The vacuum polarization loop

The ideas of Gordon in Compton scattering have helped us forward with the analysis of the vacuum polarization loop in Feynman diagrams.

                               wave of an

                                emitted photon

    electron wave         \   \

                      ---------        \   \

^ time           ---------

|

|             ____________

           \    ____________    positron wave

        \   \   ____________

     \   \   \

       \   \   \

      photon wave

1. Let us imagine that there is a positron around. The positron is a solution of the Dirac equation with no electromagnetic field.

2. A (virtual) photon causes a disturbance in the positron field. The disturbance is a source term in the Dirac equation.

3. We try to remedy the solution of the Dirac equation by using Green's functions of the Dirac equation to cancel the source term.

4. Green's functions produce an electron wave. We may interpret that the positron traveling backward in time absorbed the photon and turned into an electron. 

5. Next we imagine that there is an electromagnetic wave which corresponds to the electromagnetic wave which would be produced by the electron emitting the photon which it absorbed earlier.

6. The imagined wave disturbs the wave of the electron. The disturbance produces a positron wave which matches the original positron solution. An emitted photon wave is also produced.

The loop is complete! The positron, which we first just imagined, was "produced" by the scattering of the electron backwards in time, and the scattering also produced the emitted photon wave, which we originally only imagined to exist.

It is like trying to find solutions for the perturbed Dirac equation by assembling Lego blocks. We can use a block where an incoming photon produces an electron-positron pair.

If we turn that block around, we have a block where an incoming electron and a positron produce a photon.

As long as we can assemble a diagram which obeys certain rules, we are free to "imagine" the existence of whatever particle.

Note that in the diagram, all the waves really span the entire diagram area, and are overlapped. There is a large spatial uncertainty about the location of each particle.

What if the waves were classical waves?

Classically, we cannot just imagine the existence of any non-zero wave. In the diagram, there would be no positron wave present. The photon wave would proceed undisturbed.

What about the magnitudes of each wave? Let us use classical mechanics. Let us assume that the imagined waves do exist.

The electron flux is typically very small compared to the positron flux. It cannot "produce" the entire positron flux which exists in the diagram.

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

Baron Münchhausen told the story where he pulls himself out of a swamp by his own pigtail.

The Baron Münchhausen type trick of creating an electron-positron loop from (almost) nothing cannot work in classical mechanics if the disturbance is small. The "feedback" of the loop should be strictly equal to one, to allow a Münchhausen type of a process.

We know that pairs are produced in high-energy collisions of electrons. In quantum mechanics, a disturbance seems to have the ability to "concentrate" its effect on a very small spatial area, such that the feedback of a loop becomes strictly 1.

The diverging of the Feynman integral over a loop

The diverging of the Feynman integral indicates that something is wrong with the assumption that quantum mechanics can conjure up Baron Münchhausen type loops without any restriction. Feynman's rules allow the loop to carry any 4-momentum around, without any restriction.

In previous blog posts we developed the particle model of a photon as a rotating electric dipole.

If we assume that all the particles, including photons, obey certain restrictions of classical mechanics, then it is impossible for a loop to carry an arbitrarily large 4-momentum. No diverging of integrals is possible.

But does that restrict Feynman diagrams too much, so that they would no longer agree with empirical data?

Why does Feynman use a Green's function to describe the electric field of an electron?

In the electron-electron collision diagram, one electron sends a virtual photon, carrying some 4-momentum. The other electron absorbs this photon and receives a push.

Feynman assumes that the distribution of various 4-momenta in the photon is the Green's function for the massless Klein-Gordon equation. Why?

Let us consider the drum skin analogy of the static electric field of the electron. If I press the drum skin with my finger, it creates a depression into the skin. That depression is analogous to the static electric field of a particle.

We may imagine that instead of pressing with a constant force F, I keep tapping the skin with my finger at a very rapid pace.

The tapping creates a depression. A single tap is equivalent to applying an "impulse source" to the wave equation of the drum skin. The Green's function for the skin wave equation, by definition, is the response of the skin to that impulse.

That is, we may imagine that the static electric field of a particle consists of a very rapid pace of Green's functions emanating from the particle. The electric field does not carry energy away. There has to be a total destructive interference for the "on-shell" waves in the decomposition of the Green's function.

