^ ^ ^
| | | | | | | | |
v v v v v v
-----------> observer
The arrows denote a static electric field in a tube. An observer flies in the tube at almost the speed of light.
Alternatively, electromagnetic waves propagate in the tube and cause polarization that progresses at almost the speed of light, and the observer is static in the tube.
The observer thinks that he is flying in a medium against a coherent laser beam.
If the observer has an electron flying along with him, he will see the electron move up and down, propelled by the electric field of the laser beam.
He thinks that the laser beam consists of real photons. The photons scatter from the electron and cause it to move up or down. The (small) energy to oscillation comes from the laser beam.
The electron will slow down slightly as it collides with photons coming from the right in the diagram.
We have another observer who is static relative to the tube.
If the static observer believes in a Feynman-style model, he thinks that the static electric field consists of virtual photons that transfer momentum. Virtual photons are absorbed by the electron flying past him. Those virtual photons transfer momentum to the electron and cause it to oscillate up and down. The energy to the oscillation comes from the original kinetic energy of the electron. The electron slows down slightly.
Here we have two observers. One thinks that he is in a dynamic field of a laser beam and another thinks that it is a static electric field.
What about the vacuum polarization loop in the Feynman diagram where we have a virtual photon transferring momentum? Does it have a counterpart in the dynamic interpretation as a laser beam?
The Feynman diagram which describes an electron colliding with a photon is:
----------- e- -------------->
/ \
/ \
The electron absorbs a photon and then emits it again. We cannot put a virtual pair loop into this diagram because there is no internal photon line.
If we add a self-energy photon for the electron, then we can put a virtual pair loop into it.
The Feynman diagram which describes static electric field interaction is something like:
----------- e- --------->
|
|
----------- e- ---------->
The vertical line is a virtual photon which carries a momentum p but little energy. We can add a virtual pair loop to the virtual photon line. The Feynman integral of the loop diverges badly. There are renormalization tricks to remove the divergence.
Conjecture 1. The laser beam diagram describes the physical system as well as the static field diagram. We can do away with the diverging Feynman integral by replacing the static electric field with a laser beam of a very low frequency.
Conjecture 1 probably implies that we can discard the vacuum polarization loop altogether from a Feynman diagram where the internal virtual photon transfers just momentum but no energy. That is the case if we have two electron beams moving at opposite directions at a considerable distance:
e- e- e- e- e- ----->
<--- e- e- e- e- e-
Conjecture 1 probably implies that the Schwinger effect does not exist.
High-energy collisions
If the collision is high-energy, even a real pair can be produced.
e- ---------------------->
|
|
<------------------------ e-
Diagram A.
Above is a diagram in the case where no real particle is produced, just some momentum and energy is transferred. We draw the lower electron moving to the left to make the diagram more realistic.
/
/
e- ------------------------>
|
|
<-------------------------- e-
Diagram B.
Above the upper electron converts some of its kinetic energy to a real photon and then gives its extra momentum to the lower electron.
e- ----------------------------->
| ----------- e- --->
|/
\
------------ e+ --->
|
<--------------------------------- e-
Diagram C.
Above the upper electron gives up some of its kinetic energy to a virtual photon. The photon must carry away a lot of momentum. The photon transforms to a real pair, which transfers its extra momentum and energy to the lower electron.
Classical limit of high-energy collisions
We need to understand the high-energy collision mechanisms intuitively. Studying the classical limit is one way of understanding what is going on.
If the electrons would have a very large rest mass and a very big charge, then the collision should resemble the collision of two classical charged bodies.
Diagram A corresponds to a very mundane exchange of momentum and energy, mediated by the static electric field of the charges.
Diagram B describes the radio waves emitted in the collision. The electric field changes in a very complex way as the charges accelerate or decelerate. Since there is no positive charge present, there is no dipole and the intensity of emitted real photons should be very low or zero. The converse process is a photon absorption. If we try to absorb a photon with a system which contains just two negative charges, it is hard to conserve momentum, energy, and angular momentum.
Diagram C is the problematic one in the classical limit. Classically, pairs cannot jump out of empty space. What about a classical collision in a medium which contains elementary electron-positron dipoles which the changing electric fields can separate?
https://en.m.wikipedia.org/wiki/Electrical_breakdown
An electrical breakdown is the natural classical analogue of pair production. When colliding charges come very close together, charges of dipoles in the medium separate and try to reduce the increased electric field.
If we have a zone in space where the electric field E has a large absolute value, then, classically, the energy density is E^2. An electric breakdown will reduce the energy density and convert some of the energy of the electric field into the energy of the separated charges.
From this point of view, pair production is a tunneling phenomenon where energy of the electric field is converted into energy of the created pairs.
In a medium, the density of atoms restricts the rate of tunneling. But if we assume that empty space contains an infinite density of virtual pairs, then the tunneling rate should be infinite. This paradox is at the heart of non-causality, and probably at the heart of Feynman integral divergences.
Our goal is to show that production of pairs happens only under dynamic conditions where there is flow of energy between the kinetic energy of the system and the energy of the electric field (= potential energy). Then we probably can tame the divergences. This also implies that the Schwinger effect does not exist.
https://en.m.wikipedia.org/wiki/Electrical_breakdown
An electrical breakdown is the natural classical analogue of pair production. When colliding charges come very close together, charges of dipoles in the medium separate and try to reduce the increased electric field.
If we have a zone in space where the electric field E has a large absolute value, then, classically, the energy density is E^2. An electric breakdown will reduce the energy density and convert some of the energy of the electric field into the energy of the separated charges.
From this point of view, pair production is a tunneling phenomenon where energy of the electric field is converted into energy of the created pairs.
In a medium, the density of atoms restricts the rate of tunneling. But if we assume that empty space contains an infinite density of virtual pairs, then the tunneling rate should be infinite. This paradox is at the heart of non-causality, and probably at the heart of Feynman integral divergences.
Our goal is to show that production of pairs happens only under dynamic conditions where there is flow of energy between the kinetic energy of the system and the energy of the electric field (= potential energy). Then we probably can tame the divergences. This also implies that the Schwinger effect does not exist.
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