity

xlviii

Now, the phase θ of the wavefunction can be written either as (i) a function of the rest energy (E0) and

of time (t’) in the reference frame of the particle, or (ii) energy (Ev), time (t) and its linear momentum (p)

associated with its classical (linear) velocity and relativistic mass as a moving particle. The equation

below (and the illustration above) effectively shows how classical motion adds a linear component to

the orbital motion of the pointlike charge.

It is now easy to see how all of the information is in the phase: the Evt - pxx and E0t’ expressions have

the (physical) action dimension (Nms), and the division by ħ (Planck’s quantum of action, and also a

natural unit of angular momentum) ensures the phase comes out as a number (in radians, to be

precise). We can now either integrate or differentiate with respect to t or with respect to x. If we choose

the x-axis of the reference frame so as to coincide with the direction of linear motion (which is given by

x = vt), we can, for example, differentiate the (x, t) → −iθ = −i(Evt − pxx)/ħ function with respect to x:

This may look like a weird expression, but there is absolutely nothing weird about it: what we wrote

above is a quick and intuitive derivation of the quantum-mechanical operator. Indeed, operators

abstract away from the function they are operating on, so we just leave the wavefunction ψ out of the

expression, and we get the quantum-mechanical momentum operator:

xlvii

See, for example, our paper on the Zitterbewegung hypothesis and the scattering matrix. We may never be able to

conclusively prove the validity of the ring current or oscillator model of elementary particles (a common critical remark by

mainstream physicists), but we feel the easy explanation of all of the properties which, in mainstream physics, remain intrinsic

(read: mysterious) has quite some value.

xlviii

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

16

Is it that simple? Yes. We refer the reader to Feynman’s derivation of the other quantum-mechanical

operators (Lectures, III-11-5) and fully agree with his derogatory appreciation of these things: “All the

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

hocus-pocus.” Let us now discuss the position operator. Because it is even simpler, it is even weirder:

Applying the operator to the wavefunction, we get the following trivial identity:

Now, we promised you to show how canonical commutation relations work, so let us describe the

general logic here. Clever mathematicians thought of combining operators (preferably operators related

with so-called conjugate variables

xlix

), say operator A and B and define the following mix of products as a

commutator

l

:

Now, if A and B would be numbers or functions, AB − BA would be zero, of course, but because we are

talking different beasts here (x/x (/x)x = x/x = 1, for example), this is not necessarily the case in

quantum mechanics. We can see why when taking our new position and momentum operators as an

example. So then we have a commutator expression like [x px] = xpx − pxx and we can write it all out and

see what we get. One of the nice things about expressions like the one above is that we can relate them

to other operators, such as angular momentum operators. However, we will refer to Feynman’s Lecture

on operators here, from which we also copy the ‘typically Feynman’ comments on the [x px]

commutator expression below.

xlix

As you can see, everything is conjugated or complementary in real physics: charge, force, energy, momentum, etcetera. We,

therefore, attach relativity little value to this kind of rocket quantum mathematics which, in our not so humble view, do not add

much to truly understanding what (quantum) physics is all about.

l

The Wikipedia article on canonical commutation relations is rather sloppy on the use of the hat with the operator symbols, so

we are not ashamed of being equally sloppy. Richard Feynman does much better in terms of consistently distinguishing

between operators and operators operating on something (usually that something is the wavefunction) in his Lecture on

operators (III-20).

17

We have nothing much to add to Feynman’s exposé except, perhaps, a short reflection on the

meaningfulness of these operator expressions if, paraphrasing James Jeans

li

, we leave these operators

“hungry for something to differentiate.” Let us take the [x px] = xpx − pxx expression once more to see

what we can say about it if there is no wavefunction to operate on. We get this:

This expression looks very different from the [x px] = xpx − pxx = −iħ expression! You can now appreciate

we need to multiply both sides with some function f = f(x, t) to get some meaningful result, so we write:

And then we need to find a function f for which this equation works out, and that function is, of course,

the wavefunction f = ψ(x, t). Why? Because that is the function we started out with when deriving the x

and px operators in the first place, of course!

