UPDATE October 27, 2019: If there is just a single vacuum polarization loop, the loop length is 2, and the propagator for large k is roughly 1 / k^2. Since k can have any value in R^4, the integral diverges badly, like k^2.
But if we have a swarm of virtual photons, and some of the virtual photons create virtual pairs, and the pairs recombine in such a way that the loop length for a momentum loop is 6 or more, then the propagator for large k is 1 / k^6 or less. Then the integral converges.
If we could modify the Feynman method in such a way that the momentum loop length is always at least 6, we would get rid of the ultraviolet divergence.
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Let us assume that two electrons are colliding in a particle accelerator.
^ ^
\ p /
|~~~~|
/ \
e- e-
The simplest Feynman scattering diagram contains just one virtual photon which is exchanged between the electrons. The photon carries a four-momentum p.
The propagator of a photon is
i g_μν / p^2.
Let us think about a classical scattering of two electrons. If the distance is larger than the Compton wavelength 2 * 10^-12 m, then classical Coulomb scattering is a good approximation.
In the classical scattering, the electrons move in curved paths. If we want to use the quantum mechanical particle interpretation, the electrons exchange a large number of virtual photons on their path.
Consider then the vacuum polarization loop for a single virtual photon:
p p + k
r_0 ~~~~~O~~~~~ r_3
-k
The photon starts from a spacetime point r_0 and ends up at r_3. The electron-positron pair are born at r_1 and annihilate at r_2. The contribution in the phase of the Feynman diagram is something like:
exp(i *[(r_1 - r_0) • p
+ (r_2 - r_1) • (p + k - k)
+ (r_3 - r_2) • p
])
=
exp(i (r_3 - r_0) • p).
We see that there is a perfect constructive interference if we let r_1, r_2, and k vary.
The constructive interference causes the integral over all k to diverge badly.
We have in our blog stressed that momenta k >> p should have a negligible contribution to natural phenomena where a momentum p is the input. That is, if the process is fuzzy at a length scale L, phenomena with a length scale << L should not affect much.
But in the vacuum polarization loop, high k contribute greatly.
A possible solution is to require that in the scattering, the electrons exchange a very high number of low-momentum virtual photons.
Then the corresponding virtual electrons and positrons form large "swarms", where they can annihilate with any of a large number of opposite charges. There is no longer perfect constructive interference because an electron of a momentum p + k can annihilate with a positron of an arbitrary momentum k', and the positron was not born at the same spacetime point as the electron.
We conjecture that there is an almost perfect destructive interference for high momenta k. Only if k is of the same order of magnitude as p, is the contribution considerable.
Let the electrons pass by at a distance L.
We conjecture that the classical limit of the process is that the EM field as well as the reaction to it, the vacuum polarization electron field, are fuzzy at the length scale L. The fields have little contribution from high momentum k planar waves.
Our suggestion has an obvious problem: how do we model the creation of an energetic real photon or a real electron-positron pair in a collision? We should allow a high momentum p photon to produce a 1.022 MeV pair.
So far we have not found in literature practical examples of how large a cutoff Λ one should use in a collision of a momentum p, so that the Feynman formulas predict the outcome well. Is 2 × p a suitable cutoff? Or should it be much larger?
The Feynman diagram with just one virtual photon is a perturbation diagram where we approximate the perturbation as a single Dirac delta impulse on the other field. That sounds like a very crude way to calculate a solution for the QED lagrangian (whatever the correct lagrangian is). The divergence in the Feynman integral might be an artifact of the very crude approximation.
If we calculate with a swarm of virtual quanta, we might be closer to solving the fields non-perturbatively. That is, closer to the correct solution. If large momenta k have a very low weight in the correct solution, then there is no divergence problem. Then the solution to divergences is to calculate correct, non-perturbative solutions.
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