Does the Feynman diagram plane wave assumption produce incorrect results?

Let us assume that two electrons pass a nucleus. What is the probability that one of the electrons gains momentum p in the experiment?

We shoot the electrons from opposite directions toward the nucleus. We model the electron fluxes as plane waves with a definite momentum q or -q.

    q  e-  ------>      ● Z+      <-----  e-  -q

  

Let us draw some Feynman diagrams.

 -q  e- -----------------------

  q  e- -----------------------

                    |  p  virtual photon

      Z+ -----------------------

Above we have the basic case.

    q e- -------------------------

                       | p

   -q e- -------------------------

                   | p

     Z+ ------------------------

Above is another case.

The end result is the same in both diagrams: the nucleus and one of the electrons exchange momentum p. We measure the momenta of both electrons after the experiment.

Feynman diagrams calculate probability amplitudes in the momentum space. The amplitude is a function of the momenta of the output particles.

Feynman rules say that we must add the probability amplitudes (which are complex numbers) of the diagrams to get an approximation of the scattering probability. Typically, the probability amplitudes have a different phase. There is some destructive interference.

Adding probability amplitudes is correct if the output is a product of plane waves.

Note that the nucleus is an almost classical object. After the experiment, we can measure its path. Let us assume that the nucleus had very small momentum - it essentially stayed static in the experiment.

Electrons which gained the momentum p directly from the nucleus came from a relatively small spatial volume. They do not form a plane wave.

Electrons which gained the momentum via the other electron come from a much larger spatial volume.

Let us next do a thought experiment.

Let us assume that that we send two arbitrary waves which have a 180 degree phase shift. The wave A comes from a small spatial volume v. A weaker wave B comes from a large spatial volume V.

Why would there be a lot of destructive interference? There is not. We cannot destroy the energy that we used to create the wave A by putting energy to the wave B.

Feynman rules assume that the output of a collision is a product of plane waves. But often it is not. If the electron receives the momentum p in the first diagram, then it flew close to the nucleus. We gained knowledge about the relative position of the nucleus and the electron. Then we cannot model the output with plane waves.

We could calculate the experiment result using the Schrödinger equation because no new particles are created in it. Who thinks that a very crude plane wave model can replace the Schrödinger equation calculation? That would be a miracle.

How to fix Feynman rules? Obviously we have to be very careful if we try to add the probability amplitudes of two diagrams. If the output cannot be approximated with plane waves, then we cannot calculate any interference of the diagrams.

What about vacuum polarization? There Feynman did add the interference of two diagrams, the simplest one and the one-loop diagram. Is that an error?

What is the role of virtual particles or malformed waves?

Let us look at Coulomb scattering again.

    proton ●

                        ^

                        |

                        e-

url=https://www.unige.ch/isdc/ferrigno/index.php/download_file/50/149/

Andrii Neronov (2017)

We were not able to find from the Internet figures for the total cross section for pair production, but let us guess it is of the same order of magnitude as for 5 MeV gamma rays, if the electron is in the MeV range. That is, around 0.1 barn = 10^-29 m^2.

That suggests that the electron has to come within 1.5 * 10^-15 m from the center of the proton, to produce pair(s).

We could satisfy energy and momentum conservation even if the electron would pass the proton at a very large distance, say 2 * 10^-12 m, or the Compton wavelength of the electron. Any output from a collision satisfies conservation principles - there is no need for the particles to be born close to each other.

Tunneling, with 1 MeV of energy borrowed, is easy through a barrier of 0.1 Compton wavelengths, or 2 * 10^-11 m. The tunneling barrier cannot explain why the electron and the proton must come very close.

Let us imagine that the process creates a positron close to the electron, and an electron close to the proton. Both particles have to get their energy from the kinetic energy of the electron. To give up that kinetic energy, the electron must lose a lot of momentum. To lose that momentum, it must come very close to the proton. The Compton wavelength is way too far away. Here we have a semiclassical explanation why the cross section is ~ 0.1 barn.

The process is described using a particle model of the electron, and almost classical physics.

What are virtual electrons?

To analyze vacuum polarization we should understand what exactly are the virtual electrons in various processes. There is a virtual electron line in the Feynman diagram for pair annihilation or the converse process. There is a virtual electron in Thomson/Compton scattering.

The virtual electron in Thomson/Compton scattering is a "very real" electron that moves in interaction with the photon field(s). There is nothing unreal or mysterious in that electron.

The virtual electron in pair production

The virtual electron in pair annihilation, or the converse process, is mysterious. When we analyzed pair production from colliding photons using a the drum skin wave model, we suggested that the virtual electron is some kind of a malformed wave which transfers momentum in the Dirac field.

But later we used a particle model, and tunneling to potential wells, to explain pair production. How can we reconcile a particle model with a malformed wave?

As we have noted, the wave equation source, which creates an electron or a positron, is quite small, ~ 10^-15 m (in the MeV range). The source is as close as we get to a pointlike "particle" if we work in a wave model.