What is the point that we are trying to make here? The point is this: a lot of these arguments involving

high-brow quantum math are merely circular or – as Feynman puts it – “elementary hocus-pocus.” So

never be scared to question any logic, and surely never be impressed!

li

The reference to Jeans here is Feynman’s, though. We did not read James Jeans.

18

Annex II: Relativistic kinetic energy and momentum

The concepts of kinetic energy and momentum are related but how exactly? In a non-relativistic analysis

(low velocities – hence, mass can be treated as a constant inertia to a change in the state of motion),

momentum appears as the derivative of kinetic energy with respect to velocity. We are, perhaps, not

quite used to such derivatives (we usually differentiate with respect to time or position), so let us write

it out:

This offers a nice (intuitive) way of interpreting kinetic energy and momentum, and the relation

between these two concepts. Why are both concepts relevant? Because we will usually need both the

momentum as well as the energy conservation law to analyze and explain quantum-mechanical

phenomena.

lii

Of course, we will want to analyze things in a relativistically correct way, so let us take a

specific example. Imagine what is referred to as a relativistic spring: a mass swinging back and forth on a

spring, but with velocity high enough to cause relativistic mass increase.

Figure 3: Harmonic oscillator

Assuming a constant restoring force (Hooke’s law

liii

), the relativistically correct force law is:

F = dp/dt = –kx with p = mvv = γm0v

The mv = γm0 varies with speed because γ varies with speed

liv

:

lii

Think of the derivation of the Compton wavelength in the context of Compton scattering, to just give one example. The

concept of momentum is useful because it keeps track of directions⎯if we write it as a vector equation, of course, which is

what is done in the mentioned example (derivation of Compton wavelength), and in other arguments: p = mv. Energy, in

contrast, is just a scalar⎯not a vector. Apart from the directional extra in the momentum formula, it is also so good to have

two relations with v (and related variables) if you want to find some kind of unique equilibrium value for the whole. The

number of equations should, indeed, match the number of the degrees of freedom in the model of the system. This is, in fact,

why going through high-brow physical explanations is so nice: the model never is over- or under-determined⎯it is always just

right and complete (Occam’s Razor Principle).

liii

In our mass-without-mass models of elementary particles, we also assume a constant centripetal force keeping the pointlike

charge in place, so we do not have any issue with Hooke’s law in the context of oscillators.

liv

The t and t’ refer to the time variable in in our reference frame (t) – which we may also refer to as the coordinate time – and

the time in the reference frame of the object itself (t’) – which is usually referred to as the proper time. We may want to use

these two concepts of time in a later development, so it is good to introduce them here.

19

Hence, both m and v vary with the position and time variables. From the force law, we can derive an

energy conservation equation. We do so by multiplying both sides with v = dx/dt, and I will skip the

math, but you should be able to verify the following relation(s)

lv

:

Let us look at the terms in this energy conservation equation. We recognize the potential energy: it is

the same kx2/2 formula we get for a non-relativistic oscillator. No surprises: potential energy depends

on position only, not on velocity, and there is nothing relative about position.

lvi

However, the (½)m0v2

term that we would get when using the non-relativistic formulation of Newton’s Law is now replaced by

the γmc2 = γ2m0c2 term

lvii

, which incorporates that velocity impacts kinetic energy (and momentum) in

two ways: directly, and through the relativistic mass effect.

The equation above tells us the sum of kinetic and potential energy is and remains constant, although

their functional shape is very different from what we would get when analyzing a non-relativistic

oscillator. However, we leave it to the reader to play with a graphing tool and appreciate the weirdness:

we do not get the usual continuous curves for PE and KE. The interesting situations are, of course, when

v = c. At that point, the equation blows up because we get infinities in the energy and in the Lorentz

factor. We refer to Clark’s paper (link in footnote) for interesting numerical analysis, as well as some

analytical analysis remarks as well. All we will do here is to point, once again, to the unfamiliar kinetic

energy formula in the energy conservation equation: KE = γmc2 = γ2m0c2. What is that? We are not sure

ourselves, to be honest.

