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1

A mass-without-mass model

of protons and neutrons

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

8 February 2021

Contents

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

The electron .................................................................................................................................................. 3

The muon-electron ....................................................................................................................................... 4

The proton .................................................................................................................................................... 5

The neutron .................................................................................................................................................. 7

Electron, proton, and neutron magnetic moments ...................................................................................... 9

Conclusion ................................................................................................................................................... 10

Introduction

‘Mass without mass’ models analyze elementary particles as harmonic oscillations whose total energy –

at any moment (KE + PE) or over the cycle – is given by E = ma22. One can calculate the radius or

amplitude of the oscillation directly from the mass-energy equivalence and Planck-Einstein relations, as

well as the tangential velocity formula⎯interpreting c as a tangential or orbital (escape) velocity.

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

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

electron orbitals

1

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

understood as a quantization of spacetime.

1

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

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

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

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

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

2

Figure 1: Circular and elliptical orbital motion

2

The model is based on the assumption of a pointlike charge

3

with no other properties but its charge

(zero rest mass). However, this zero-mass point charge acquires an effective mass which accounts for

half of the energy of the elementary particle: the other half of the energy is in the (electromagnetic or

nuclear) field which sustains the motion of the charge. As such, the pointlike Zitterbewegung (zbw)

charge is photon-like but, unlike a photon, it carries (electric) charge.

The motion is not necessarily circular: one may imagine elliptical orbitals, such as depicted by the polar

rose in the illustration.

4

The r() = a0·cos(k0 + 0) equation gives us the radial distance r as a function of

the phase = (t) = t of the oscillation. Thinking of r as a vector in 2D space (the plane of motion), we

get a wavefunction:

For an electron, we get the following energy-mass calculation:

This yields the following equation:

This represents the E = KE + PE energy conservation equation. The velocity v is an orbital or tangential

velocity

5

and the mv2/2 formula for the kinetic energy is, therefore, relativistically correct.

We get the radius or amplitude of the oscillation from the E = ma22 equation:

2

Illustrations made from Wikipedia templates. For the orbital equations, see the MIT OCW reference course on

orbital motion.

3

Pointlike but not of zero dimension. See our explanation of the anomaly in the magnetic moment of an electron

in, for example, our paper on the basics.

4

The Rutherford-Bohr model considers circular orbitals only. Schrödinger’s wave equation adds non-circular (non-

spherically symmetric) orbitals as solutions.

5

This velocity is sometimes referred to as the escape velocity, but the terms are not to be used interchangeably

because they may refer to subtly different things (e.g. the velocity component with right angle to the semi-

major/minor axis of the ellipse). Needless to say, m is the relativistic mass.

3

We may interpret the positive and negative root of ħ2/m2c2 as the two possibilities that correspond to

the direction of angular momentum, which distinguishes an electron from a positron.

6

This formula misses the 1/2 factor of the effective mass m, which is half the total mass (m = E/c2) of the

elementary particle (m or E), and which explains why elementary (charged) particles are spin-1/2

particles, as shown by the calculation below:

The momentum is, of course, orbital angular momentum only. As such, we are essentially modeling

spin-zero (zero spin angular momentum) particles. We should, of course, note that a moving charge is a

current, which explains the magnetic moment

7

:

The electron

The radius formula works perfectly well for the electron. It yields the electron’s Compton radius a = rC:

The idea here is that an orbital cycle of the pointlike charge in its Zitterbewegung does not only pack the

electron’s energy (E = m·c2) but also Planck’s quantum of action (S = h).

8

For an electron, we also get the

following cycle time and electric current:

We can also calculate the electric current:

These values look rather phenomenal (we have a household-level current (almost 20 ampere) at the

sub-atomic scale here), but they are what they are and far from the radius-energy values one gets for

black holes (Schwarzschild radius).

6

The antimatter counterpart of an elementary particle has opposite angular momentum but shares the same form

factor. This explains why a proton and an electron are not a matter-antimatter pair: their form factors are

different. Positive and negative charge remain separate concepts, however: two electrons will, therefore, not

annihilate each other but coexist as an electron pair through a coupling of their magnetic moments, thereby

lowering total energy (which explains their stability as a pair). Formally, opposite angular momentum may also be

modelled by inversing the time sign, but time goes in one direction only:

7

The small anomaly in the magnetic moment may be explained by assuming the pointlike charge has a tiny (non-

zero) dimension itself. See our paper on the essentials.

8

The idea of an oscillation packing some amount of physical action may not be very familiar. In the context of our

model we think of physical action as the product of (i) the force that keeps the zbw charge in its orbit, (ii) the

distance along the loop, and (iii) the orbital cycle time.

4

We get the classical electron radius from the formula above:

This illustrates the interpretation of the fine-structure constant as a scaling parameter: re = rC.

