What would happen if the fine structure constant were set to 1?

Question

While reading up on magnetic monopoles, I have been led to understand that, due to S-duality, the magnetic equivalent of the fine-structure constant, $\alpha_M$ must be related to the reciprocal of $\alpha$--thus, much larger than 1. Thus, magnetic matter would be qualitatively different from electrically-bound atomic matter, due to magnetism being strongly coupled rather than weakly coupled such that perturbative calculations don't converge.

But suppose that $\alpha$ were set to 1 (by reducing $c$ to keep atomic physics as unperturbed as possible, as described in this answer). Is 1 a large enough value to screw up QFT calculations? And either way, am I correct in thinking that this would result in magnetic and electric charges being comparable in strength, such that, however matter ends up behaving in this hypothetical universe, magnetic and electric atoms would be similar?

Answer 1

The perturbative expansion in $\alpha$ won't converge very quickly at all if $\alpha=1$. Actually you'd expect every term in the expansion to be equally important. (In fact the perturbation series doesn't even converge at all -- the perturbative QFT expansion is really an asymptotic series in $\alpha$ -- so in principle even non-perturbative effects will be just as important as any given term in the perturbative expansion).

Someone may correct me if I'm wrong, but I think you also need to assume the existence of magnetic monopoles for $S$-duality to apply to electromagnetism coupled to charged matter. If magnetic monopoles exist, and if you somehow could dial $\alpha$ to a value near 1, then electric and magnetic monopoles would behave similarly, but both in some way difficult to describe with perturbation theory.