Aharonov, Popescu, Rohrlich (2021): On conservation laws in quantum mechanics

Yakir Aharonov, Sandu Popescu, and Daniel Rohrlich have an interesting paper On conservation laws in quantum mechanics (January 5, 2021, PNAS):

https://www.pnas.org/content/118/1/e1921529118

They have noticed a problem similar to the one which we in this blog have called the "length scale problem".

http://meta-phys-thoughts.blogspot.com/2021/06/the-number-of-photons-may-change.html

In our June 3, 2021 blog post we wrote that if a laser beam is reflected or refracted by a very small object, then a moving observer may see a very high energy photon when he moves past the object. That is because the electromagnetic waveform close to the object has very fine detail, and the Fourier decomposition of the wave then will contain very high frequencies.

                  photon

       | ~~~~~~~~~~~~~ |

                     box

Aharonov et al. accomplish the same by preparing a photon into a box, in a pure state which has carefully chosen Fourier components. Then in the middle of the box, the wave function locally seems to have a wavelength much shorter than any of the Fourier components.

Aharonov et al. open a small window in the box for a short time. The photon may escape through the window, having a frequency much higher than any of the Fourier components had. We have a paradox.

Aharonov et al. call for a new energy conservation principle in quantum mechanics.

Instead of a single photon, we could prepare many coherent photons in the Aharonov et al. box. Then we would have a standing classical wave in the box, whose apparent wavelength in the middle of the box would be very short. An observer could then measure a very high-frequency wave, and consequently, a very high-energy photon in the box, even though we only put low-energy photons inside. Classically, there is no paradox. The observed high-frequency wave draws its energy from the low-frequency waves in the box.

Aharonov et al. present a thought experiment where we have many boxes and a single photon in each of them. If we measure a high-energy photon in one of the boxes, could it be that it draws its energy from the photons in other boxes?

The interaction with the measuring apparatus destroys the high-frequency wave in the middle?

Could it be that the measuring apparatus disturbs the wave in the middle of the box so much that it destroys any high frequencies?

Probably not. If we put a classical electromagnetic wave in the box, we certainly can measure the the high-frequency wave in the middle. The apparatus does not destroy high frequencies.

Possible solutions for the paradox

http://meta-phys-thoughts.blogspot.com/2021/02/a-possible-solution-to-length-scale.html

In our February 1, 2021 blog post we suggested that the paradox can be solved with a particle model, where it is the path integral which introduces wavelike properties into the system. Then one can only observe low-energy photons in the Aharonov et al. box.

But the solution does not work if we have a classical coherent wave. Classically, we can certainly observe a high-frequency wave in the middle of the box. It would be very strange if the classical high-frequency wave does not consist of photons.

In the June 3, 2021 blog post we suggested that we must drop the notion of a fixed number of quanta in an electromagnetic wave. We gave the following motivation: in a quantum mechanical experiment, one should assume the minimum of things about a photon. For example, one must not assume any definite path for a photon. A step further is that in a coherent wave, one must not assume any fixed number of photons.

Let us try to outline a solution for the paradox:

If we have just a single low-energy photon in the box, then energy conservation dictates that we cannot observe a high-energy photon. A particle model with a path integral approach may be a suitable way to model this.

But if we have a whole laser beam of coherent photons bouncing around in the box, then we believe that classical physics is the correct way to model the process. Then one can observe a high-energy photon. Energy conservation has to be enforced in the classical way. One may observe a high-energy photon, but it draws its energy from a bunch of low-energy photons which were inserted into the box.

Does the energy come from the preparation of the experiment or from the measuring device?

Aharonov et al. discuss the possibility that an observed high-energy photon might draw its energy from the measurement operation. They conclude that it is not possible.

Let us analyze this further. How do we put a low-energy photon into a box? We might put an excited hydrogen atom into the box and let it decay.

To confine a wave function in a fixed location, we need to "cut off" the fringes of the wave function. Could it be that this cutting procedure introduces high frequencies into the wave function and explains the birth of a high-energy quantum? For example, the hydrogen atom may be accelerated to a very high speed, and the photon which it emits may have very high energy.

A measurement is like preparation with time reversed. Could it be that the measurement supplies the energy? In the case of a classical laser beam wave, it is hard to see how the measurement could supply the energy in high-frequency waves. Clearly, high frequencies draw their energy from low-frequency waves.

Question. If large-energy photons get their energy from low-energy photons, how many low-energy photons we must have in the box so that this can happen? One photon is not enough. How about ten?

