As much as I understand the renormalization group transformation and the concept of relevant/irrelevant operators, I'd say that if we push the reasoning of only looking at relevant operators when we follow a "renormalization group orbit" (a curve in the space of the parameters) to the infra-red regime the other way, so going to the U.V., we should keep all the operators this time (as they are no more suppressed by huge factors), or there is a infinite number of them, depending on an infinite number of the derivatives, and we end up with a non-local theory. As this reasoning can be applied to every QFT, does this mean that they are all non-local in the U.V.? or the reasoning fooled somewhere (breakdown of the QFT framework somewhere for example...)?
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Quite on the contrary, every genuine and genuinely consistent QFT is exactly local in the UV; it must converge to a scale-invariant theory, a fixed point. One may obtain nonlocalities in the IR for most theories (through the higher-derivative corrections with arbitrarily many derivatives) – if we compute the effective field theory description of the dynamics, either directly from the exact UV starting point, or otherwise. |
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