Aharonov, Rohrlich, and the energy-time-position uncertainty principle

Yakir Aharonov and Daniel Rohrlich argue in their book Quantum Paradoxes:

https://onlinelibrary.wiley.com/doi/book/10.1002/9783527619115

that the energy-time uncertainty principle does not hold in all cases.

Aharonov and Rohrlich present their argument in chapters 7 and 8. They assume a time-dependent coupling constant g(t), which differs from zero between times t = 0 and t = T, and couples the system under measurement and the measuring device.

If one can measure something with an arbitrary precision, and have T arbitrarily small, then they say the measurement can be done impulsively.

An uncertainty principle for preparation of a wave packet

A measurement is related to preparing a quantum system to a certain state.

How quickly we can prepare a photon whose energy E is known with a great precision?

We need to create a wave packet whose spectrum in the Fourier decomposition is narrow and close to E. Such a wave packet necessarily is very long. How can we create such a wave packet?

If a hydrogen atom decays to a lower energy state, it sends a photon from a very small spatial volume, compared to the wavelength of the photon (λ is around 100 nm, while the diameter of the atom is only 0.1 nm). This shows that we can create a required wave packet in a small spatial volume if we have a long time available to create the packet.

On the other hand, we may imagine a very long device which creates a very long electromagnetic wave packet almost instantaneously.

               

      ---------------------------------------------------

                                     ^

                                     one finger disturbs Δt

                          v         v        v

      ---------------------------------------------------

                                ^        ^ 

                         multiple fingers disturb Δx

A classical analogue is a string. We can create a wave packet either by disturbing one location of the string for a long time, or by disturbing a long segment of the string for a very short time.

Energy-time-position uncertainty principle for a massless particle. For a prepared photon the following holds:

       ΔE * max(Δt, Δx / c) >= h,

where ΔE is the uncertainty of the energy, Δt is the time used in preparation, and Δx is the size of the device used in preparation.

Question. Does the same relation hold for measuring the energy of a photon?

What about preparing a wave packet for a massive particle like the electron?

The position-momentum uncertainty relation for the wave packet of a massive particle is

       Δx Δp >= h,

or

       Δx Δ(m v) >= h

<=>

       Δx / v * Δ(m v² / 2) >= h / 2

<=>

       Δt ΔE >= h / 2,

where Δt is the time it takes the electron to move the approximate length of the wave packet. This is very much analogous with the classical string example. Either we need to spend the time Δt to disturb at one location, or we need to spend a very short time to disturb along a long segment Δx.

Let us ignore the factor 1/2 in h / 2 above. We are working with fuzzily defined uncertainties and can ignore factors that are of the order 1.

Energy-time-position uncertainty principle for a massive particle.  The following holds:

       ΔE * max(Δt, Δx / v) >= h,

where ΔE is the uncertainty in energy, Δt is the preparation time, and Δx is the length of the preparation spatial volume. The velocity of the prepared particle is v.

The relation is less strict than for a photon. If v is, for example, 0.01 c, we can make Δx very small.

A practical experiment to prepare electrons with precise energy in a very short time

What practical method can prepare an electron with a sharp momentum p in a very short time in a short distance?

   source          shutter                    shutter

                             --------------------------     

       e- ------>       |   electric field    |        ----> 

                             --------------------------         p + Δp

                                          Δx                      angle α

We have a source of electrons located far away, so that we know that the momentum to the y and z directions is almost zero. We want to select electrons whose momentum in the x direction is very accurately p with a precision Δp. That corresponds to some uncertainty ΔE in the energy.

The position-momentum uncertainty relation gives the length Δx of the wave packet of a selected electron:

       Δx Δp >= h.

Let us assume that the incoming electrons have the momentum very roughly equal to p. Let us have a uniform electric field along a distance Δx. The field is only present for a very short time Δt.

We have shutters which only open the area of the electric field for electrons for some time ΔT. A shutter is a very high repulsive potential which we can create very quickly.

Then we can apply the electric field for a very short time Δt. All the electrons in the area will get the same impulse from the field. The shutter at the far end is opened and the electrons continue their journey. Electrons which are deflected to a certain angle α are the selected ones.

What should ΔT and Δx be? Almost all of the wave packet of an electron must fit in Δx. Otherwise the shutters would exert significant forces on the electron, changing its momentum significantly during the process.

The electric field changes quickly in the time Δt in a spatial volume of the length Δx. That will create, among others, real photons whose energy is up to

       h c / Δx

and momentum up to h / Δx = Δp.

An electron departing to the angle α may have bumped into such a real photon, boosting the momentum of the electron. The boost might be up to 2 Δp, quite significant. However, we can reduce the number of such photons by making the electric field smaller. Thus, the photons do not pose a problem.

Our experiment satisfies the uncertainty relation

       ΔE Δx / v >= h,

which can be calculated like in the previous section.

Using shutters really is cheating. They, too interact with the electron, and they are manipulated over a much longer time ΔT >> Δt.