On the other hand, waves carrying just linear momentum p, can progress. Those waves apparently are responsible for the static depression in the drum skin or the static electric field of a particle.

The decomposition for the various p obeys the decomposition of the Green's function.

If there is a planar wave describing another electron nearby, the photon waves for various p disturb the free Dirac equation of that other electron. That is, the equation no longer is equal to zero, but a (small) source term appears.

Each wave p creates a source term. If we perturb the planar wave solution to find a more accurate solution for the source term associated with p, then another wave appears. That wave is interpreted as the wave of an electron which absorbed the photon with a momentum p.

Relationship to the classical scattering from a static Coulomb potential

If we calculate the scattering distribution, assuming that the electrons are charged particles of classical mechanics, the result is the same, or almost the same as when we use the Feynman diagram formula.

Classically, the momentum p which the electrons exchange is roughly proportional to 1 / r, where r is the minimum distance between the electrons. The number of electrons receiving a push > |p| is proportional to

       1 / |p^2|,

which is derived from the fact that the area for passing at a distance < r is proportional to r^2.

There is probably some general mathematical theorem which shows that an 1 / r potential for an incoming flux of particles can be implemented through the absorption of quanta of the Green's function for the massless Klein-Gordon wave equation.

If a photon is an orbiting virtual electron-positron pair, does that explain Compton scattering?

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

Thomson scattering means that a low-energy photon is scattered by an electron at rest.

Compton scattering is the same phenomenon with a high-energy (> 511 keV) photon.

http://www.mso.anu.edu.au/~geoff/HEA/5_Compton_Scattering_I.pdf

The cross section of Thomson scattering is of the order of the electron classical size. The classical electron radius is 3 * 10^-15 m. That is also the distance where the potential energy of two close electrons is equal to 511 keV, that is, the mass of the electron.

The cross section of Compton scattering is of the order of the electron classical size divided by the energy of the photon (given in units of 511 keV).

Let us assume that a "photon" moves in a medium of coupled electron-positron dipoles. Oscillation of such a dipole spreads to the neighbor dipole through the electric force. The photon is really a phonon of this medium. We do not assume the existence of any electromagnetic waves. The oscillation is strictly in the dipoles.

Suppose then that we have a free electron in the medium. What is the cross section of its collision with a phonon?

We may model a phonon as a moving oscillation of a single dipole. The oscillation of a single dipole jumps to the neighboring dipole at (almost) the speed of light. The phonon moves fast through the medium.

If the free electron happens to be within 3 * 10^-15 meters from the positron or the electron in the oscillating dipole of the phonon, then there is very strong interaction between the free electron and the phonon. This might explain why the cross section of a photon-electron collision is of the order of that length.

The free electron robs energy and momentum from the oscillation of the dipole.

We may assume that the dipole has before the collision assumed an equilibrium position in the electric field of the electron.

When the dipole starts to oscillate, what is the effect on the free electron? If the electron is not close to the ends of the dipole, the momentum transfer is inversely proportional to the distance to the ends of the dipole, and the periodically changing field probably cancels away most of the momentum transfer to the free electron.

Why is the cross section inversely proportional to the energy of the photon in Compton scattering?

The history of the Klein-Nishina formula

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5709540/

In 1928, Klein and Nishina were able to derive the correct differential cross section formula for Compton scattering, based on the brand-new Dirac equation. Yuji Yazaki in the link (2017) tells about the history of the discovery.

In 1926, Dirac treated scattering as a state transition of the system electron & an oscillating electromagnetic field. The apparent "collision of a photon" is a state transition which happens at a certain probability per second. Dirac derived the correct formula for a "spinless" electron. Klein and Nishina included the magnetic field of the electron in the formula.

We need to find out what is the relationship between the Feynman approach to scattering and the Klein-Nishina approach.

https://arxiv.org/abs/1501.06838

Waller and Tamm (1930), and in unpublished notes, Ettore Majorana, modified the Klein-Nishina semiclassical approach to a quantum field theoretical framework. It turned out that the electron goes through intermediate states. Thomson scattering is produced by negative-energy, that is, positron, intermediate states.

