The Zitterbewegung hypothesis
and the scattering matrix
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
21 March 2021
This paper offers a comprehensive presentation of the Zitterbewegung hypothesis and shows how it
generates all of the intrinsic properties of the electron, proton, and neutron (including the anomalies). It
also shows how it can be used to explain antimatter. Finally, it relates the wavefunction to what we
think of as a reinvigorated S-matrix program (Bombardelli, 2016). However, based on the standard
physical explanation of Compton scattering in terms of energy and momentum conservation, we must
note that the S-matrix representation seems to lose track of the (linear) momenta (magnitudes as well
as direction) of the incoming and outgoing particles, which we think of as a major disadvantage of the
The Zitterbewegung interpretation of an electron ....................................................................................................... 1
The Zitterbewegung interpretation of a proton ............................................................................................................ 3
Recap: the oscillator model for stable and unstable (elementary) particles ............................................................ 3
The strong (nuclear) force ......................................................................................................................................... 4
Magnetic moments ................................................................................................................................................... 7
The neutron as a composite particle ............................................................................................................................. 8
The Zitterbewegung hypothesis and the scattering matrix ......................................................................................... 10
Form factors and the nature of quarks ........................................................................................................................ 11
Conclusion ................................................................................................................................................................... 12
References ................................................................................................................................................................... 13
Annex I: Elementary wavefunction math .................................................................................................................... 14
The wavefunction with decay factor ....................................................................................................................... 14
Interpretation of the wavefunction of stable particles ........................................................................................... 14
The use of natural units........................................................................................................................................... 16
The wavefunction as a description of motion ......................................................................................................... 17
Explaining force and momentum from the radius of the oscillation. ..................................................................... 18
Quantum-mechanical operators ............................................................................................................................. 21
Calculating the velocity and acceleration vector from the stationary wavefunction ............................................. 23
Annex II: The wavefunction and special relativity ....................................................................................................... 24
Annex III: The wavefunction and general relativity ..................................................................................................... 30
Annex IV: An S-matrix representation of Compton scattering? .................................................................................. 31
The Zitterbewegung interpretation of an electron
The Zitterbewegung interpretation of quantum mechanics goes back to Erwin Schrödinger advancing a
trivial solution to Dirac’s wave equation for an electron (1927): a local oscillatory motion of the charge.
Erwin Schrödinger referred to this motion as Zitterbewegung
, and Dirac highlighted its significance at
the occasion of his Nobel Prize lecture:
“It is found that an electron which seems to us to be moving slowly, must actually have a very high
frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us.
As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light.
This is a prediction which cannot be directly verified by experiment, since the frequency of the oscillatory
motion is so high, and its amplitude is so small. But one must believe in this consequence of the theory,
since other consequences of the theory which are inseparably bound up with this one, such as the law of
scattering of light by an electron, are confirmed by experiment.” (Paul A.M. Dirac, Theory of Electrons and
Positrons, Nobel Lecture, December 12, 1933)
Dirac fails to distinguish the concepts of the electron and the (electric) charge here. The Zitterbewegung
model of an electron effectively combines the idea of a charge and its motion so as to account for both
the particle- as well as the wave-like character of matter-particles. It also explains the magnetic moment
of the electron, further building on the ring current or magneton model of an electron⎯which had
already been proposed early on (in 1915, to be precise) by the British chemist and physicist Alfred Lauck
Parson (1915) to do exactly that.
Below we show how this model generates the intrinsic properties of
the free electron. This model imagines the free electron as a pointlike charge in an electromagnetic
orbital oscillation, whose radius (which is nothing but the Compton radius) is the effective radius for
interaction or interference between a photon and the electron. Interpreting the velocity of light as a
non-linear velocity in circular or elliptical orbits effectively yields the Compton radius of an electron:
Paraphrasing Prof. Dr. Patrick LeClair
, we can now understand this distance as “the scale above which
the electron can be localized in a particle-like sense”, and it also clarifies what Dirac referred to as the
law of (elastic or inelastic) scattering of light by an electron”⎯Compton’s law, in other words.
Zitter refers to a rapid trembling or shaking motion in German.
The magnetic properties of the electron had just been discovered. We may refer to Ernest Rutherford’s remarks
on Parson's ‘électron annulaire’ (ring electron) and the magnetic properties of the electron in his lecture on ‘The
Structure of the Electron’ at the 1921 Solvay Conference, which illustrate quantum physicists had already generally
accepted the idea.
See: http://pleclair.ua.edu/PH253/Notes/compton.pdf, p. 10.
We think Compton scattering may be explained conceptually by accepting the incoming and outgoing photon are
different photons (they have different wavelengths so it should not be too difficult to accept this as a logical
statement: the wavelength pretty much defines the photon⎯so if it is different, you have a different photon). This,
then, leads us to think of an excited electron state, which briefly combines the energy of the stationary electron
and the photon it has just absorbed. The electron then returns to its equilibrium state by emitting a new photon.
The energy difference between the incoming and outgoing photon then gets added to the kinetic energy of the
electron through Compton’s law:
is equal to the elementary charge (qe) times the frequency (f) and the calculation yields a
household value: about 19.8 A (ampere). A rather astonishing value but, combining this with radius of
the loop (the Compton radius a = ħ/mc), this yields the magnetic moment:
Instead of using the f = c/2πa formula for the frequency, we can also use the Planck-Einstein relation to
get the same result:
The calculation yields μ = 9.27401…10−24 J·T−1. The CODATA value – which represents the scientific
consensus on the measured value for the magnetic moment – is slightly larger: μCODATA =
9.2847647043(28)10−24 J·T−1. The difference is the so-called anomaly, which explains 99.85% of
Schwinger’s factor (α/2π = 0.00116141…):
The discrepancy can easily be explained by noting that the formulas for the theoretical value of the
magnetic moment ( = qħ/2m) assume all of the charge is concentrated in one mathematical
(infinitesimally small) point. This is a mathematical idealization only: we must distinguish between a
theoretical and an effective radius of the electron. Figure 1 shows why: if the pointlike charge is whizzing
in the Zitterbewegung electron at the speed of light, then its center of charge will not coincide with a
point on its orbit. The effective radius of the orbit, which we denote by r, will be slightly larger than the
theoretical radius a.
Figure 1: The effective and geometric center of a charge in orbital motion
We can now apply the usual formulas for the magnetic moment to get the effective radius of
interference of the free electron:
This physical law can be easily derived from first principles (see, for example, Patrick R. Le Clair, 2019): the energy
and momentum conservation laws, to be precise. More importantly, however, it has been confirmed
The motion of the charge may be chaotic, but it must be regular. Otherwise, the concept of frequency (and,
therefore, energy) makes no sense.
Of course, this is a circular argument: we equated the anomaly to 1 + α/2π here. This is, therefore, a
first-order approximation only: when using the CODATA value for μr, we get a μr/μa ratio that is equal to
99.99982445% of 1 + α/2π. We think that is good enough to validate our model and further iterations
should yield an even better result.
The above suggests that the negative charge (electron) may have a fractal structure.
measurements of the proton radius and its magnetic moment may or may not confirm the positive
charge has a fractal structure too. We offer some reflections on that in the next section.
Before we do so, the readers should note the equations above also naturally relate the intrinsic
properties (energy, mass, and magnetic moment) of the muon-electron. The muon’s lifetime – about 2.2
microseconds (10−6 s) – is quite substantial and we may, therefore, consider it to be a semi-stable
particle (as opposed to the tau-electron, which we consider to be a mere resonance rather than a
transient). We, therefore, get a sensible result when using the Planck-Einstein relation to calculate its
frequency and/or radius. Inserting the 105.66 MeV (about 207 times the electron energy) for its rest
mass into the formula for the zbw radius, we get:
The CODATA value for the Compton wavelength of the muon is 1.17344411010−14 m
0.00000002610−14 m. Dividing this by 2 to get a radius instead of a linear length, one gets the same
value: about 1.8710−15 m. One can also calculate the magnetic moment and one will find it is in
agreement with its CODATA value.