A source so small will produce also electrons in the GeV range, unless we use a cutoff, or sum over many different source locations, to eliminate waves which have too high a 4-momentum.

A possible model:

1. The virtual electron in pair production is just a symbol for the (complex) process which happens, in a particle model, in a volume of a size 10^-15 m.

This is analogous to the fact that the virtual photon which transfers momentum in Coulomb scattering is just a symbol for Coulomb interaction.

2. We external observers cannot discern details which are much smaller than the Compton wavelength 2 * 10^-12 m. For us, the electron and the positron just pop up in a volume ~ 2 * 10^-12 m.

An external observer may interpret the 2 * 10^-14 m scale process in the Feynman way: a positron coming from the future was scattered by a photon into a virtual electron, and that virtual electron was scattered by a second photon to a real electron.

This model reconciles the almost pointlike character of the photons and the pair, with the larger scale view of waves scattering from each other.

Collapse from a wave to a particle: the teleportation model

The "collapse" of a wave to a particle can be explained by the teleportation model which we introduced in an earlier posting. The original particle system which produced the wave is magically teleported close to the system which observes the particle. The original system is (in the MeV range) ~ 10^-15 m in size. This explains why the particle appears quite pointlike.

The Wheeler-Feynman absorber theory had some similarity to the above idea. In the absorber theory, a mysterious mechanism simultaneously decides the emitter and the absorber of a photon. It is like the teleportation process.

The Bethe-Heitler 1934 paper about bremsstrahlung and pair production

Let us look at the 1934 paper by Hans Bethe and Walter Heitler.

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

The paper is freely readable at the link.

                     photon k

                     _______________

                  /

  p_0  e- ------------------------------

                           |   virtual

                           |   photon q

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

When an electron meets a nucleus, Bethe and Heitler consider the above Feynman diagram (though not called a Feynman diagram yet in 1934). They also consider the alternative diagram where the photon with the momentum k is emitted after the virtual photon q line.

Above, after the electron has emitted the photon k, the electron is in an intermediate (virtual) state until it collides with the nucleus and gives up the surplus momentum it had.

Formula (4) in the paper is somehow derived from the density of final states, probably Fermi's golden rule.

The classical limit of the process is a very heavy particle, with the same charge / mass ratio as the electron, meeting the nucleus. If the particle is very heavy, we can consider it a classical charged particle.

Classically, the Larmor formula gives an estimate of the radiated electromagnetic energy in the bypass.

Our rubber plate model qualitatively explains classical radiation.

A. First the nucleus pulls the electron as the electron approaches. The rubber plate lags behind. The released potential energy V is channeled into:

1. kinetic energy of the electron;

2. the stretching and kinetic energy of the rubber plate.

B. Then the electron recedes. Now the rubber plate is moving ahead of the electron and pulls it away from the nucleus.

Kinetic energy of the electron and the rubber plate is channeled into:

3. the potential energy of the electron;

4. the new stretching of the rubber plate.

The end result is that the rubber plate gained oscillation energy (waves), the electron lost kinetic energy, and the extra momentum was absorbed by the nucleus.

We assume that someone has calculated that the Bethe-Heitler formula agrees with Larmor, if a suitable cutoff is made.

What is the significance of the two Feynman diagrams? Do they correspond to phases A and B above?

The Feynman diagrams are an abstract way of describing the end result. In our classical rubber plate model, momentum first flows to the electron, and later flows out. The Feynman diagram just shows the net effect - no details of the process.

What is the virtual electron in the Feynman diagrams?

What is the meaning of the virtual electron in the Feynman diagram? Classically, it is an electron which is interacting both with the Coulomb field of the nucleus, as well as its own Coulomb field.

Bethe and Heitler write in their paper that the electron is simultaneously interacting with the nucleus and the "radiation field" H.

The interaction of the electron with its own field eventually produces the electromagnetic wave. The electromagnetic wave is a "wrinkle" in the electromagnetic field of the electron.

Could we say that there is no electromagnetic field associated with empty space? Electromagnetic waves are just wrinkles in the fields of individual charges? This hypothesis would free us from the hypothetical infinite energy of oscillations of the vacuum electromagnetic field!

Pair production

Bethe and Heitler write that pair production is handled "similarly" to a photon emission.

They write that it is a photoelectric effect. An electron is raised from a -511 keV state to a +511 keV state. The hole that is left behind has the same positive energy 511 keV. But they say that the positron itself has an energy -511 keV.

What does it mean that the positron has a negative energy but the hole has a positive energy? We cannot separate the positron from the hole. This is obscure.

Bethe and Heitler consider the interaction of the created electron and positron. They write that the "matrix element" (18) which denotes a transition from an electron to a positron under their mutual Coulomb interaction, vanishes unless the sum of their 4-momenta is zero. The zero sum would mean that they annihilate but leave no photons behind.

But what if the combined energy of the pair is < 1.022 MeV? Then the pair will annihilate almost certainly. Maybe Bethe and Heitler assume that the combined energy is above 1.022 MeV?

Is there sense in a pair whose combined 4-momentum is zero?