We need to compare the ideas of Waller, Tamm, Majorana, and Feynman.

If Navier-Stokes allows simulation of a Turing machine, then Gödel may make existence of a solution undecidable

Let us consider the Navier-Stokes equation of perfect fluid, with no atoms or other type of a cutoff at very short distances.

Solutions of the Navier-Stokes equation easily develop turbulence. Turbulence is a fractal-like phenomenon.

Can we harness turbulence to do complex digital calculations, like on a Turing machine?

If yes, then we might get analogues of the Gödel incompleteness theorem for solutions of the Navier-Stokes equation.

Suppose that a proof of a contradiction from the Peano axioms is equivalent to proving that a certain solution of the Navier-Stokes equation develops a singularity. Then it would be an undecidable problem if a singularity appears.

https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

Terence Tao in his 2007 blog post refers to a possible connection of problems of complexity theory (e.g., P = NP) to solutions of the Navier-Stokes equation. Turbulence develops complex, pseudorandom structures. (See his note "6. Understanding pseudorandomness".)

If we can build a digital computer from turbulence, then complexity theory will pop up.

Real fluid has a cutoff at the atomic scale. We do not expect a turbulence-based Turing machine to have any relevance in the real physical world.

Quantum fields probably have a cutoff at the scale of the Planck length, because mini black holes may turn up. It is unlikely that we can harness microscopic quantum fields to make a Turing machine, but this deserves further thought.

A swarm of virtual quanta: a possible solution to the renormalization problem of QED

UPDATE October 27, 2019: If there is just a single vacuum polarization loop, the loop length is 2, and the propagator for large k is roughly 1 / k^2. Since k can have any value in R^4, the integral diverges badly, like k^2.

But if we have a swarm of virtual photons, and some of the virtual photons create virtual pairs, and the pairs recombine in such a way that the loop length for a momentum loop is 6 or more, then the propagator for large k is 1 / k^6 or less. Then the integral converges.

If we could modify the Feynman method in such a way that the momentum loop length is always at least 6, we would get rid of the ultraviolet divergence.

---

Let us assume that two electrons are colliding in a particle accelerator.

         ^                   ^
             \     p      /
              |~~~~|
             /             \
         e-                 e-

The simplest Feynman scattering diagram contains just one virtual photon which is exchanged between the electrons. The photon carries a four-momentum p.

The propagator of a photon is

       i g_μν / p^2.

Let us think about a classical scattering of two electrons. If the distance is larger than the Compton wavelength 2 * 10^-12 m, then classical Coulomb scattering is a good approximation.

In the classical scattering, the electrons move in curved paths. If we want to use the quantum mechanical particle interpretation, the electrons exchange a large number of virtual photons on their path.

Consider then the vacuum polarization loop for a single virtual photon:

         p           p + k
   r_0 ~~~~~O~~~~~ r_3
                       -k     

The photon starts from a spacetime point r_0 and ends up at r_3. The electron-positron pair are born at r_1 and annihilate at r_2. The contribution in the phase of the Feynman diagram is something like:

       exp(i *[(r_1 - r_0) • p
                    + (r_2 - r_1) • (p + k - k)
                     + (r_3 - r_2) • p
               ])
       =
       exp(i (r_3 - r_0) • p).

We see that there is a perfect constructive interference if we let r_1, r_2, and k vary.

The constructive interference causes the integral over all k to diverge badly.

We have in our blog stressed that momenta k >> p should have a negligible contribution to natural phenomena where a momentum p is the input. That is, if the process is fuzzy at a length scale L, phenomena with a length scale << L should not affect much.

But in the vacuum polarization loop, high k contribute greatly.

A possible solution is to require that in the scattering, the electrons exchange a very high number of low-momentum virtual photons.

Then the corresponding virtual electrons and positrons form large "swarms", where they can annihilate with any of a large number of opposite charges. There is no longer perfect constructive interference because an electron of a momentum p + k can annihilate with a positron of an arbitrary momentum k', and the positron was not born at the same spacetime point as the electron.

We conjecture that there is an almost perfect destructive interference for high momenta k. Only if k is of the same order of magnitude as p, is the contribution considerable.

