The divergence in the vacuum polarization Feynman integral is due to the fact that in the loop we can have an arbitrary value of momentum p circling around.
The value of p is not restricted in any way by the input particles that enter the scattering experiment. This is "acausal" physics.
We may interpret the momentum p making circles as "vacuum fluctuations" of the Dirac field.
If we allow arbitrary field configurations in a path integral, then we end up calculating "vacuum fluctuations", and many quantities turn up to have infinite values. An example is the vacuum energy. A standard calculation yields an infinite value per cubic meter, or something like 10^100 joules per cubic meter if we put a cutoff at the Planck scale.
If we try to calculate the polarizability of vacuum under an electric field, it is hard to restrict the number of virtual electric dipoles in the vacuum, and polarizability tends to be infinite.
The problem is in quantum field theory, and its origin is in the fact that we allow arbitrary field configurations in a path integral.
The traditional n-particle quantum mechanics does not suffer from the problem of infinities.
The vacuum polarization diagram is somewhat analogous to the following setup in traditional quantum mechanics:
A particle with kinetic energy E enters a test area. There it can either fly freely or bump into a potential wall which is higher that E. The particle may bounce from the wall or tunnel through the wall. At the end, the fluxes from all these routes are combined and form an interference pattern.
The analogues traditional particle <-> Feynman diagrams of a scattering experiment are:
1. particle flies freely <-> no virtual or real electron-positron pair is formed;
2. particle entes the wall but does not tunnel <-> a virtual electron-positron pair is formed and it annihilates;
3. particle tunnels through the wall <-> a real electron-positron pair is formed.
In case 2, the traditional setup does not suffer from infinities, but the Feynman setup does. Why?
If we try to estimate using a discrete grid and a path integral, the traditional setup keeps the probability of seeing the particle inside the wall small, while the Feynman method makes the probability of creating a high-energy virtual pair to grow when we make the grid finer. That is because the number of different high-energy paths grows faster than the number of low-energy paths. We assume that also energies are discretized. If there is more energy, we can distribute it to the grid in more ways.
A high-energy virtual pair corresponds to the particle entering deep in traditional setup.
A possible solution: in the grid method, combine the paths of case 1 and case 2 to a single path "universe". When we make the grid finer, the number of paths for 1 grows as fast as for case 2 and keeps the probability of high-energy paths in 2 low.
In the traditional setup, when we make the grid finer, the paths of case 1 balance the effect of paths 2. A finer grid allows more paths in 1.
In quantum field theory, we might only allow "causal path integrals" where the input to the scattering experiment keeps the probability amplitude of high-energy virtual pairs small. Then there would be no infinities and we could say that the input "causally controls" the process. But how do we implement this?
A huge problem of physics is that the infinite energy of the vacuum would cause the space to collapse gravitationally. To avoid the gravitational collapse, we should be able to show that the vacuum does not have much gravitational effect on a test mass.
UPDATE Sept. 7, 2018. It turns out there is prior research in the "causal" direction:
https://en.m.wikipedia.org/wiki/Causal_perturbation_theory
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