We have been working hard to find a plausible interpretation for the vacuum polarization integral in QED.
The vacuum polarization integral above is from Section 12.16 of the book by Hagen Kleinert. The Ward identity has already been used in the integral to remove the badly divergent part. The integral above is only logarithmically divergent.
e- ------------------------------------
| virtual
| photon q
|
-k O k + q virtual pair
|
| virtual
| photon q
-------------------------------------
proton
Above we have the relevant Feynman diagram.
Interpretation of the vacuum polarization integral.
Case of the free electron. Let the Feynman diagram describe an electron moving essentially freely, far away from the proton. We set q = 0. We claim that there is total destructive interference of the virtual pair wave function in the diagram. That is because there is no reason why the loop should be born at a position x of the electron rather than at a slightly later position x'.
Alternatively, if the electron is static, there is no reason why the loop should be born at a time t rather than at a slightly later time t'.
When we sum the probability amplitudes of all histories, there is total destructive interference of the virtual pair wave function.
We claim that the full integral
Π_μν(0)
describes the virtual pair in the case that there is no momentum exchange between the electron and the proton. That integral is totally wiped out by destructive interference.
e- --------------------------------------
-k O k virtual pair
---------------------------------------
proton
The actual Feynman diagram is above. The virtual pair floats freely in space, undisturbed.
Case of momentum exchange q != 0. In this case, the free floating of the virtual pair is disturbed by the momentum exchange q between the electron and the proton.
There is still almost total destructive interference for large |k|. The Feynman integral in this case
Π_μν(q)
still describes the part of the full integral which is wiped out by destructive interference. The difference
Π_μν(0) - Π_μν(q)
is the part of the full integral which survives destructive interference. The rest of the full integral "transmits" the momentum q from the electron to the proton. Note that the "propagator" of transmitting q over the virtual loop does not depend on q. Why is that? In an earlier blog post we sketched the idea that the virtual loop is a "dipole particle" and the momentum q is transmitted over the dipole "simultaneously".
| spatial momentum q
|
____|___ e+ 4-momentum k
/ | \
\___ ●___/ e- 4-momentum -k
|
|
| spatial momentum q
The diagram is above. At the dot ●, the momentum q flows simultaneously between the lower and upper momentum q lines. The flow does not affect the electron propagator.
Why is the formula Π_μν(q) the exact description of the part which is wiped out by destructive interference? Why not some other formula? The model of the next section would explain that.
Another interpretation of the full integral: the vacuum polarization effect on the electron is "bremsstrahlung" of virtual pairs
Let us first assume that the electron flies freely.
k
~~~ O ~~~
q / -k \ q
e- -------------------------------------
In the diagram, the electron sends itself arbitrary 4-momentum q as a virtual photon. The virtual pair loop receives and sends q simultaneously. That is why q does not appear in the virtual pair 4-momenta k, -k or the propagators of the pair.
The full integral of the process above is Π_μν(0). The full integral is completely wiped out by destructive interference.
Suppose then that the flight of the electron is disturbed by a proton, between sending and receiving q. In that case, the virtual pair cannot receive and send q simultaneously. The 4-momentum q has to spend some time in the loop. Now q appears in the propagators of the loop.
The electron can still absorb most of the wave that it sent, especially the high frequencies. The integral for the absorbed wave is
Π_μν(q).
The electron sent a wave described by Π_μν(0), but was only able to absorb Π_μν(q) back. What happened to the rest of the wave? It was absorbed by the proton. In a sense, the disturbance made part of the wave Π_μν(0) to get "detached" from the electron and be absorbed by the proton.
The sharp hammer model and the part of the wave which is not absorbed
Recall that our "sharp hammer" model of the static electric field of an electron explains bremsstrahlung as the part of the wave which gets "detached" from the electron in the wave created by hits of the sharp hammer (a hit is a Dirac delta source to the massless Klein-Gordon wave equation).
The static electric field is created by quickly repeating hits with a sharp hammer. If the electron changes its course, it cannot absorb the entire wave back. Part of the wave (some real photons) escapes as bremsstrahlung.
We have claimed that the vertex function is a classical electromagnetic process where part of the static electric field of the electron "lags behind". This sounds a lot like our ideas for bremsstrahlung. Part of the wave gets "detached" but is eventually absorbed by the electron itself.
In a forthcoming blog post we will analyze once again the QED vertex function (correction), and how we can explain it with our sharp hammer model.
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