Suppose that the perturbative Dyson series of Feynman diagrams converges for a weak repulsion between electrons. It produces nice, well behaved results.
Let the coupling constant be a small positive number
e² > 0.
The series is analytic. We can then analytically continue the series to small negative numbers
e² < 0,
and the perturbative series still converges.
A negative value of e² means that electrons attract electrons and positrons attract positrons.
e- ------------------- e-
|
------------ e+
|
------------ e-
|
e- ------------------- e-
--> t
But then the system can tunnel into a lower energy state where we have a large number of electrons close to each other, and elsewhere in space an equal number of positrons close to each other. The attractive potential of these gatherings is hugely negative, more than the mass-energy of the electrons and the positrons. The system has tunneled into a lower energy state where a huge number of electron-positron pairs came into existence.
In the diagram above, we have a pair born spontaneously.
Such tunneling should be visible in the perturbative series, and should cause divergences in the series.
*** WORK IN PROGRESS ***
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