Let us have a schematic Dirac electron:

      ψ ~ exp(-i (E t - p x)).

The discussion above brought up the following problem: if we have an electron wave ψ (energy E) in the Dirac equation, we might interpret it also as a positron wave with a negative energy (observed from the outside) -E.

If we create an electron, and a positron whose 4-momentum is opposite to the electron, then the particles have an identical wave function? What is the sense in this? In an earlier blog post we remarked that if a positron has a negative mass-energy, it will in an electric field move like an electron, and the electric field cannot pull it apart from the electron.

The wave-particle duality - why the electron in the hydrogen atom does not radiate photons?

We brought up the length scale problem in the previous blog post. The particle model of Coulomb scattering says that relativistic ~ 1 MeV electrons scatter in a distance ~ 10^-15 m, but the Compton wavelength is ~ 10^-12 m, and the radiated photons have a wavelength longer than 2 * 10^-12 m.

How do we explain the discrepancy? What hides the fine 10^-15 m detail in the process?

Classically, a relativistic charge which makes a sharp turn within 10^-15 m will produce at least some electromagnetic waves whose wavelength is ~ 10^-15 m. What prevents us from seeing such 1 GeV photons?

        sensor for the

        scattered electron

        -------------------

                                    \  digital camera

                                     \   for photons

                                      \

     Z+ ● 

                     ^

                     |

                     |

                     e-

               

            | b   | impact parameter

Let us again look at Coulomb scattering of an electron from a nucleus. We may initially model the incoming electron as a wave packet whose size is > 2 * 10^-12 m. The encounter we have to calculate using a particle model. The electron may pass the nucleus at a distance ~ 10^-15 m. An electromagnetic wave is produced by the close encounter.

The measured result is an excitation in a cell of a digital camera, and an excitation in a cell of an electron detector.

https://en.wikipedia.org/wiki/Bremsstrahlung

The Wikipedia page gives a heuristic way to calculate the amount of energy radiated: we put a minimum value for the possible impact parameter b. The minimum value is the de Broglie wavelength of the electron.

Suppose that the impact parameter b is very small, ~ 10^-15 m.

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

When the electron turns abruptly at the nucleus, it will send an electromagnetic wave which looks like the Edward M. Purcell diagram at the top of the linked page above.

                ______  electric field line

  _______|               

         sharp

         turn at time t

Let W(t) be the produced classical electromagnetic wave if the electron passes the nucleus at a time t.

If we assume that the digital camera will see a sum of waves W(t) for a range of different t, then the sum will have less sharp turns in the electric field lines. That is, there is destructive interference which reduces the energy of the wave.

Classically, the camera would see the wave for one value of t.

When should we use a wave model and when a particle model?

Let us recapitulate what we have found out in the past few blog postings.

A. Coulomb scattering of an electron. We can use either the wave model or the classical particle model to calculate the deviation of an electron which passes a nucleus. 

B. Photon produced in Coulomb scattering. A classical particle model predicts photons of a very short (10^-15 m) wavelength, which is wrong. We have to use a wave model where the electron is a wave packet of the size of de Broglie wavelength.

We may simply cut off the short wavelengths, or calculate a sum of of waves where destructive interference removes the short wavelengths. A path integral might be one way to calculate the sum.

Demanding energy conservation of quanta is one way to cut off short wavelengths.

C. Pair production. We do not know how to model the sharp 1 / r Coulomb potential of an electron if we represent the electron as a wave. The energy 1.022 MeV which is needed in pair production requires very steep potential wells. We have to use a particle model.

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

We need to check how Bethe and Heitler (1934) handled this.

A specific problem is how wide a wave packet we should use in various cases?

Equivalently, what should be the cutoff wavelength if we use a cutoff?

Or, how many paths we should sum if we use a path integral?

In case B, we may model the electron as a continuous charge distribution which flows past the nucleus. Some of the infinitesimal charges are scattered by the nucleus. If the wave packet is very large, say 1 meter across, it is like an electric current flowing past the nucleus. The electromagnetic field would be almost static. There would be essentially no photon radiation. This is clearly a wrong model.

Feynman diagrams in case B are a wave model where we use energy and momentum conservation to implement a cutoff. The Feynman recipe sounds the most rational.

But if there is a loop in the diagram, then we need some other method.

Why the electron in the 1s state in the hydrogen atom does not radiate photons?

https://physics.stackexchange.com/questions/20003/why-dont-electrons-crash-into-the-nuclei-they-orbit

The most common explanation is the uncertainty principle. If we try to restrict the electron to a volume which is smaller than the Bohr radius, we need to use large momenta p to build the wave packet, and the total energy of the electron becomes larger than on the 1s orbital.

Let us try to apply the observations of the previous section to this problem. Let us send an electron slowly toward a proton. We may model the electron as a wave packet whose size is the de Broglie wavelength of the electron.

The wave packet passes by the proton and loses some 13.6 eV of energy as radiation. After that, the wave packet is wrapped around the proton.

