Gaining Insight into Scalar Field Theory Questions

[Kalimaa23]

Greetings,

I stumbled across two question that I have no idea on how to answer them.

1) The interaction term in a scalar field theory is $$-\frac{\lambda}{4!} \phi^4$$

Why should lambda be positive? (they say look at the energy of the ground state...)

2) Write down the Feynman rules for phi^4

I have no clue as to how you get the two intersecting lines that give is -i lambda vertex. I see where the propagator comes from though.

Any thougths?

 

[dextercioby]

For #2,the vertex is the 4 point OPI Green function,namely

$$(2\pi)^{4}\delta^{4}\left(p_{1}+p_{2}+p_{3}+p_{4}\right)\tilde{\Gamma}_{1}^{(4)}\left(p_{1},p_{2},p_{3},p_{4}\right) =-\lambda (2\pi)^{4}\delta^{4}\left(p_{1}+p_{2}+p_{3}+p_{4}\right)$$
$$\Longrightarrow \tilde{\Gamma}_{1}^{(4)}\left(p_{1},p_{2},p_{3},p_{4}\right) =i\cdot \ \mbox{vertex}$$

Consider the function $V(\phi)=\frac{m^{2}}{2}\phi^{2}+\frac{\lambda}{4!}\phi^{4}$.

Get its minimum & see what constraint needs to be imposed on $\lambda$ to ensure the minimum to be 0 or greater.

Daniel.

 

[Kalimaa23]

Great! Thanks a lot!

 

[dextercioby]

I hope you know how to get the other rules,using the connected Green functions in various orders of perturbation theory.

Daniel.