Some further links:

https://arxiv.org/abs/2005.00138

Chiara Marletto and Vlatko Vedral (2020) discuss another thought experiment where a box contains a photon in an energy eigenstate, and the box is suddenly made longer.

http://meta-phys-thoughts.blogspot.com/2021/02/sean-m-carroll-and-jackie-lodman-about.html

On February 2, 2021 we wrote about the paper of Sean Carroll and Jackie Lodman where the expectation value of energy changes in a measurement.

A running coupling constant breaks the classical limit and energy conservation?

UPDATE August 20, 2021: In the discussion below we did not analyze the fact that the fine structure constant α, too, changes if we change the charge e of the electron. The fine structure costant is approximately 1 / 137, and is proportional to e². If we increase e by a factor 12, then α will be larger than 1, and corrections will start to dominate quantum electrodynamics, making many formulas non-convergent. This requires a more thorough analysis.

The classical radius of the electron is 1 / 137 of the reduced Compton wavelength λ / (2 π) of the electron. Has Nature made α small enough, so that quantum mechanics hides the "inner field" of the electron closer than the classical radius?

----

It is strange that people have not noticed the problems with a running coupling constant. If the force felt by a charge in a force field depends on the motion of the charge, that will break energy conservation. It also breaks the classical limit (Bohr's correspondence principle) of the quantum system, a fact about which we have already written in this blog.

https://www-thphys.physics.ox.ac.uk/people/FrancescoHautmann/Ralhep/index.html

Francesco Hautmann explains in the link the running of the coupling constant in QED.

       A ●  -------->

                                     impact paramer b

                      ● heavy charge

Let us have a charged particle A which passes another, very heavy charge. In QED, a running coupling constant means that the force felt by A depends on the amount of momentum it exchanges with the other charge. The coupling constant α is a function of the exchanged momentum q:

        α(q²).

If we make A go slower, then q² is larger, and A will move along a path which is different from what classical mechanics predicts. This breaks the classical limit.

Also, energy conservation is broken if the strength of a force felt by a charge depends on the motion of the charge. In QED, the coupling constant increases with |q|.

If we move the charge A slowly to a lower potential, there is a large exchange of momentum, and A will feel a strong force. We can then extract energy E. Then we move A quickly back to its original position. There is less exchange of momentum. The force is now weaker and we need to spend less energy E' < E. We have constructed a perpetuum mobile.

There are numerous problems with a running coupling constant. We have not seen any discussion of the problems in literature. As if researchers would have forgotten about conservation laws in physics.

Vacuum polarization, vertex function, self-energy: are there classical analogues?

UPDATE August 29, 2021: In our August 27, 2021 post we found a classical analogue for vacuum polarization. If we have a solid, for which electric polarization is superlinear on the field strength, then that material "conducts" strong electric fields better than weak fields.

----

In our blog we have claimed that the vertex function (correction) is mostly a classical phenomenon.

The vertex function

                         virtual photon

                         ~~~~~~~~

                       /                     \

       e- -----------------------------------------

                                  |

                                  | virtual

                                  | photon

      Z+ -----------------------------------------

The static electric field of the electron lags behind in sudden movements. Some of the inertia in the electric field is temporarily shaken off the electron. The effective mass of the electron appears then smaller than 511 keV.

The reduced mass of the accelerating electron explains how a radio transmitter can function. Some of the work to accelerate the electron goes to bending the electric field lines rather than giving kinetic energy to the electron. The deformation energy can escape as vibration: electromagnetic waves or bremsstrahlung.

If the field of the electron would be absolutely rigid, then its mass would always appear as 511 keV and no electromagnetic waves could escape. There would be no bremsstrahlung. There would be no vertex correction either.

Electron self-energy

The Feynman diagram for electron self-energy is like for the vertex function, but this time there is no external disturbance caused by the nucleus.

                          virtual photon

                           ~~~~~~~~

                         /                     \

          e- ----------------------------------------

The process does not appear at all for a classical electron. If there is no external disturbance, the static electric field flies along the electron and nothing happens.

Does self-energy have any classical analogue? If we interpret that the static electric field of the electron consists of virtual photons, then we could say that the field of the electron is the self-energy.