For starters, we may want to explore the difference between non-relativistic and relativistic kinetic

energy by expanding the equation for the total or combined energy (E = mc2) as a power series using the

binomial theorem (with a fractional power

lviii

):

lv

We are grateful to a bright BA student, Dylan B. Clark, for making things so simple. Most papers on relativistic oscillators are

very hard to read, but his BA thesis on relativistic springs (2011) is a true delight. In case you do not have access to the arxiv.org

papers, he repackaged it in a paper on academia.edu (2012), which you can find here. For the reader who cannot get through

the math, we do add the detail at the end of this annex.

lvi

You may want to think about this because the distance between two points can vary because of relativistic length

contraction.

lvii

We note that previous versions of this paper had an error: they did not have the Lorentz factor. We apologize. It is the main

reason we revised this paper.

lviii

See: Feynman’s Lectures, I-15-8, and I-15-9 (relativistic dynamics. The total energy is given by E = mc2 = m0c2, so the term we

expand is m0:

This is multiplied with c2 again to obtain the series in the text (note that we might denote relativistic mass by m = m = m0 for

improved readability, but we think the context is clear enough). To make it easy for the reader to verify calculations, we insert

the formula for the binomial expansion for negative integral or fractional powers:

Note that such binomial expansion is only possible if −1 < x < +1, which is the case here: 0 < v2/c2 < 1 −1 < −v2/c2 < 0. This is a

strict inequality so v cannot quite reach c. This refers to our remark in footnote above about the singularity when v = c.

20

We recognize rest energy (m0c2), non-relativistic kinetic energy (m0v2/2), and higher-order terms which

alternate in sign, so we do not infinite values for the energy. We will leave it to the reader to interpret

these terms in the context of this discussion.

lix

One idea is to use this equation to differentiate this new

relativistic concept of kinetic energy (γmc2) with respect to the position or velocity variable. That will

probably be easier using the relative velocity = v/c. So, we should work out derivatives like this:

The reader may try other derivations and, perhaps, he or she will find meaningful interpretations. Of

course, you may wonder: are such power series relevant? Again, we are not sure: all we want to do here

is to trigger the imagination of the reader and encourage him to think this through for him- or herself.

Being able to play with power series is, effectively, another important tool in the toolbox of a (quantum)

physicist.

lx

Another (related or not) interesting exercise related to power series would be to see whether we can

relate qe2 or even 1/r in the formula for the energy of a charge in a (static) Coulomb field (given below)

to some power series

lxi

:

Why? Because all energy has an equivalent mass, and so we may uncover some new relationship

between charge and mass perhaps, so we might write the total energy of some charged particle as E =

m0c2 + U(r) and then look at these power series (but make sure all constraints for meaningful

development of power series are respected) and recombine and see what comes out.

lxii

We do not

expect you to find anything new, but that should not stop you from trying!

By way of conclusion, we should probably say a few more words about the relativity of magnetic and

electric fields. Feynman offers an interesting analysis of what he refers to as the relativity of electric and

magnetic fields, in which he tells us we should look at electromagnetic fields as a whole because the way

it gets ‘cut up’ – so to speak – between E and B vectors depends on the reference frame. So, perhaps,

we might choose another reference frame in which we do not see any magnetic field but an

electrostatic or electrodynamic field only? And, perhaps, we would then not have to worry about why

lix

Feynman (reference above) gives an example of how the heat inside of a body might increase because of the classical m0v2/2

term: “When the temperature increases the v2 factor increases proportionately, so we can say that the increase in mass is

proportional to the increase in temperature.”

lx

For an overview of various mathematical techniques, see our All of Quantum Math paper, but these will not help much: one

just has to calculate and get through the grind, and we have not done that yet so, yes, we do invite the reader to try his or her

hand at it.

lxi

U(r) = V(r)·qe = V(r)·qe = (ke·qe/r)·qe = ke·qe2/r with ke 9109 N·m2/C2. Potential energy (U) is, therefore, expressed in joule (1 J