Let the electrons pass by at a distance L.

We conjecture that the classical limit of the process is that the EM field as well as the reaction to it, the vacuum polarization electron field, are fuzzy at the length scale L. The fields have little contribution from high momentum k planar waves.

Our suggestion has an obvious problem: how do we model the creation of an energetic real photon or a real electron-positron pair in a collision? We should allow a high momentum p photon to produce a 1.022 MeV pair.

So far we have not found in literature practical examples of how large a cutoff Λ one should use in a collision of a momentum p, so that the Feynman formulas predict the outcome well. Is 2 × p a suitable cutoff? Or should it be much larger?

The Feynman diagram with just one virtual photon is a perturbation diagram where we approximate the perturbation as a single Dirac delta impulse on the other field. That sounds like a very crude way to calculate a solution for the QED lagrangian (whatever the correct lagrangian is). The divergence in the Feynman integral might be an artifact of the very crude approximation.

If we calculate with a swarm of virtual quanta, we might be closer to solving the fields non-perturbatively. That is, closer to the correct solution. If large momenta k have a very low weight in the correct solution, then there is no divergence problem. Then the solution to divergences is to calculate correct, non-perturbative solutions.

What is the bare charge of an electron?

The scattering experiment in our previous blog post can be interpreted as a way to measure the electric repulsion between two electrons, and therefore, their charge.

Some people think that the electron is surrounded by a cloud of virtual electron-positron pairs which screen part of the negative bare charge of the electron. For an unknown reason, there is less screening if real electrons come close to each other. That would explain the stronger coupling constant. But this does not help much because we have to refer to an "unknown reason".

We in this blog have the philosophy that only measurable quanta exist as "particles" and the rest is classical fields which obey classical field equations.

We do not know the correct QED lagrangian. Let us assume that Feynman diagrams indirectly describe the lagrangian correctly, whatever it is.

The trigger to create a virtual pair in the one-loop Feynman diagram is the exchange of the large momentum virtual photon between the colliding electrons.

The reaction in the electron field is to the momentum exchange. That suggests that the reaction is a dynamic phenomenon.

Suppose that we somehow attach two electrons very close to each other. They exchange a lot of momentum in a second. Is there a similar virtual pair loop present in this case?

We need to analyze the Feynman formulas for the loop. What kind of a reaction do the formulas describe in the electron field?

Scattering of two electrons is a classical field phenomenon - is renormalization really needed?

In last fall we promised in this blog that we will show that the divergences in the Feynman loop diagrams are an artifact, which is a result of a wrong integration order.

Thus, no renormalization is needed if the scattering amplitudes are calculated in the correct way.

Our analysis of of the QED lagrangian brought this question up again.

Scattering of two electrons with low energy

Let us consider the scattering of two electrons which possess much less than 1.022 MeV of kinetic energy. Let us assume the bounce is symmetric.

    ^                ^
      \            /
        \        /
         |~~~| virtual photon
        /         \
      /             \
   e-                e-

There is electric repulsion between the two electrons which makes them to bounce off each other. The virtual photon marks the repulsion. The virtual photon is not a particle in any way.

An analogue for the repulsion is a spring:

  e-  ---|\/\/\/\/\/\/\/\/|---  e-

The electrons push each other with a rod containing a spring.

What is analogous to the virtual electron-positron loop in the Feynman diagram for the virtual photon?

       ~~~~O~~~~
             loop

An analogue is that the spring can give way not just in the spring /\/\/\ part but also in the straight rod parts ---.

If we want to calculate the bounce very precisely, we need to take into account all other degrees of freedom where the kinetic energy of electrons can be stored temporarily, not just the electric field.

In the case of the spring, the straight rods could store a little energy. It is similar for a virtual electron-positron loop: it can make the repulsive potential between the two electrons a little less steep. The loop will store some energy for the time when the electrons pass by. The loop will return the energy back to the kinetic energy of the electrons when they start to recede.

The virtual loop, which is also called vacuum polarization, might create a temporary charge distribution like this:

       -   e-  ++  e-    -

The positive charge density between the electrons makes the repulsion a little weaker in the bounce.