The inner field of the electron at distances

       < 1.4 * 10⁻¹⁵ m = r₀ / 2,

where r₀ is the classical radius of the electron, poses a problem in classical physics. Why does the inner field appear to be massless even though its energy density

       1/2 ε₀ E²

integrated down to the radius zero yields an infinite result? Above, ε₀ is vacuum permittivity and E is the strength of the electric field.

We have not found a solution for this problem of classical physics. One can claim that it constitutes a classical regularization and renormalization problem.

Can quantum mechanics solve the classical problem? Imagine that the Planck constant h is much smaller (but it cannot be > 137 times smaller since then α > 1 and many QED formulas diverge) than the current value. Then the electron and its inner field will be mostly classical objects. It looks like quantum mechanics will not help us. 

Another question is if we have to "correct" Feynman diagrams somehow for the self-energy. Classically, if the electron is not under an external disturbance, its electric field follows it without any effect on the electron. Thus, there is no correction needed.

In the Feynman diagram above, there is a problem in conservation of the speed of the center of mass. Suppose that the electron is initially static. It then emits a virtual photon carrying momentum k. The electron starts to move. The electron subsequently absorbs the photon. The electron moved from its initial position even though no external force was present.

In our "sharp hammer" model of the electron electric field this problem does not exist because the hammer sends a shock wave symmetrically to all directions. The sharp hammer model is classical. In quantum mechanics we are used to treating individual quanta. An individual virtual photon would breach the conservation rule.

Vacuum polarization

       e- ----------------------------------

                            |    virtual photon

                            |

                         /    \

                  e-  |        | e+  virtual pair 

                         \    /

                            |

                            |    virtual photon

       Z+ ---------------------------------

We have not found a classical analogue for vacuum polarization and, consequently, suspect that the phenomenon does not exist at all in quantum electrodynamics.

In the Feynman diagram, a virtual photon is "reflected" from a virtual pair. The phase of the reflected photon changes 180 degrees and the photon has the opposite effect to a photon which would move unhindered.

The process is suspicious. A particle is reflected from itself, or from "nothing". By nothing we mean that the virtual pair without input from the photon would have zero energy and zero momentum.

https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma#/media/File%3AMuenchhausen_Herrfurth_7_500x789.jpg

In classical physics, such a Baron Munchausen trick cannot happen. If an object has zero energy and zero momentum, it cannot affect anything. And a particle cannot be reflected by itself.

In the Feynman diagram above, the virtual photons contain no energy, just spatial momentum. It is an elastic process - the nucleus does not give any energy to the electron. Could it be that the virtual pair somehow temporarily loosens the attraction between the nucleus and the electron, but the whole process eventually ends up being elastic?

Classical electric polarization in a solid

In a solid polarizable material, classical electric polarization does exist. Let us analyze the basic mechanics.

Let the nucleus Z+ be immersed in the material, as well as the electron at some distance away.

Polarization of the material at the distance of the electron puts some negative charge between the nucleus and the electron.

     polarization             polarization

           +       -                   +       - 

      <----- f                          -f ----->

           e- ●                         ●  Z+

               F ------->     <------ -F

There is an attractive force F which pulls the electron toward the nucleus. Some of that force is offset by a polarization force f on the electron.

The electron, in turn, exerts a force -f on the polarized material close to it. Where does the momentum from this force go? It is balanced by an opposite force which the nucleus exerts on polarized material close to the nucleus.

There has to be pressure, or rigidity, in the material for the balancing of the forces to occur.

If there is no pressure, then the polarized material close to the electron will obtain momentum.

If we replace the solid with a vacuum, then either there has to be rigidity in the vacuum, or the part of the vacuum close to the electron will obtain momentum. Both of these hypotheses sound strange. This suggests that vacuum polarization does not exist.

Vacuum polarization as failed pair production

If a forming pair receives less than 1.022 MeV of energy, it will annihilate. Could this process be vacuum polarization?

Classically, the pair has to receive at least some energy to start the process of separation. But in the vacuum polarization diagram above, no energy is contained in the virtual photons.

It might be that the electron reabsorbs the energy which it initially gave to the forming pair. What would then be the net effect? The electron eventually does not lose any energy, and receives some momentum. It is like a rubber band between the nucleus and the electron, which almost - but not quite - breaks. The net effect would be that the Coulomb force appears slightly weaker if the electron passes the nucleus very close.

We need to think about this. The classical process is quite complicated. Can we somehow map it to a quantum process?

Our analysis thus far suggests that in Feynman diagrams, loops which only have two lines out do not make sense in classical physics. There has to be an external disturbance to make something to happen.

