The Klein-Gordon equation with a potential

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    Klein-gordon Potential
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Discussion Overview

The discussion revolves around the Klein-Gordon equation modified by a potential term, specifically of the form \(\delta V= \lambda \Psi^{*}\Psi\). Participants explore the implications of this potential in the context of quantum field theory, particularly regarding self-interactions and the interpretation of coupling constants.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question the nature of the term \(\delta V= \lambda \Psi^{*}\Psi\) as a potential, suggesting it may represent a non-linear interaction rather than a traditional potential.
  • There is a discussion about the meaning of the coupling constant \(\lambda\), with some suggesting it represents the strength of self-interactions in the field.
  • One participant points out that the term \(\lambda \Psi^{*}\Psi\) can be interpreted as a self-interaction, where both \(\Psi\) and \(\Psi^{*}\) refer to the same spacetime point.
  • Another participant provides an example of how perturbative expansions around small \(\lambda\) can be performed, although the absence of a source term \(J\) in the current context is noted as a point of confusion.
  • Some participants express uncertainty about how the term leads to interactions, contrasting it with more familiar forms of interaction terms in quantum electrodynamics (QED).
  • There are multiple interpretations of the interaction term, with some suggesting it should be expressed differently to align with known forms of self-interaction in quantum field theory.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the potential term or its implications for self-interactions. Multiple competing views and interpretations remain, particularly regarding the nature of the coupling constant and the form of the interaction term.

Contextual Notes

Some participants note limitations in their understanding of the relationship between the potential term and the Lagrangian formulation, as well as the absence of a source term in the current discussion.

  • #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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