Let us assume that the wave packet is completely symmetric around the proton at this stage.

Let us use a path integral to calculate outgoing radiation from a point x close to the proton. If in a path, there is a point particle electron with a velocity vector v, there is another path with the exactly opposite velocity vector -v there. The sum of the radiation from these two paths is zero.

Another way to look at this is that the charge density at x stays constant, and there is no reason why the current at x would change either, as a function of time. This implies that the electromagnetic field of the various paths stays constant. There is no radiation.

A basic assumption in our argument is that we should sum all paths and their effects to the electromagnetic field.

Why does the hydrogen atom appear electrically neutral to the outside world? If we must calculate the electric field using the above method of  path integral sums, and the wave packet is symmetric around the proton, we arrive to the conclusion that the electric field is zero far away from the atom.

Note that in this case we can treat the electron like its charge would be spread over a relatively large volume, and the charge density would be continuous. We may claim that the electron does not have a particle nature in this situation.

But if we shoot the hydrogen atom with a photon or another electron, the particle nature of the orbiting electron becomes evident again.

Can we somehow explain stationary states of the electron with these observations? If the orbital would be time-dependent, then there obviously would be electromagnetic radiation, and the electron would decay into a lower energy state.

But why does a stationary state 2s or 2p eventually decay into 1s? What causes the electromagnetic radiation?

The length scale problem: how to make a tiny wave model of a particle?

Our previous, extensive, post brought up the old problem about the wave-particle duality. When should we model particles as particles and when as waves, and what size should the wave packet be?

The Compton wavelength of the electron, 2 * 10^-12 m is way too large to model a close encounter with a positron, where the distance at 1 GeV energies can be as small as 10^-18 m. A modest 1 MeV energy corresponds to a distance of 10^-15 m.

A possible way to model tiny waves is to use sources. If we have a 30 cm long radio antenna, it can easily produce 300,000 km long waves if the charges in the antenna oscillate at 1 Hz.

The "dipole" model for photons, which we introduced in the previous blog post makes use of the fact that a tiny source in a wave equation can produce very long waves.

A teleportation model of the photon

We may imagine that a photon is a way to "teleport" the source of an electromagnetic wave very close to the receiving system, for example, a hydrogen atom.

When a hydrogen atom falls from the state 2s to 1s, it produces a photon whose wavelength is 122 nm. That is 1,220 times the diameter of the hydrogen atom, 0.1 nm.

When another hydrogen atom absorbs that photon, the receiving atom kind of sees all the energy in the photon concentrated to a tiny 0.1 nm volume.

If the atoms were adjacent, with a distance 0.1 nm, it would be easier to understand how all the energy is concentrated into a tiny volume.

The teleportation/dipole model makes a collision of two photons like a close encounter of two electrons and two positrons.

Pair production using the drum skin model with sources

The drum skin model of the previous blog post is an attempt to find a wave description for pair production. What happens if we use the dipole model to describe the colliding photons? Then we have sources to the Dirac equation, and the sources are separated only by ~10^-15 m.

A hit with a sharp hammer to a drum skin will produce all frequencies. A better model is to put the sharp hammer to touch the skin, and let the hammer move up and down smoothly. We can then produce sine waves of a desired length with a sharp, pointlike source.

How can we use a hamiltonian to justify the production of a pair? The electric field of the produced pair should reduce the potential energy of the oscillating electric field in the tiny, 10^-15 m volume. How do we model the Coulomb field of the produced pair?

As long as we treat the produced pair as particles, the Coulomb field can be modeled with the familiar 1 / r potential. But also in that case, we need to decide when exactly is the Coulomb field born, and how quickly it spreads to the environment.

What is the role of our massive Klein-Gordon drum skin then? The massive Klein-Gordon equation neatly explains why low-frequency photons cannot create pairs. In the particle model, the same restriction follows from conservation of energy.

           e-                e+

           |                 |              ^

           |                 |              |  oscillation

           |                 |              v

           e+               e-

      dipole        dipole

Let us then attempt a new hybrid model of a photon collision. We put two electron positron dipoles at a 10^-15 m distance from each other. The dipoles make one oscillation. At some point, the electric potential is so steep that a creation of a 1.022 MeV pair would reduce the total potential energy of the system.

Let us try to estimate how hard it is for a pair to tunnel into existence. There is a potential wall whose height is ~ 1 MeV and width is ~ 10^-15 m.

We have to "borrow" 1.6 * 10^-13 J for a time ~ 10^-23 s. The product 1.6 * 10^-36 is much less than the Planck constant. The energy-time uncertainty relation says that tunneling is very easy.

That is, if a new pair can reduce the potential energy of the electric field > 1.022 MeV, the pair will almost certainly be created.

In the massive Klein-Gordon equation, or the Dirac equation, we may create the new pair with two, almost pointlike, sources. The distance of the sources is ~ 10^-15 m.

How should we model the attraction between the newly created particles? It is hard to model that in the wave model. We should continue using the particle model until the distance of the pair is much more than 2 * 10^-12 m, i.e., the Compton wavelength. From then on, a sharp wave packet will work.

