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I was initially confused about which way to choose the sign of the momentum, since it gives rise to different exponential momentum combinations and thus different deltas for momentum conservation. I came to the conclusion that it doesn’t matter as long as you are consistent, which I found to be correct when I came across Sign ambiguity when going from position to momentum space evaluating Feynman diagrams.

My question pretty much is about the definition of consistency here. There are two things to consider:

  1. having some consistent pattern to decide which external vertices you point toward a vertex, and which away from a vertex

  2. what system you choose to denote positive and negative relative to, i.e right and left, up and down, or into a vertex and out of a vertex.

My choice I like to use is: 1. all point in the same direction - e.g. right 2. + or - given by whether in to a vertex or out

Are the choices 1) and 2) which I talk about sufficient to obtain consistency? Is it even correct to referring to what you may denote by plus or minus using left / right or should it always be into and out of a vertex?

Regarding number 1, I feel as though the easier option is to label everything in same direction, but say I am considering variations of a diagram by keeping the external points fixed (1,2,3,4). I believe another way of consistency would be to always associate the same direction with the same external point - these could be all the same, three same, one not, two-two - it does not matter. Am I correct in thinking this is a valid move?

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  • $\begingroup$ Hi, I edited your post rather heavily, but I still can't quite tell what "the momentum" in your first sentence refers to. Are you talking about momentum vectors in the Feynman diagram? What do you mean by "right and left" for such a diagram, given that the diagram really are just graphs and have no axes? $\endgroup$
    – ACuriousMind
    May 10, 2019 at 16:39

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If I understood correctly your question, I think the two choices are equivalent. In both ways you are able to identify which lines will have the negative momentum. It is more a matter of words saying that a momentum line is ingoing with negative momentum or outgoing with positive momentum.

Often it is simpler to forget about the distinction between ingoing and outgoing and just fix the sign of the momenta using the conservation at the vertex. This is especially handy when you want to study a diagram independently on which of the external legs are taken to be initial states and which are considered final states.

However it seems that in the beginning you were referring to the sign of the phases in the fields you write down. Here the consistency follows from the conventions in the fields definition: you always associate to creation operators the same sign of the momentum in the exponential, opposite to annihilation operators. For different fields you could still change convention but every time the same field appears in your perturbation theory it is important that it is written following the same convention in all the instances.

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