All of physics
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
Oscillator math .............................................................................................................................................. 1
The electron and proton mass as constants of Nature................................................................................. 2
Wavefunction math, the Lorentz force, and electromagnetic theory .......................................................... 3
The proton model and quaternion math ...................................................................................................... 4
The nuclear (or strong) force ........................................................................................................................ 5
Non-equilibrium states and composite particles .......................................................................................... 5
The concepts of mass and force are related – or defined, one might say – through Newton’s force law: F
= ma, or m = F/a. Mass is the inertia to a change in the state of motion of… What? A particle? The idea
of a particle is a philosophical or ontological concept and we will, therefore, avoid it and prefer to speak
of charge, which is something we can measure.
The acceleration vector may be centripetal.
We can, for example, imagine a centripetal force holding a
pointlike (but not necessarily infinitesimally small) charge in a planar or spherical oscillation. This is just
an extension of the linear oscillator idea, whose energy is given by the E = ma2ω2/2 equation. Add two
perpendicular oscillations in harmony with each other (no phase difference) and the total energy in the
system is twice the energy of the individual linear oscillators: E = ma2ω2. We will soon present an
oscillation in three dimensions also.
The frequency of a linear oscillator is given by the ω2 = k/m relation: think of k/m as the elasticity per
unit mass. The factor k is an essential feature of the system. It models the restoring force F = −kx.
that the (square of the) frequency is expressed as something (k) per unit mass. It will be useful to (also)
express the force in terms of mass or energy units. If the oscillation is linear, the force and the
acceleration will vary. In a circular orbital, the centripetal force and acceleration will be constant.
However, we have the same equation for both linear as well as orbital oscillations:
Linear motion adds a linear component to the (orbital) velocity vector.
Of course, the nature of this elasticity (we admit we use the term rather loosely here) will depend on the
physicality of the system: think of the stiffness of a mechanical spring, the impedance of an electrical circuit,
This equation relates force (F), acceleration (a) and mass (m), and it also relates energy (E) to frequency
(ω) and displacement (x), but what gives us the energy? The Planck-Einstein relation tells us that the
ratio of the energy and the frequency equals Planck’s quantum of action: E/f = h or E/ω = ħ. This puts an
additional restriction on the allowable energy levels.
The electron and proton mass as constants of Nature
In Nature, we have composite systems and elementary systems. We think of the electron and the
proton as elementary oscillations, whose frequency and amplitude (the radius of the oscillation) is
determined by their energy or, what amounts to the same, their mass. Indeed, using the mass-energy
equivalence relation, we get the following fundamental equation for the electron:
This is the equivalent of the ω2 = k/m relation for the electron oscillation. We express the frequency (per
mass or energy unit) as a ratio of two given constants (c2 and ħ).
However, the mass factor is not random. It cannot be just anything: it is the electron mass. We may,
therefore, consider the electron mass (or energy) to be a constant of Nature too, and write the electron
frequency as ω = mc2/ħ = E/ħ.
This oscillator model of an electron is, essentially, a mass-without-mass model, because we think of it as
a real oscillation in space. More specifically, we think all of the mass of an elementary particle is the sum
of (1) the relativistic mass of the pointlike charge (whose rest mass we assume to be zero, and it,
therefore, moves about at lightspeed) and (2) the equivalent of the energy in the oscillation. Assuming a
, this gives us the effective radius of an electron:
An electron interferes (absorbs and emits) photons (e.g. Compton scattering
) and we, therefore, think
the nature of the oscillation is electromagnetic. This is consistent with the ring current model of an
electron (Parson, 1905; Breit, 1928, Schrödinger, 1930; Dirac, 1933; Hestenes, 2008, 2019).
We may think of the motion being elliptical or chaotic but the concept of an orbital frequency implies the motion
must be regular enough so as to be able to apply the orbital velocity formula c = aω. We will usually want to write
this formula as a vector cross-product c = aω, and already note that ω is an axial vector, which we may also write
as the vector cross-product of the two axial vectors c and 1/a.