The classical limit

Let us imagine physics without gravity. Then point particles can be very heavy without being black holes. We can model a close encounter (10^-15 m) of two unit charges using a wholly classical description. The speed of the encounter can be much less than the speed of light. We can use newtonian physics.

The encounter will produce electron-positron pairs.

This suggests that the particle model actually is the right way to model close encounters. A wave model would be a wrong way.

Where do we need a wave model? To calculate interference patterns, for example. To calculate the electron orbitals of hydrogen. We could, in theory, calculate these with path integrals, which would be closer to a particle model.

An external observer does not see the sharp details of a particle collision

In the particle model, electrons can make sharp turns in a distance of 10^-15 m, much less than the Compton wavelength. But an external observer can never directly observe such detail. Whatever the collision will produce, will have the smallest detail size of 2 * 10^-12 m, the Compton wavelength.

One might speculate that the sharp detail does not exist at all - it is just some symbolic mechanism which we use to calculate probability amplitudes for particles coming out of the collision. Who knows.

The goal of our analysis of pair production has been to cast light on vacuum polarization. Let us look next at vacuum polarization, again.

A new mechanical model of the massive Klein-Gordon equation

The string B in our previous blog post does not model the Dirac field well. The string B can have oscillations of arbitrary frequency, that is, quanta with arbitrarily small energies.

The way to force a minimum frequency to oscillation is to use a harmonic oscillator:

https://en.wikipedia.org/wiki/Klein–Gordon_equation

       d^2 ψ / dx^2 - m^2 ψ = d^2 ψ / dt^2

                  drum skin B

       ======================   

             |    |    |    |    |    |    |

     springs that resist displacement

     of the skin

The massive Klein-Gordon equation does the job. If we ignore the term d^2 ψ / dx^2, we have the differential equation for a harmonic oscillator.

B is like an array of harmonic oscillators which are coupled by the drum skin.

The impulse response of the massive Klein-Gordon equation

Let us imagine a Dirac delta term

       δ(t, x)

summed to the Klein-Gordon equation at a certain point in spacetime t, x. How does the equation react to such delta source? An answer is a Green's function.

The Klein-Gordon equation can be interpreted as calculating the vertical acceleration of a a small square of the drum skin B. The term d^2 ψ / dt^2 is the acceleration. The term d^2 ψ / dx^2 is the vertical force which results from the bending of the tense skin. The term m^2 ψ is the force exerted by the spring of a harmonic oscillator.

A Dirac delta impulse is like hitting the drum skin with a sharp hammer.

Obviously, a single hit with a hammer would make B to oscillate and create sine waves, that is, real particles. A single hit is like the creation operator in quantum field theory.

Modeling particle creation in the massive Klein-Gordon equation

Let us assume some unknown coupling between drum skins B and C. In our previous blog post, the coupling was a thin rubber membrane (between strings).

In our previous blog post we found out that colliding waves in C can create particles in B. A transient standing wave in C will create sine waves in B.

Let the resonant frequency of the harmonic oscillators in B be f_B. We may imagine that f_B corresponds to 511 keV.

If colliding waves in C have the frequency f_B, they will act as an oscillating source in B. The oscillation creates waves which slowly spread to the environment. If we use the circular polarization framework in B, opposite polarizations might correspond to an electron and a positron.

              |        |       springs

              ●      ●

            /    \   /    \     drum skin

          ●      ●      ●

         |         |         |    springs

           ● = oscillator

If the frequencies in C are larger than f_B, then the oscillators in B may raise their resonant frequency by using the tension in the drum skin, like in the diagram above. The black dots are the oscillators. If the angles in the tense drum skin are steep, the tension in will make the oscillators to oscillate faster. These waves correspond to electrons and positrons with kinetic energy > 0 in QED.

Modeling the Coulomb force

Let us have a plane wave in B. If the wavelength is shorter in some direction, then the wave tends to turn to that direction.

We can make the wavelength shorter in some area by making the springs looser there. The wave has to keep its frequency. It has to compensate the loosening of the spring by steepening the curves in the string - the wavelength has to become shorter.

If a wave packet has positive potential energy, it is located in an area where the springs are tight: this corresponds to a large m in the Klein-Gordon equation.

Recall that we solved the Klein paradox a couple of years ago by letting potential energy add to the inertial mass of the particle.

We have a model for the Coulomb potential: a high electric potential A in some area, through some mechanism, tightens the springs in B in that area, a low potential loosens the springs.

The bare charge of the electron is positive under its electric field?

The interaction term in the QED lagrangian is

       -e ψ-bar γ^μ A_μ ψ.

If we assume that the electron has a positive electric charge, then creating a potential well at the location of the electron will probably lower the total potential energy of the system. The cost is increased potential energy in the electric field component of A, but we save potential energy of the charge of the electron.

There would be no direct Coulomb force between charges. The repulsion or the attraction would be a result of the potential pit (or hill) created by the other charge.

