virtual pair
__
/ \
e- \__/ e+
● Z+ nucleus
We have never seen a detailed explanation how exactly this would happen.
Feynman diagrams contain vacuum polarization loops. If the incoming particles have very low kinetic energy, then we may view electric fields as almost static.
virtual λ / 2 phase
photon shift
Z+ ~~~~~~~~~~O~~~~~~~~~~ e-
vacuum polarization
loop
In a Feynman diagram, a vacuum polarization loop is quite different from the one in the naive diagram at the top of this blog post. Feynman thought that the loop reduces the effect of the electric field, because the virtual photon is scattered (is reflected) from the loop and suffers a half a wavelength λ / 2 phase shift relative to a photon which flew directly.
The λ / 2 phase shift causes destructive interference to the loopless diagram, reducing the probability amplitude of significant scattering.
In our previous blog posts we have argued that reflection from a null-energy object, like from the vacuum polarization loop, cannot change the phase of a photon. That looked quite clear if the photon carries energy. But what about photons which carry just momentum, no energy?
A photon which carries just momentum does not proceed in the time dimension, just in the spatial dimensions. It "moves" infinitely fast.
If the electron and the positron in the loop both have zero energy, they "move" just in space, not in time. But then we have problems understanding how they end up at different distances from the nucleus Z+, and can cause polarization.
A particle model
Let us then treat the other case. We assume that the electron in the loop has positive energy E, and the positron negative energy -E.
In the naive diagram, the nucleus pulls the electron. Let it receive a momentum p toward the nucleus. The positron receives a momentum -p.
The electron moves closer to the nucleus. How does the positron move? It moves closer to the nucleus, too! Since the mass-energy of the positron is negative, it moves to the opposite direction from what one would naively expect.
<--- e-
<--- e+
Z+ nucleus
The movement has to be this way. Otherwise, the center of mass of the system would move.
The annihilation is easy to understand: the particles are at the same position. They do not need to jump anywhere to annihilate.
Now it is obvious that the virtual pair cannot cause any polarization, because the particles are at the same position.
Thus, if we have a static setup, then virtual pairs cannot gain energy from anywhere, and there cannot be any vacuum polarization.
Note that in this analysis we apply the "Schrödinger method" to assess the effect of a pair: we take into account the electric field of both the electron and the positron. The net effect is zero.
Feynman thought that only one particle in the pair needs to interact with other particles. We coined the word "minimal quanta method" to describe the assumption that the other particle can be ignored.
Thomson scattering with a classical model
For Thomson scattering, there exists a very good classical analogue. The electric field E of the incoming wave tries to make the electron to oscillate in space. But since the electron has inertia, it lags behind and emits an electromagnetic wave which has a half a wavelength phase shift relative to the incoming wave.
weight
-------------------●------------------
tense string
It is just as with a tense string where an extra weight is attached. The weight will act as a source for the wave equation, with a 180 degree phase shift.
Suppose then that we attach two weights to the string. One has mass m, and the other has a negative mass -m.
If the weights are at the same position, then their effect obviously will be zero.
E p
-e ---->
< ---- +e
-p -E
Suppose then that we have a virtual pair e- e+, whose total energy is zero. We may assume that the electron originally has the energy E and the spatial momentum p. The positron has (-E, -p).
The electron moves to the direction of p. Since the positron has negative energy, it moves to a surprising direction - it moves to the same direction as the electron! They stick together.
If there is an electric field present, it will give the electron some extra momentum q, and the positron the opposite momentum -q. But again, the particles will move to the same direction. They stick together.
The electron and the positron act as sources of electromagnetic waves as they oscillate. Since the particles are at the same position, their waves totally cancel each other. There is no Thomson scattering.
It is not a surprise: how could a real wave bounce back from a zero-mass object, the virtual pair? Zero mass means that the pair is essentially "nothing". One cannot bounce back from nothing.
This means that vacuum polarization loops cannot occur for a free photon. Vacuum polarization has no effect at all for a photon which flies freely.
What about virtual photons for which the momentum is off-shell? The classical analogue of a virtual photon is a malformed electromagnetic wave, whose E and p do not match. It is hard to see how the system of an electron and a positron at the same position could have any effect on the malformed wave.
Thomson scattering with a malformed electromagnetic wave
^
| rubber plate
-----------●-----------------•-------
charge weight m
Our rubber plate model can be used to illustrate a malformed wave. Let us move upward the charge in the diagram.
The rubber stretches, but the stretching is not in the right form to leave as a sine wave (= real photons). The stretching is a malformed wave.
The weight m in the diagram has the role of a positive energy electron. But at the very same position, there is a weight with a negative mass -m. Together, these weights have a zero effect.
A static electric field in the rubber model is modeled as charges pushing or pulling horizontally on the (very stiff) rubber plate. What is the effect of an extra weight attached to the plate then? Zero. This suggests that virtual pairs have no effect whatsoever on static electric fields.
Collisions with energy > 0
If a static electric field induces no vacuum polarization, what happens in a dynamic setup where particles collide?
Collisions do produce real pairs of electrons and positrons. It is very likely that collisions cause vacuum polarization, too. If the collision "almost succceeds" in creating a real pair, the pair probably exists for a while as virtual, and will cause vacuum polarization.
We need to study the collision process with the rubber plate model as well as with a particle model.
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