The language of physics
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
22 March 2021
Symbols, logic, and the arrow of time ........................................................................................................................1
The stationary wavefunction ......................................................................................................................................4
Velocity and acceleration vectors ..............................................................................................................................6
The ring current model and the scattering matrix .....................................................................................................6
Form factors and the nature of quarks ......................................................................................................................8
The wavefunction and special relativity .....................................................................................................................8
The wavefunction and general relativity ................................................................................................................. 14
An S-matrix representation of Compton scattering? .............................................................................................. 15
References ............................................................................................................................................................... 18
Symbols, logic, and the arrow of time
With a stationary wavefunction, we mean the wavefunction in the reference frame of the elementary particle
itself: there is only elliptical motion⎯no linear motion, i.e. no classical velocity or momentum. We imagine a
pointlike charge, whose rest mass is zero but which acquires relativistic mass because of its (relativistic) velocity.
The situation is depicted below for elliptical and circular motion, respectively.
We will stick to an analysis of circular motion for the time being. We can, therefore, treat the radius of the loop
a, and all related variables, such as the velocity, as constants. They depend on Nature’s constants: Planck’s
quantum of action (ħ) and the speed of light (c), principally (we will introduce the fine-structure constant ()
later). It is essential to understand the language of physics:
1. The imaginary unit (i) represents a rotation in a plane. We will consider the plane of oscillation of the
pointlike charge as the reference plane and we can, therefore, represent the position of the pointlike charge by
either (i) the position vector r (which we may also write as the radius vector a
), (ii) its rectangular or polar
coordinates (x, y) and (r, θ), or (iii) the wavefunction ae−iθ = cosθ − isinθ.
2. The argument (θ) of the wavefunction is the polar angle, and its unit is the radian
: the i rotation corresponds
to a rotation by /2 radians. In 3D space, we may also imagine a rotation along an azimuthal angle φ or ϕ (phi).
Such rotation may be perpendicular to the reference plane, in which case we may denote it by another
imaginary unit j.
Let us immediately give an all-important example of our rotation operator. The components of an
electromagnetic oscillation can be described in terms of an electric and magnetic field vector and, using our
rotation operator j, we can write the magnetic field vector (B) and electric field vector (E) as B = jE/c. Why do
we use j instead of i? Because of Lorentz’ force law: F = qE + q(vB): the magnetic force is perpendicular to the
plane of oscillation of the electric field vector.
3. Then we have scalars and vectors. You already saw we denote the latter by boldface (r or v, for example).
And we also have operators, such as derivatives, or integrals (definite or indefinite
)⎯and we may combine
If we talk about the position vector, we are interested in the terminal point of the vector, whereas the radius vector a is
associated with its length and, therefore, with the coefficient of the wavefunction . However,
we may also write the length as . Occasionally, we will use a capital A for the coefficient of the
wavefunction. The reader will also know all about amplitudes. This is an ambiguous term, however, because it alternative
refers to the wavefunction as a whole or, alternatively, to its coefficient only. We think of the amplitude as the (maximum)
amplitude of an oscillation.
We may occasionally use or refer to the old degrees when talking about angles. The use of degrees goes back to the base-
60 numerical system of the Mesopotomians, which also informs our system of dividing hours into 60 minutes and minutes
into 60 seconds. is a more natural unit for expressing angles because of the 2r and r2 formulas for the circumference
and surface area of a circle, respectively. You might think that we should redefine the unit for angles in terms of a fraction
or multiple of i: +i, −2i, or i/, for example. The 2r and r2 formulas would then be written as 4ir and 4ir2, and the
reduced form of Planck’s quantum of action h = ħ/2 would then be written as ħ/4i. However, you would get into trouble
because 4 rotations by i bring you back to the zero point and, hence, we would have weird identities such as 4i = 0 and,
therefore, 4ir = 0. Hence, it is preferable to keep to denote a length and i to denote a rotation: everything has a physical
We will be using a lot of Greek letters, so it is useful to remember them: φ (also written as Φ) and ϕ are phi, Ψ and ψ are
psi, Θ and θ are theta, Δ and δ are delta, Ω and are omega (Φ, Ψ, Θ, Δ, and Ω are capital letters). We will soon be using
more, so you may to check the Greek alphabet.
We must agree on a convention for the plus or minus sign of j here: should we write B as B = jE/c or as B = −jE/c? Both
could be considered: it depends on your perspective on the plane of oscillation. You might say it depends on the spatial
relation between our reference frame and that of the particle, but that is not exactly true: we must define a convention
even then! We must do so by deciding on a right- or left-hand rule (best, of course, to opt for another righthand rule). The
reader should note that this particular sign business has got nothing to do with the fact that the phase of the magnetic field
vector always lags the phase of the electric field by vector by 90 degrees: that is taken care of by the play of the cosine and
sine in Euler’s formula, i.e. the sine always lags the cosine by 90 degrees. Now, we may imagine an antiforce, for which the
magnetic field vector would precede the electric field vector – and we actually think such antiforce is the force
corresponding to antimatter – but that can be modelled by putting a minus sign in front of the coefficient of the whole
function, i.e. by taking the negative of the whole wavefunction!
