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

petergreat
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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ansgar
have you tried work it out for yourself?

petergreat
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.

councilmage
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.