This is like pressing a drum skin with a slippery finger to make a pit. Two fingers tend to slide toward each other because together the fingers can press the skin down more: potential energy is freed.

How does an oscillating electric charge produce electromagnetic waves?

                 drum skin B

     ====================== 

           |     |     |     |     |     |    springs

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

      very tense lightweight

      drum skin C

The drum skin C takes the role of the electromagnetic field in the following. Pits and hills in C control the tightness of springs in B.

In our model, if we through some mechanism force a wave packet (charge) to move back and forth in B, it will force the "Coulomb field" in C to move accordingly. If the pit or hill in C would not move along with the wave packet in B, then the potential energy of the charge would increase unnecessarily.

It is intuitively clear that a pit moving back and forth in the drum skin C will produce sine waves far away.

The wave-particle duality: whenever we measure the relative position of particles, we have to use a particle model or a wave packet model

Collisions of particles are hard to model with plane waves. How can low-amplitude waves spread over a large volume of space cause scattering to large angles?

For example, if an electron and a positron collide, they will change their direction significantly.

A possible solution is to treat the electron and the positron as a single particle moving in 6-dimensional space, and treat their interaction potential as an external potential which affects the single 6D particle. But then it is complicated to explain electromagnetic radiation sent by the particles.

When particles scatter from each other, we gain precise information about their relative position. That suggests that we should use a particle model or a sharp wave packet model to model the process.

It is like a photon hitting the image sensor of a digital camera. We have to treat a photon as a particle to explain why it hit just a single pixel cell.

When a hydrogen atom decays to a lower energy state, it for a short time acts like a dipole antenna. When the atom is excited to a higher energy state, the converse process happens. In this case, it is natural to threat the photon as a wave packet.

In our mechanical drum skin model this means that we have to model scattering events using sharp wave packets. If we try to model scattering with waves spread over a large area, the mechanical model fails.

What about the double-slit experiment where the distance of the slits is L? Can we use sharp wave packets to model it? Not very sharp. The packets have to be of the size ~L to form an interference pattern on the screen.

How do colliding waves in the electromagnetic field create waves in the Dirac field?

This is a difficult question. In our previous string model we assumed that there is an ad hoc rubber membrane which couples the two strings. We cannot resort to ad hoc mechanisms now, but have to derive, in some way, the behavior from the QED lagrangian.

Since the collision of two photons measures their relative position, the previous section of this blog post suggests that we should model the photons with sharp wave packets or particles.

    photon

    ~~~~~  --------------------- e-

                | virtual

                | electron

    ~~~~~  --------------------- e+

     photon

In a Feynman diagram, a photon scatters a positron which "travels backward in time", into a virtual electron. Another photon scatters that virtual electron "forward in time" into a real electron.

The diagram is full of mysteries:

(a) What is the energy and the charge of the virtual electron?

(b) Do the electron and the positron attract each other? If yes, when does the attraction start?

(c) From where does the virtual electron appear? Should we take seriously the claim that the the positron arrives from the future?

(d) We can create a 5 million volt potential difference with a van de Graaff generator. Electron-positron pairs do not form spontaneously to neutralize the charges. Why not? Is there some maximum distance that the virtual electron can travel?

An oscillating dipole model of the photon: we can make the spatial dimensions much smaller than the wavelength

Should we model the photons with sharp wave packets whose size is of the order of their wavelength, or 10^-12 m? The wave packet is too large and its energy way too diluted in space.

Compare this to a hydrogen atom whose radius is 5 * 10^-11 m and which absorbs a photon whose wavelength is 122 nm. The wavelength is 1000X the atom diameter. The way to model the photon is some kind of a hybrid particle-wave model, where the oscillation is restricted to an area much smaller than the wavelength of the photon.

          e+

          |          ^

          |          |   oscillation

          |          v     

          e- 

A possible way to model a photon is to use an imagined electric dipole which has an electron and a positron at a close distance. The dipole can be, say, 10^-15 m long, but represent a wave whose wavelength is 10^-12 m. The dipole oscillates much slower than the speed of light.

A linearly oscillating dipole represents a linearly polarized photon. A rotating dipole models a circularly polarized photon.

A drum skin wave model of pair production

When our two photons collide, they form a standing wave where a strong electric field oscillates, e.g., up and down.

            ^        

            |        e+       created

            |

            |        e-        pair

       electric

       field of

       photons

Everything happens in an area whose size is 1 / 1000 of the photon wavelength, or less than 10^-15 m.

Apparently, the electric field in that tiny area is strong enough to produce a new electron-positron pair.

What is the role of the virtual electron? It is a wave for which the energy E is typically small or zero, and the momentum p is substantial. It is an off-shell electron.

Let us look at a wave model.

An analogous construct in a drum skin is a pit which we make by pressing the skin with an elongated object. 

                  object

       ____    #####    ____   skin

        |     \________/     |

        |     |   pit       |       | springs

As long as we hold the object still, there is no oscillation relative to time. The "energy" E in an oscillator is zero:

      exp(-i (E t - p x)).

