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Feynman diagrams and relativistic propagators

  1. Nov 26, 2012 #1
    Hey again,

    I have a question on a couple of things related to feynman diagrams but also the relativistic scalar propagator term.

    First of all, this interaction:

    feynman2.png

    The cross represents a self-interaction via the mass and characterised by the term: -im^2, is this just some initial state then self-interacting with itself via the mass, with nothing changing and it entering into a final state the same as the initial state? Can anybody explain what exactly is happening in this interaction?

    My second question is on the propagator for a relativisitc scalar particle, I believe it has form:

    [tex]\frac{i}{E^{2}-\mathbf{p}^{2}-m^{2}}[/tex]

    My professor said that this is where this form comes from, he said to imagine summing up all the possible number of self-interaction from 0 self interactions to (presumably) and infinite number of interactions, so :

    Self_Interaction_Sum.png

    So the first line has no mass interaction so m=0 and the factor for the first one is :

    [tex]\frac{i}{E^{2}-\mathbf{p}^{2}}[/tex]

    then the second line has one mass interaction so the factor associated with it is :

    [tex]\frac{i}{E^{2}-\mathbf{p}^{2}}(-im^{2})\frac{i}{E^{2}-\mathbf{p}^{2}}[/tex]

    and the third line has 2 mass interactions and so the factor is :

    [tex](\frac{i}{E^{2}-\mathbf{p}^{2}})^3(-im^{2})^2[/tex]

    And so we sum all these factors up (to the maximum number of self-interactions) and can make factorisation below:

    [tex]\frac{i}{E^{2}-\mathbf{p}^{2}}(1+\frac{m^{2}}{E^{2}-\mathbf{p}^{2}}+\frac{m^{4}}{(E^{2}-\mathbf{p}^{2})^2}+\cdots )[/tex]

    We use identity:

    [tex](1+x)^{-1}=1+x+x^{2}+x^{3}+\cdots\: ,\: x=\frac{m^{2}}{E^{2}-\mathbf{p}^{2}}[/tex]

    and thus obtain the relativistic scalar propagator:

    [tex]\frac{i}{E^{2}-\mathbf{p}^{2}-m^{2}}[/tex]

    How does this work? Is the scalar interaction just the sum of all the self-interactions by mass?

    Thanks,
    SK
     
  2. jcsd
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