Applying LSZ reduction - scattering particles - quantum theory

[binbagsss]

Homework Statement

Hiyah

I was initially confused about which way to choose the sign of the momentum , since it gives arise to different exponential ^ momentum combinations and thus different deltas for the momentum conservation . I came to conclusion that it doesn’t matter as long as you are consistent . ( which I found this I Be correct when I came across this https://physics.stackexchange.com/questions/452539/sign-ambiguity-when-going-from-position-to-momentum-space-evaluating-feynman-dia# ).

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

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

2) what system you choose to denote positive and negative 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 - eg right
2) + or - given by whether in to a vertex or out

My question is are my 1) and 2) which I talk about for consistency correct ? E.g is it even correct to consider you may denote plus or minus by 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 woulda 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 ‘ ?

Many thanks !

Relevant Equations

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