Question

I'm trying to understand a statement of my teacher of TQFT: he said that when expanding the effective action in terms of proper vertices, we can choose a new theory with only tree diagrams in order to eliminate divergences since the diagrams of Γ

are the same of the original action as a property of Legendre transformation. Then, in order to find those proper vertices he presented an alternative way to renormalize the theory (that he said comes from Bogoliubov but I didn't find it) in which we don't need to introduce renormalization constants but we just impose renormalization conditions. I don't understand this procedure: do the renormalization conditions fix the proper vertices at tree level? And if so, why? Could you explain this procedure better to me?

Answer 1

In TQFT, expanding the effective action in terms of proper vertices and choosing a new theory with only tree diagrams can eliminate divergences. Renormalization conditions can fix the proper vertices at tree level to ensure meaningful physical predictions.

Step-by-step explanation:

In the study of Topological Quantum Field Theory (TQFT), it is often necessary to expand the effective action in terms of proper vertices. To eliminate divergences, one approach is to choose a new theory with only tree diagrams, since the diagrams for the new theory and the original action are the same as a property of Legendre transformation.

In order to find the proper vertices, your teacher presented an alternative way to renormalize the theory, which comes from Bogoliubov's method. This method allows you to impose renormalization conditions instead of introducing renormalization constants. The renormalization conditions can fix the proper vertices at tree level, which means that they determine the values of the vertices in the new theory.

By imposing appropriate renormalization conditions, the divergences in the theory can be eliminated and meaningful physical predictions can be made.