# Is the contribution of diagrams without vertices taken into account in calculating the process amplitude?

It seems I have finally decided on the question. When expanding the interaction process in a perturbation theory series, it is also necessary to take into account the contribution of the diagrams in which no interaction occurs? As in the first two diagrams in the figure.

• I think it depends on what you're trying to compute. Usually we calculate the S-matrix without the trivial part corresponding to no scattering, but of course for the "full" S-matrix they should be included (this point is sometimes glossed over) Jan 21, 2021 at 8:58

Consider the Dyson series for the S-matrix: $$S = \lim_{t\to\infty} U(-t, +t) = \mathcal{T}\exp\left(-i\int_{-\infty}^{\infty}\mathrm{d}^4 x \ \mathcal{H}(x)\right)$$ Expanding out the first few terms, we see that this is $$S = \mathcal{T}\left[\color{red}1-i\int \mathrm{d}^4x \ \mathcal{H}(x) + \frac{-i^2}{2!}\iint \mathrm{d}^4x \ \mathrm{d}^4y \ \mathcal{H}(x)\mathcal{H}(y)+\dots \right]$$

Usually, when people talk about computing the S-matrix, they are really talking about the transfer matrix $$T$$, in which the trivial $$1$$ part is discarded, corresponding to ignoring the processes in which there is no interaction (zero vertex diagrams). In fact, most people take it a step further and really talk about evaluating the "amplitude" $$\mathcal{M}_{fi}$$, which is defined as $$S_{fi} = \langle f|S|i\rangle = \delta_{fi}-iT_{fi} = \delta_{fi}-i(2\pi)^4\delta^4(p_f-p_i)\mathcal{M}_{fi}$$

For the "full" S-matrix defined by the Dyson formula though, the trivial diagrams must be included.

The answer is no, there is no meaning to the first two , because in the crossection for the interaction ,the term "interaction" means there is energy-momentum exchange. The Feynman diagrams are a pictorial representation of how that exchange happens, breaking that down to the perturbative terms. This link may help you

Each Feynman diagram represents a term in the perturbation theory expansion of the matrix element for an interaction.Normally, a full matrix element contains an infinite number of Feynman diagrams.

There is no term with no vertex.

• You have to start with free field theory. Only then can you turn on the interaction and read the corrections. And, of course, a diagram without interaction has its own amplitude, since ALL stories must be considered. It’s obvious to me that you’re not good at theory. Jan 21, 2021 at 7:41
• I gave you a link for my statements, where is yours? Free field theory describes nothing, it is mathematics, if it exists at all. Physics is about measurements and obervations. Jan 21, 2021 at 8:19
• physics.stackexchange.com/questions/606461/… Jan 21, 2021 at 9:08
• this answer is about non interacting, if there is no interaction there is no measurement to check , I can find no zero order here web.mit.edu/tabbott/www/papers/feynman-diagrams.pdf feynman diagram,. Here it talks of "unlinked diagrams" eduardo.physics.illinois.edu/phys582/… might enlighten you , if you can wade through it. Jan 21, 2021 at 9:42
• @anna v To calculate the total scattering matrix, it is also necessary to take into account the contribution of the non-interacting component. Jan 21, 2021 at 10:12