9

The

harmonic oscillator model can be used to show that the elasticity or stiffness parameter k or, expressed

per unit mass, k/m in the F = −kx formula is equal to:

This equation shows various oscillatory modes are possible: these modes are characterized by the

frequency (or its square

10

) which, in turn, depends on the speed of light (c) and the radius or range

parameter (a).

We can also easily calculate the magnitude of the centripetal force F= F from Newton’s force law

11

:

This is a huge force at the sub-atomic scale: it is equivalent to a force that gives a mass of about 106

gram (1 g = 10−3 kg) an acceleration of 1 m/s per second! It is, however, an electromagnetic force only.

The muon-electron

The same formulas apply to the muon-electron as well but suggest the centripetal force is of an entirely

different nature. The muon carries 105.66 MeV (about 207 times the electron energy) and has a (mean)

lifetime which is much shorter than that of a free neutron

12

but longer than that of other unstable

particles: about 2.2 microseconds (10−6 s).

13

This may explain why we get a sensible result when using

the Planck-Einstein relation to calculate its frequency and/or radius.

14

9

The 2019 revision of the system of SI units incorporates these new physics, which amount to what we refer to as

a realist interpretation of quantum physics. Needless to say, the fine-structure constant has other interpretations

as well. See our paper on the meaning of the fine-structure constant.

10

The energy in an oscillation is proportional to the square of the frequency (and the square of its amplitude too).

11

See p. 14-15 of our paper on the electron model for the formula for the centripetal acceleration (ac = a·ω2 =

vt2/a). In the same paper, we also comment on the rather particular behavior of the momentum function p = mv,

which resembles the mathematical particularities of the Dirac function.

12

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

13

The tau-electron is just a resonance (as opposed to a transient particle) with an extremely short lifetime

(2.910−13 s only). Hence, the Planck-Einstein relation does not apply: it is not an equilibrium state. We think the

conceptualization of both the muon- as well as the tau-electron in terms of particle generations is unproductive.

Likewise, charged pions (π±) has no resemblance whatsoever with the neutral pion (see footnote 14).

14

The longevity of the muon-electron should not be exaggerated, however: the mean lifetime of charged pions,

for example, is about 26 nanoseconds (10−9 s), so that is only 85 times less. As for the 1.87 fm value, this is a radius

5

The ratio of the centripetal forces which keep the charge in its orbital for the electron and muon

respectively is equal to:

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

referred to as a strong force, right? Is it the nuclear force? It underscores the point about the modes of

the elementary oscillation depending on the radius or amplitude a = ħ/mc of the oscillation. We may,

indeed, rewrite the force ratio as

15

:

The proton

The proton mass is about 8.88 times that of the muon, and it is about 2.22 times smaller: we have a rather

mysterious 1/4 factor here, which needs explaining. Indeed, when applying the a = ħ/mc radius formula

to a proton, we get a value which is 1/4 of the measured proton radius: about 0.21 fm, as opposed to the

0.83-0.84 fm charge radius which was established by Professors Pohl, Gasparan and others over the past

decade.

16

The 1/4 factor is puzzling, and there may be no obvious way to explain it. However, geometry offers a way

out. We have a 1/4 factor between the volume of a sphere (V = 4πr2) and the surface area of a circle (A =

πr2) and, hence, one might intuitively think of an oscillation in three rather than just two dimensions only.

In other words, the oscillator would be driven by two (perpendicular) forces rather than just one. We can

model this by thinking of two oscillators which, according to the equipartition theorem, should each pack

half of the total energy of the proton. This spherical view of a proton – as opposed to the planar picture

of an electron – fits nicely with packing models for nucleons.

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

and, hence, should be multiplied by 2 to get the CODATA value (more or less) for the Compton wavelength of the

muon (1.17344411010−14 m 0.00000002610−14 m).

15

We did not find any easy interpretation of this ratio in terms of the fine-structure constant, however. Hence, the

mass or energy of the elementary particles may be considered to be fundamental constants of Nature themselves.

16

For the exact references and contextual information on the (now solved) ‘proton radius puzzle’, see our paper

on it: https://vixra.org/abs/2002.0160, in which we also make some remarks on the (anomalous) magnetic

moment of the proton.

6

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

17

This, then, gives us the

experimentally established value for the proton radius:

The force along one of the two axes or planes of oscillation inside of a proton is equal to:

Hence, we get a force of 4,532 N inside of a muon and a force of 89,349 N inside of a proton. Compensating

for the 1/4 factor (which we loosely refer to as the different form factor of the proton oscillation), we find

the force inside of a proton is almost 5 times stronger than the force inside of a muon. Hence, we may

conclude that the force inside of a muon-electron and a proton (and neutron, which we think of as a

proton-electron combination

18

) are of the same nature. However, a muon-electron is, clearly, not the

antimatter counterpart of a proton !