If we do not know the initial state of the system, we can only integrate over a cycle. If we do not know the initial state of
the system (e.g. the exact value of the phase θ at time t), then we must use indefinite integrals over many cycles, so as to
be able to calculate average values for the variables (observables) that we calculate from the wavefunction.
these operators with the rotation operator and scalar or vector functions, and so then we have words and
statements, so to speak: all of the language of physics is here!
So let us now apply all of these notions. Before we do so, we must note something else about the rotation
operator i. We can rotate the reference frame itself. For example, if we use the x-axis for the position and the y-
axis for time, then we can rotate the y-axis into the x-axis, and the x-axis will then rotate into the former y-axis.
Applying this to the physical dimension of the velocity vector, which we write as [v], we get:
Let us give another example. The physical dimension of the electric field vector is newton/coulomb (N/C). The
physical dimension of the magnetic field vector B = jE/c is, therefore, equal to
Note that s/m factor is the physical dimension of the 1/c factor, so the rotation operator i acts on the 1/c
factor ⎯not on E. This shows the rotation operator is intimately related to our notions of space and time. In this
regard, we must note one more thing with regard to time: unlike spatial directions, we can only imagine time as
going in one direction only. We could invoke an argument referring to the laws of thermodynamics (if there is
friction, entropy must increase) but, at the smallest level, there is no friction: processes are reversible
entropy concept is of no use. The arrow of time is defined logically.
It is this: spacetime trajectories – or, to put it more simply
, motions – need to be described by well-defined
functions. That means that for every value of t (time), we should have one, and only one, value of x (space).
The reverse, of course, is not true: a particle can travel back to where it was (or, if there is no motion, just stay
However, the reader should note something funny happens here: the positive direction of the y-axis (time-axis) will now
coincide with the negative direction of the former x-axis. We think this rather deep fact is explained by the negative sign of
the general argument of the wavefunction ae−iθ: i is a counterclockwise rotation but the hand of the clock of our
elementary particle does move clockwise! Note, however, that we may put another minus sign to model opposite spin.
Hence, we will generally write the wavefunction of matter as aeiθ, which requires a consistent convention for both the
direction of i as well as the direction of time. The logic of these plus/minus signs is complicated and, hence, we request our
reader to double-check!
We use the letter E to refer to energy, but it is also used to refer to the magnitude of the electric field vector E. Feynman
uses an Ԑ for the electric field when it is easy to confuse both, but we will assume the reader is smart enough to know what
is what depending on the context of the expression.
Physicists refer to this as the hermeticity condition for equations involving wavefunctions.
We do not like the use of the term spacetime because it is usually not very clearly defined. We may use it as a shorthand
to refer to four-vector algebra.
We may generalize to position vectors in three-dimensional space, of course: x = (x, y, z). However, we may consider
motion in one dimension only, or choose our reference frame such that the direction of motion coincides with the x-axis.
That is done quite often to simplify the calculations. The result can usually be generalized quite easily to also encompass
two- and three-dimensional motion. There is no such thing as four-dimensional physical space. Mathematical spaces may
have any number of dimensions but the notion of physical space is a category of our mind, and it is three-dimensional: left
or right, up or down, front or back. Time and space are surely related (through special and general relativity theory, to be
precise) but they are not the same. Nor are they similar. We do, therefore, not think that some ‘kind of union of the two’
will replace the separate concepts of space and time any time soon, despite Minkowski’s stated expectations in this regard
back in 1908. Grand statements and generalizations are not always useful in physics.
where it is). Hence, it is easy to see that the concepts of motion and time are related. Logic imposes the use of
well-behaved functions to describe reality.
This is illustrated below: a pointlike particle which moves like what is show on the right-hand side cannot exist
because there are a few occasions here where the particle occupies multiple positions in space at the same
point in time. This is logically impossible.
Figure 1: A well- and a not-well behaved trajectory in spacetime
The stationary wavefunction
Let us immediately apply the definitions above. The wavefunction of an electron in free space is this:
This is a ring current model
: we imagine the free electron as a pointlike charge in an electromagnetic orbital
oscillation, whose radius is the Compton radius. Paraphrasing Prof. Dr. Patrick LeClair
, we can understand this
distance as “the scale above which the electron can be localized in a particle-like sense.” Note that the sign in
the argument of the wavefunction distinguishes the up and down direction of spin (angular momentum).
Interpreting an electron as a pointlike charge in an electromagnetic oscillation clarifies what Dirac, in his Nobel
Prize lecture (1933), referred to as “the law of (elastic or inelastic) scattering of light by an electron”: Compton’s
law, in other words.
