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
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.
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
We acknowledge the 21st anniversary of the 9/11 events.
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
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
) 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
). 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
) – 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
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
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.
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
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.
but also one unit of ħ. The photon that is emitted packs both. The energy is given by the Rydberg
We have the fine-structure constant here
, 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.
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.
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.
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.
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.
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.
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.
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.
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
), 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
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
This remark will become very relevant in a few moments.
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
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
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).
(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
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
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
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.
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.
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
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
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.
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.
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
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
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
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
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.
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’.
See our paper on the nuclear force hypothesis, which considers non-conservative potentials, which we end up dismissing.
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)
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.
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.
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.
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.
However, we keep digressing from the matter at hand. What is that we can say now about it? It is this:
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.
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).
Our previous papers make it abundantly clear that we do not think highly of modern quantum field theory.
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.
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.
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
complementarity principle, makes a lot of sense to us.
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
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
, we can combine as:
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.
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.
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.
The c2/h ratio is equal to 1.356392489652131050 (not approximately but exactly), which is a rather
humongous number, as you can see.
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
– 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
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
, 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.
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).
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.
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.
The Planck-Einstein relation can thus be associated with the (in)famous uncertainty principle, which we
should probably always write as two complementary relations
h = 6.62607015×10−34 Nms
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:
E = /0
E = −B/t
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
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.
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
: 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.
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
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.
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.
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.
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
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
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.
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
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
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.
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!
Annex I: The Complementarity and Uncertainty Principles
We did quite a few more ontological or epistemological papers on physics already
, 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
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)
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
(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
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
See, for example, Ontology and Physics, February 2021.
On the usefulness of quaternion math in physics, see our paper on the nuclear force and quaternion math.
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).
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.
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
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
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:
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.
We borrow this illustration from G. Vassallo and A. Di Tommaso (2019).
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
), say operator A and B and define the following mix of products as a
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.
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.
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
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
, 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!
The reference to Jeans here is Feynman’s, though. We did not read James Jeans.
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
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
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
), 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
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).
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.
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.
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)
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.
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
, 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
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.
You may want to think about this because the distance between two points can vary because of relativistic length
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.
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.
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.
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)
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
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.
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
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.”
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.
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.
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.
we do not see any variations and why the electron – as an oscillation of a pointlike charge – is not losing
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.
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
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!
illustration of how potential and kinetic energy effectively vary and add up over a full cycle of an
Figure 4: Kinetic (K) and potential energy (U) of an oscillator
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.
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.
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
You will find this diagram in many texts, but we took this one from the https://phys.libretexts.org/ site—excellent hub for
Also, we would probably want to use other units than the joule (J) or the meter (m).
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.
In case you want to check your calculations, we once again recommend the online (free) LibreTexts Physics textbook.
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.
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
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.
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
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
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:
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).
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.
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.
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.
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.
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.
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
– 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
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
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!
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.
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.
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.
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.
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,
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
so that might change the −A/t term in the equation above. Perhaps it becomes something like a
and add to the usual electrostatic potential instead of neutralizing it.
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
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?
Of course, the meaning of left- and right-handed depends, once more, on convention. We must apply them consistently
when doing such analyses.
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).
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.
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
– 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
): 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
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.
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.
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).
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?