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I've been reading Tony Zee's "Quantum Field Theory" and I'm really enjoying it. However, on p. 45 I came across what I think is an inconsistency. The sentences in question are:

The rules go something like this:

(1) diagrams are made of lines and vertices at which four lines meet;

...

(3) for each line assign 1/m2;

Here's the thing: if a vertex is a point at which four lines meet, then most of his figures are wrong. Here are a few examples:

  • Fig. 1.7.1b is listed as having four lines and one vertex, but if a vertex is a point where four lines meet then it has five lines.

  • Fig. 1.7.1c is listed as having four lines and one vertex, but if a vertex is a point where four lines meet then it has six lines.

  • Fig. 1.7.2b is listed as having six lines and two vertices, but if a vertex is a point where four lines meet then it has seven lines.

  • Fig. 1.7.2c is listed as having six lines and two vertices, but if a vertex is a point where four lines meet then it has eight lines.

  • Fig. 1.7.3b is listed as having seven lines and two vertices, but if a vertex is a point where four lines meet then it has eight lines.

The problems I noted disappear if you consider the loops I examined to count as one line when assigning the mass terms. However, this is not what Zee says. Is he wrong or am I missing something?

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1 Answer

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Feynman diagrams are graphs, in the sense of graph theory. The concepts of an edge (line) and a vertex are given precise definitions in graph theory, such that an edge is a link which has a vertex at each end. One line in the diagram always corresponds to one edge, even if the vertex at one end of the line happens to be the same as the vertex at the other end. The point is, a loop counts as one line. If Zee says something to the contrary, it's incorrect (although I would find it quite surprising if he made such a basic error).

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Thanks for the feedback and the edit. However, I don't understand why you write "If Zee says something to the contrary ...". In my question I already quoted what Zee says: "vertices at which four lines meet". – jackj Dec 21 '11 at 6:35
Yes, but that doesn't imply that four different lines meet at a vertex. I can see how it is a little unclear, though. If you want to be precise, you can say that a vertex is where four endpoints meet; those four endpoints may, but don't necessarily have to, belong to four different edges (lines). It is possible that two different endpoints of the same edge are "attached" to the same vertex. – David Zaslavsky♦ Dec 21 '11 at 7:01
Thanks again for the clarifications. I don't really agree, however, that point (1) "doesn't imply that four different lines meet". If we took that approach then point (3) might possibly also not imply "each different line" and then all hell breaks loose: the rules would become useless, since he hasn't explained how to interpret them. Even so, in your answer and comment you reinforced my impression that loops count as one line, so I'm going to go ahead and accept your answer. – jackj Dec 21 '11 at 16:56
Thanks... it's really a language issue at play here, namely that words are imprecise. The thing is, when reading about physics (or other technical subjects), what matters is not so much what the reader thinks the words imply, but rather what the author meant them to imply. This sort of thing is a common source of confusion in technical writing. Bottom line, though, loops do count as one line. – David Zaslavsky♦ Dec 21 '11 at 22:33

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