# Non-Perturbative QFT: A Loophole?

I’ve been musing on yesterday’s post, and in particular a potential loophole in my argument. Recall that the whole shebang hinges on the fact that $e^{1/g^2}$ is smooth but not analytic. We were interpreting $g$ as the coupling constant in some theory, say QED. But hang about, surely we could just do a rescaling $A_\mu \mapsto A_\mu/g^2$ to remove this behaviour? After all the theory is invariant classically under such a transformation.

But it turns out that this kind of rescaling is anomalous in (most) quantum field theories. Recall that renormalization endows quantum field theories with a $\beta$ function, which determines the evolution of coupling constants as energy changes. The rescaling will only remain a symmetry if $\beta(g)$ is globally zero. Otherwise the rescaling only superficially eliminates the non-perturbative effect – it will reappear at different energies!

This raises a natural question: can you have instantons in finite quantum field theories? By definition these have $\beta$ function zero. Naively we might expect scale invariance to kill non-perturbative physics. A popular finite theory is $\mathcal{N}=4$ SYM, which crops up in AdS/CFT. A quick google suggests that my naive thinking is wrong. There are plenty of papers on instantons in this theory!

There must be a still deeper level to non-perturbative understanding. Sadly most physics papers gloss over the details. Let’s keep half an eye out for an explanation!