The vertex function has the virtual photon line to the nucleus, and is very sensible in classical physics.

Electron self-energy is a loop with no external influence and does not make much sense classically.

The vacuum polarization loop occurs in the middle of a virtual photo line. We have failed to make sense of it classically.

    e- ------------------------------------------------------

                            |                            /   virtual

                            |    e+                  /     photon

                            |-----------           /

                            |              \___/

                            |              /

                            |-----------

                            |    e-

                            |

    Z+ ------------------------------------------------------

Above we have a Feynman diagram of failed pair production. The electron and the nucleus kick components of a zero-energy pair. The pair fails to make it to the outside world, annihilates, and the electron absorbs the result of the annihilation.

In the rubber band analogy, the band almost breaks, but eventually the electron absorbs whatever energy and momentum there were in the band.

Bethe and Heitler (1934) and production of pairs instead of bremsstrahlung

Let us then analyze pair production in the Bethe and Heitler 1934 paper:

https://royalsocietypublishing.org/doi/10.1098/rspa.1934.0140

Pair production from a virtual photon

This time the encounter of the electron with the nucleus does not produce a real photon like in bremsstrahlung. It produces a virtual (off-shell) photon which excites another electron in a negative energy state to become a real electron. The "hole" which is left behind the electron is the positron.

It looks like we can conjure up a negative energy electron at will - there is no shortage of them. A suitable virtual photon will always produce a pair.

                                               --------- e-

                                            /

                                         /  \

                                       /       --------- e+

                                     /

                                   /    virtual photon

         e- --------------------------------------

                            |

                            | virtual

                            | photon

         Z+ --------------------------------------

Above is the Feynman diagram of the process.

How would we interpret this classically?

The virtual photon has to contain a lot of energy compared to its momentum because it has to create the rest masses of the electron and the positron. The electric field of such a photon oscillates mainly in time and the oscillation does not move at the speed of light.

The obvious classical candidate for such a photon is the stretching of the electric field lines of the electron as it makes a sharp turn close to the proton. The far field of the electron lags behind and the electric field lines stretch and bend. Creating a pair breaks the stretched field lines and reduces the energy of the electric field.

Pair creation would mean that the "rubber plate" of the electron electric field is torn apart. The edges of the torn rubber are the new electron and the positron.

An alternative classical interpretation is that the rapidly changing electric field of the electron tears apart a zero-energy pair which initially is at a distance 1.4 * 10⁻¹⁵ m or half the electron classical radius from each other.

Let us assume that the incoming electron has the energy ~ 2 MeV.

If the new created electron has the energy ~ 511 keV, then its wavelength is ~ 2.4 * 10⁻¹² m. The new electron is born in a volume and time which is ~ 3 * 10⁻¹⁵ m. It has problems emerging from such a small volume of spacetime. We expect the cross section to be inversely proportional to h, just as in the case of bremsstrahlung of a large photon.

Does the positron have problems emerging from the small volume? Let us check the Bethe-Heitler formula.

The formulae (21) and (22) in their paper contain the familiar factor 1 / 137. Thus, the cross section is inversely proportional to h.

What is the intuitive reason why the positron has no problem emerging from a small spacetime volume? Maybe it really is a "hole" and not a particle at all?

Pair production "between" the colliding particles

Bethe and Heitler do not cover this process in their paper.

        e- ------------------------------------

                         | virtual photon

                         |

                         |----------------------- e-

                         |

                         | virtual electron

                         |

                         |----------------------- e+

                         |

                         | virtual photon

        Z+ -----------------------------------

We assume that the positron is available at will. If we reverse time, the nucleus scatters the positron into a virtual (off-shell) electron. We then restore time to the normal order. The electron scatters the virtual electron into a real (on-shell) electron.

Another interpretation is that the nucleus excites an electron in the negative energy state. The incoming electron further excites the newly created electron, so that it becomes on-shell. The hole which is left behind is the positron.

What is the classical analogue for this peculiar process?

We have an obvious candidate. When the incoming electron comes very close to the nucleus, the common electric field of the particles becomes very strong between them. Then it may reduce energy to create a new pair. The electric field lines break.

Alternatively, the colliding particles kick the components of a pre-existing zero-energy pair.

Bremsstrahlung: why a large value of the Planck constant makes it weak?

Let us continue our study of bremsstrahlung.

We think we have found the explanation why a big Planck constant h makes the cross section small for the case of the electron losing most of its kinetic energy in the emission of a large photon.

