Why we know the renormalization work to all order in renormalizable theory?

[ndung200790]

Please teach me this:
In renormalizable theory,a finite divergent constants being absorbed by counterterms diagrams which are straight line counterterms and vertex counterterms.But I do not understand why in higher order of perturbation series the absorption still work.
Thank you very much in advanced.

 

[ndung200790]

At first time,e.g in Phi4 theory there are only two divergent diagrams which are two leg and four leg diagrams with three divergent constants.With two counterterm diagrams that are straight line counterterm diagram and vertex counterterm diagram,I thought that the ''absorption'' for renormalization working for all diagrams(so for all order in perturbation series.But after that I realized in higher order diagrams,there is the interplay of different couterterms.So I suspect the working of ''the absorption'' for renormalization for all order of diagrams.Because my simple thinking before is not still correct.

 

[Dickfore]

Actually, your condition that renormalization works is a definition of a renormalizable theory, so your question is a tautology.

 

[ndung200790]

It seem to me that is not tautology,because the definition of renormalizable theory is: Only a finite ber of amplitudes superficially diverge;however,divergences occur at all orders in perturbation theory.

 

[ndung200790]

Sorry,having some spell mistakes.The definition of renormalizable theory is: Only a finite number of amplitudes superficially diverge;however,divergences occur at all orders in perturbation theory.

 

[ndung200790]

At the moment,I have known that the answer for the question is the BPHZ(Bogoliubov-Parasiuk-Hepp-Zimermann) theorem.Any one could be pleasure to help me to give the proof of BPHZ theorem?

 

[A. Neumaier]

At the moment,I have known that the answer for the question is the BPHZ(Bogoliubov-Parasiuk-Hepp-Zimermann) theorem.Any one could be pleasure to help me to give the proof of BPHZ theorem?

try http://www.math.ru.nl/~landsman/CMP1.pdf