I have a question about renormalization in QFT. Why does a renormalizable theory require only a finite number of counterterms?

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    $\begingroup$ As Josh answered a theory is called renormalizable if it requires only a finite number of counter terms for canceling infinities. Initially it was thought that non-renormalizable theories are in some sense bad in that they have no predictive power. However nowadays such theories too are seen with respect and are studied as "effective field theories". But still most physicists believe that a fundamental theory of nature should have the property of renormalizability. Here by fundamental I mean a theory written in terms of fundamental entities i.e those which can not be broken into smaller ones. $\endgroup$ – user10001 Mar 13 '13 at 19:42
  • $\begingroup$ Classical Electrodynamics of a point charge only needs one counter-term to get rid of the mass correction, but the reminder $\dddot{r}$ is still unphysical - it leads to runaway solutions. In this sense CED is non renormalizable. It needs further "development" - replacing the unknown radiation reaction term $\dddot{r}$ with a known function of time $f(t)$. $\endgroup$ – Vladimir Kalitvianski Mar 14 '13 at 13:51

The necessity of a finite number of counterterms to cancel all divergences is often taken as the definition of renormalizability. See for example this page in Coleman's Aspects of Symmetry.

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    $\begingroup$ thank you very much for your answers, but my question was what is the reason or need for finite not infinite number $\endgroup$ – kara Mar 13 '13 at 23:04
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    $\begingroup$ Infinite number of counter-terms = infinite number of coupling constants you have to measure = infinite number of experiments you have to do before you can make exact predictions. You can still make predictions to some given order of accuracy if you can guarantee that all but a finite number of couplings are small and the higher dimensional couplings vanish at least as fast as dimensional analysis would imply. This is the sense in which non-renormalisable theories are used as "effective" field theories. $\endgroup$ – Michael Brown Mar 14 '13 at 0:27
  • $\begingroup$ @kara I see. In that case, I would rephrase the question as "Given that a renormalizable theory is defined as a theory requiring a finite number of counterterms to cancel all divergences, why are non-renormalizable theories not "good" some some sense." Then I would point to Michael Brown's nice comment directly above this one :) Also, you'll find this informative I think physics.stackexchange.com/questions/4184/… $\endgroup$ – joshphysics Mar 14 '13 at 2:10
  • $\begingroup$ so this means that the coupling constants have non-negative mass dimension then it leads to renomalizable theory $\endgroup$ – kara Mar 14 '13 at 12:43

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