# Finding the interaction vertices

Given a Lagrange density $$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{m^2}{2}\phi^2 - \frac{\lambda_3}{3!}\phi^3 - \frac{\lambda_4}{4!}\phi^4$$ where $$\phi$$ is a scalar field, I want to find the interaction vertices.

I started by noting that the first two terms form the density for the Klein-Gordon field, so the last two terms form the interaction Lagrangian density we need. From this I am quite confused on how to proceed. I know it has to do with the $$S$$-matrix and its expansion. The first term of the expansion will be $$1$$, so we must look at the second term. This is given by $$S^{(1)} = -i \int d^4 x \mathcal{T}\left\{ :\frac{\lambda_3}{3!}\phi^3 + \frac{\lambda_4}{4!}\phi^4 : \right\}.$$ I am confused by this because the Time ordering operator and the normal operator are not linear. So how can I proceed from here?

I'm really new to QFT, so I definitely have not seen everything.

• Can you give a source for this expression? I am confused by the normal ordering in the interaction term, where does it come from at all? Commented Nov 28, 2022 at 18:25
• @Photon I'm using Mandl & Shaw's book on QFT, there they derive the following expression for the $S$-matrix $$S = \sum^{\infty}_{n=0} \frac{(-i)^n}{n!} \int \dots \int d^4x_1 d^4 x_2 \dots d^4 x_n \mathcal{T}\left\{H_I(x_1) \dots H_I(x_n) \right\}$$ Commented Nov 28, 2022 at 18:42