If we smoothen the 1 / r potential at the origin so that there is no deep well, then the Fourier decomposition of the new potential does not contain large momenta k. This shows that it is the deep well which contributes the large momentum k exchanges with the electron. This is just like in classical physics: the electron has to go very close to the nucleus for the electron to gain a large change in the momentum.

We see that it is the close encounters which produce large photons in the QED framework.

A close encounter happens quickly. The distortion in the electron wave function lasts for a very short time.

          __          __        

        /     \___/     \___/   photon

          __

        /     \   distortion

If the Planck constant h is small enough that the distortion "fits" in half a cycle of a large photon, then the "projection" of the distortion on the photon wave is large. There is a large possibility that the photon gets emitted in the process. Having a small h makes the process look like the classical limit. In the classical limit all 1 MeV electrons which come within 10^-14 m from the nucleus (a proton) will lose most of their kinetic energy in the encounter.

       ___________

     /                       \____________/  photon

                __

              /     \    distortion

But if the value of h is big, then the large photon has a long wavelength and a long cycle. The projection of the distortion on the photon wave is small: there is a low probability that a large photon gets emitted.

The wavelength of a 250 keV photon is 5 * 10^-12 m. To emit such a photon, the 1 MeV electron has to come within 10^-14 m of the proton. We conclude that in our universe the value of the Planck constant h is "big", and few large photons are emitted in bremsstrahlung.

This means that most close encounters are elastic or almost elastic. In a classical universe no almost elastic close encounters would happen.

The classical limit and what happens if we increase the value of the Planck constant

A way to analyze bremsstrahlung is first to look at it as a classical process.

If we set h to 1 / 100 of its usual value, then the de Broglie wavelength of a 1 MeV electron is roughly 10^-14 m. A close encounter to within 10^-14 m of a proton is "almost classical".

Increasing h back to its normal value suppresses the emission probability of a large photon by a factor 1 / 100.

What about our "rubber plate" model of the electron electric field? We explained the vertex correction with the classical model. Does our explanation work if we increase h?

It is hard to produce a large real photon of energy E in a small volume in a short time. That is what our analysis showed.

But there is no problem in exchanging a similar amount of momentum k in that small volume and short time. Coulomb scattering is not affected if we raise the value of h. As if pushing or pulling is not affected by a large h, but producing a vibration (a real photon) is hampered by a large h.

The effect of the rubber plate in the vertex correction is to push and pull on the electron. We conjecture that the rubber plate model works in the classical way even if h has a large value.

We need to check if the vertex function, or correction, in QED depends on the value of h.

The length scale problem is solved!

We have spent a lot of time wondering how the very sharp turn which the 1 MeV electron makes close to the proton can produce a relatively long wave photon, the wavelength ~ 5 * 10^-12 m and energy ~ 250 keV.

Now we have the solution. The sharp turn in rare (1 / 100) cases is projected into a long-wave photon. In other cases, the encounter is elastic or almost elastic.

If the Planck constant would be 100 times smaller, then the encounter would in most cases produce a short wavelength photon, ~ 5 * 10^-14 m. The energy of the photon would still be ~ 250 keV.

A long wavelength photon has problems going through a small hole in spacetime

This setup is clearly related to the analysis above. The bremsstrahlung photon whose wavelength is ~ 5 * 10^-12 m is "born" in the area of encounter, and that area is only ~ 10^-14 m in size and ~ 10^-14 m / c in time. The would-be photon has to make it through a small hole in spacetime, in order to break out into the external world.

Which scattering cross sections depend on the value of the Planck constant?

The 1934 bremsstrahlung paper by Bethe and Heitler is freely readable at 

https://royalsocietypublishing.org/doi/10.1098/rspa.1934.0140

Bethe and Heitler seem to use the principle later known as Fermi's Golden Rule:

https://en.wikipedia.org/wiki/Fermi%27s_golden_rule

where the "density" of possible end states is counted.

      incoming         distorted      outgoing

 e- |   |   |   |   |     /    /    /    /     |        |        |

                  

                                                       ~~~~~~~~~~~

                                                       photon

                                      ● Z+ nucleus

Bethe and Heitler assume a plane wave of the incoming electron, which is perturbed (distorted) for a short time dt by the Coulomb potential of the nucleus Z+.

Then they take the projection of the distorted wave function to an off-shell electron plane wave. An off-shell electron plane wave is one where the energy-momentum relation

       E² = p² + m²

does not hold.

