# QED perturbation series convergence versus exact solutions

1. Jan 16, 2013

### Claustral

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.

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?

2. Jan 16, 2013

### The_Duck

I'm sure others know much more about this, but I think a rule of thumb may be that we expect there to be non-perturbative effects which are suppressed by a factor that looks like $\exp(-1 / g^2)$ where g is the coupling constant. For example, the BPST instanton in Yang-Mills theory has action $8 \pi^2 / g^2$ and so should be suppressed by $\exp(-8 \pi^2 / g^2)$. These instantons are completely invisible in perturbation theory, and you can see why if you try to construct a power series for $\exp(-8 \pi^2 / g^2)$ around g = 0. So I'd expect problems with perturbation theory to show up at least by the time you reach this level of precision.

Regarding your second question, I think the Landau pole is generally taken to mean that QED is probably inconsistent, at least as a standalone theory: http://en.wikipedia.org/wiki/Landau_pole . I think this would preclude the existence of any exact solutions.

Last edited: Jan 16, 2013