= 1 N·m), while potential (V) is expressed in joule/Coulomb (J/C). Since the 2019 revision of the SI units, the electric, magnetic,

and fine-structure constants have been co-defined as ε0 = 1/μ0c2 = qe2/2αhc. The CODATA/NIST value for the standard error on

the value ε0, μ0, and α is currently set at 1.51010 F/m, 1.51010 H/m, and 1.51010 (no physical dimension here), respectively.

lxii

One of the results of such modeling should be some q/m ratio (charge per unit mass, expressed in C/kg). Unfortunately,

electron and proton mass are both associated with the unit charge, but their mass is very different. For some preliminary

reflections on this, see our paper on the nuclear force hypothesis.

21

we do not see any variations and why the electron – as an oscillation of a pointlike charge – is not losing

any energy?

The answer to the first question should be: yes, of course! However, the answer to the second must be

negative: if there are tiny variations in field energy, then they would show up in whatever reference

frame you would want to use.

lxiii

Addendum:

We promised to give you the detail of the derivation of the relativistic energy conservation equation in

the context of the relativistic oscillator. Indeed, the author of the paper (Clark, 2012) on it does not give

it to the reader, so we do it here. The first step is to multiply the relativistically correct force law with v

on both sides. So, we had this:

F = dp/dt = –kx with p = mvv = γm0v

Multiplying with v yields:

So now we need to prove the following two identities:

The second equation is easy: just calculate the derivative, and you see it is fine. Let us prove the first

equation by working backwards too. We just calculate the derivative:

Done! The point to note is, once again, that we do not have γm0 in the expression, but γm = γγm0 = γ2m0.

We have seen this expression before, but you may want to look at it more in depth!

[…]

OK. Let me help you in your search to make sense of this by reminding you of two other formulas, one

involving potential energy, and another relating the force to potential energy, as opposed to kinetic

energy. Before we do that, let us remind ourselves that kinetic and potential energy add up to the total

energy and, as such, we must think of them as each other’s mirror image, so to speak. Let us insert an

lxiii

We just apply plain logic here. Again, we did not work out the math but invite the reader to do so! Feedback is welcome on

my ResearchGate site!

22

illustration of how potential and kinetic energy effectively vary and add up over a full cycle of an

oscillation:

Figure 4: Kinetic (K) and potential energy (U) of an oscillator

lxiv

Of course, you should be worried about the relativistic velocities and how these may or may not blow up

these graphs. Indeed, when v would equal c or, worse, would be superluminal, then we will be in deep

trouble because of singularities and/or discontinuities. However, let us not worry about that now. Just

make sure you can relate Figure 3 and Figure 4: imagine the spring oscillating back and forth between

±A, and the point charge or mass turning at those points and reaching maximum velocity (possibly v = c)

at the equilibrium point which, of course, is no longer an equilibrium point of the oscillation. [Needless

to say, we hope that you inferred that we think of a point charge rather than some mass here.

lxv

]

Now, there are several ways to relate mass, force, and energy. The simplest is the law of motion itself,

according to which mass is the ratio of force and acceleration. However, we will use the one we started

out with, which is force as the time rate of change of the momentum.

lxvi

So let us write the relativistic

force law once again, which defines a force as that what changes the (relativistic) momentum of an

object: F = dp/dt.

Now, if we take F to be the force in the direction of motion – which is the case here – then we can write

this in terms of the magnitudes of F and p (F = dp/dt), and we avoid vector math, including vector

operators. Then we have simple derivatives d/dx or ∂/∂x instead of the gradient or vector differential

operator = (∂/∂x, ∂/∂y, ∂/∂z), for example. The derivation is somewhat less straightforward than you

might expect at first but by using the product and chain rule, and with some re-arranging, you should be

able to prove the formula

lxvii

:

lxiv

You will find this diagram in many texts, but we took this one from the https://phys.libretexts.org/ site—excellent hub for

open-access textbooks.

lxv

Also, we would probably want to use other units than the joule (J) or the meter (m).