But the skin is deformed: the "momentum" p is not zero.

There certainly is energy in the pit: we had to deform the drum skin to make the pit. The E in the above formula is not the true energy of the deformation.

The elongated object is really the interaction of the electromagnetic field with the Dirac field. The interaction deforms the Dirac field.

Once we remove the elongated object (or the interaction), we get two sine waves which start to spread to every direction in the drum skin. These are analogous to the created real electron and the real positron.

Why would Nature create these particles? Nature tries to minimize the potential energy in the hamiltonian of the system. When the photons are very close to each other, creating a pair apparently helps to reduce potential energy.

Local conservation of energy, momentum, and charge requires that a Dirac wave cannot suddenly appear or disappear in empty space where there are no other fields that interact with the Dirac field.

The role of the virtual electron line might be to ensure some kind of "continuity" between the created real electron and the real positron. Feynman speaks about the electron "scattering backward in time or forward in time". Some kind of continuity of waves is required by all wave equations.

1. The created sine waves carry some momentum to every direction in the skin when the elongated object is removed. The pit is kind of a "spring" which pushes the waves and gives them the momentum. This is just like the virtual electron line in the Feynman diagram. The line transfers momentum between the created particles.

2. The energy as well as the momentum in the created sine waves completely come from the deformation of the drum skin and the springs in the pit.

3. The virtual electron in the Feynman diagram "moves" faster than light. It has more momentum p relative to the energy E than the energy-momentum relation allows. The created electron and the positron do not initially see each other.

4. The virtual electron is not a solution of the Dirac equation because it does not honor the energy-momentum relation. That is no surprise: the virtual electron is created by the interaction between the Dirac field and the electromagnetic field. It cannot be a solution of the free Dirac equation. Rather, it is a solution of the Dirac equation with a source - the impulse which the interaction gives.

5. Why created pairs do not neutralize the charge in a 5 megavolt van de Graaff generator? The elongated object in our diagram would in that case be very long. Apparently, the energy to make the very long pit to the drum skin would be too large: we would not recover that energy from the reduction of the electric field.

6. The virtual electron (the pit) may at some point of time contain all the energy to create the pair. The charge of the virtual electron is not really relevant because it very quickly decays into a real electron and a real positron, or is a combination of them all the time. We could say that the "charge" of the virtual electron is an electric dipole.

7. The virtual electron, the electron, and the positron are all created from the interaction of the electromagnetic field with the Dirac field. There is no positron which "arrives from the future".

A particle model of pair production

Imagine an electron and a positron placed at a 1.4 * 10^-15 m distance from each other. The sum of the masses of the particles is 1.022 MeV, but the electric potential is -1.022 MeV. The total energy of the system is zero.

The electric field of the electron at the distance 1.4 * 10^-15 m is

      E = k e / r^2 = 7 * 10^20 V/m.

If the colliding photons (modeled with dipoles) produce an electric field which is stronger, it is able to pull the pair apart.

If the colliding photons have > 1.022 MeV energy, they might produce a pair.

The Schwinger limit for pair production is 1.3 * 10^18 V/m. We need to check why it differs from E calculated above.

Is this a perturbative model?

If we treat the positron as a particle traveling backward in time, then we could say that it absorbs a photon and is scattered into a virtual electron which moves "sideways" in time, or superluminally.

One can say that this is a perturbation: the photon only scatters a small portion of positrons traveling backward in time. Most positrons continue undisturbed.

If we look forward in time, most photons will not collide. Only a very small number will produce pairs since the cross section is very small.

However, if we assume that the two photons will collide, then the process definitely is not just a perturbation: it is a completely different process and not some fine-tuning.

How to reconcile different length scales?

Almost all the energy in the electric field of the electron is contained within a 10^-14 m radius from the center. If we want to use the electron to "fill" a pit in the electric potential, the pit has to be of this size.

However, the Compton wavelength of the electron is 2 * 10^-12 m, much larger. If we want to use a wave packet model, we cannot make the electron smaller than that.

We encounter the same length scale problem in Coulomb scattering. A relativistic electron and a positron have to pass within 10^-14 m for the momentum exchange to be substantial. How can we use wave packets to model that?

In the case of an electron scattering from a nucleus, we may assume the nucleus to be a classical point charge, and be static. Then we can model the electron as a wave. The large wavelength of the electron is not a problem in that case, because the nucleus is a particle.

It looks like in a close encounter, we have to treat one or all the particles using a particle model. Our drum skin model cannot handle such cases well, as the model is a pure wave model.

The Schwinger limit of 1.3 * 10^-18 V/m seems to assume that a pair can easily form at the distance of the Compton wavelength 2 * 10^-12 m from each other. If that were the case, would the process create electrons close to atomic nuclei, and make nuclei to radiate positrons? We need to check how Sauter and Schwinger calculated.

The length scale problem is glaring when a photon excites a hydrogen atom. We need to check if anyone has comments on this.

The famous wave-particle duality is central in analyzing particle collisions. Feynman diagrams use plane waves - they are a wave model.