The orbital energy equation for the nuclear field is given by

19

:

Can we calculate a out of this? Maybe. How? Perhaps by evaluating potential and kinetic energy at the

periapsis, where the distance between the charge and the center of the radial field is closest. However,

the limit values vπ = c (for rπ → 0) and rπ = 0 (for vπ → c) are never reached and should, therefore, not be

used: neither the kinetic nor the potential energy seems to reach the zero value and we can, therefore,

probably not simplify any further.

20

We, therefore, prefer the simpler zbw approach as outlined above:

We may now substitute this value for a in the orbital energy equation

21

:

17

This explanation is similar to our explanation of one-photon Mach-Zehnder interference, in which we assume a

photon is the superposition of two orthogonal linearly polarized oscillations (see p. 32 of our paper on basic

quantum physics, which summarizes an earlier paper on the same topic).

18

See our paper on the nuclear force and the neutron hypothesis.

19

The reader can/should check the physical dimensions:

Note that we have a plus (+) sign in the equation because the potential energy in the orbital energy equation is

zero at the center and, therefore, always positive. For more details, see our paper on the nuclear force.

20

One can, however, calculate other interesting properties of the orbitals, such as the eccentricity (see the above-

mentioned MIT OCW reference course on orbital motion).

21

We use the definition (cf. the 2019 revision of SI units) of the fine-structure constant:

formula.

7

Re-arranging yields:

It is a nice formula. In the next section, we will find a similar formula for the neutron.

The neutron

We think of the (free) neutron as a composite (non-stable) particle consisting of a ‘proton’ and an

‘electron’. However, we will soon qualify this statement: the reader should effectively think in terms of

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

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

in a nuclear as well as in an electromagnetic oscillation. In order to interpret v as an orbital or tangential

velocity, we must, of course, choose a reference frame. Let us first jot down the orbital energy equation

for the nuclear field, however

22

:

Figure 2: Two opposite charges in elliptical orbitals around the center of mass

23

22

A dimensional check of the equation yields:

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

23

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

motion.

8

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

24

, which is the sum of the

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

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

25

and may be

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

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

are being kept separate by the nuclear force.

26

The electromagnetic force which keeps them together is

the Coulomb force:

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

therefore, write:

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

27

Dividing both sides

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

yields:

It is a beautiful formula, and we could/should probably play with it some more by, for example,

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

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

never reached and should, therefore, not be used. We are sure one of our readers will find ways to get a

specific value for the radius a, which should be, hopefully, very near to 0.84 fm (the proton/neutron

diameter. It should, in fact, be slightly larger because of the energy difference between a proton and a

neutron, which is of the order of about 1.3 MeV, which is about 2.5 times the energy of a free electron.

28

24

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

25

This relates to the point we made in regard to the nature of the ‘proton’ in the neutron: it is not like the massive

proton at the center of the hydrogen atom, with the electron orbiting around it.

26

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

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

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

positive potential energy here as well.

27

One easily obtains the keqe2 = ħc identity from the

formula.

28

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

measuring the radius of a decaying neutral particle. As for the instability of the free neutron, its lifetime is very

9

Till then, we must assume we may apply the mass-without-mass formula for the proton radius to the

neutron too:

Electron, proton, and neutron magnetic moments

The magnetic moment of an elementary ring current is the current (I) times the surface area of the loop

(πa2). The current is the product of the elementary charge (qe) and the frequency (f), which we can

calculate as f = c/2πa, i.e. the velocity of the charge

29

divided by the circumference of the loop. We

write:

Using the Compton radius of an electron (ae = ħ/mec), this yields the correct magnetic moment for the

electron

30

:

The CODATA value for the magnetic moment of a proton is equal to:

μ = 1.4106067973610−26 J·T−1 0.00000000060 J·T−1

The electron-proton scattering experiment by the PRad (proton radius) team at Jefferson Lab measured

the root mean square (rms) charge radius of the proton as rp = 0.831 ± 0.007stat ± 0.012syst fm.

When applying the a = μ/0.24…10–10 equation to the proton, we get the following radius value:

We interpret this to be the effective mm (magnetic moment) radius because we think of the proton

oscillation as a spherical (3D) oscillation⎯in contrast to the electron oscillation, which is a plane (2D)

long as compared to the muon-electron, so we may effectively assume that the oscillation must very nearly pack

one unit of Planck’s quantum of action.