The Zitterbewegung interpretation of a proton
Recap: the oscillator model for stable and unstable (elementary) particles
The ring current or mass-without-mass model of elementary particles considers particles as harmonic
oscillations whose total energy – at any moment (KE + PE) or integrated over the cycle – is given by E =
ma22. One can then calculate the radius or amplitude of the oscillation directly from the mass-energy
equivalence and Planck-Einstein relations, as well as the tangential velocity formula⎯interpreting c as a
tangential or orbital (escape
) velocity, as we did above.
The model gives a sensible explanation of the quantization of spacetime. Particles are effectively
interpreted as finite quanta: their energy/mass is finite, and they pack a finite amount of physical action.
Stable particles pack one or multiple units of ħ (angular momentum): E = nħ = nhf = nh/T. We do
See, for example, Oliver Consa, Helical Solenoid Model of the Electron, August 2018.
The concepts of orbital, tangential and escape velocity are not always used as synonyms. For a basic but
complete introduction, see the MIT OCW reference course on orbital motion.
not model particles as wavepackets but as definite orbital oscillations of the (electric) charge.
therefore, write the wavefunction of stable particles as:
Applying Occam’s Razor Principle (i.e. matching all mathematical possibilities in the expression with
experimentally verified (physical) realities), we think of the sign of the coefficient A as modeling
matter/antimatter, while the sign of the complex exponent (iEt/ħ) captures the spin direction of
The wavefunction of unstable particles (transients) involves an additional
decay factor :
Light-particles differ from (stable) matter-particles because they carry no charge. Their oscillation (if
photons are electromagnetic oscillations, then neutrinos must be nuclear oscillations) is, therefore, not
local: they effectively travel at the speed of light.
The strong (nuclear) force
The assumption, so far, is that the oscillation is electromagnetic, or planar (2D) in nature. The
components of the field are electric and magnetic. For matter, the phase of the magnetic field vector
lags that of the electric field vector by 90 degrees. The minus sign of the coefficient of the wavefunction
yields a preceding phase by the same amount (90 degrees). Antimatter should emit or absorb
antiphotons and is, therefore, hard to detect, which explains why we think of dark matter in the
Universe as antimatter.
Particle physics experiments – especially high-energy physics experiments – provide evidence for the
presence of a nuclear force. We think of a nuclear oscillation as an orbital oscillation in three rather than
just two dimensions: the oscillation is, therefore, driven by two (perpendicular) forces rather than just
one, with the frequency of each of the oscillators being equal to = E/2ħ = mc2/2ħ. Each of the two
perpendicular oscillations would, therefore, pack one half-unit of ħ only.
The = E/2ħ formula also
incorporates the energy equipartition theorem, according to which each of the two oscillations packs
half of the total energy of the nuclear particle⎯a proton, to be precise. This spherical view of a proton
fits nicely with packing models for nucleons and yields the experimentally measured radius of a proton:
The 4 factor is, of course, the same factor 4 as the one appearing in the formula for the surface area of a
sphere (A = 4r2), as opposed to that for the surface of a disc (A = r2).
The calculated radius is about 3.35 times smaller than the calculated radius of the pointlike charge inside
Both electromagnetic and nuclear forces act on the electric charge and we, therefore, will simply refer to the
electric charge as charge.
See Annex IV, V and VI of our paper on ontology and physics.
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).
of an electron (re = rC 2.82 fm). In other words, we cannot find any obvious relation between the radii
here, which leads us to conclude that the nature of the positive charge appears to be quite different
from that of the negative charge.
We may illustrate this by thinking of the pointlike charge being held in its orbit by a centripetal force and
comparing the magnitudes of this force for the various particles. For the electron, we get a magnitude
equal to about 0.106 newton (N)
This is, without any doubt, 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!
However, the force
calculation for the muon (whose energy is about 105.66 MeV, so that is about 207 times the electron
energy) yields a much larger value.
The ratio of the force for the electron and the muon is equal to:
If a force of 0.106 N is pretty humongous, then a force that is 42,753 times as strong, may surely be
called a strong force, right? We may now use the same formula to calculate the force, and the force
ratio for the muon-electron and proton respectively, for the proton:
Forces of 4,532 N inside of a muon and a force of 89,349 N inside of a proton are humongous, but the
reader may quickly calculate the corresponding Schwarzschild radii, which show that we are not modeling
some kind of black hole here.
We should also note that the humongous difference between the magnitude of the forces here is related
to scale and the 2/3D form factor only. This can be illustrated by considering orbital energy equations. If
we denote electromagnetic mass as mC and nuclear mass by mN, one can write the orbital energy equation
We have derived this formula elsewhere. The ½ factor is there because we think of the zbw charge as having an
effective (relativistic) mass that is 1/2 (half) of the total electron mass.
We can also calculate field strengths and other magnitudes, but we do not want to be too long here.
for the electromagnetic and nuclear force using the standard parameters and respectively
We may now compare the order of magnitude of the electromagnetic and nuclear force may be
compared by using the same numerical values for mC = mN. Evaluating the potential energy functions
above at r = a and taking their ratio, we get:
This should not surprise us: we define the range parameter here as the distance r = a for which the
magnitude of the two forces (whose direction is opposite) is the same. The form factor and, hence, the
nature of the two forces, is very different, though.
Let us get back to the numbers distinguishing the proton and the muon. The 4 factor is easily explained
by distinguishing between a charge in a planar as opposed to a spherical orbital. In other words, we think
of the force inside of the muon as a 2D nuclear force, as opposed to the 3D nuclear force inside of the
proton. However, the 8.88 and 2.22 numbers (ratio of masses and radii respectively) remain inexplicable,
and strongly suggest that the nature of the positive charge is very different of that of the negative charge:
while they carry the same unit charge, the electron and muon on the one hand, and the proton on the
other, cannot be considered to be each other’s opposite in a physical sense. In other words, the muon
The Greek letter is used for a variety of physical concepts here: magnetic moment, muon-electron, and
standard parameter. The context should make clear what is what. The nuclear force keeps like charges together,
while the Coulomb force repels them, and the reader may, therefore, question the sign for the potential energy.
However, this sign depends on the reference point for the potential energy: the U = 0 point may be chosen at r =
or at r = 0. We choose it at r = 0, which is not the usual convention.
We use a modified Yukawa potential to model the nuclear force.
We may compare this with the energy equation for gravitational orbitals, which follows from Kepler’s laws for
the motion of the planets:
It should be noted that the kinetic and potential energy (per unit mass) add up to zero instead of c2 (nuclear and
electromagnetic orbitals), which is why a geometric approach to gravity makes eminently sense: massive objects
simply follow a geodesic in space, and there is no (gravitational) force in such geometric approach. We can now
compare the standard parameters by equating m to mC (in practice, this means using the mass and charge of the
electron in the equation below) and, once more, equating r to a:
Hence, the force of gravity – if considered a force – is about 1042 weaker than the two forces we know
(electromagnetic and nuclear).
and proton are associated with the same force but with different charges!
Before we move on, we should quickly highlight the consistency of the calculations of the magnetic
moment of the proton.
We recall our calculations for the electron. 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 pointlike charge
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
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 radius for the calculation of the magnetic moment because we
think of the proton oscillation as a spherical (3D) oscillation⎯in contrast to the electron oscillation,
which is a plane (2D) oscillation. Indeed, 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.
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.
The standard error on this measurement is consistent with the theoretical 0.84 fm value. Hence, we cannot be
sure whether or not there is an anomaly in the magnetic moment of the proton. As mentioned above, there is
current no evidence for a possible fractal structure of the positive unit charge⎯as opposed to the negative charge,
which is associated with multiple radii (classical electron radius, Compton radius, and Bohr radius), which are
related through the fine-structure constant, which appears as a scaling constant here.
For a discussion of the anomalies in the magnetic moment, see our paper on the topic.