lxvi

When several directions are involved, the mere concept of mass does not work very well in a relativistic or even non-

relativistic analysis of forces because one loses track of the directional element. That is evidenced, for example, by Einstein’s

rather convoluted argument about ‘longitudinal’ versus ‘transverse’ mass (think of some kind of mass vector, with a mx, my and

mz component) for a slowly accelerating electron in his 1905 article on special relativity: On the electrodynamics of moving

bodies, p. 22. The Wikipedia article on relativistic mass or on the (kinetic) energy concept offers a good overview of the history

of ideas here, including Einstein’s intuition that mass – as a pure measure of inertia – is not only variable (cf. the dependence on

the reference frame) but some kind of 3D concept as well. The idea is not ridiculous, as we will show in the next footnote(s). It

just does not make all that much physical or mathematical sense.

lxvii

In case you want to check your calculations, we once again recommend the online (free) LibreTexts Physics textbook.

23

This relativistically correct definition of a force does away with the definition of a force as the product of

mass and acceleration (and of mass as the ratio of force and acceleration), which we get by substituting

the rest mass m0 by the relativistic mass m0 in Newton’s F = m·a formula.

lxviii

It, therefore, confirms the

definition of mass as a simple scalar quantity measuring the inertia of a charge to a change in its state of

motion.

lxix

Let us go back to the matter at hand. The force formula above has a squared Lorentz factor too, so it is

tempting to try to relate our KE = γm = γγm0 = γ2m0 formula to. In fact, we actually have a cube Lorentz

factor in that force formula: F = γ2mva = γ3m0, so that suggests a higher or lower order of derivation.

Another formula comes to mind here: we can always calculate the force as the (negative) derivative of

the potential energy. Why? Because we define or measure potential energy as the work that is done

against the force over some distance.

lxx

The formula is given in the graph in Figure 4 (let us go with the U

symbol for potential energy here, only because PE is a bit cumbersome and ambiguous to use in

formulas):

How can we relate this to kinetic energy? We can always choose a new reference point for the potential

energy: a new U = 0 reference. For example, if we denote the total energy (KE + PE) by C, then we can

shift the energy scale or axis (look at Figure 4 once more) by the total energy C by C/2. In that case, KE

and PE add up to zero, always. The operation may come across as somewhat weird, because it implies

that both potential as well as kinetic energy can have negative values – but you should just go along

with it. It is quite consistent with the basic energy conservation equation we derived, shown below once

again:

We can integrate this to find a whole family of functions which will only differ from each other by some

constant, so our shifting of the energy scale should be OK (from a mathematical point of view, at least).

If we do that, then PE and KE add up to zero and, hence, we get this:

So, the derivatives, with respect to the position x, of both functions should be the same. Let us calculate

them. First, the potential energy term kx2/2:

lxviii

From the above, it should be clear this is not the correct force law: it yields F = mv·a= m0·a. This equation does not have the

required cube of the Lorentz factor (F = 3m0·a).

lxix

It is obvious that, if we would write m as a vector quantity (m), we would have a vector dot product m·v = mxvx + myvz + mzvz

= px + py + pz. Hence, it is quite convenient we may consider mx, my, and mz to be equal to one and the same scalar quantity mv,

because then we can simply write px, py, and pz as px = mvx, py = mvy, and pz = mvz, respectively.

lxx

We know that the path does not matter when conservative forces are involved, which is the case here. However, this remark

does not matter because we consider simple linear motion here.

24

That looks good because it is, of course, what we started off with: an oscillator or spring with a force

that obeys Hooke’s law. Let us now try to find the derivative of the kinetic energy with respect to the

position variable. We must apply the chain rule here once again: the Lorentz factor is a function of

velocity (v) and velocity is, in turn, a function of the position (x). Let us do it:

We can see easily see that the dγ/dv factor can be calculated, but we must find the velocity function as a

function of x to calculate the dv/dx factor. In other words, we must write v as v = v(x). That is not easy:

we may think of the acceleration here, but acceleration is a derivative of velocity with respect to time: a

= dv/dt, and it is not clear how we can bring the time variable in. Perhaps through that F = dp/dt formula

once again? Not sure. Let us think about that later, and first calculate the dγ/dv factor:

That does not look easy. We need to find that v(x) function. We know the two derivatives must be the

same, so let us write out the equality that should be there (we replace the = sign that should be there by

a sign to highlight that we did not manage to establish the equivalence, yet):

As you can see, it is not easy. We have a differential equation, and it is not an easy one even if we only

have one derivative here: dv/dx. Solving this involves integration and then various techniques may be

applied to establish the equivalence. We must probably go back to Clark’s article to see how we can find

the velocity function in order to double-check this identity. It must be correct, but we were not able to

show why because the relations between the position, time and velocity variables are, obviously, much

more complicated for a relativistic oscillator than for a non-relativistic oscillator. :-/

To be honest, we do not want to dwell on this problem because we think we have an easier approach to

the model of a relativistic oscillator. The gist of that is presented in a section dedicated to the oscillator

model for elementary particles in our paper on the geometry of the wavefunction. For those who want a

more elaborate exploration, I also put a few graphs in an early but lengthy manuscript on these

intuitions. These show what happens when v = c or when m0 = 0 (which we think is the case and,

therefore, makes classical calculations difficult or – formally speaking – not applicable because we are

outside of the domain range to which they apply). Unfortunately, the manuscript has typos and errors

here, which we should correct. We are, therefore, hesitant to refer the reader to it now.

25

Annex III: Static potentials and spacetime curvature

The reflections in this paper triggered another question in our mind which, in previous versions of this

paper, we treated as part of the main text. However, we have now separated out in this Annex.

lxxi

The question which was triggered is this: could static potentials and the curvature of spacetime

possibly be related? Again, we repeat that this question has nothing much to do with the questions

which we answered in this paper (and which are also further expanded in the first and second annex to

this paper), but we deal with it there because, with all the knowledge that the reader has gathered

above, he or she might be interested to consider it. So let us go and try to deal with it.

We already wrote a couple of times that we think of a potential being caused by a charge. Static

potentials must, therefore, be caused by static charges. However, we also suggested the idea of a static

charge cannot correspond to anything real: matter is charge in motion. So, what about the idea of a

static potential? When one starts studying physics and the math that comes with it, it is hard to acquire

an intuitive understanding of oscillating or time-varying fields and dynamic concepts such as vector

potentials but, after a while, one finds thinking in terms of dynamics is the easy part. For example, we

readily accept a field oscillation will travel at lightspeed and that we, therefore, need to introduce

retarded time in field equations. But what about static fields? If we put a charge in empty space, will all

of space immediately change because of it? Will the static potential that we associate with it be there,

not as some retarded effect but instantaneously?

The common-sense answer to that question is: charge cannot be created out of nothing and, therefore,

the question must be non-sensical.

lxxii

Indeed, if we put a charge in empty space, it needs to come from

somewhere: we will have to move it there from some point r1 to point r2. We must, therefore, always

have a dynamic effect.

The analogy with gravity comes to mind here: general relativity theory tells us gravitation is not a real

force⎯we must think of it in terms of spacetime curvature and there is, therefore, no point in thinking

in terms of retarded effects, except when the configuration of masses and energies changes. When that

happens, we talk of gravitational waves or ripples – such as those that were detected by the LIGO lab in

2015

lxxiii

– and we think it is only logical these travel at the speed of light. But so we are talking a change

in the curvature of spacetime here. The curvature itself just is.

We are firm believers in the curvature of spacetime along the lines of Einstein’s general relativity theory,

which tells us to think of gravitation as a pseudo-force

lxxiv

resulting from the curvature of spacetime. We

think this is the only possible explanation for the accelerating expansion of our Universe: there must be

other Universes out there, beyond our time horizon. Because these Universes are beyond our horizon

(the distance between our Universe and these other (hypothetical) Universes, measured in lightyears, is

lxxi

We should restructure all of our 30 papers on ResearchGate, but time is limited – not in theory, but in practice: a man’s life is

short, after all!

lxxii

Gilbert Ryle (The Concept of Mind, 1949) would refer to it as a ‘category mistake’, which sounds much better than

nonsensical, but we are not into philosophical niceties here.

lxxiii

We may doubt whether quarks or gluons are more than just mere mathematical form factors, but we do not share the

doubts that some physicists continue to entertain in regard to the reality of the gravitational wave.

lxxiv

When we say pseudo-force, it is not like a centripetal or centrifugal force, which are real because they produce real effects.

