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.)

 

[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.