It is well known due to the famous argument by Dyson that the perturbation series for quantum electrodynamics has zero radius of convergence. Dysons argument essentially goes like that: If the power series in α had a finite (r>0) radius of convergence it also would converge for some small negative α (fine structure constant). This, however, leads to a contradiction since for negative α like charges would attract each other and this essentially renders the theory unstable for negative α.(adsbygoogle = window.adsbygoogle || []).push({});

I have three questions in connection with this:

First of all, is it possible to estimate at which order n the series will begin to diverge? Do we have a limit beyond which it does not make sense to use perturbation theory for QED any more and is this limit wihtin reach?

Secondly, even though we cannot explicitly state an exact solution, can we at least prove or do we know that there is an exact solution for this theory or can it be the case that mathematically there is no exact solution to QED at all?

Thirdly, it is well known that the different terms of the perturbation series are visualized via Feynman diagrams and that these diagrams lead to the picture of virtual particles being exchanged. Now, in case of an exact solution: Can we make any statement about a particle concept there? Are a finite number of particles involved? Does it make sense to talk about particles at all? Will there be things like virtual particles? Or do we simply know nothing at all with respect to exact solutions?

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# QED perturbation series convergence versus exact solutions

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