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!


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