The Klein-Gordon equation with a potential

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The discussion revolves around the Klein-Gordon equation modified by a potential term of the form δV = λΨ*Ψ, which is linked to self-interactions in scalar field theory. Participants clarify that λ is a coupling constant representing the strength of the interaction, and the term Ψ*Ψ is not a conventional potential but rather a non-linear interaction term. The conversation highlights confusion regarding the nature of interactions depicted in Feynman diagrams and the role of perturbative expansions in quantum field theory. It is noted that the correct form of the equation for self-interaction should include terms that lead to three or four-particle interactions, depending on how the states are defined. Overall, the thread emphasizes the need for a deeper understanding of the mathematical framework underlying quantum field theory and its applications to particle interactions.
  • #31
Hey,

My professor says an equation of this form:
(\frac{\partial^2 }{\partial t^2}-\bigtriangledown^2+m^2)\Psi=\lambda\Psi_{f'}^{*} \Psi_{i'}\Psi
Will give a 4 particle interaction where I need to make the 'f'' state on the rhs an external (not sure what this means yet) and this useful for looking at the Higgs.

Whereas an equation of form:

(\frac{\partial^2 }{\partial t^2}-\bigtriangledown^2+m^2)\Psi=\lambda\Psi_{f'}^{*} \Psi_{i'}

Is a 3 particle interaction where they meet at a junction and some internal scalar particle is propagated.
 
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  • #32
sure, I don't have a problem with that. But I don't think that is a consequence of a potential of the form V=\lambda \Psi^{\dagger}\Psi. I think the potential must have a different form if the r.h.s of the equation looks like that.
 
  • #33
Well he included a delta sign next to the potential i.e. it was δV=λψ*ψ, are you supposing it should be δV=λψ*?
 

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