____
photon e+ / \
~~~~ ~~~~
e- \_____/
| virtual photon
| which acts on e+, too,
| because of Furry's theorem
Z+ ----------------------------
The momentum of the new photon differs from the incoming one because the Coulomb field of the nucleus interacts with the pair. Furry's theorem requires that there must be an even number of photon vertices in a fermion loop. We have only drawn one photon line, to avoid cluttering the diagram.
B. Kunwar (2011) writes about Delbrück scattering.
This scattering forms a good setting for studying vacuum polarization. Earlier, we have looked into pair production. Here we have a pair which never gains enough energy to become real. Nevertheless, the pair produces a measurable scattering effect on the photon.
Besides Delbrück scattering, there is Rayleigh scattering from the electrons of an atom, and scattering caused by the nucleus. Empirical tests must take into account these effects.
Experimental tests of scattering have been performed in the photon energy range 0.89 MeV - 1 GeV. According to B. Kunwar, the agreement of theoretical calculations and measurements is good, often within a few percent.
We should check if various authors allow an arbitrary momentum k to circulate in the virtual pair loop. If yes, then the integrals easily diverge. Maybe they use some cutoff for k?
A "reversed time" model for a photon or a wave function collapse
When a pair annihilates, it typically emits two photons. The converse reaction would convert the entire energy of the two photons into a pair.
How can we use a particle model for the produced pair and erase the entire electromagnetic field of the two photons?
The only way to make such a process seems to be running the annihilation in reverse.
We have the same problem when a hydrogen atom is excited by a photon. How can the electric dipole formed by mixed 1s and 2p states absorb the entire electromagnetic field of the photon?
We need to change our "teleportation model" of the photon. The new model is a reversed time model. A photon (or photons) is magically able to run in reverse a process which could emit such a photon.
Classically, it would be a miracle that a complex process runs in reverse because it would dramatically reduce the entropy of a complicated electromagnetic wave.
In quantum mechanics, such a miracle is the norm.
The miracle enables life to exist in our universe. Classical electromagnetic waves become soon too diluted to excite anything enough. Entropy grows too fast in classical physics. Photosynthesis would be difficult in a classical universe.
The reversed time model has similarity to the absorber theory of Wheeler and Feynman. The electromagnetic field of a photon is just a mechanism for relaying a "message" to the absorbing system that it is allowed to make a certain state transition, that is, absorb a certain photon.
Our teleportation model brought the emitting and the absorbing system very close, to explain why the electromagnetic field is not diluted. Our reversed time model augments this by saying that a transition in the emitting system can induce a transition in the absorbing system, such that the entire emitted energy is absorbed.
A classical model where the created pair absorbs the whole momentum of the photon
In a blog posting a couple of years ago we noted that an electron which is close to a positive charge may have the inertial mass large, even though m + V is zero. Here m is the electron mass and V is its negative potential close to the positive charge.
That is because the electron makes a "hole" in the electric field of the positive charge. If we move the electron, the energy content of the electric field moves to the opposite direction.
If m + V is negative, the inertial mass of the electron might be m + |m + V| = -V.
A positron may have an inertial mass m + V', where V' is the positive potential it has close to the nucleus.
4 * 10^-14 m 3 * 10^-15 m
Z+ ● • e- • e+
^
| photon E
Let us put the particles in a line, such that the total energy of the pair equals the energy of an incoming photon, say, E = 511 keV. The separation of the pair is about 3 * 10^-15 m.
The inertial mass of the electron in the diagram may be, for example, 1.022 MeV. If Z+ is iron, its charge is 26 e, and the distance to the electron 4 * 10^-14 m, so that the potential V of the electron in the field of the nucleus is -1.022 MeV. The potential in the field of the positron affects the inertial mass of the electron, too, probably increasing it somewhat.
If we let the electron move downward in the diagram at the speed ~ c / 2, the electron can absorb the whole momentum of the incoming photon. Then the electron will have some kinetic energy, too. We need to readjust the distance between the electron and the positron so that the total energy
2 m + potential + Mv^2 / 2= E,
where M is the inertial mass of the electron and v is its (nonrelativistic) speed.
There we have a classical model where we have a "virtual" pair absorbing the whole energy and the whole momentum of the photon.
The pair is not very "virtual" in any mysterious sense, but quite normal particles which interact. We have stressed that interacting particles do not need to obey the energy-momentum relation. They may be off-shell.
Our classical model suggests that Delbrück scattering would be possible quite far away from the nucleus. We need to look at calculations of Delbrück scattering.
The relevance of the classical model in real pair production
____________ e-
/
~~~~
\____________ e+
| virtual photon p
|
Z+ -----------------------------
Above is a typical diagram for real pair production from a photon hitting atomic matter.
The virtual photon gives the excess momentum p to the nucleus.
The positron is in the mysterious virtual state until it gets rid of the extra momentum. Our classical model shows that the virtual state might not be so mysterious, after all.
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