μp = (1.99489926410−26 J·T−1)/2 = 1.4106067973610−26 J·T−1
Let us now turn to the neutron.
The neutron as a composite particle
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 effectively think of a neutron as a composite particle: n = p + e, which is stable only inside of a
nucleus. We think it combines an electromagnetic and nuclear oscillation.
We propose an orbital
energy equation using a modified Yukawa potential with a range parameter to represent the nuclear
The mass factor mN is the equivalent mass of the energy in the oscillation
, 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 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.
The electromagnetic force which keeps them together is
the Coulomb force:
We think the neutron’s instability outside of the nucleus confirms rather than disproves our model of a neutron
as a composite particle.
We will use the subscripts xN and xC to distinguish nuclear from electromagnetic mass/energy/force. There is
only one velocity, however⎯which should be the velocity of one charge vis-á-vis the other.
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, which we introduce in the following paragraph, 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.
The total energy in the oscillation is given by the sum of nuclear and Coulomb energies and we may,
The latter substitution uses the definition of the fine-structure constant.
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 leave it to the reader to further play with it 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. Here, we are only interested in the formula because it gives us an order of
magnitude for the nuclear range parameter a. This order of magnitude may be calculated by equating r
to a in the formula above
The ħ/mc constant is, obviously, equal to the classical electron radius re 2.818 fm (10−15 m)⎯which is
of the order of the deuteron radius (about 2.128 fm) and which is the usual assumed value for the range
parameter of the nuclear force. We think it is a significant result that the lower limit for the range
parameter for the nuclear force must be at least twice at large.
An upper limit for this range parameter must be based on the experimentally measured value for the
radius of atomic nuclei. The scale for these measurements is the picometer (10−12 m). The nucleus of the
One easily obtains the keqe2 = ħc identity from the
formula. See the rationale of the 2019 revision of SI
The ħ/mc factor is the classical electron radius. Needless to say, the a in the formula(s) above is the range
parameter of the nuclear force, which is not to be confused with the Zitterbewegung (zbw) radius or the
coefficient A of the wavefunction!
The range parameter is usually defined as the distance at which the nuclear and Coulomb potential (or the
forces) equal each other. See: Ian J.R. Aitchison and Anthony J.G. Hey, Gauge Theories in Particle Physics (2013),
section 1.3.2 (the Yukawa theory of force as virtual quantum exchange).
very stable iron (26Fe), for example, is about 50 pm.
The radius of the large (unstable) uranium (92U) is
about 175 pm. The fine-structure constant may be involved again: 5.536 fm times 1/ yields a value of
about 77 pm. We think this is a sensible value for the (range of) the upper limit for the (nuclear) range
parameter, which will, of course, depend on the shape (eccentricity) of the actual orbitals.
Of course, the stability of the nucleus of an atom is determined by other factors, most notably the
magnetic coupling between the nucleons and the electrons in the atomic (sub)shells. This should,
somehow, explain the ‘magic numbers’ explaining the (empirical) stability of nuclei, but the exact science
behind this seems to be beyond us.
The Zitterbewegung hypothesis and the scattering matrix
Based on the considerations above, one can analyze the rather typical K0 + p → 0 + + decay reaction
and write it as follows
The minus sign of the coefficient of the antikaon wavefunction reflects the point we made above:
matter and antimatter are each other opposite, and quite literally so: the wavefunctions –AeiEt/ħ and
+AeiEt/ħ add up to zero, and they correspond to opposite forces and different energies too!
precise, the magnetic field vector is perpendicular to the electric field vector but – instead of lagging the
electric field vector by 90 degrees (matter) – it will precede it (also by 90 degrees) for antimatter, and
the nuclear equivalent of the electric and magnetic field vectors should do the same (we have no reason
to assume something else).
Indeed, the minus sign of the wavefunction coefficient (A) reverses both
the real as well as the imaginary part of the wavefunction.
This is Feynman’s calculated radius of a hydrogen atom, but the measured radius of the hydrogen nucleus is
about half of it. To be precise, the empirical value is about 25 pm according to the Wikipedia data article on atomic
radii. We leave it to the reader to think about the 1/2 factor and the fine-structure constant as a scaling parameter.
See the Wikipedia article on magic numbers (nuclei).
Of course, there are further decay reactions, first and foremost the 0 + + → − + p + + reaction. We chose the
example of the K0 + p reaction because Feynman uses it prominently in his discussion of high-energy reactions
See our previous remarks on the lag or precession of the phase factor of the components of the wavefunction.
Needless to say, masses and, therefore, energies are positive, always, but the nature of matter and antimatter is
We think this explains dark matter/energy as antimatter: the lightlike particles they emit, must be
antiphotons/antineutrinos too, and it is, therefore, hard to detect any radiation from antimatter. See our paper on
However, it is immediately obvious that the equations above can only be a rather symbolic rendering of
what might be the case. First, we cannot model the proton by an AeiEt/ħ wavefunction because we think
of it as a 3D oscillation. We must, therefore, use two rather than just one imaginary unit to model two
oscillations. This may be solved by distinguishing i from j and thinking of them as representing rotations
in mutually perpendicular planes. Hence, we should probably write the proton as
In addition, the antikaon may combine an electromagnetic (2D) and a nuclear (3D) oscillation and we
may, therefore, have to distinguish more than two planes of oscillation.
Last but not least, we should note that the math becomes even more complicated because the planes of
oscillation of the antikaon and the proton are likely to not coincide. We, therefore, think some modified
version of Hamilton’s quaternion approach may be applicable, in which case we have i, j and k rotations.
Furthermore, each of these rotations will be specific to each of the particles that go in and come out of
the reactions, so we must distinguish, say, the iK, jK, kK, from the i, j, k rotations.
The j and k rotations may be reserved for the two perpendicular (nuclear) rotations, while the Euler’s
imaginary unit (i) would model the electromagnetic oscillation (not necessarily perpendicular to any of
the two components of the nuclear oscillation). In addition, we must note these planes of rotations are
likely to rotate in space themselves: the angular frequency of the orbital rotations has a magnitude and
a direction. If an external field or potential is present, then the planes of oscillation will follow the
regular motion of precession. In the absence thereof, the angular rotation will be given by the initial
orbital angular momentum (as opposed to the spin angular momentum).
Form factors and the nature of quarks
All that is left is to wonder what the S-matrix and the coefficients s11, s12, s21, and s22 actually represent.
We think of them as numbers – complex or quaternion numbers – but sheer numbers (i.e. mathematical
quantities rather than ontological/physical realities) nevertheless.
This raises a fundamental question in regard to the quark hypothesis. We do not, of course, question the
usefulness of the quark hypothesis to help classify the rather enormous zoo of unstable particles, nor do
we question the massive investment to arrive at the precise measurements involved in the study of
high-energy reactions (as synthesized in the Annual Reviews of the Particle Data Group).
However, we do think the award of the Nobel Prize of Physics to CERN researchers Carlo Rubbia and
Simon Van der Meer (1984), or – in case of the Higgs particle – Englert and Higgs (2013) would seem to
We use an ordinary plus sign, but the two complex exponentials are not additive in an obvious way (i j). Note
that t is the proper time of the particle. The argument of the (elementary) wavefunction a·ei is invariant. We refer
to Annexes II and III of this paper for an analysis of the wavefunction in the context of SRT and GRT.
The K and subscripts denote the (neutral) antikaon and lambda-particle, respectively. We use an underbar
instead of an overbar to denote antimatter in standard script (i.e. when not using the formula editor).
have awarded 'smoking gun physics' only, as opposed to providing any ontological proof for the reality
of virtual particles.
