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In QFT particles are considered point-like. If we integrate over all internal virtual momenta of a closed loop, in the case of a point-like particle, [because $\Delta x$ can be arbitrarily small, which isn't the case if particles have an extended structure, (by which I don't mean string-like particles), they can have infinite momenta], you have to integrate from 0 to infinity. So a cut-off momentum is introduced. If the moment can't get above a certain value because $\Delta x$ of the particle has a value greater than 0, and thus a momentum that's finite, though big because of the smallness of $\Delta x$, the integrals don't blow up either (the cut off momentum is the momentum associated with the finite $\Delta x$. Or do I have a wrong understanding of the renormalization procedure?

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    $\begingroup$ for details have a look, it is more complicated imperial.ac.uk/media/imperial-college/… $\endgroup$
    – anna v
    Commented Feb 7, 2017 at 18:28
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    $\begingroup$ In the 1940s before renormalization techniques were developed this possibility was considered. The technique was called "regularization." One of the propenents of this idea was Podolsky. The approach led to problems and was not pursued further after renormalization techniques were developed. $\endgroup$
    – Lewis Miller
    Commented Feb 8, 2017 at 1:29

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