Quantum Field Theory – Definition and Proof of Symmetry Factor of Feynman Diagram

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Studying QFT, I was told that symmetry factor is defined by:

if there are $m$ ways of arranging vertices and propagators to give identical parts of a diagram (keeping the outer ends of external lines fixed and without cutting propagator lines) we get a factor $D_i = m$. The symmetry factor is given by the product of all symmetry factors
$D =\Pi_i D_i.$

and then, when using normalized pertubation, the factor of contribution of some diagram (which include all the combinatoricall ways to create this diagram both in the numerator and denominator of the term:

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and the normalization) is just given by $1/D$. (below I will refer to this as "The Statement".)

I found in Wikipedia this definition for symmetry factor:

The symmetry factor theorem gives the symmetry factor for a general diagram: the contribution of each Feynman diagram must be divided by the order of its group of automorphisms, the number of symmetries that it has.

An automorphism of a Feynman graph is a permutation $M$ of the lines and a permutation $N$ of the vertices with the following properties:

  1. If a line $l$ goes from vertex $v$ to vertex $v′$, then $M(l)$ goes from $N(v)$ to $N(v′)$.
  2. If the line is undirected, as it is for a real scalar field,
    then M(l) can go from N(v′) to N(v) too.
  3. If a line l ends on an
    external line, M(l) ends on the same external line.
  4. If there are different types of lines, M(l) should preserve the type.

My questions are:

  1. Are those definitions equivalent?

  2. How do we prove The Statement I mentioned above?

Best Answer

Regarding #2, due to the indistinguishability, each amplitude (or diagram) must have the factor of permutation $1/D$ when superpositioning. A Feynman diagram is a representation of the equivalence class of an interaction, which means that there are permutations of that interaction behind it. Counting $D$ is a combinatorial matter and out of the scope of the question.

Regarding #1, the first 'definition' is just a dumbed-down version of graph isomorphism, which Wikipedia's stating.

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