A general principle of interacting, non-turbulent, fields: you cannot produce arbitrarily fine detail

NOTE January 3, 2021: the heavy string B is not like the Dirac field, but just another model of the massless Klein-Gordon equation. A better model for a Dirac field is the massive Klein-Gordon equation, where the term m^2 ψ can be interpreted as springs attached to the string, to make it kind of a harmonic oscillator.

----

Feynman diagrams are based on the QED lagrangian, which is an equation between fields. The fields are the electromagnetic field and the Dirac field.

The precise meaning and interpretation of the QED lagrangian is fuzzy at some points, as we have remarked in this blog earlier.

Let us study a classical field equation of two fields A and B, to make the setup concrete.

            massive string

      ===========================   B

            |    |    |    |  rubber membrane

      -----------------------------------------------------   A

      lightweight, very tense string

Let us have a heavy, moderately tense string, and a lightweight, very tense string. They are coupled with a thin, easily stretchable rubber membrane.

The heavy string B is like the Dirac field, with the non-zero mass of the electron. The lightweight string A is like the electromagnetic field. Waves travel at a very great speed in the lightweight string.

The rubber membrane tries to transmit waves between the strings, which is not easy as the strings behave in quite different ways. The rubber membrane is the interaction.

Let there be a wave in the field A. The interaction causes disturbance in the field B. We estimate the impact on field B by using a Green's function of B.

Each spacetime point acts as a Dirac delta source in the B field equation. The intensity of the source depends on A at that point.

Let a wave in B be of the form

       sin(E t - k x),

where we call E the energy and k the momentum.

Intuitively, the Green's function method should produce some kind of an estimate for the disturbance in B. But applying the Green's function at just a single spacetime point of B does not make sense. The Green's function contains arbitrarily high momenta k, which corresponds to an arbitrary small wavelength in B.

We must consider the sum of Green's functions at various points, to get the estimate of the disturbance in B. Destructive interference cancels then high momenta k.

The size of "features" or "detail" in the disturbance in B is roughly the same as in A. A typical interaction does not magically create fine detail in a disturbance.

An exception is turbulence, where fine detail spontaneously appears in a field.

We may further use Green's functions to calculate what effect the disturbance in B has on A. Since B behaves in quite a different way from A, there is little "resonance", and B generally returns the disturbance completely back to A.

We may interpret that a "real particle" in A is converted to a "virtual particle in B", and is returned to A as a real particle.

What this means for vacuum polarization?

In a Feynman diagram, a virtual pair whose components have arbitrarily high momenta k appears. If we try to integrate over all k, there is divergence.

There might well be divergence in our string example if we would not take into account destructive interference in B.

This suggests that one should take into account destructive interference in vacuum polarization. It should cancel high momenta whose wavelength is less than the smallest "feature size" of the electromagnetic field in question.

If there is no such destructive interference in vacuum polarization, then the physics of the phenomenon is quite strange: the physics cannot be modeled with a typical field equation.

Is there need for a high-momentum cutoff?

In our string example, we may imagine that the strings consist of atoms. If we would have problems with high momenta k in our analysis of the strings, maybe we should use the atom hypothesis to cut off high momenta k?

Maybe that would work, but first we have to sort out destructive interference. If we still have divergences, then we might resort to atoms.

Regularization and renormalization in QED is like resorting to the atom hypothesis without first checking what is the effect of destructive interference on, e.g., a vacuum polarization loop.

An analogue of virtual pair production

In out string example, let us classify waves using the circular polarization framework. A wave can rotate clockwise or counter-clockwise, as seen from the left. A sine wave is a sum of these circular polarization states.

If A has a sine wave, it will tend to create transient vertical oscillation in B. It always creates a clockwise and counter-clockwise transient wave in B at the same time. This reminds us the production of an electron and a positron. They are Dirac waves that rotate to opposite directions.

If there is an external electric field present, then the electron and the positron accelerate to opposite directions. Simulating that in a string is hard. A clockwise wave should repel other clockwise waves? 

                      -->                        <--

  A   _________/\_______________/\_________ 

                    wave                  wave

What about a collision of waves in A? Could it somehow create a pair in B?

Waves in B tend to move slowly, and a collision creates a transient standing wave in A whose frequency is some f. If there is some nonlinearity in the system A & B & interaction, the (strong) standing wave does create a wave in B of the same frequency f. This works nicely.

However, we cannot model conservation of charge in our string model. Any oscillation in the massive string B quickly leaks into oscillation in A. That would mean that an electron spontaneously decays into photons.

An electron in B, viewed from the left, consists of clockwise rotation of the string. A positron is counter-clockwise rotation. The interaction (i.e., the rubber membrane) should be such that a pure circularly polarized wave cannot leak from B to A. Then we could ensure charge conservation.

Our string model illustrates the following rules:

1. A single massless particle cannot decay into a massive particle, but two colliding massless particles can do that.

2. A massive particle can decay into massless particles.

A massless particle is a wave in the lightweight string A, and a massive particle is a wave in the massive string B.