In this regard, we should also note Richard Feynman's discussion of reactions involving kaons, in which
he – writing in the early 1960s and much aware of the new law of conservation of strangeness as
presented by Gell-Man, Pais and Nishijima – also seems to favor a mathematical concept of strangeness
or, at best, considers strangeness to be a composite property of particles rather than an
In fact, Feynman's parton model
seems to bridge both conceptions at first, but closer examination
reveals the two positions (quarks/partons as physical realities versus mathematical form factors) are
mutually exclusive. We think the reinvigorated S-matrix program, which goes back to Wheeler and
, is promising because – unlike Feynman’s parton theory – it does not make use of
perturbation theory or other mathematically flawed procedures (cf. Dirac's criticism of QFT in the latter
half of his life).
We think this paper sets a firm basis for a renewed examination of Dirac’s concluding remarks of his 4th
edition of The Principles of Quantum Mechanics:
“Quantum mechanics may be defined as the application of equations of motion to particles. […] The
domain of applicability of the theory is mainly the treatment of electrons and other charged particles
interacting with the electromagnetic field⎯a domain which includes most of low-energy physics and
Now there are other kinds of interactions, which are revealed in high-energy physics and are important
for the description of atomic nuclei. These interactions are not at present sufficiently well understood to
be incorporated into a system of equations of motion. Theories of them have been set up and much
developed and useful results obtained from them. But in the absence of equations of motion these
theories cannot be presented as a logical development of the principles set up in this book.
We are effectively in the pre-Bohr era with regard to these other interactions. It is to be hoped that with
increasing knowledge a way will eventually be found for adapting the high-energy theories into a scheme
based on equations of motion, and so unifying them with those of low-energy physics.” (Paul A.M. Dirac,
The Principles of Quantum Mechanics, 4th edition (1958), p. 312)
We hope this paper offers a contribution to this objective.
The rest mass of the Higgs particle, for example, is calculated to be equal to 125 GeV/c2. Even at the speed of
light - which such massive particle cannot aspire to attain – it could not travel more than a few tenths of a
femtometer: about 0.310-15 m, to be precise. That is not something which can be legitimately associated with the
idea of a physical particle: a resonance in particle physics has the same lifetime. We could mention many other
See: Feynman’s Lectures, III-11-5.
See, for example: W.-Y. P. Hwang, Toward Understanding the Quark Parton Model of Feynman, 1992.
See D. Bombardelli, Lectures on S-matrices and integrability, 2016. We opened a discussion thread on
ResearchGate on the question.
The reference list below is limited to the classics we actively used, and publications of researchers whom
we have been personally in touch with:
⎯ Richard Feynman, Robert Leighton, Matthew Sands, The Feynman Lectures on Physics, 1963
⎯ Albert Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik, 1905
⎯ Paul Dirac, Principles of Quantum Mechanics, 1958 (4th edition)
⎯ Conseils Internationaux de Physique Solvay, 1911, 1913, 1921, 1924, 1927, 1930, 1933, 1948
(Digithèque des Bibliothèques de l'ULB)
⎯ Jon Mathews and R.L. Walker, Mathematical Methods of Physics, 1970 (2nd edition)
⎯ Patrick R. LeClair, Compton Scattering (PH253), February 2019
⎯ Herman Batelaan, Controlled double-slit electron diffraction, 2012
⎯ Ian J.R. Aitchison, Anthony J.G. Hey, Gauge Theories in Particle Physics, 2013 (4th edition)
⎯ Timo A. Lähde and Ulf-G. Meissner, Nuclear Lattice Effective Field Theory, 2019
⎯ Giorgio Vassallo and Antonino Oscar Di Tommaso, various papers (ResearchGate)
⎯ Diego Bombardelli, Lectures on S-matrices and integrability, 2016
⎯ Andrew Meulenberg and Jean-Luc Paillet, Highly relativistic deep electrons, and the Dirac
⎯ Ashot Gasparian, Jefferson Lab, PRad Collaboration (proton radius measurement)
⎯ Randolf Pohl, Max Planck Institute of Quantum Optics, member of the CODATA Task Group on
Fundamental Physical Constants
⎯ David Hestenes, Zitterbewegung interpretation of quantum mechanics and spacetime algebra
(STA), various papers
⎯ Alexander Burinskii, Kerr-Newman geometries (electron model), various papers
⎯ Ludwig Wittgenstein, Tractatus Logico-Philosophicus (1922) and Philosophical Investigations
⎯ Immanuel Kant, Kritik der reinen Vernunft, 1781
Annex I: Elementary wavefunction math
The wavefunction with decay factor
The math of transients is not so difficult: it suffices to multiply the wavefunction (let us refer to our
unstable particle as U, so we can denote something stable as S) with a real-valued negative exponential:
The illustration below shows how this works: both the real and imaginary part of the wavefunction –
think of the electric and magnetic field vector here, for example – lose amplitude and, therefore, energy.
Where does the energy go? It cannot get lost, so we must assume it goes into the field, where it
contributes to progressively building up another oscillation. The combined particle-field combination
will, therefore, be something stable (S) that conserves energy (and, therefore, mass):
We may apply the usual interpretation to the and factors:
1. The in the e−t decay function gives us the mean lifetime of the unstable particle ( = 1/) and,
as Feynman points out, such mean lifetime will be of the order of 10−9 to 10−12 seconds.
2. The in the Ae−t decay function is equal to E/ħ and will generally be a frequency (its dimension
is s−1) that is much larger than . The frequency of an electron, for example, can be calculated
As we can see, we have a difference of 10 orders of magnitude (1010) between and here, and an
electron is not very massive as compared to a proton! Of course, this explains that transient or resonant
particles do not last very long, but still pack like 1010 cycles during their short lifetime!
Interpretation of the wavefunction of stable particles
A stable (elementary) particle is stable. We must, therefore, represent it by a mathematical function
which evolves in time and, at the same time, does not. The exponential function is its own derivative
(det/dt = et), so that suits the need.
Time is measured as the motion of the hand of a clock (you should think of an old-fashioned stopwatch
with one hand only here). Such motion combines rotation and a number, so one might write time as −it
and measure time in (circular) radians. The aspect of orbital motion is thus captured by the d(−it)/dt =
−i expression. The second derivative is the derivative of a constant (the imaginary unit is a mathematical
constant) and is, therefore, nil: d2(−it)/dt2 = d(−i)/dt = 0. We consider this to be the pendant to the
det/dt = et relation, and will, therefore, take it as a definition of time.
According to mathematical convention, the imaginary unit i represents a counterclockwise rotation.
Such convention only makes sense when establishing a line of sight between the subject (observer) and
the object (particle) and agreeing on what is up/down, left/right, and front/back.
Now, we represent a particle by a complex exponential (eiθ = eit) with a real-valued argument (t or t).
The frequency appears as a scaling constant of time. It disappears when choosing a natural time unit T
= 1/ = ħ/E (Nms/Nm = s). We will soon be using natural units to simplify the expressions.
To derive a complex-valued function with respect to a real variable, we must take the derivatives of the
real and imaginary parts, respectively
Using Feynman’s convention and writing the wavefunction as e−iθ = e−it, we get
The derivative of the wavefunction with respect to time is, therefore, equal to the same wavefunction
but with real and imaginary parts rotated (clockwise rather than counterclockwise) over 90 degrees. We
have no immediate meaningful explanation of the −ie−iθ expression. There can be no time reversal: time
goes in one direction only.
The direction of spin (angular momentum) has not reversed either, so we
have no parity reversal either. The minus sign of the wavefunction reminds us of antimatter, but the
expression may more resemble the B = −iE/c equation from electromagnetic theory. We have no
clue⎯for the time being, that is (we will give you a logical interpretation in the next section of this
Annex, however). For the moment, we should only highlight that it is rather significant that the second
time derivative is zero.
In the context of real variables (e.g. energy), time appears a linear feature. Hence, one might say one can take
the derivative of e−iθ in two complementary ways⎯linear and orbital, so to speak: de−iθ/dθ = −ie−iθ and de−iθ/d(iθ)
= e−iθ. We think this reflects the sloshing back and forth of the kinetic and potential energy of the particle. See our
remarks on the physical dimension of the wavefunction at the end of this annex.
