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If ultraviolet divergences in Feynman diagrams involve arbitrarily short time periods, approaching infinity, then can a finitist approach to time (assuming, perhaps, a limit to the time lengths that would avoid these divergences) possibly help to justify renormalization? Is there a clear reason why this would or would not be plausible?

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I'm pretty sure a completely finitist theory would be far too weak to even describe the problem of renormalization. You'd disregard much of real number theory long before you speak of spinors taking values in $\mathbb C^n$. – NikolajK Jul 5 '14 at 17:03

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