They further calculate the transition probability of this off-shell wave to an on-shell outgoing electron wave under a perturbation of the field of the emitted photon. If we look backward in time, it is the emitted photon which causes a state transition of the outgoing electron to an off-shell electron.

Besides the process described above, there is another - less intuitive - process where the electron emits the photon before the electron meets the nucleus.

                                          real photon

                                          ~~~~~~~~~~

                                        /

     e-  ---------------------------------------------

                             |  virtual

                             |  photon

                             |  k

     Z+ ---------------------------------------------

In the corresponding Feynman diagram, the electron absorbs a virtual photon which is sent by the nucleus Z+ and carries 4-momentum k. The electron goes off-shell. The electron then emits a real photon and returns to on-shell.

The probability of the electron receiving a virtual photon whose 4-momentum is k, is governed by the photon propagator. Richard Feynman in his famous paper derives the propagator from the Fourier decomposition of the Coulomb potential 1 / r. Another way to derive it is from the Green's function of the Klein-Gordon equation.

An aside: why is the Fourier decomposition of the Coulomb potential similar to the Green's function? Our "sharp hammer" hypothesis in this blog is a manifestation of this relationship.

Which processes depend on the value of the Planck constant h?

Coulomb scattering is equivalent in classical physics and quantum field theory. Cross sections for it do not depend on the value of h.

Cross sections in Compton scattering depend on the classical radius of the electron and on the mass-energy of the photon. They do not depend directly on the value of h.

But in bremsstrahlung, the cross section for emitting most of the kinetic energy in a photon is inversely proportional to h.

What is the difference in these processes? An obvious difference is that in bremsstrahlung the number of particles grows by one. Bethe and Heitler count the number of possible final states in a unit volume.

The ratio of the number of final states and the number of input states probably depends on h. If we make h smaller, the ratio will probably grow. This may explain why in bremsstrahlung making h smaller will increase the cross section.

In Coulomb scattering and Compton scattering the ratio probably does not depend on h because the process is symmetric.

Hypothesis. QED processes which conserve the number of particles happen like in classical physics - if one takes into account that the photon is a particle. Their cross sections do not depend on the Planck constant h. But if the number of particles grows in the process, then the cross section is reduced by quantum physics, and a smaller h increases the cross section.

Our hypothesis claims that the "mechanics" of a process is classical. The value of the Planck constant enters the stage when we calculate the number of possible states in a unit volume.

The hypothesis explains the coincidence that Coulomb scattering is equivalent in classical and quantum physics.

Our hypothesis smashes the hopes in our previous two blog posts: the fine structure constant is not determined by the geometry of the electron static electric field.

This is because calculating the number of possible states in a unit volume is not connected to the geometry of the electron electric field.

Or could it be? Are there "resonances" in vibrations of the static electric field of the electron? If the electron orbits a proton, could it be that these resonances enforce the Bohr orbits? We need to study this.

The fine structure constant: an update

In our previous blog post we claimed that in a classical model an electron might lose most of its kinetic energy if it passes the proton at a distance

       < 0.9 * 10^-15 m.

The idea was that all of the inertia of the electron is located in its static electric field farther than

       r₀ / 2 = 1.4 * 10^-15 m

from the pointlike electron. There

       r₀ = 2.8 * 10^-15 m

is the classical radius of the electron.

A more detailed analysis reveals that it is enough that a half of the inertia of the electron "lags behind" in the movement. Imagine that a half of the mass of the electron is attached to it with an elastic rubber band. The rest of the mass is rigidly fixed to the pointlike electron.

              ● half of the electron mass

               |

               |  rubber band

               |

              ● e- electron

              v  ------->

Let the initial velocity vector of the system be v.

If the pointlike electron suddenly bounces back in the field of a proton, so that its velocity vector becomes -v, then all of the original kinetic energy of the system will go to stretching the rubber band. That is, the kinetic energy is totally converted to vibration or electromagnetic waves.

Where is the inertia of the electron located? We again meet the mystery of the "inner field" of the electron. We do not know the inertia distribution close to the classical radius of the electron. The Larmor formula suggests that in the far field of the electron, the inertia is in the mass-energy of the static electric field.

In our previous blog post, the cross section, which we calculated to be 25 millibarn, could be off by a factor 100, depending on the distribution of the inertia of the electron. That is bad news for our claim that the fine structure constant is determined by the geometry of the electron electric field.