See, for example, Y.D. Chong, Complex Methods for the Sciences (course MH2801), 2020.
Time goes in one direction only and, hence, we must use one or the other convention here. However, one can
show that Feynman’s choice makes sense physically: Maxwell’s equation imply the magnetic field vector lags the
electric vector (by 90 degrees, i.e. /2) and we must, therefore, write B = −iE/c. The sign of the imaginary unit
can then be used to capture the spin direction of the (elementary) particle. We model antimatter particles using
negative wavefunctions −Aeit, again with the sign of the imaginary unit modeling up or down spin, respectively.
The minus sign of the coefficient A amounts to modeling an antiforce. In case of the electromagnetic force, we will
have a magnetic field vector whose phase will precede the phase of the electric field vector by the same amount
(/2 radians). The sin(θ) = icos(θ) and cos(θ) = −isin(θ) can easily be explained geometrically or using the usual
This is inherent to the notion of motion or velocity: v = ds/dt. A particle may go back and forth, but a particle
cannot be at two different places at the same time.
This is only because the orbit of the pointlike charge is circular. For elliptical orbitals, the situation would be
The use of natural units
We may use the cycle time T as a natural time unit. The Planck-Einstein relation tells us that the cycle
time is equal to the energy of the particle divided by Planck’s quantum of action: T = E/ħ = 1/, and,
when using the cycle time as a natural unit, the numerical value of T will be equal to 1 radian (note the
frequency is an angular frequency: = 2f). The particle wavefunction Ae−iTt then reduces to Ae−it.
Let us write it all out:
The coefficient A is the Compton radius of the (free) particle and is equal to A = ħ/mc = ħc/E = c/T = c.
When using T as the natural unit, the A = c/T reduces to A = c, which is quite remarkable. Now, if we
have a natural time unit (T = 1), we can also choose a natural distance unit so as to ensure the numerical
value of the speed of light will also equal one (of course, its physical dimension remains that of a linear
(radial) or tangential velocity). Hence, we write: T = 1 (radians) and c = 1, and A = 1/T, and our function
above reduces to:
An unstable particle will lose energy and, hence, we can no longer treat the energy and, therefore, the
cycle time T, as a constant. A = 1/T will, therefore, also no longer be constant. We, therefore, have an
additional decay factor in the wavefunction. However, we may still treat T0 = E0/ħ as a constant. Using
natural units once more, we may write:
The time derivative can, therefore, be written as:
The function differs, once more, only by the decay factor = 1/τ. We are, once again, not sure how to
interpret the result, but we note this is not just the particle’s antiparticle with a phase shift of 90
degrees (/2). Only for very large values of τ (τ → ) do the stable and unstable particle resemble each
other, but that is stating the obvious.
Perhaps the following reflections on the physical dimension(s) of the wavefunction help to interpret the
results in regard to the derivatives. We suggest the oscillations of the real and imaginary part of the
wavefunction capture potential and kinetic energy sloshing back and forth. Their physical dimension
must, therefore, be that of an energy density: J/m3 = Nm/m3 = N/m2⎯a force (N) per unit area (m2). The
time derivative of the wavefunction must, therefore, somehow model how the particle energy interacts
with the field(s) that sustain its oscillation.
Perhaps we should think of de−iθ/dθ = −ie−iθ derivative as representing a (stable) virtual field oscillation:
a virtual or field particle which, for all practical purposes, serves as the particle’s field counterpart? For
unstable particles, we would have an additional
term which would then, somehow, model
how energy dissipates away in space⎯not feeding back into the particle oscillation so as to keep it
Yes? No. There is a simpler solution. Let us elaborate the point geometrically.
The wavefunction as a description of motion
We may try to visualize the above. A wavefunction may describe both the position (r = a = e−iθ = cosθ +
isinθ) of the pointlike charge on its orbit – in terms of its coordinates x = (x, 0) = (cosθ, 0) on the real
axis and y = (0, y) = (0, sinθ) on the imaginary axis (y = ix) – or, alternatively, in terms of the force F = Fx +
Fy which keeps the pointlike charge in place. The force is a centripetal force, so it is equal to −r.
Figure 2: The Zitterbewegung model of a charged elementary particle
Now, we also have the momentum vector p = mc. The initial point of the position vector r = e−iθ is the
zero point of the reference frame, while the initial point of the momentum vector p (i.e. its point of
application) coincides with the (moving) terminal point of the position vector. Denoting vectors in the
negative x- and y-direction as −x and −y respectively, we can now easily relate the two components of
the momentum vector to the x and y components of the position vector:
px = −iy and py = −ix
We can, therefore, effectively consider the wavefunction to describe the position r of the pointlike
charge, while its time derivative describes the momentum vector. We, therefore, write:
r = e−iθ
p = −ie−iθ
Note that it is tempting to write the imaginary unit as vector quantity too: it has a magnitude (90
degrees or /2 radians) and, as a rotation, a direction too (clockwise or counterclockwise). However, its
direction depends on the plane of oscillation and we, therefore, write it in lowercase (i instead of i).
As for the mechanism behind, the equations below illustrate how one can imagine the sloshing back and
forth of the energy between the real and imaginary axes, respectively.
Energy is a force over a distance (1 joule is 1 newton times 1 meter) and it is, therefore, tempting to think of
energy as having some direction too. However, the reader must remember the force components depend on the
reference frame. As such, they are mathematical objects only.
Figure 3: The wave propagation mechanism
We do not expect the reader to have an Aha-Erlebnis here because the equations above require an
intuitive understanding of vector differential operators (gradient, divergence, and curl), which our
readers may not possess. The reader should, at the very least, recognize (fragments of) Schrödinger’s
equation (the first set of equations) as well as Maxwell’s equations in free space (the second set of
In any case, we hope our readers will appreciate the similarities between the two sets of
Note that the momentum vector has the same direction of the velocity vector v = c. We can also write it
as p = meffc = mc/2.
What about the force? The relativistically correct formulation of Newton’s force
law then yields the following:
However, the derivation is quite complicated because the change in p is a change in direction⎯not in
magnitude. We must, therefore, refer to the force calculation in the main body of this paper.
For the time being, this is all we can reasonably say about the wavefunction. For a more detailed
analysis, we refer to our rather critical review of Feynman’s argument on the Hamiltonian equations.
Note that we do not necessarily assume the motion of the pointlike charge is deterministic: we only
assume its motion is sufficiently regular so as to give meaning to the concept of (orbital) frequency.
When everything is said and done, these physical theories are and will forever remain bootstrap
Explaining force and momentum from the radius of the oscillation.
We are not sure if this approach using the derivative is the right one. Let us try another one. Let us recap
the essential: the wavefunction r = ae−iθ represents the position of a pointlike charge in orbital motion.
Kinetic and potential energy slosh back and forth. For (non-circular) elliptical orbitals, the distance a will
be a function of θ as well, so we write the position vector as: r = a(θ)e−iθ. The phase θ is a function of
the rest energy (E0) and of time (t’) in the reference frame of the particle, and of energy (Ev), time (t) and
Free space means we have no (other) currents. For a more comprehensive exploration of the geometry, we refer
the reader to The Wavefunction as an Energy Propagation Mechanism. The reader may also want to check out our
tentative Geometric Interpretation of Schrödinger’s Equation.
The concept of effective mass is one of those concepts which, surprisingly, Feynman did not seem to quite get
right. We talk about it in our manuscript.
Jean Louis Van Belle, Feynman’s Time Machine, June 2020.
its linear momentum (p) associated with its classical (linear) velocity and relativistic mass as a moving
particle. The equation below, and the illustration, shows how classical motion adds a linear component
to the orbital motion of the pointlike charge (see Figure 7).
