What Are the Feynman Rules for Momentum Space Calculations?

In summary, the conversation discusses using Feynman rules to write down a mathematical expression for a diagram representing a 2-point Green's function. The rules involve assigning momenta and using factors for external and internal lines and the vertex. The final expression needs to be multiplied by a combinatorial factor of 12, which represents the different ways the lines can be connected. It is recommended to draw out all possible diagrams and use resources such as the book "Quantum Field Theory and the Standard Model" for better understanding.
  • #1
jeebs
325
4
I'm trying to do the following problem:
feynmandiagram1.jpg


ie. I'm trying to use the Feynman rules for momentum space to write down the mathematical expression that the diagram is supposed to represent. However, I don't feel very confident in what I've managed so far.

Now as I understand it, these diagrams represent expressions that are equal to n-point Green's functions, where n is the number of external lines. So, looking at that diagram, I would think that the 2 straight lines coming off that single vertex are the external lines, so n=2.

So, according to the Feynman rules, I assign them each a momentum pk each one gets a factor of [itex]\frac{i}{p_k^2 - m^2 +iε}[/itex].

Also, another of the rules states that each internal line (which I take it means that single loop here) has a momentum kj and contributes a factor of [itex]∫\frac{d^4k_j}{2\pi^4}\frac{i}{k_j^2 - m^2 +iε}[/itex], and the vertex (the dot, right?) contributes a factor [itex] \frac{-iλ}{4!}2\pi\delta(\sum momenta)[/itex]. This gives:

[tex]G_2(p_1,p_2) = \frac{i}{p_1^2 - m^2 +iε}\frac{i}{p_2^2 - m^2 +iε}∫\frac{d^4k}{2\pi^4}\frac{i}{k^2 - m^2 +iε}\frac{-iλ}{4!}2\pi\delta(p1+p2) = -\frac{\lambda}{4!}\frac{\delta(p_1 + p_2)}{(p_1^2 - m^2 +iε)(p_2^2 - m^2 +iε)}∫\frac{d^4k}{k^2 - m^2 +iε} [/tex]
[tex] = -\frac{\lambda}{4!}\frac{1}{(p_1^2 - m^2 +iε)^2}∫\frac{d^4k}{k^2 - m^2 +iε}[/tex] since for the delta function to be equal to 1, we need p2 = -p1 (which ensures momentum is conserved at each vertex).

Now this looks OK to me so far, but apparently I am supposed to also multiply this by a "combinatorial factor", which I have heard is supposedly 12, but I do not know how to get this, or what its significance is.
Am I on the right lines here or what?

Also, if anyone knows any good resources on the net (or elsewhere) that could help me understand this stuff better, I would appreciate it - the notes I'm learning from aren't the greatest. Ideally I could do with seeing some example problems with solutions, if you happen to know where any might be found.

Thanks.
 
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  • #2
Yes, you are on the right lines. The combinatorial factor 12 comes from the fact that there are 6 ways to connect the two external lines to each other and two ways to connect them to the internal line. This means that there are 12 possible diagrams for this particular configuration. To understand this better, it might be useful to draw out all of the possible diagrams and calculate the corresponding expressions. That way, you can see how the combinatorial factor works in practice. As for resources, I would recommend the book "Quantum Field Theory and the Standard Model" by Matthew D. Schwartz. It has a lot of example problems with solutions which should help you get a better grasp of these concepts.
 

FAQ: What Are the Feynman Rules for Momentum Space Calculations?

1. What are Feynman rules in momentum space?

Feynman rules in momentum space are a set of mathematical rules used to calculate the probability amplitudes for particle interactions in quantum field theory. They are named after physicist Richard Feynman who developed them in the 1940s.

2. How are Feynman rules different from other methods of calculation?

Feynman rules are different from other methods of calculation because they use diagrams to represent particle interactions and involve the use of momentum space, instead of position space. This allows for a more intuitive understanding of the physical processes and simplifies calculations.

3. What are the steps for using Feynman rules in momentum space?

The steps for using Feynman rules in momentum space are as follows: 1) Identify the initial and final states of the particles involved in the interaction. 2) Draw the corresponding Feynman diagram. 3) Assign momentum labels to each line in the diagram. 4) Write down the Feynman rules for each vertex. 5) Apply conservation of energy and momentum at each vertex. 6) Simplify and calculate the probability amplitude.

4. What information can be obtained from using Feynman rules in momentum space?

By using Feynman rules in momentum space, one can calculate the probability amplitude for a specific particle interaction, including the possible outcomes and their associated probabilities. This can provide insight into the behavior and properties of particles and their interactions.

5. What are the limitations of using Feynman rules in momentum space?

One limitation of using Feynman rules in momentum space is that they are only applicable to certain types of interactions, specifically those involving elementary particles. They also do not take into account the effects of quantum corrections, which can be important in certain situations. Additionally, the calculations can become very complex for interactions involving more than a few particles.

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