We actually mean no force, which is why we think of gravitational orbitals as having no energy (for more detail, see our short

paper on cosmology.

26

beyond the age of our Universe), they can only tear our Universe apart, so to speak, if the curvature of

spacetime is simply there⎯so there is no question of effects traveling from here to there in space and in

time. The (static) gravitational effect must, therefore, be instantaneous: it is only when the curvature of

spacetime changes, that we need to invoke (special) relativity and think in terms of perpetuations (at

lightspeed) of that change.

lxxv

Let us get back to the question: regardless of what is real and not-so-real in the equation above, would

it, perhaps, be possible to relate gravitation to electromagnetic fields? Let us phrase it more precisely:

could there, perhaps, be some way to relate the nature of gravitation⎯or the nature of the curvature of

spacetime, we should say⎯to small asymmetries in the electromagnetic fields?

Indeed, we know that (1) the magnitude of the electrostatic field is c times that of the magnetic field (B

= E/c) and (2) that, because matter is composite (it consists of positive and negative charges in small

tightly knit combinations), it appears as neutral matter above the atomic and nuclear scale. However,

we legitimately continue to wonder whether the asymmetry in the charge distribution (negative charge

orbits around massive positively charged nuclei) might explain the curvature of spacetime and,

therefore, gravity.

One may argue such line of reasoning raises as much questions as it might solve (and, as you will see in a

minute, we would agree with that, but let us explain why). We believe, for example, that the dark

matter in the Universe is antimatter, and the charge distribution in antimatter should be opposite to

that of matter and, therefore, neutralize or even over-compensate any asymmetry in the electrostatic

potential of matter, right?

Maybe. Maybe not. We noted in our paper(s) on the nature of antimatter (and antiphotons and

antineutrinos) – that the electromagnetic antiforce appears as left-handed (instead of right-handed

lxxvi

),

so that might change the −A/t term in the equation above. Perhaps it becomes something like a

+A/t term

lxxvii

and add to the usual electrostatic potential instead of neutralizing it.

lxxviii

However, such considerations do not solve the basic question: if antimatter consists of antiprotons and

positrons (anti-electrons), we will still think of the antiprotons (and antineutrons) as massive nuclear

lxxv

Of course, another explanation than 'other Universes beyond our horizon' might be that there is something like an absolute

space, and that we are spinning around in it, so then we have another pseudoforce (Mach’s principle) explaining the

acceleration in the pace of expansion of our Universe, but such explanation is mathematically equivalent to the 'other Universes

explanation' (so also no need to think of gravity in terms of EM potentials and gravitational waves or fields propagating at

lightspeed) and then it's a matter of taste: can we define an absolute space without reference to 'other objects' (the other

Universes) beyond the horizon defined by the age of our Universe, i.e. about 13.8 billion lightyears? If your answer is yes to that

question, then there is no need to hypothesize other Universes. But this is a question resembling: 'Does God exist?' Again, I take

Gilbert Ryle's pragmatic (1949) answer to that kind of questions: such questions may involve a category mistake, so they might

not have any (practical) meaning. Ryle would probably rephrase the question like this: we know Mach's principle applies to a

spinning bucket of water on Earth - but can we still apply it when we replace the bucket by all of the Universe?

lxxvi

Of course, the meaning of left- and right-handed depends, once more, on convention. We must apply them consistently

when doing such analyses.

lxxvii

A full analysis should involve an analysis of the impact of such signature change on Maxwell’s equations and the associated

Lorenz gauge. It may or may not be simple. Perhaps all that is needed is to change the sign in Maxwell’s E = −B/t = 0

equation (we may put a plus instead of a minus sign).

lxxviii

The idea of anti-gravity or negative gravity/mass (matter and anti-matter repelling rather than attracting each other) might

have crossed your mind briefly (there is more antimatter in our Universe than matter, so it would be a convenient explanation

of the accelerating pace of expansion of the Universe (a discovery for which a Nobel Prize in Physics (2011) was awarded), but

there is no evidence for that at all, and it would be hard to reconcile with the reality of matter-antimatter pair annihilation.

