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Phi^4 theory

  1. Jun 12, 2005 #1
    Greetings,

    I stumbled across two question that I have no idea on how to answer them.

    1) The interaction term in a scalar field theory is [tex]-\frac{\lambda}{4!} \phi^4[/tex]

    Why should lambda be positive? (they say look at the energy of the ground state...)

    2) Write down the Feynman rules for phi^4

    I have no clue as to how you get the two intersecting lines that give is -i lambda vertex. I see where the propagator comes from though.

    Any thougths?
     
  2. jcsd
  3. Jun 12, 2005 #2

    dextercioby

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    For #2,the vertex is the 4 point OPI Green function,namely

    [tex] (2\pi)^{4}\delta^{4}\left(p_{1}+p_{2}+p_{3}+p_{4}\right)\tilde{\Gamma}_{1}^{(4)}\left(p_{1},p_{2},p_{3},p_{4}\right) =-\lambda (2\pi)^{4}\delta^{4}\left(p_{1}+p_{2}+p_{3}+p_{4}\right)[/tex]
    [tex]\Longrightarrow \tilde{\Gamma}_{1}^{(4)}\left(p_{1},p_{2},p_{3},p_{4}\right) =i\cdot \ \mbox{vertex} [/tex]

    Consider the function [itex] V(\phi)=\frac{m^{2}}{2}\phi^{2}+\frac{\lambda}{4!}\phi^{4} [/itex].

    Get its minimum & see what constraint needs to be imposed on [itex]\lambda [/itex] to ensure the minimum to be 0 or greater.

    Daniel.
     
    Last edited: Jun 12, 2005
  4. Jun 12, 2005 #3
    Great! Thanks alot!
     
  5. Jun 12, 2005 #4

    dextercioby

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    I hope you know how to get the other rules,using the connected Green functions in various orders of perturbation theory.

    Daniel.
     
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