Figure 4: The Compton radius of a particle decreases with increasing velocity
Let us first analyze things in the inertial reference frame (t = t’ and x = px = 0). There is, then, only orbital
or tangential momentum, which we will denote by pθ or just p, and we will use the x and y coordinates
to describe the position of the pointlike charge in its plane of oscillation once more (so just forget about
the x = 0 above for the time being). The pointlike charge has no rest mass and, therefore, its (tangential)
velocity always equals c (as shown in Figure 7). The relations between the radius a = ħ/mc = ħc/E, the
momentum p = mc and the centripetal force F which keeps the pointlike charge in its orbit.
Figure 5 shows that the components of the velocity vector v = c are orthogonal to the radius vector a.
We can use the imaginary unit to represent a rotation and, therefore, we get the following vector
relation from the a = ħ/mc equation:
Figure 5: The Zitterbewegung model of a charged elementary particle
We can, therefore, write the tangential momentum as:
We borrow this illustration from G. Vassallo and A. Di Tommaso (2019).
We can easily verify this by re-substituting:
The reader should note this embodies the (second) de Broglie equation p = h/λ, but we must (i) interpret
λ as the circumference of the loop which, for a circular orbit
, is equal to λ = 2a, and (ii) appreciate the
geometry of the situation using i and j rotational operators
, which we use in conjunction with the
angular momentum vector h. We write:
We must now address the question of the force. The relativistically correct F = dp/dt law does not apply
here, because it concerns linear motion only. The relation we should use here is the E = Fλ F = E/λ =
relation: energy is force over a distance (joule = newton times meter). Here, the length λ must be
interpreted correctly: it is the circumference of the loop, the radius of the loop, or – what we think is the
case and which we will, therefore, use as an operational definition – twice the radius of the loop (2a)?
We think the latter is the case based on our analysis of the effective (or relativistic) mass of the pointlike
charge, which is half the mass of the elementary particle. This is reflected in Schrödinger’s wave
equation for an electron in free space
Taking into account that the direction of the centripetal force is opposite to the direction of the radius
vector a, we get:
This is the formula we used to calculate the forces inside of an electron and a proton in this paper. What
remains to be done is a dimensional analysis of the equations (easy), and explain why the quantum-
mechanical operators (especially the position, momentum, and angular momentum operators) give the
results we get from them. Finally, the results above need to be generalized to non-circular elliptical
orbitals. The complexity of this is illustrated below (Figure 6) and shows this may not be very obvious.
Note there is, as yet, no simple analytical formula for the circumference of an ellipse. Integration is required to
The imaginary unit i is a (counterclockwise) rotation within the plane of motion/oscillation. We use j to represent
a rotation in a plane that is perpendicular to the plane of motion/oscillation. The plus/minus () sign of the rotation
should be defined by the usual righthand rules (the sign of i is a clockwise/counterclockwise rotation within the
plane of motion). The i and j rotations are, obviously, perpendicular in 3D space, and, therefore, the condition i2 =
j2 = −1 must apply. We admit we skip a few steps here. We leave it to the reader to work out the geometry. We
should warn him/her, however: the jħ/a = −iħ/a is quite deep⎯and, therefore, not very straightforward.
See the annex to our paper on de Broglie’s matter-wave.
Figure 6: Position and angular/radial velocity vectors (elliptical orbitals)
So what about that derivative of the wavefunction, then? The derivative of a position vector gives us it
velocity: dr/dt = v. the geometry is a bit particular here: r and v do not have the same direction, but dr
and v, yes! No problem! So here you go:
Sounds weird? Check it:
What about probability amplitudes? Energy is proportional to the square of the amplitude: E = ma22.
All the rest follows from that. We add amplitudes – fields and forces – to model interference, and that is
all there is to quantum mechanics. It is a lot, and then not.
We will conclude this annex by saying a few word about quantum-mechanical operators.
Understanding the invariance of the argument θ = (Evt − pxx)/ħ = E0t’ of the wavefunction is crucial to
understanding how quantum-mechanical operators work. It is easy to see 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). The argument of the wavefunction is a complex
function, however. Indeed, we write the wavefunction as ψ= ae−iθ.
We can now understand the linear component of the velocity of the pointlike charge as a drift velocity
(see Figure 4), and it easy to see we can 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:
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:
Is it that simple? Yes. We could refer the reader to Feynman’s derivation of the other quantum-
mechanical operators (Lectures, III-11-5): “All the complicated theories that you may hear about are no
more and no less than this kind of elementary hocus-pocus.” Can we show that?
The position operator is simple and weird at the same time:
Applying the operator to the wavefunction, we get the following trivial identity:
This looks trivial. What does it mean? If we interpret the position x as the position of the pointlike
charge (r), then xψ is equal to r2, which is a square whose surface area is equal to a2 = ħ2c2/E2. This
must, of course, have something to do with the probability of finding the charge, which is proportional
to the squared energy densities.
What about the energy operator? Feynman (III-20) writes it as:
Applying it to the wavefunction, we get the righthand-side of Schrödinger’s equation for the hydrogen
We know the solutions to Schrödinger’s equation yield definite energy states, but Feynman uses the
energy operators also to calculate average energy over a range of states, and shows this average is
weighted by the probabilities of the system being in one of these possible states. More generally
speaking, however, we should note that the context of Schrödinger’s equation is quite specific. We,
therefore, prefer to define another energy operator:
This matches better with the momentum operator:
We cannot add much more here in terms of interpretation. Operators are a convenient shorthand to
write down complicated equations, and that is probably the only reason why we developed this section
on their logic here. In fact, in the next section, we will show we can consider all operators to be
quantum-mechanical operators, using the example of the first- and second-order time derivative.
Calculating the velocity and acceleration vector from the stationary wavefunction
The velocity vector v is the (tangential) velocity v = c of the pointlike charge⎯not of the elementary
particle, which we think of as being at rest. The position vector r = ae−it can then be derived with
respect to time to yield the velocity vector v = c:
This is fine: the magnitude of the velocity vector is c, and its dimension is that of a velocity alright (m/s).
Let us now calculate the acceleration vector a (there should be no confusion with the amplitude a or the
radius vector r, here):
We find that the magnitude of the (centripetal) acceleration is constant and equal to −a.
This is a most
To conclude, we can show this also work for Bohr-Rutherford electron orbitals. Their radius is of the
order of the Bohr radius rB = rC/, and their energy is of the order of the Rydberg energy ER = 2mc2, with
The velocity and accelerations are, therefore, equal to:
We get the classical orbital velocity v= c, while the magnitude of the acceleration equals c2/a,
which has the right physical dimension (m2/s2)/m = m/s2. As you can see, there is nothing magical or
mysterious about quantum-mechanical operators: /t and 2/t2 are quantum-mechanical operators
The minus sign is there because its direction is opposite to that of the radius vector r.
If the principal quantum number is larger than 1 (n = 2, 3,…), an extra n2 or 1/n2 factor comes into play. We refer
to Chapter VII (the wavefunction and the atom) of our manuscript for these formulas.
Annex II: The wavefunction and special relativity
Particles are finite quanta: their energy/mass is finite, and they pack a finite amount of physical action.
Stable particles pack one or multiple units of ħ (angular momentum): E0 = ħ = hf = h/T. For unstable
particles, the Planck-Einstein relation is not valid. The wavefunction of unstable particles involves an
additional decay factor :
The sign of the coefficient A captures the difference between matter and antimatter, while the sign
of the complex exponent (iEt/ħ) captures the direction of spin (angular momentum).
differ from matter-particles because they carry no charge. Their oscillation (if photons are
electromagnetic oscillations, then neutrinos must be nuclear oscillations) is, therefore, not local: they
effectively travel at the speed of light.
The energy in the wavefunction is the rest energy of the particle, which we think of as a wavicle: its
essence is an oscillating pointlike charge. We, therefore, think of the elementary wavefunction to
represents the motion of the pointlike charge by interpreting r = A·eiθ = A·ei·(E·t − k·x)/ħ as its position
vector. The coefficient A is then, equally obviously, nothing but the Compton radius A = rC = ħ/mc. The r
= A·eiθ = A·ei·(E·t − k·x)/ħ expression shows how classical motion adds a linear component to the argument of
the wavefunction (see Figure 7).