The calculated radius is the Compton radius of an electron: it is the radius of interference with photons. Hard
(inelastic) scattering occurs because we think the pointlike charge is small but not infinitesimally small: it is of the
order of the classical electron radius (Thomson or Lorentz radius) re = rC. The small but finite size of the pointlike
charge also explain the small anomaly (/2) in the magnetic moment of an electron.
Wavefunction math, the Lorentz force, and electromagnetic theory
We may interpret the elementary wavefunction as representing a radius or position vector
ψ = r = a·e±iθ = a·[cos(±θ) + i · sin(±θ)]
According to the energy equipartition theorem, half of the total mass of the electron is kinetic, while the
other half is (potential) field energy. This gives rise to the concept of the effective mass of the pointlike
charge (meff = m/2).
If the electron oscillation is electromagnetic, the centripetal force is nothing but the
Lorentz force (F), which we may divide by the mass factor so as to relate (unit) charge and (unit) mass:
We use a different rotation operator or imaginary unit here (j instead of i) because the plane in which
the magnetic field vector B is rotating differs from the E- plane.
The gyromagnetic ratio is defined as the
factor which ensures the equality of (1) the ratio between the magnetic moment (which is generated by
the ring current) and the angular momentum and (2) the charge/mass ratio:
The angular momentum is measured in units of ħ, and the Planck-Einstein relation tells us it must be
equal to one unit of ħ. In contrast, the magnetic moment is usually measured in terms of the Bohr
magneton, which involves an additional ½ factor (B = qħ/2m). The g-ratio of an electron is, therefore,
equal to 1/2, which gives rise to the idea of spin-1/2 particles
The reader can easily verify the first- and second-order derivative with respect to the position of the pointlike
charge yield the (orbital) velocity and (centripetal) acceleration vector, respectively. We write the cos and sin term
in boldface because we think of the two terms as vectors, which are related to one another by the rotation
operator i (imaginary unit).
We may also invoke a geometric argument here, or refer to an analysis of relativistic oscillators.
Whether we consider the magnetic field vector to lag or lead the electric field vector depends on us using a right-
or left-hand rule. Using a consistent convention here, we believe the magnetic field vector will lead the electric
field vector when considering antimatter. Antimatter may, therefore, be modeled by putting a minus sign in front
of the wavefunction (see our paper on the Zitterbewegung hypothesis and the scattering matrix). We believe the
dark matter in the Universe to consist of antimatter, and it is dark because the antiphotons and antineutrinos they
emit and absorb are hard to detect with matter-made equipment. In regard to cosmology, we must add we think
gravity is not a force but a geometric feature of physical space (i.e. space with actual matter/energy in it). We
believe this is a rational belief grounded in the accelerating expansion of our Universe: the horizon of our Universe
is given by the speed of light. Our Universe may, therefore, be torn apart by other Universes that are beyond
observation, but this is only possibly when such gravitational effects are instant (i.e. part of the geometry of
We get the same result when using the effective mass concept. We think the use of the gyromagnetic ratio and
the Bohr magneton confuses rather than enlightens the analysis, but we wanted to show the analysis is consistent
with standard theory. The same spin-1/2 property (which we consider to be rather artificial as it depends on one’s
definition of the magneton) can be derived for the muon-electron and the proton. The magnetic moments of the
neutron (which we consider combining a positive and a negative charge) and the deuteron model (two positive
charges and one negative charges) can, likewise, be explained. See our paper on the nuclear force hypothesis. For
the convenience of the reader, here are the formulas for muon-electron and proton (we add a mysterious 2 and 4
The ring current model of an electron generates all of the empirically measured properties of an
electron (notably its radius and magnetic moment) and conveniently explains Compton scattering. The
q/m or q/2meff ratio(s) relate(s) the charge and (electromagnetic) mass concepts: these models maybe
referred to as mass-without-mass models, but not as charge-without-charge models!