29

We assume a pointlike charge with zero rest mass. Its (orbital/tangential) velocity is, therefore, equal to the

speed of light (c). This ring current model, which was first proposed by the British chemist and physicist Alfred

Lauck Parson (1915) and which was mentioned rather prominently in the Solvay Conferences, amounts to a mass-

without-mass model of elementary particles (John Wheeler, 1957, 1960, 1961). It analyzes them as harmonic

oscillations whose total energy – at any moment (KE + PE) or over the cycle – is given by E = ma22. One can then

calculate the radius or amplitude of the oscillation directly from the mass-energy equivalence and Planck-Einstein

relations, as well as the tangential velocity formula⎯interpreting c as a tangential or orbital velocity, indeed:

30

The calculations do away with the niceties of the + or – sign conventions as they focus on the values only. We

also invite the reader to add the SI units so as to make sure all equations are consistent from a dimensional point

of view. The CODATA values were taken from the NIST website.

10

oscillation. If we multiply this effective radius with 2, we get a value for the proton radius which fits

into the 0.831 0.007 interval:

Conversely, if we divide the magnetic moment that is associated with the above-mentioned radius value

(0.83 fm) by the same factor 2, we get the actual magnetic moment of a proton.

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μp = 1.4106067973610−26 J·T−1 = (1.99489926410−26 J·T−1)/2

Let us now turn to the neutron. The CODATA value for the neutron magnetic moment is:

μn = −0.9662365110−26 JT−1

The difference (2.3768433073610−26 JT−1) corresponds to a radius of about 1 fm:

This is the right order of magnitude, and we think the difference with the actual neutron radius may,

once again, be explained by the 2 factor: the negative charge inside of the neutron will be in a three-

dimensional oscillation itself, and the effective radius of the electron inside of the neutron (which we

denote as ae) will, therefore, be smaller:

We take this to confirm our neutron model.

Conclusion

Mass-without-mass models of elementary particles model the oscillation of a pointlike charge. Potentials

and forces depend on and/or act on a charge: the elementary charge e± (e, e, ep) or its antimatter

counterpart. We think there are only two forces/potentials: electromagnetic and nuclear⎯or some

combination thereof.

32

We also think forces/potentials/particles have a field- or light-particle counterpart: the photon or the

neutrino (as applicable to proton/neutron Verwandlung reactions).

Of course, we are very much aware that we offer a non-conventional analysis here which breaks away

from common ideas on several critical points. Most importantly, perhaps, we think leptons do partake in

nuclear interactions, which explains deep electron orbitals

33

, our model of the muon as a (potentially)

pure nuclear oscillation (neutrinos carry the (excess) energy of a muon decay reaction) and, perhaps, why

low-energy nuclear reactions (transmutation of nucleons by laser irradiation) can possibly take place.

We effectively think the classification of particles into generations or into baryons and leptons are too

general to be useful: we just have two forces/potentials, and combinations thereof. Incidentally, we also

31

For a discussion of the anomalies in the magnetic moment, see our paper on the topic.

32

We develop a model for the deuteron nucleus in the above-mentioned paper too!

33

See, for example, Andrew Meulenberg and Jean-Luc Paillet, Deep Electron Orbitals and the Dirac Equation,

January 2020.

11

think the quark hypothesis might not be very useful: at best, they are temporary non-equilibrium states

and, as such, mathematical abstractions. We get the following table of elementary matter- and light-

particles

34

:

Table 1: Elementary particle classification according to form factors

2D oscillation

3D oscillation

electromagnetic force

e (electron/positron)

orbital electron (e.g: 1H)

nuclear force

(muon-electron/antimuon)

p (proton/antiproton)

composite

pions (π/ π0)?

e.g: n (neutron),

D+ (deuteron)

corresponding field particle

(photon)

(neutrino)

This is nice and complete: each theoretical/mathematical/logical possibility corresponds to a physical

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

So what is our theory of reality, then? We think physic reality and our logical representation of it blends

as part of the sensemaking process. Wittgenstein was wrong on language, but his intuition was quite

correct: Die Welt ist die Gesamtheit der Tatsachen, nicht der Dinge. (Wittgenstein, TLP, 1.1)

We, therefore, stick to classical quantum physics or, as we refer to it, to a realist interpretation of it.

Indeed, we think the criticism of H.A. Lorentz of the new theories, before he left the scene, was quite

apt:

“I would like to draw your attention to the difficulties in these theories. We are trying to

represent phenomena. We try to form an image of them in our mind. Till now, we always tried

34

We think of the tau-lepton as a resonance or a very short-lived transient. It is, therefore, not an elementary

particle in our view but only an intermediary reaction product.

12

to do using the ordinary notions of space and time. These notions may be innate; they result, in

any case, from our personal experience, from our daily observations. To me, these notions are

clear, and I admit I am not able to have any idea about physics without those notions. The image

I want to have when thinking physical phenomena has to be clear and well defined, and it seems

to me that cannot be done without these notions of a system defined in space and in time.”

35

We conclude with Wittgenstein’s last and final proposition in his Tractatus Logico-Philosophicus (TLP):

“Wovon man nicht sprechen kann, darüber muss man schweigen.”

Brussels, 8 February 2021

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See our brief history of quantum-mechanical ideas.