QED perturbation series convergence versus exact solutions

In summary, Dyson's argument states that the perturbation series for quantum electrodynamics has a zero radius of convergence. This is due to the contradiction that arises when considering small negative values of the fine structure constant. It is not possible to estimate at which order the series will begin to diverge, and there may be non-perturbative effects that are beyond the reach of perturbation theory. The existence of an exact solution for QED is still unknown and it is possible that there may not be one. The concept of particles in an exact solution is also uncertain. The Landau pole suggests that QED may be inconsistent and therefore an exact solution may not exist.
  • #1
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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?
 
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  • #2
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.
 
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1. What is QED perturbation series convergence?

QED perturbation series convergence refers to the idea that the perturbation series used in quantum electrodynamics (QED) to calculate physical quantities may not always converge to the exact solution. This means that the calculated values may deviate from the true values, especially at higher orders of perturbation.

2. How do we know if a QED perturbation series converges?

The convergence of a QED perturbation series can be determined by looking at the ratio of successive terms in the series. If this ratio is less than 1, then the series is said to be convergent. However, this does not guarantee that the series will converge to the exact solution.

3. Why is the convergence of QED perturbation series important?

The convergence of QED perturbation series is important because it affects the accuracy of calculated values. If the series does not converge, then the calculated values may deviate significantly from the exact solution, leading to incorrect predictions and interpretations.

4. How can we improve the convergence of QED perturbation series?

There are several methods that can be used to improve the convergence of QED perturbation series. One method is to resum the series, which involves summing up certain terms in the series to improve its convergence. Another method is to use higher-order perturbation terms, which can lead to a more accurate approximation of the exact solution.

5. Are there any cases where QED perturbation series converges to the exact solution?

Yes, there are cases where QED perturbation series can converge to the exact solution. This is more likely to occur in simpler systems with fewer interacting particles. In these cases, the perturbation series can provide a good approximation of the exact solution.

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