But the running coupling constant makes the electron repulsion stronger at short distances. What can cause that?

We remarked in our earlier post about a radio transmitter that if an EM wave is created by a disturbance of the EM field, and the wave is guaranteed to get absorbed soon again by another disturbance, then the wave may store a lot of momentum relative to the energy. A free plane wave in an EM field stores less momentum per energy.

The same is probably true for a short-lived wave in the electron field. We would need a correct QED lagrangian to analyze this in detail. The temporary field is born by interaction from the rapidly changing electric field between the bouncing electrons. We should show from the correct QED lagrangian that the electron field, indeed, is disturbed by the rapidly changing electric field, and stores some energy and momentum for a short time.

Let us assume that a temporary wave in the electron field is able store a little energy and considerable momentum. The temporary field does not need the 1.022 MeV of energy which would be needed for a real pair.

Is our analysis of the waves fully classical? Where does quantum mechanics enter the picture?

Note that our qualitative analysis did not refer to quanta anywhere except that the colliding electrons in the pictures were assumed to be particles.

A more precise analysis would assume that the electrons are waves obeying the Dirac equation when moving free of interactions.

The waves are smooth. The fuzziness of the waves in space it at least of the order of the Compton wavelength of the electrons.

If we solve the waves fully classically, where does quantum mechanics enter the picture? Maybe only at the measuring device. It will measure particles whose probability distribution can be derived from the classical wave solution.

What is the divergence in the Feynman loop diagram?

The well-known problem in the Feynman diagram formulas is the divergence of the calculation of the loop contribution. If we integrate over all possible momenta carried by the loop, the integral diverges.

How does that divergence show up in our classical analysis?

It does not because when two classical waves of a wavelength λ meet, we do not need to consider detail much smaller than λ in the reaction of the electron field. We can use a cutoff at λ, and intuitively know that finer detail has very little effect.

Using large momenta in the electron field would involve fine spatial detail in the electron field.

A Feynman diagram calculates all possible paths of the bouncing electrons. Electrons are point particles in the Feynman diagram and can come to a very close distance from each other. Very fine detail in the reaction of the electron field to that close encounter does have an effect on the Feynman calculation formula. If we try to calculate an intermediate result after the loop, the result may well diverge.

However, that intermediate result is not what we measure from the experiment. It makes no sense to calculate such.

If we only calculate the end results of the experiment, and if our intuition that fine detail has a vanishing effect is right, then the end results will not diverge.

If we are right, the divergence in Feynman diagrams is just an artifact from a wrong integration order. We must not calculate the diverging intermediate result.

The general problem of divergences in partial differential equations

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

A Millennium problem is to prove the existence and the smoothness of solutions for the Navier-Stokes equation. The problem is in turbulence. Does its infinitely fine detail have a large effect on the solution?

The QED classical wave equations might have a similar problem. We need to prove that no turbulence-like phenomenon can appear. If we cannot prove the existence and smoothness, then the divergence of Feynman intermediate results is a symptom of a real mathematical problem.

If we can prove the smoothness and existence of QED wave solutions, then the divergence is just an artifact.

We have in this blog post shown the connection between the existence of smooth solutions for a classical partial differential equation and the need to renormalize a Feynman diagram calculation.

To recapitulate:

1. If the classical equation has smooth solutions, then renormalization is really not needed. The apparent need for renormalization is a result of trying to calculate nonsensical intermediate results.

2. If the classical equation does not have smooth solutions, then it is not a proper physical model. We need to modify it, for example, by introducing a cutoff, which is equivalent to renormalization with a cutoff.

We need to look at the literature about the existence of smooth solutions for various partial differential equations. Coupled field equations are probably nonlinear in most cases. Little is known about the existence of solutions for nonlinear equations.

Linearized general relativity is not renormalizable. How does that show up when we try to solve the classical Einstein equations?

Conclusions

Renormalization may be unnecessary in QED.

The need for renormalization, or new physics at the Landau pole scale or the Planck scale, is connected to the existence of solutions for the corresponding classical field equations.

If smooth solutions exist, no renormalization nor any new physics is required. In such a case, the concept of an effective field theory is unnecessary.