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Why in QFT are $\phi^n$, where $n>2 $, terms in your lagrangian density interpreted as interaction terms?

so $\phi^4$ is considered a self-interaction term.

Similarly for two different fields $\phi,\chi$ one would say $\phi^2 \chi^2$ is interpreted as the interaction between the two fields.

But I do not understand why they are interpreted as such.

Qmechanic
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1 Answers1

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  • Let us introduce the notation that $S_n$ denotes action terms that depend on the $n$th power of the fields.

  • A quadratic action $S_2$ (and hence linear EL equations) corresponds to a free (=non-interacting) theory.

  • The quadratic part $S_2$ is related to the propagators, or equivalently, the lines in Feynman diagrams, cf. e.g. my Phys.SE answer here.

  • $S_{\geq 3}$ is the interaction part. The Feynman rule for $S_n$ with $n\geq3$ is an $n$-vertex. In this way $n$ fields interact, cf. OP's title question.

  • For more details, see e.g. this Phys.SE post.

Qmechanic
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