The proton model and quaternion math
Two imaginary units (or rotational directions) will do for modeling electromagnetic oscillations (E and B
field vectors), but not when trying to model the proton oscillation. We think of the proton oscillation as
an orbital oscillation in three rather than just two dimensions. We, therefore, have two (perpendicular)
orbital oscillations, with the frequency of each of the oscillators given by ω = E/2ħ = mc2/2ħ (energy
equipartition theorem), and each of the two perpendicular oscillations packs one half-unit of ħ only
(such half-units of ħ for linearly polarized waves also explains the results of Mach-Zehnder one-photon
interference experiments). Such spherical view of a proton fits with packing models for nucleons and
yields the experimentally measured radius of a proton:
The 4 factor here is the one distinguishing the formula for the surface of a sphere (A = 4πr2) from the
surface of a disc (A = πr2).
We may now write the proton wavefunction as a combination of two
factor in the denominator and numerator of the formula for the proton, which the reader can check from our
proton model, which we will present in a moment):
The 4-factor also appears in the CODATA value for the Compton wavelength (2 times the radius) of the proton
(and neutron) for the same reason (spherical (3D) instead of planar (2D) oscillation). Finally, a 2 must be applied
to a spherical ring current to calculate the effective magnetic moment. We trust the reader will be able to do the
calculations. If not, he can check against the paper in which we do this kind of easy things. For a discussion of the
photon (a zero-charge spin-1 particle), see section 1-5 and 1-6 of our elementary introduction to quantum physics.
This is a reference to John Wheeler’s geometrodynamics program. Our models are more modest: mass is defined
as an inertia to a change in the state of motion of charges. Both concepts (mass and charge) are, therefore,
essential in the language of physics.
We also have the same 1/4 factor in the formula for the electric constant, and for exactly the same reason
The solutions to the wave equation will combine various combinations (products) of i- and j-terms, and
products thereof. Therefore, the use of quaternion algebra is required when working out when deriving
the scalar and vector potentials or when modeling nuclear wave equations.
The nuclear (or strong) force
There is no such thing as a ‘nuclear’ charge, nor is there a nuclear equivalent of the electric constant.
We might, therefore, say there is no such thing as a nuclear or a strong force: we only have two very
different fundamental oscillations in spacetime – and electron and a proton oscillation – and
combinations thereof. Such combinations, however, can all be explained by standard electromagnetic
theory, including electromagnetic dipole theory. We, therefore, tentatively agree with Di Sia’s
conclusion: the 80-year-old problem of the nuclear force has been resolved.
However, we do think of neutrinos as the counterpart of photons when dealing with nuclear reactions.
In this sense, neutrinos may be said to carry a nuclear force, even if its essence is electromagnetic too!
Also, the proton mass is, obviously, of a different nature than the electron mass, because the two
oscillations are very different (different mass and different geometry). Because the proton and the
neutron, which we think of as a composite particle but with similar mass) make up all of the matter
inside of a nucleus, the reference to a nuclear force comes in handy, although we should not use in
juxtaposition to the electromagnetic force. Ultimately, all force is electromagnetic, including binding
Non-equilibrium states and composite particles
Non-stable particles are non-equilibrium states, to which the Planck-Einstein relation does not apply.
They may be modelled by adding a transient factor to the wave equation.
Modeling composite particles (stable or non-stable) require the combination of potentials or, when
modeling particle reactions, the use of advanced vector and matrix algebra (scattering matrices).
Brussels, 20 April 2021
We interpret the 2019 redefinition of SI units as confirming this hypothesis: over the past 100 years, no
specifically nuclear-related new constant in Nature has appeared, and our current knowledge of fundamental
constants incorporate all laws of physics.
Paolo Di Sia, A solution to the 80-year-old problem of the nuclear force, 2018.
We think of photons and neutrinos as lightlike particles which carry energy (force over a distance), but they do
not carry charge, which is why they travel at lightspeed⎯as opposed to inertial charged (matter-)particles.
We refer to the above-mentioned paper on the nuclear force hypothesis.
See our paper on the Zitterbewegung hypothesis and the scattering matrix.
See the reference(s) above.