What's the classical picture of phi^4 scalar field theory?

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Discussion Overview

The discussion centers on the classical picture of phi^4 scalar field theory, particularly regarding the nature of interactions between particles—whether they attract or repel each other. Participants explore theoretical implications and mathematical interpretations related to this field theory.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant notes a lack of discussion on classical phi^4 theory in standard field theory texts, prompting curiosity about particle interactions.
  • Another participant suggests that particles may repel each other due to the phi^4 potential, which grows faster than phi^2, making large amplitudes energetically unfavorable.
  • A different viewpoint proposes that the nature of the interaction (attractive or repulsive) depends on the sign of the delta potential derived from the scattering amplitude, which in turn is influenced by the coefficient of phi^4 in the Lagrangian.
  • One participant encourages others to work through the problem independently, implying that personal exploration may yield insights.

Areas of Agreement / Disagreement

The discussion reflects multiple competing views on the nature of interactions in phi^4 theory, with no consensus reached on whether particles attract or repel each other.

Contextual Notes

Participants express uncertainty regarding the mathematical details and implications of the phi^4 potential, as well as the dependence on specific parameters in the Lagrangian.

petergreat
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I know that the classical picture of QED is Coulomb interaction, magnetic interaction etc. But what does the classical phi^4 theory look like? In particular, do particles attract or repel each other in this theory?

P.S. I'm surprised that my field theory books never discuss this. (At least in Peskin & Schroder's introductory chapter on scalar field theory.)
 
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have you tried work it out for yourself?
 
My intuition is that particles will repel each other to separate apart, because clumping together and forming large amplitudes is energetically unfavorable due to the phi^4 potential which grows faster than phi^2. Is this correct? I'll try to do some math when I have time.
 
For \phi\phi \rightarrow \phi\phi scattering, I think the scattering amplitude translates into a delta potential between the two mesons.

Whether the interaction is attractive or repulsive will depend on the sign of the delta potential which then depends on the coefficient of \phi^4 in the Lagrangian.



I'm pretty sure that I saw this somewhere in a (non-advanced) field theory text. But correct me if what I said was wrong.
 

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