Symmetry factors-Srednicki

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Taylor expansion correspond to different symmetry factors.I hope this summary helps clarify the concept of symmetry factors in Feynman diagrams. Just remember, they account for the overcounting of terms in the dual Taylor expansion and help us accurately calculate the amplitudes.
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kexue
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I am trying to understand Feynman diagrams and especially symmetry factors. I'm learning from Srednicki's QFT book page 60 to 63, or in the online version page 72 to 74.

Take formula 9.11 . (P-propagator, V-vertice, E-external source and E=2P-3V)

Srednicki says:
a, there are (2P)!/(2P-3V)! combinations that the 3V functional derivatives can act on the 2P sources

b, the number of terms for the RHS of 9.11 that result in one particular diagram is given by a -counting factor of 3! for each V
-a counting factor of V!
-a counting factor of 2! for each P
-a counting factor of P!

c, the counting factors cancel the number from the dual Taylor expansion in 9.11, but sometime we overcount the number of terms that give identical results, the factor by which we overcount is called symmetry factor

My questions
1. What is meant by statement b,? Do we multiply all these counting factors?
2. How do the (2P)!/(2P-3V)! combinations match the combinations that we got from drawing all the individual Feynam diagrams for a given E, P, V?
3. Do all the individual diagrams for a given E, P, V give the same number of possible terms for the RHS of 9.11? ( As by statement b, implied)
4. Why then different symmetry factors? How do they cancel the number from the dual Taylor expansion in 9.11?

My approach: -Pondering the above.
-I played for three hours with E=0, P=6, V=4 which corresponds to figure 9.2 and the diagrams showing there to see if something would make sense, but it did not.

Any help gratefully appreciated
 
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Hello,

I can understand your confusion with the concept of symmetry factors in Feynman diagrams. Let me try to explain it in a simpler way.

First, let's understand what a symmetry factor is. In the context of Feynman diagrams, a symmetry factor is a number that accounts for the overcounting of terms in the dual Taylor expansion of the amplitude. This overcounting arises due to the fact that in the dual Taylor expansion, we consider all possible ways of arranging the terms, whereas in the Feynman diagrams, some of these arrangements may give the same result.

Now, let's go through the statements in your post one by one.

1. Statement b says that the number of terms for the RHS of 9.11 (the dual Taylor expansion) that result in one particular diagram is given by multiplying the counting factors. So, yes, we do multiply all the counting factors.

2. The (2P)!/(2P-3V)! combinations that the 3V functional derivatives can act on the 2P sources correspond to all the possible ways of connecting the external sources to the internal vertices in a Feynman diagram. This is because each functional derivative corresponds to one external source, and there are 2P external sources and 3V vertices in total. So, these combinations match the combinations we get from drawing all the individual Feynman diagrams for a given E, P, V.

3. Yes, all the individual diagrams for a given E, P, V give the same number of possible terms for the RHS of 9.11. This is because each diagram corresponds to one particular way of arranging the terms in the dual Taylor expansion.

4. Different symmetry factors arise due to the different ways in which we can arrange the terms in the dual Taylor expansion. For example, consider a simple diagram with two vertices and two external sources. In the dual Taylor expansion, we can have two terms: one with the two vertices connected to the two external sources, and one with the two vertices connected to each other and the two external sources connected to each other. These two terms give the same result in the Feynman diagram, but they are counted separately in the dual Taylor expansion. The symmetry factor accounts for this overcounting and cancels out the extra terms. In your example of E=0, P=6, V=4, you can try drawing the diagrams and see how the different arrangements of the terms in
 

1. What are symmetry factors in the context of Srednicki's text?

Symmetry factors in Srednicki's text refer to the mathematical factors that account for the redundancy in the Feynman diagrams of quantum field theory. These factors are used to eliminate overcounting of diagrams when calculating scattering amplitudes.

2. How are symmetry factors calculated?

Symmetry factors are calculated by counting the number of ways a particular diagram can be rearranged without changing its overall structure. This number is then divided by the total number of equivalent diagrams, giving the symmetry factor for that particular diagram.

3. Why are symmetry factors important in quantum field theory?

Symmetry factors are important because they ensure that the calculated scattering amplitudes are correct. Without taking into account the redundancy of Feynman diagrams, the final result would be overcounted and incorrect. These factors also help to simplify complex calculations in quantum field theory.

4. Can symmetry factors be negative?

Yes, symmetry factors can be negative. This can occur when the diagrams being counted have a reflection or rotation symmetry, resulting in a negative factor. However, in the final calculation, these negative factors will cancel out with other positive factors.

5. Are symmetry factors used in other areas of physics?

Yes, symmetry factors are used in other areas of physics, such as statistical mechanics and condensed matter physics. In these fields, symmetry factors are used to calculate partition functions and to account for degeneracy in systems with multiple states.

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