Figure 7: The Compton radius must decrease with increasing velocity
The relativistic invariance of the argument of the wavefunction is then easily demonstrated by noting
that the position of the pointlike particle in its own reference frame will be equal to x’(t’) = 0 for all t’.
We can then relate the position and time variables in the reference frame of the particle and in our
frame of reference by using Lorentz’s equations
See Annex IV, V and VI of our paper on ontology and physics.
We borrow this illustration from G. Vassallo and A. Di Tommaso (2019).
We can use these simplified Lorentz equations if we choose our reference frame such that the (classical) linear
motion of the electron corresponds to our x-axis. See Feynman’s Lectures, I-15-2.
When denoting the energy and the momentum of the electron in our reference frame as Ev and p =
m0v, the argument of the (elementary) wavefunction a·ei can be re-written as follows
Besides proving that the argument of the wavefunction is relativistically invariant, this calculation also
demonstrates the relativistic invariance of the Planck-Einstein relation when modelling elementary
Needless to say, the plane of the local oscillation is not necessarily perpendicular to the direction of
(linear) motion, nor must we assume the local oscillation is necessarily planar. For a proton, one must
apply an extra factor (4) to calculate its Compton radius:
The 4 factor is the 4 factor which distinguishes the formula for the surface area of a sphere (A = 4πr2)
from the surface area of a circle (A = πr2).
We effectively think of an oscillation in three rather than just
two dimensions only here: the oscillation is, therefore, driven by two (perpendicular) forces rather than
just one, and the frequency of each of the two oscillations is equal to = E/2ħ = mc2/2ħ: each of the two
perpendicular oscillations would, therefore, pack one half-unit of ħ only
, and – applying the
equipartition theorem – each of the two oscillations packs half of the total energy of the proton. This
spherical view of neutrons (and protons) – as opposed to the planar picture of an electron – fits nicely
with packing models for nucleons.
Let us analyze the argument of the wavefunction more in detail. We wrote it as:
The momentum of a photon (and, we must assume, a neutrino
) is equal to p = mc = mc/c2 = E/c, with
E = Ev = Ec. The equation above is, then, equal to:
We use the relativistically correct p = mv equation, and substitute m for m = E/c2.
The relativistic invariance of the Planck-Einstein relation emerges from other problems, of course. However, we
see the added value of the model here in providing a geometric interpretation: the Planck-Einstein relation
effectively models the integrity of a particle here.
Cf. the 4π factor in the electric constant, which incorporates Gauss’ Law (expressed in integral versus differential
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).
We think a neutron consists of a positive and a negative charge, and combines an electromagnetic as well as a
nuclear oscillation. See the above-mentioned paper on ontology and physics.
We think of the neutrino as the light-particle of the nuclear force: just like a photon, it does not carry charge, but
it carries nuclear energy.
We can, therefore, see that the argument of the wavefunction for a particle traveling at the speed of
light vanishes! This is not easy to interpret. It is not like time has no meaning anymore but relativistic
time dilation becomes absolute: in our frame of reference, we think of the clock as the photon as
standing still. To put it differently, all of its energy is in its motion, and it derives all of its energy from its
For particles that are not traveling at the speed of light, we still have the two terms:
The dimensional analysis of the Ev/ħ and the p/ ħ is rather instructive and shows the argument (of
phase) of the wavefunction has no physical dimension:
This makes sense because the phase of the wavefunction is measured in radians which can be used both
as distance as well as time units. One can appreciate this idea when re-writing the phase as:
The p = mv = Ev/c2 relation allows us to rewrite the argument of the wavefunction also as:
This relation, too, can be easily verified
The point is this: an elementary particle packs one unit of physical action (ħ) – per oscillation cycle, that
is – and, when in motion, we think of this as expressing itself as a combination of (i) angular momentum
(and, therefore, rotational energy) and (ii) linear momentum.
Now, the functional behavior of the t’ = (t − vx/c2) function may not be immediately obvious: goes
from 1 to infinity () as v goes from 0 to c, and time dilation may, therefore, not be immediately
understood. Hence, a graph may be useful. To produce one, we write x as a function of t: x(t) = vt. The t’
function can, therefore, be rewritten as:
The −1 factor is the inverse Lorentz factor, and its function (for positive v) is the arc of the first quadrant
of the unit circle, as illustrated below. It is, therefore, easy to see that, for any velocity v (0 < v < c), t’ will
be smaller than t, which illustrates the point.
We use the
Figure 8: The inverse Lorentz factor (−1) as a function of
Likewise, the behavior of the = (Evt − px)/ħ function may also not be immediately obvious, but
rewriting it as = (E0t)/ħ – and taking what we wrote about the t’ = −1t function – shows that the
phase of the wavefunction shows the same time dilation.
Note: The reader should not think we established a non-heuristic logical proof of special relativity based
on the reality of the wavefunction. If anything, we only showed that quantum mechanics is fully
consistent with special relativity (and, as we will show in the following annex, with general relativity).
We do think, however, that we did show what the relativistic invariance of the argument of the
wavefunction actually means, and that quantum mechanics and relativity theory mutually confirm each
That does not amount to an intuitive understanding of special relativity, of course. Understanding
(special) relativity theory intuitively may not be possible, but the following considerations may or may
not help the reader to play some more with it.
When observing a object which is moving sideways with velocity v, we may think of its velocity v as a
Figure 9: Tangential velocity
Of course, you will say that most objects are not moving sideways only, but also towards or away from
us. However, such motion along the line of sight (which we will refer to as the radial velocity) can be
determined from the red- or blueshift of the light we use to determine the position of the object (in
order for us to able to track the position of an object – in what we refer to as the inertial reference
frame – it has to emit or reflect light). Hence, if we can determine both the tangential as well as the
radial velocity, we can add the two velocity components to get the combined velocity vector.
It is good to specify what is relative and what is not here: the distance between us, the observer, and the
object is not relative: there is no length contraction along the line of sight. Also, in the reference frame
of the object (which we will refer to as the moving reference frame), the (tangential) velocity of our
reference frame will be measured just the same: v. Finally, the speed of light does not depend on the
reference frame, either. Clock speeds, however, will depend on the reference frame, which gives rise to
the distinction between t and t’.
Because there is no length contraction along the line of sight, its length will be measured the same in
the inertial and moving reference frame. Lightspeed is used as the yardstick in both reference frames
and we must, therefore, conclude this distance must be measured using non-moving clocks. In other
words, we must assume the same clock is used here.
In contrast, the relative velocity of the reference
frames is measured using moving clocks:
When combining this with the t’ = −1t relation (which establishes time dilation
), we get the relativistic
length contraction equation:
We get the same graph (Figure 8): for any velocity v (0 < v < c), ds’ will be smaller than ds, and s’ will,
therefore, be smaller than s
, which illustrates the point.
There is little to add, except for a few remarks on geometry perhaps:
1. If the distance between the origin of the inertial reference frame and the s = s’ = 0 point is equal to a
(the same in both reference frames, remember!), then we may measure that distance in equivalent time
units by dividing it by the speed of light. This amounts to measuring the distance a as a time distance. Of
course, we can always go back to measuring a as a distance by multiplying the time distance by c again:
we then get the distance expressed in light-seconds, i.e. as a fraction or multiple of 299792458 m.
In fact, we think a good understanding of the absolute nature of the speed of light, and a deeper
understanding of the equivalence of using time and spatial distances may be all what can be provided in
terms of a more intuitive understanding of relativity theory. Indeed, when everything is said and done,
we are always measuring things in one specific reference frame: swapping back and forth between
reference frames is a rather academic exercise which does not clarify all that much: the laws of physics
(mass-energy equivalence, Planck-Einstein relation, force law, etcetera) are the same in every reference
This is not a matter of synchronization: we must assume the clock that is used to measure the distance from A to
B does not move relative to the clock that is used to measure the distance from B to A. It is one of these logical
facts which makes it difficult to understand relativity theory intuitively: clocks that are moving relative to each
other cannot be made to tick the same. An observer in the inertial reference frame can only agree to a t = t’ = 0
point (or, as we are talking time, a t = t’ = 0 instant, we should say). From an ontological perspective, this entails
both observers can agree on the notion of an infinitesimally small point in space and an infinitesimally small instant
of time. Indeed, both observers also have to agree on the s’ = s = 0 point!