What about removing the shutters completely? A small number of electrons will receive the impulse from the electric field partially, which will spoil our preparation to some extent. But the shutters themselves cause similar spoiling.

Let us remove the shutters. Then the interaction really is on for a very short time Δt.

What about preparing an electron in a very short distance Δx in a time Δt = h / ΔE?

Let us try to reuse the setup above. This time we let Δx be very small. But the geometry of the electric field is not suitable for our purposes. What to do?

We can create a photon with precise energy in a very small spatial volume if we can use a time Δt = h / ΔE. Then we can let the photon to free an electron from an atom with the photoelectric effect. An atom takes a very small spatial volume. In this indirect way we can prepare an electron in a small spatial volume and have the uncertainty of the energy very small.

Is there a direct way to prepare the electron in a very small spatial volume?

Conclusions

We believe that the energy-time uncertainty principle should be replaced with an energy-time-position uncertainty principle. This is also in line with special relativity where one cannot separate time and space.

In this blog post we studied preparation of a particle with sharply defined energy. Measuring the energy precisely is a related problem, but it should be studied separately.

We should also find out what is the role of "compensating forces" in the book of Aharonov and Rohrlich, in chapter 8. The authors assume an interaction which lasts for a very short time. But how large is the spatial volume where the interaction acts?

The piston quantum experiment of Aharonov and Rohrlich

Let us continue reading the quantum paradox book of Yakir Aharonov and Daniel Rohrlich (2005).

https://onlinelibrary.wiley.com/doi/book/10.1002/9783527619115

There is the following thought experiment in the book. A particle is in a cylinder of a length L, closed with a piston. The particle is in an energy eigenstate, so that

       L = (N / 2) λ

for some natural number N > 0 and the de Broglie wavelength λ of the particle. The piston is quickly pulled a length ΔL less than λ, so that L increases.

A. Projecting eigenfunctions

The discrete Fourier transform of the old wave function in wavelengths of the form

       (L + ΔL) / (n / 2),

where n > 0, will contain Fourier components whose wavelength is smaller than λ. This suggests that the particle may end up in a higher energy eigenstate. Is this really possible?

B. Interpretation as a scattering experiment

Let us look at this as a kind of a scattering experiment of the particle from the piston. Then it is clear that the particle cannot pick up speed if it bounces back from a receding piston.

The "wave function" of the particle inside the cylinder really is a path integral of particle paths. Once the piston is moved, these paths will form in a new way. Energy conservation makes sure that any new stationary state has energy less or equal to the old stationary state. Thus, projecting the old wave function to new eigenfunctions is not the right way to calculate the new wave function.

C. Classical coherent wave

If we instead of a particle, confine a classical resonant coherent electromagnetic wave inside the cylinder, and pull the piston quickly, then the new waveform will probably contain Fourier components which have a shorter wavelength than the original wave.

It might be that projecting the old wave to new resonant waves (eigenfunctions) is a good approximate way to calculate the new wave.

Classically, the bouncing wave inside the cylinder pushes the cylinder. If we pull the piston, then the wave loses energy. Classically, high-frequency waves must drain their energy from the original wave.

What if we have just a single photon bouncing in the cylinder? Can it end up in a higher energy state, somehow draining energy from the receding piston?

Suppose that it is the photon itself which moves the piston farther. Energy conservation says that the photon cannot end up in a higher energy state.

D. Electron gas inside the cylinder

We cannot make a coherent electron wave. Let us assume that we have a gas of electrons bouncing inside the cylinder.

A path integral of N electrons might be the right way to calculate (though not practical). The piston cannot increase the energy of an electron in the path integral.

A possible resolution of the problem

If there is a classical electromagnetic wave confined in the cylinder, then we believe that classical physics is correct. When the piston is pulled, the wave loses energy. The resulting new wave will contain some higher frequencies than the original wave. The photons in the higher frequencies drained their energy from photons of the original wave.

If there is just a single particle in the cylinder, then we believe that a scattering experiment is the way to model the behavior. The wave function of the particle really is a path integral of various paths. The particle cannot end up having higher energy than it originally had.

https://arxiv.org/abs/2005.00138

Chiara Marletto and Vlatko Vedral in their 2020 paper The quantum totalitarian property and exact symmetries recommend using a path integral approach.

How does a high-energy photon drain energy from low-energy photons?

The piston must be an electric conductor to reflect the electromagnetic wave inside the cylinder. An electron in the piston can be accelerated through absorbing many low-energy photons. It is possible that the frequency of the electron oscillation is much higher than the frequency of the any of the absorbed photons. Then the electron will radiate high-energy photons.

It is an individual electron which converts energy in a bunch of low-energy photons into energy of high-energy photons.

Can we assume that a classical electromagnetic wave has a fixed number of photons?

Could it be that we, after all, can assume a fixed number of photons in a classical wave? It is the measuring device which converts energy in low-energy photons to a high energy photon?

A challenge in assuming a fixed number of photons is what happens if we measure a photon from an accelerating frame of reference. Suppose that a laser falling freely on Earth sends a single photon to space.

