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Indeed, the QFT notion of locality is that observables at space-like separation commute, i.e. $$(x - y)^2 < 0 \implies [\mathcal{O}_1(x),\mathcal{O}_2(y)] = 0$$ for all local observables $\mathcal{O}_1,\mathcal{O}_2$, which are generically polynomials in the fields and their derivatives. This is our notion of locality because, classically, we know that ...