We get the time dilation equation from writing s as a function of t: s(t) = vt and substituting in the Lorentz
See footnote 61: observers need to agree both on the t = t’ = 0 as well as on the s = s’ = 0 point!
frame and, hence, students should probably consistently focus on understanding these rather than
relativity, as relativity is just a logical consequence of these laws!
In any case, let us agree on writing a – which is, of course, the length of the base of the triangle in Figure
9 – as a spatial distance but assume all spatial distances are measured in light-seconds. This also implies
that we can write the velocities v, vt, and vr as relative velocities , t, and r, respectively.
Let us, indeed, introduce the radial velocity again now. We can then write the velocity vector as = t +
r, with t = ds/dt = ds’/dt’. The length of the hypotenuse will, therefore, be equal to a + rt.
Pythagoras’s Theorem then gives us the following equation:
(a + rt)2 = a2 + (tt)2
a2 + r2t2 + 2art = a2 + t2t2
(t2 − r2)t = 2ar
Multiplying both sides with c2, yields an equation in terms of the usual velocities measured in m/s:
(vt2 − vr2)t = 2acvr
It is a nice equation, but there is probably not all that much we can do with it.
2. Figure 9 introduces the concept of the phase (), which we measure in radians, and the angular
frequency , whose dimension is s−1. The two are related through the = t equation and, also using
the v = a equation, it will be easy for the reader to verify the following relation:
We leave it to the reader to establish the relations for the variables in the moving reference frame.
The reader will probably know Pythagoras’s Theorem does not apply to curved spacetime, but here we are
talking about special relativity only. Note that the ac factor gives us a radial distance expressed in meter again (not
in light-seconds). We are a little bit puzzled to what this expression might mean geometrically, so any suggestion
and/or correction of our readers is most welcome!
Annex III: The wavefunction and general relativity
We know a clock goes slower when placed in a gravitational field. To be precise, the closer the clock is to
the source of gravitation, the slower time passes. This effect is known as gravitational time dilation.
This cannot be explained by writing the argument of the wavefunction as a function of its energy Ev and
its momentum p. We will, therefore, distinguish (i) the rest energy of the particle outside of the
(gravitational) field (E0) and (ii) the potential energy it acquires in the field (Eg). The total energy as
measured in the equivalent of the inertial frame of reference (which is the reference frame without
gravitational field, i.e. empty space), and the argument of the wavefunction, can therefore be written as:
E = E0 + Eg E0 = E − Eg
This effectively shows the frequency of the oscillation is lower in a gravitational field. At first, the
analysis looks somewhat counterintuitive because the convention is to measure potential energy (PE) as
negative (the reference point for PE = 0 is usually taken at infinity, i.e. outside of the gravitational field).
However, when noting extra energy must be positive (i.e. when taking the reference point for PE = 0 at
the center of the gravitational field, or as close to the source as possible
), all makes sense. We hopes
this provides a more intuitive understanding of gravitational time dilation based on the elementary
The reader should note this analysis is also valid for an electromagnetic or nuclear potential, or for any
potential (which may combine two or all three of the forces
). We may refer the reader here to
Feynman’s rather excellent analysis of potential energy in the context of quantum physics in his
Lectures, in which he also explains the nature of quantum tunneling.
However, we think Feynman’s
analysis suffers from a static view of the potentials involved.
We think one should have a dynamic view of the fields surrounding charged particles. Potential barriers
– or their corollary: potential wells – should, therefore, not be thought of as static fields: they vary in
time. They result from two or more charges moving around and creating some joint or superposed field
which varies in time. Hence, we think a particle breaking through a ‘potential wall’ or coming out of a
potential ‘well’ is just using a temporary opening corresponding to a very classical trajectory in space
and in time. We, therefore, think there is no need to invoke an Uncertainty Principle.
See, for example, the Wikipedia article on gravitational time dilation.
A gravitational field comes with a massive object which is usually taken to have a (finite) radius.
We are not aware of any successful attempt proving the gravitational force may be analyzed as some residual
force resulting from asymmetries or other characteristics of the two forces which we consider to be fundamental
(electromagnetic and nuclear). The jury is, therefore, still out on the question of whether or not we should think of
the gravitational force as a pseudoforce. We, therefore, still think of Einstein’s geometric approach to gravity
(curved spacetime) as an equivalent analysis. The question may be entirely philosophical: it should be possible to
also come up with a geometric interpretation of the electromagnetic and nuclear forces but, because of their
multidimensional character (2D/3D, respectively), this may not be easy.
See: Feynman’s Lectures, Potential energy and energy conservation (III-7-3).
Annex IV: An S-matrix representation of Compton scattering?
Compton scattering is a scattering process too. Can we represent the scattering event in terms of the S-
matrix? It should be possible: we have two particles going in (the electron at rest and the incoming
photon) and two particles going out (the moving electron and the outgoing photon). Let us, therefore,
give it a try. We will use the analysis of Compton scattering by prof. Dr. Patrick LeClair
to try to shed
some light on the equations. The geometry of the situation is shown in Figure 10.
Figure 10: Compton scattering
The (linear) momentum conservation law (considered along the horizontal and vertical axes) gives the
following equations for the angles ϕ and θ:
We multiply the second identity with the imaginary unit (i) and add both (ei0 = 1):
The Compton radius, of an electron and a photon respectively, is given by
See: http://pleclair.ua.edu/PH253/Notes/compton.pdf. We found this exposé quite enlightening and, therefore,
borrow quite a lot from it. We assume the subscript f (in the pf expression) refers to the changed frequency of the
outgoing photon. We will use the symbol to refer to a photon in general, but substitute by i or f when denoting
the incoming and outgoing photon specifically.
We use the pc = Ev/c relation here, which reduces to E = pc for the photon ( = v/c = 1). It should be noted that
the electron acquires momentum only through the interaction. Before the interaction, the classical velocity of the
electron is zero. We distinguish the rest energy of the electron from the energy of the moving (outgoing) electron
by denoting them as E0 and Ee, respectively.
We can, therefore, rewrite the equation as follows
Are these wavefunctions? No. The wavefunctions of the photon and electron respectively are given by:
LeClair (2019) defines three dimensionless parameters by taking the ratios of (1) the energies of the
incoming, outgoing photon, and the scattered electron respectively, and (2) the energy of the electron
at rest, which we will denote as E0 so as to distinguish it from the energy of the electron after the
interaction (Ee) . These are, effectively, frequency ratios and, therefore, dimensionless numbers:
We should note that, in LeClair’s argument (which we will further follow here), Ee is redefined as the
kinetic energy of the moving electron only: it no longer includes the rest mass of the electron. We
further refer to LeClair (2019) for the derivation of Compton’s law from the usual conservation laws
(energy and momentum), and will just write down the results:
What happened to the other angle ϕ? We refer, once more, to LeClair (2019) to show one can calculate
ϕ from calculating θ from the relation(s) above:
Our exercise failed. Of course, we could use the wavefunctions above to rewrite the Compton scattering
process as a system of equations using the S-matrix, but there is no obvious relation between the
standard equations that we have presented above, and the S-matrix representation, which we write
We might have substituted p for p = mv straight away, but we wanted to remind the reader of the physicality of
the interaction by mentioning the Compton radii.
It should be possible to relate the Compton equations to this set of S-matrix equations, but we do not
see immediately how.
We note that the S-matrix representation seems to lose track of the (linear) momenta (magnitudes as
well as direction) of the incoming and outgoing particles, which we think of as a major disadvantage of
Any solutions proposed by our readers will be read with interest.