Nonconvergence of Series
An argument by Freeman Dyson shows that the radius of convergence of the perturbation series in QED is zero. The basic argument goes as follows: if the coupling constant were negative, this would be equivalent to the Coulomb force constant being negative. This would "reverse" the electromagnetic interaction so that like charges would attract and unlike charges would repel. This would render the vacuum unstable against decay into a cluster of electrons on one side of the universe and a cluster of positrons on the other side of the universe. Because the theory is 'sick' for any negative value of the coupling constant, the series do not converge, but are an asymptotic series. This can be taken as a need for a new theory, a problem with perturbation theory, or ignored by taking a "shut-up-and-calculate" approach.
Read more about this topic: Quantum Electrodynamics
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