Our previous blog post left open what exactly is the "virtual photon" which in a Feynman diagram carries momentum p from the nucleus Z to the electron which is flying by.
Let us assume that we have a heavy nucleus Z which is static and an electron flies past it quite far. The kinetic energy of the electron is almost exactly conserved (the radiation of real photons is negligible), but its momentum is slightly changed by an amount dp downward.
electron ------------------->
Z nucleus
The wavelength of the electron stays the same after the flyby because the kinetic energy is the same. But to receive the extra momentum dp, the electron has to dive into the potential well of the nucleus, and during that dive, the wavelength of the electron is shorter.
Thus, the phase exp(i φ) of the electron wave function gets a boost, which is larger if the change in the momentum dp is larger.
Let us now look at the Feynman diagram of the process.
e- ----------->-------------->
| virtual photon
|
Z ------------>--------------->
The probability amplitude is calculated by multiplying the value for each line. In the diagram, the energy of the electron and the nucleus stays exactly the same. The flyby does not affect their phase exp(i φ) at all. But we know that the phase of the electron got a boost in the flyby. The role of the virtual photon is to add that boost exp(i φ) to the multiplied value of the diagram. When calculating the interference pattern of various electron paths, we need to have that boost taken into account.
Conjecture 1. The virtual photon in the Feynman diagram is just an obscure and misleading way to add the boost exp(i φ) to the phase of the electron.
UPDATE Sept 14, 2018: Our Conjecture 1 is wrong! See the blog post Sept 13, 2018. Also, since the non-relativistic Schrödinger equation assumes interaction which travels superluminally, also the prediction of the phase shift calculated using that equation is obviously wrong.
Since the wavelength of the electron is shorter close to the nucleus Z, we have the obvious optical analogue for the electron wave function.
Theorem 2. The vicinity of a nucleus acts like an optically dense area for the wave that describes a passing electron. QED.
UPDATE Sept 14, 2018. Optical theory gives wrong predictions for the the electron phase after the flyby! See our blog post Sept 13, 2018.
The behavior of an electron wave function that passes by heavy nuclei (but not too close) is like waves of light passing and entering grains of optically dense material. The grains have a higher optical density at the center of the grain.
Question 3. Is there such a thing as a "virtual photon"? We can define a virtual electron as a real electron which has entered a high potential wall and has a negative kinetic energy. The phase exp(i φ) of the virtual electron is frozen inside the wall. However, the phase of the "virtual photon" in the Feynman diagram changes because it is a device to add the boost to the electron phase.
A photon has a zero rest mass and can shed all its energy by redshift when it climbs up a potential wall. A photon coming from the surface of a neutron star is an example. In what way we could create photons of a negative kinetic energy?
Our simple example above does not contain anything about the possible vacuum polarization that the passing electron might cause around the nucleus. If the virtual photon is just a misleading way to add the boost to the phase of the electron, why would the virtual photon decay into a virtual electron-positron pair for a short while?
Is there experimental evidence that a virtual photon can decay into a virtual pair?
Our blog post about pair creation as tunneling suggests that a dynamic process is needed to create real or virtual pairs. There is a dynamic process in the flyby. It will produce real photons as bremsstrahlung and these real photons may decay into virtual pairs for a while. When the path of the electron changes, its electric field no longer moves at a constant speed in the rest frame of the nucleus Z. Also, the electric field comes closer to the nucleus electric field in the flyby. Thus, there is ample dynamic behavior present. We need to find out in what way that can cause vacuum polarization.
In the blog posts of spring 2018 we conjectured that electromagnetic waves are phonons in a sea of virtual electron-positron pairs, or the Dirac sea of negative energy electrons. A real photon would be a real phonon. Do virtual phonons exist in solid state physics?
Let us investigate the flyby of an electron in greater detail. The electron path will dive into the potential well of the nucleus Z:
_____ _____
\_____________________/
potential energy
electron pathlength in meters -->
Let us designate the potential of a far-away electron with 0. In the diagram above the potential is proportional to 1 / r, where r is the distance from the nucleus. The integral over the path length in meters will tell us the boost dφ that the electron got to the phase of its wave function exp(i φ) in the flyby.
If the electron is slow enough not to be relativistic, and the potential well is not deep, then there is only a marginal change in the speed of the electron as it passes by the nucleus. Then the time is approximately linear in the diagram above. The impulse force that acts on the electron is proportional to 1 / r^2 and the direction of the force changes as the electron flies past. The momentum dp which the electron receives is proportional to the integral of potential^2 ~ 1 / r^2 during the flyby.
If the electron passes closer to the nucleus, the diagram above contains a deeper potential well. In that case, dp would grow faster than dφ as we make the potential well steeper.
