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

In summary, the conversation discusses the concept of renormalization in renormalizable theory, where finite divergent constants are absorbed by counterterm diagrams. The question is raised as to why this absorption still works in higher orders of the perturbation series. The discussion also touches on the definition of a renormalizable theory and the BPHZ theorem, with the request for help in understanding and proving the theorem.
  • #1
ndung200790
519
0
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.
 
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  • #2
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.
 
  • #3
Actually, your condition that renormalization works is a definition of a renormalizable theory, so your question is a tautology.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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?
 
  • #7
ndung200790 said:
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
 

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

1. What is renormalization in a renormalizable theory?

Renormalization is a process in theoretical physics that involves removing divergences in a theory by introducing new parameters with the same units as the original parameters. In a renormalizable theory, this process can be done to all orders, meaning that the theory can be made finite for any desired accuracy.

2. Why is renormalization necessary in a renormalizable theory?

Renormalization is necessary in a renormalizable theory because without it, the theory would produce infinite results. This is due to quantum fluctuations and interactions that cause the theory's parameters to behave differently at different energy scales. Renormalization allows us to take into account these fluctuations and produce finite results.

3. How does renormalization work to all orders in a renormalizable theory?

Renormalization works to all orders in a renormalizable theory by iteratively removing divergences at each order of perturbation theory. This is done by introducing counterterms, which are new parameters that absorb the divergences and allow the theory to produce finite results. This process can be repeated to any desired order, making the theory finite to all orders.

4. Can renormalization fail in a renormalizable theory?

Yes, renormalization can fail in a renormalizable theory if the theory is non-renormalizable. This means that the theory has infinitely many divergences that cannot be removed even with an infinite number of counterterms. However, in a renormalizable theory, renormalization can always be done to all orders, making the theory finite.

5. What are the implications of renormalization in a renormalizable theory?

The implications of renormalization in a renormalizable theory are significant. It allows us to make precise predictions and calculations in quantum field theories, which are essential for understanding the fundamental forces of nature. Without renormalization, these theories would produce nonsensical and infinite results, making them useless in describing the physical world.

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