27

particles, combining with positrons to form anti-hydrogen and other, more massive anti-atoms or – at

lower temperatures – anti-molecules in the anti-matter parts of the Universe. The charge distribution

would, therefore, effective be opposite, and the related dipole fields would also have an opposite

orientation (or symmetry, if you prefer that word).

Apart from the more mundane objections above, the more fundamental questions are these:

1. Any variations in the fields surrounding stable charge configurations – both in matter as well as in

anti-matter – should also respect the ΔE·Δt = Δp·Δx ħ or h principle and would, therefore, be too

small to add to the energies of the (macro-)fields outside of these stable micro-configurations.

2. Even if the reasoning above would not be valid, asymmetric charge distributions result in dipole

fields, following a 1/r2 potential and a 1/r3 (inverse-cube instead of inverse-square) force law.

Moreover, the field energy conservation principle tells us such fields would not be spherically

symmetric⎯just like electric and magnetic dipole fields. We do not see how such non-symmetric

dipole fields – if they are there

lxxix

– could, somehow, magically add up to yield a nice spherically

symmetric 1/r2 force field.

So, yes, if we would have unlimited time, we would probably try to work on alternative models for

gravity, but time is what it is: eternal in theory, but in short supply practically speaking. And then we are

not very motivated either: from what we wrote above, it should be clear we are rather skeptical such

attempts might work. We are, therefore, happy to just consider gravity to be an entirely different beast,

so to speak⎯one that has absolutely nothing to do with electromagnetic theory and, therefore, can

never be reduced to EM/QM. In fact, something inside of us says talking of gravitational potential in very

much the same way as electromagnetic potential might be very misleading⎯worse: it might be entirely

meaningless. We do understand our human mind always wants to unify things somehow, but we should

be mindful of the philosophical warning of Aristotle (and Aquinas

lxxx

): quia parvus error in principio

magnus est in fine (a small error in the beginning (of a theoretical argument) can lead to great errors in

the conclusions).

Having said this, we of course do not want to discourage amateur as well as academic physicists who

continue to try their hand at this.

lxxxi

lxxix

We already mentioned that we are very skeptical in this regard: any asymmetric electrostatic dipole fields that would be

there, would have been measured already. Of course, we can think of the polarity of H2O atoms and many other examples of

polarized molecules, but (1) such molecules line up and, therefore, neutralize each other, so to speak, and (2) molecules make

up a tiny part of the Universe’s mass: most is electrically neutral hydrogen or hydrogen plasma, or – an even larger part – plain

radiation energy. We should also add another skeptical note here: there has been a lot of research on the chemical composition

of stars and planets, but none of them finds variations – based on different mass composition – in the gravitational constant. If

gravitation could be explained in terms of (residual) dipole moments, such dipole moments would effectively depend on the

matter at hand⎯literally.

lxxx

Thomas Aquinas starts his de Ente et Essentia (on Being and Essence) with this phrase, but it is a quote from Aristotle’s De

Caelo et Mundi (On the Heavens and the Earth).

lxxxi

It may be an interesting exercise to ‘translate’ or ‘transform’ electrostatic potentials (not varying in time) into a

mathematically equivalent geometric model and see what the electromagnetic equivalent of the gravitational constant (G)

would be. Such analysis should be easy to do based using the ubiquitous qe/m ratio⎯but what ‘equivalent’ mass should we

use: the electron mass, the proton mass? Simple questions like this become complicated quite rapidly, so that is why we should

just accept Einstein’s general relativity theory and not treat gravity as a force or a (residual or separate) field/potential. Also, we

are not well versed in general relativity theory but, if we are not mistaken, Einstein solely uses special relativity and Mach’s

Principle to derive the principles of GRT⎯so why would we try a puzzle that has been solved already?