The Feynman propagator for the (virtual) photon has a phase factor exp(i p (x - y)) where x is the position of the electron and y is the position of the nucleus. When we go closer to the nucleus, (x - y) is shorter and the phase change is smaller. This agrees with our analysis.
In the Feynman diagram, the virtual photon may temporarily decay into a virtual pair
_____ e+
/ \
~~~~~~~~~ ~~~~~~~~~~~~~~
\_____/ e-
The effect of a virtual photon is to advance the phase of the electron by dφ, and the effect of vacuum polarization should generally make the effect of an electric field weaker. In Feynman diagrams, typically the next level diagram works in the opposite direction to the first level diagram.
Vacuum polarization effects are larger close to the nucleus. We thus conjecture that in a close encounter, the phase shift is less than we would expect.
Let us consider the flyby process from the viewpoint of the energy of the electric field. The combined energy of the electric fields of the nucleus Z and the electron e- can be calculated by integrating E^2 over the whole space, where E is the electric field at a point in space.
When the electron flies by, then part of the energy in the electric field is converted to the kinetic energy of the electron. We pump some energy from the electric field and convert it temporarily into energy of the Dirac field (that is, energy of the electron field).
Vacuum polarization would mean that a virtual electron-positron pair will compete with the electron for that released energy The electron would use the energy to advance its speed while the virtual pair would use the energy in a vain effort to materialize as a real pair. If the flyby is far away, then the virtual pair will fail in their effort to materialize because there is not enough energy.
How to calculate the effect of a virtual pair if we do not use the framework of a Feynman virtual photon?
For a real photon, we assumed that the energy of the electromagnetic wave will try to tunnel into a pair with a probability which is proportional to the path length of the photon and the coupling constant. What is the corresponding probability in the case of an electron flyby?
An electron flyby might create real pairs or attempt creation of virtual pairs through the following process:
1. a virtual pair is born close to the flying electron;
2. the electric field of the flying electron pulls the virtual electron and positron apart; they gain some energy by moving in the electric field of the electron;
3. the flying electron accelerates and speeds away without claiming back the energy it gave to the virtual pair.
If the electron would be flying at a constant speed in empty space, then at step 3 the virtual pair would typically lose the energy it gained by moving in the electric field of the electron. There is an exception though: if the positron of the pair reaches the (real) electron, then annihilation will free enough energy to make the virtual electron real. The annihilation may be the origin of the peculiar zitterbewegung of wave packets constructed in the Dirac equation.
Our discussion above suggests that the primary mechanism of vacuum polarization in the flyby is through the acceleration of the electron in the field of the nucleus.
People tend to think that there is some kind of static vacuum polarization around the nucleus. That would require vacuum fluctuations to exist and we would end up with the problem of the vacuum containing an infinite amount of energy. Our approach is to explain vacuum polarization through dynamic processes only. The dynamic process we are considering is the flyby.
How does vacuum polarization reduce the phase shift of the electron in the flyby? It temporarily takes some of the kinetic energy of the electron away, which makes its phase shift smaller. The Feynman diagram approach suggests that vacuum polarization actually takes all gained kinetic energy to the virtual pair, since the pair replaces the virtual photon for a while. That would be strange. Maybe the Feynman approach approximates the potential with a step function? Then all acceleration will happen at the step, and it might be a good approximation that the virtual pair steals the energy at the downward step? This reminds us of the Klein paradox at a potential wall.
The magnetic field of an electric current tries to prevent changes in the current. When the electron approaches the nucleus, its velocity increases and some of its kinetic energy is stored in the magnetic field. The field pays back the energy when the electron starts receding from the nucleus.
Energy stored in the magnetic field might be considered a "real" photon which is emitted in the speedup and absorbed in the slowdown. After some thought, the magnetic field cannot be considered a "real" photon. It is just an expression of the static electric field and follows from special relativity.
If we can show that the electron flyby can be modeled with:
1. the electron moving at a constant speed until it absorbs a real photon;
2. the real photon is similar to the virtual photon of the Feynman diagram; it can decay temporarily to a virtual pair.
Then we can replace the strange virtual photon with a more tangible real photon. We can then do as in previous blog posts and show that the production of virtual pairs is "causal". We can replace the spike functions with smooth functions in their description and can show that an energy cutoff gives the best estimate of the process. Thus, we get rid of the divergence in the Feynman integral.
There are a lot of questions about the conservation of momentum and energy in the flyby. These may bear on our model:
As the speed of light is finite, how do we make sure that momemtum and energy are conserved? Is there a kind of a database transaction commit at the end of the flyby where nature makes sure that its bookkeeping of momentum and energy stays in balance?
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