A measuring device floating freely in space would see the wave as a "chirp". The measuring device may absorb a photon from the chirp. There may be soft photons left over from the absorption, and the soft photons will escape to space.

In summary, the number of photons may be fixed in the laser beam in the inertial frame of the laser. But if one wants to absorb the energy of these photons in an accelerating frame, the number and energy of absorbed photons is not predetermined.

If the laser sends a full classical wave, then the measuring device in space will see a classical chirp wave. The measuring device will absorb photons of various frequencies. The measuring device cannot be fully resonant with a chirp. The Fourier decomposition of the remaining wave, after the measuring device, will be very complex. We may interpret that the remnant contains soft photons (and also some very high-frequency photons).

Einstein's clock-in-the-box and the reply of Bohr to it

Yakir Aharonov and Daniel Rohrlich have written a very interesting book Quantum Paradoxes (Wiley, 2005):

https://onlinelibrary.wiley.com/doi/book/10.1002/9783527619115

One of the paradoxes is Albert Einstein's famous 1930 clock-in-the-box thought experiment. There are photons in the box, as well as a clock. A mechanism opens a shutter for a very short time Δt when the clock says it is noon. Some photon(s) escapes, and the total energy of the box is reduced by some amount E.

In the morning we can use a very long time to weigh the box, so that we know its total energy extremely precisely. In the afternoon we can repeat the weighing procedure.

We can determine E as accurately as we like. We also know the approximate time t₀ = noon when the box lost this energy E. Does this contradict the uncertainty principle

       ΔE Δt >= h / (4 π)

?

Niels Bohr came up with a sort of a counter-argument. He uses a complicated procedure to weigh the box. There is a spring scale and known counterweights. After we have hung the smallest counterweight, there is still an uncertainty Δx in the vertical position of the box.

General relativity tells us that the clock runs at different speeds at different heights. Bohr calculated the uncertainty ΔT in the current time (proper time) shown by the clock. (Note that this is not the same as Δt.) We assume that we do not read the clock in the box, but try to determine the time from the laboratory clock. He proved that the uncertainty relation above holds for that particular uncertainty ΔT.

Bohr's argument did not prove much

However, Bohr uses a very slow procedure to weigh the box. The uncertainty principle allows us to perform the weighing much faster, in a duration t = h / (ΔE * 4 π). If we use a faster procedure, the uncertainty ΔT in the reading of the clock inside the box is probably much less than in Bohr's procedure.

Let us then analyze how one could entirely circumvent the uncertainty in the box clock reading. It is quite easy. Instead of the box energy, let us measure the energy of the photons which left the box through the shutter. We can use an arbitrarily long time for measurement, so that ΔE is extremely small.

We can fix the box statically to the laboratory frame. Then the clock inside the box runs at the same speed as the laboratory clock. The uncertainty ΔT in the reading of the clock in the box is very small. We found ΔE and ΔT which do not satisfy the uncertainty relation.

If one tries to squeeze energy E into a wave packet to whose duration is roughly t, then

       ΔE t >= h / (4 π).

This is a typical example of an energy-time uncertainty principle. However, Einstein's clock-in-the-box is about the history of events and uncertainty of the time when an event happened. Why would there be an uncertainty relation about history? We do not see why that should be necessary.

Bohr's argument did not prove the energy-time uncertainty principle, but something about uncertainties in a certain complicated physical experiment.

Analysis in literature

Let us look at the literature. Does anyone claim that Bohr proved something about the uncertainty principle?

https://link.springer.com/article/10.1007/BF01100322

H.-J. Treder (1975) writes that the box argument has no bearing on the fourth Heisenberg relation. We agree.

https://arxiv.org/abs/1203.1139

Hrvoje Nikolić (2012) has written about EPR before EPR: a 1930 Einstein-Bohr thought experiment revisited.

Nikolić says that neither Einstein nor Bohr was right. In section II A Nikolić writes that Einstein wanted to show that

       ΔE Δt >= h / (2 π)

does not hold.

We in this blog think that Einstein was right. We can produce a photon in a short interval of time Δt, and later measure the energy of the photon extremely precisely (very small ΔE), using a long time t for the measurement. The correct energy-uncertainty principle says that

      ΔE t >= h / (4 π).

In section III D Nikolić writes that Einstein did not realize that measuring the mass-energy of the box can influence "ΔE (or any other property) of the photon".

We do not understand the claim. It was well known in 1930 that a photon is a quantum of light of a definite frequency f and definite energy E = h f. We can reduce the uncertainty in E by measuring the energy of the box or of the photon itself. Conservation of energy was taken for granted in 1930, as it is today.

After the shutter is opened and closed, then the box and the photon(s) are, of course, entangled. Measuring the energy of the box makes the wave function to "collapse" to certain energy E of the photon(s). However, talking about a "collapse" does not affect the analysis of the process in any way.

Conclusions

Albert Einstein's thought experiment about the clock-in-the-box does not concern the energy-time uncertainty principle at all. Neither does Niels Bohr's counter-argument.

It is wrong to present the debate as a "proof" that the uncertainty principle holds.

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.