The Klein-Gordon equation with a potential

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    Klein-gordon Potential
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SUMMARY

The discussion centers on the Klein-Gordon equation with a potential represented as δV = λΨ*Ψ, where λ denotes the coupling constant. Participants clarify that this interaction term induces self-interactions within the wave function Ψ, which is crucial for understanding phenomena like the Higgs boson. The equation is analyzed in the context of perturbative expansions, with emphasis on how the potential influences particle interactions in quantum field theory. The correct form of the Klein-Gordon equation is established as (\frac{\partial^2 }{\partial t^2}-\nabla^2+m^2)Ψ = λΨ_{f'}^{*}Ψ_{i'} for three-particle interactions.

PREREQUISITES
  • Understanding of the Klein-Gordon equation and its applications in quantum field theory.
  • Familiarity with perturbative expansions and coupling constants in quantum mechanics.
  • Knowledge of Feynman diagrams and their role in particle interactions.
  • Basic concepts of scalar fields and their Lagrangian formulations.
NEXT STEPS
  • Study the derivation of the Klein-Gordon equation from the Lagrangian formalism.
  • Learn about perturbative techniques in quantum field theory, focusing on the role of coupling constants.
  • Explore Feynman diagrams and their interpretation in particle physics, particularly for scalar fields.
  • Investigate the implications of self-interaction terms in quantum field theories, especially in the context of the Higgs mechanism.
USEFUL FOR

This discussion is beneficial for students and researchers in theoretical physics, particularly those focusing on quantum field theory, particle physics, and the mathematical foundations of the Klein-Gordon equation.

  • #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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