This model imagines the free electron as a pointlike charge in an electromagnetic orbital
It is also referred to as the magneton model because the British chemist and physicist Alfred Lauck Parson proposed it in
1915 to explain the magnetic properties of an electron. The term magneton now refers to the Bohr magneton, which is
nothing but the magnetic moment (mm) of an electron. We can explain the mm, and the small anomaly in it (anomalous
mm), perfectly well based on Parson’s original model. For more details, see, for example, our paper on the Zitterbewegung
hypothesis and the scattering matrix.
See: http://pleclair.ua.edu/PH253/Notes/compton.pdf, p. 10. LeClair’s exposé on Compton scattering is the best that we
have come across, although he stops short of an explanation of what might actually be happening: interference between
the electric and magnetic field of a photon and an electron which then results in a temporary disequilibrium state.
Equilibrium is restored because stable particles respect the Planck-Einstein relation (E = hf = ħ).
In our paper on ontology and physics, we show that one can apply Occam’s Razor Principle (according to which each
mathematical possibility must correspond to a physical reality) to also use a sign for the coefficient (a) of the
wavefunction to describe matter and anti-matter respectively. See our remarks in footnote 4 also.
Compton scattering of photons by an electron is referred to as inelastic scattering because – unlike elastic interactions –
the incoming and outgoing photon have different wavelengths. The interaction temporarily leads to a disequilibrium state:
the electron is in an excited (energy) 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. The energy difference of
the photons results in two different wavelengths and ’ ( = hc/E) and the difference is given by Compton’s law, which
was first established experimentally by Arthur Holly Compton (1923):
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 an orbital or tangential
velocity in circular or elliptical orbits effectively yields the Compton radius of an electron:
The E = ma22 is the energy of a harmonic oscillator but – if you ever studied these – you will note we are
missing a ½ factor here. This ½ factor can be explained by the concept of the effective or relativistic mass of the
pointlike charge (meff = me/2), and also because the total energy of the electron consists of two parts: kinetic
energy and field energy. The oscillation of the wavefunction, therefore, models the sloshing back and forth
(Richard Feynman’s expression) of potential and kinetic energy.
We can use the same formal language to describe a proton. However, the positive charge inside of a proton is
part of a nuclear oscillation, which we think of an oscillation in three dimensions. We must, therefore, use two
rather than just one imaginary unit to model this. This is solved by distinguishing i from j and thinking of them as
representing rotations in mutually perpendicular planes. Hence, we write the proton as
The 4 factor in the coefficient (and the ½ factor in the complex exponentials) is there because the oscillation is
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
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⎯the proton, to be precise.
view of a proton fits nicely with packing models for nucleons and (also) yields the experimentally measured
radius of a proton:
See our paper on the meaning of uncertainty and the wavefunction.
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. We have
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 can analyze the muon-electron as a nuclear oscillation too, but a planar one. See our paper on ontology and physics
for force calculations and other remarks.
Velocity and acceleration vectors
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 fine-
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 too!
The ring current model 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 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.
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 (Feynman, III-11-5).
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!
To be 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.
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
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 quite different.
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
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).
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
You may wonder what the S-matrix and the coefficients s11, s12, s21, and s22 actually represent. We think of them
as numbers – complex or some sort of quaternions – 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 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 existential/ontological concept.
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 Heisenberg
, 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).
Let us now focus on relativity theory, and show how all of the above is consistent with both special as well as
general relativity theory.
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 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 examples.
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 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). Light-particles 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 2).
Figure 2: 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
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 particles.
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.
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.
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 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 momentum.
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:
Cf. the 4π factor in the electric constant, which incorporates Gauss’ Law (expressed in integral versus differential form).
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.
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
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.
Figure 3: 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
We use 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 other. 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 tangential
Figure 4: 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
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!
When combining this with the t’ = −1t relation (which establishes time dilation
), we get the relativistic length
We get the same graph (Figure 3): 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 frame and, hence, students should probably
consistently focus on understanding these rather than relativity, as relativity is just a logical consequence of
In any case, let us agree on writing a – which is, of course, the length of the base of the triangle in Figure 4 – 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
We get the time dilation equation from writing s as a function of t: s(t) = vt and substituting in the Lorentz
See footnote 39: observers need to agree both on the t = t’ = 0 as well as on the s = s’ = 0 point!
It is a nice equation, but there is probably not all that much we can do with it.
2. Figure 4 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 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 wavefunction.
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
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!
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
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
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.
An S-matrix representation of Compton scattering?
Compton scattering is a scattering process. 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 5.
Figure 5: 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):
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).
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
The Compton radius, of an electron and a photon respectively, is given by
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:
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 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.
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 below:
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 the approach.
Any solutions proposed by our readers will be read with interest.
Brussels, 22 March 2021
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 equation, 2020
⎯ 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),
⎯ 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