# Irreducible graphs and srednicki's book

• geoduck
In summary, Srednicki's QFT book uses scalar φ³ as an example of a QFT, where to make a calculation to all orders, you calculate all 1PI graphs for 2 external lines and the self-energy, then the 1PI graphs for 3 external lines and the 3-point vertex function, and finally n>3 vertex functions by drawing 1PI graphs using the 3-point vertex and exact propagators. The sum of all 1PI diagrams with 3 external lines includes loop corrections to internal propagators. The 2-point vertex function is not 1PI, but the 3-point vertex function is constructed from 1PI diagrams. The self-energy is the sum of 1PI
geoduck
Srednicki's QFT book uses scalar $$\phi^3$$ as an example of a QFT. To make a calculation to all orders, Srednicki claims (chpt 19) you calculate all 1P1 graphs for 2 external lines, giving the self-energy. He then says you calculate all the 1p1 graphs for 3 external lines, giving the 3-point vertex function. Then you calculate n>3 vertex functions by drawing 1p1 graphs, but using the 3-point vertex for vertices and exact propagators for propagators. Then any amplitude can be calculated by tree level graphs using these n-point vertices and exact propagators.

I think this is mostly correct, but doesn't Srednicki need to say that you have to calculate the 3-point vertex function using the exact propagator? If not, you'd be missing a lot of diagrams that can contribute to a scattering process.

geoduck said:
doesn't Srednicki need to say that you have to calculate the 3-point vertex function using the exact propagator?
No, because the sum of all 1PI diagrams with 3 external lines includes the loop corrections to the internal propagators.

1 person
Avodyne said:
No, because the sum of all 1PI diagrams with 3 external lines includes the loop corrections to the internal propagators.

I see. So in ø4, the sum of 1PI with 4 external lines already includes loop corrections to internal propagators. Or more generally, in øn, saying that the n-point vertex is the sum of 1PI diagrams already takes into account loop corrections to internal propagators.

I'm a little confused about the meaning of the vertex function and 1PI with regards to the 2-point function however. The self-energy is the 1PI diagram, but the 2-point vertex function is actually defined as the full 2-point Green's function, divided by the external lines (each represented by a full 2-point Green's function), so the renormalized 2-point vertex is:

1/ΔR ΔR 1/ΔR=1/ΔR =p2-mR2-∑R(p2R)

whereas if you just take

1/ΔRRRRΔR)1/ΔR=p2-mR2+∑R(p2R)

If you keep careful track of i's, then you get the same result, that the two expressions differ in the sign of the renormalized self energy ∑R.

It would seem to me that the former is not 1PI, but the latter is.

So is it safe to say that the 2-point vertex (the former expression) is not 1PI? Only the 3-point vertex you can say is constructed from 1PI? I mean, you took ΔR, which is not 1PI (only ∑R is), and you cut two of the ends, so everything in between remains the same, and everything in between wasn't 1PI.

Your 2nd expression has no meaning.

The self-energy ∑ is the sum of 1PI diagrams with two external lines (and no propagators for those lines). The exact propagator is then given by a geometric series that sums up to Δ as given by your first expression.

## 1. What are irreducible graphs?

Irreducible graphs are graphs that cannot be broken down into smaller subgraphs. In other words, all paths between any pair of vertices in an irreducible graph are connected and cannot be separated into smaller components.

## 2. Why are irreducible graphs important in physics?

Irreducible graphs play a crucial role in understanding the behavior of physical systems, particularly in quantum field theory. They represent the fundamental building blocks of interactions between particles and can be used to calculate properties such as scattering amplitudes.

## 3. What is Srednicki's book about?

Srednicki's book, "Quantum Field Theory", is a comprehensive introduction to the mathematical and conceptual foundations of quantum field theory. It covers topics such as symmetries, Feynman diagrams, and renormalization, and is widely used as a textbook for graduate courses in theoretical physics.

## 4. Is Srednicki's book suitable for beginners in quantum field theory?

While Srednicki's book is a popular choice for graduate-level courses in quantum field theory, it may not be the best option for beginners. The book assumes a certain level of mathematical and physical background knowledge, so it may be more challenging for those without prior experience in the field.

## 5. Are there any other resources that can supplement Srednicki's book?

Yes, there are many other resources available for those interested in learning about irreducible graphs and quantum field theory. Other textbooks, online lectures, and research papers can provide additional explanations and examples to complement Srednicki's book.

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