Developing the Feynman diagram expansion

In summary, in section 7.2 of "Introduction to Many Body Physics" by Piers Coleman, the author introduces Feynman diagrams by first transforming the generating functional into a pictorial form. He then proceeds to calculate the n=1, m=1 term, followed by the n=1, m=2 term. The calculation of the n=1, m=2 term involves a simple application of the chain rule, as explained by the author. The conversation also includes a quote from Kipling's "Gunga Din" and the admission that the speaker is rusty in their understanding of these concepts.
  • #1
ShayanJ
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Trying to figure out Feynman rules from expansion of the generating functional.
I'm reading "introduction to many body physics" by Piers Coleman. In section 7.2 he's trying to introduce Feynman diagrams by expanding the generating functional. But first he transforms it into this pictorial form:
16005897616301785018040.jpg

Then he calculates the n=1, m=1 term like below:
1600589855368-1948804519.jpg

Which I understand. But I have no idea how he calculates the n=1, m=2 term:
1600590043927788924167.jpg


Can anybody help?
Thanks

PS
d1 and 1 mean ##dt_1 dx_1## and ##(t_1,x_1)##
 
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  • #2
I figured it out. It was a simple application of the chain rule. I'm really rusty!
$$ \int d1 U(1) \frac{\delta^2}{\delta\alpha \delta\bar \alpha}\left( \int dX dY \bar\alpha(X) G(X-Y) \alpha(Y)\right)^2 = \\ 2 \int d1 U(1) \frac{\delta}{\delta\alpha} \left[\left( \int dX dY \bar\alpha(X) G(X-Y) \alpha(Y)\right) \left( \int dY G(1-Y) \alpha(Y)\right)\right] $$
And so on and so fourth!
 
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  • #3
As Kipling wrote "You're a better man than I, Gunga Din!" :smile: I like that you wrote it out and explained what you found. I am extremely rusty. The chain rule is something I need to read up on more, I think it would help me sort of get the grasp of many of the equations, especially with expansions, etc.
 

FAQ: Developing the Feynman diagram expansion

1. What is the Feynman diagram expansion?

The Feynman diagram expansion is a mathematical tool used in quantum field theory to calculate the probability of particle interactions. It was developed by Nobel Prize-winning physicist Richard Feynman in the 1940s.

2. How does the Feynman diagram expansion work?

The Feynman diagram expansion works by representing particle interactions as a series of diagrams, with each diagram representing a possible outcome of the interaction. These diagrams are then used to calculate the probability of each outcome, which is then combined to give the overall probability of the interaction.

3. What are the advantages of using the Feynman diagram expansion?

One of the main advantages of the Feynman diagram expansion is its ability to simplify complex calculations in quantum field theory. It allows for a visual representation of particle interactions, making it easier to understand and calculate the probabilities involved.

4. Are there any limitations to the Feynman diagram expansion?

While the Feynman diagram expansion is a powerful tool, it does have some limitations. It is most effective for calculating interactions involving a small number of particles, and becomes increasingly difficult to use as the number of particles involved increases.

5. How has the Feynman diagram expansion impacted the field of physics?

The Feynman diagram expansion has had a significant impact on the field of physics, particularly in the study of quantum field theory. It has allowed for more accurate calculations of particle interactions, leading to a better understanding of the fundamental forces and particles in the universe.

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