# Nonlinear superposition and self-interaction in classical field theory [duplicate]

I am learning QFT (in a path integral formalism) and one thing I'm struggling with is that self-interaction is supposed to be a quantum phenomenon, not apparent in classical non-linear field theory. I have read and understood how a perturbative construction of nonlinear field solutions give rise to tree-level (-only) diagrams (compare Tree level QFT and classical fields/particles) but I fail to link it to my intuitive model of waves superimposing nonlinearly.

Let's calculate the 2-point function $$\left<\phi(x_1)\phi(x_2)\right>$$ of a classical theory. In a free theory, you'd calculate the fundamental solution to the field equation and evaluate it at $$x_2$$. That is, you apply a $$\delta$$-shaped source term at $$x_1$$ and calculate how the field evolves starting from there.

Going to a nonlinear theory, the prementioned construction no longer yields a fundamental solution as there is no superposition principle. Still you can can calculate how the field evolves starting from a point source (where I mean "point" both in space and time) and that should yield the 2 point correlator, no?

Now, my mental model is the following: the point source produces have a spherical wavefront that (freely) propagates for a bit. Now Huygens' principle kicks in and every point of the wavefront is again the origin of a new spherical wavefront. Only now we have no linear, but nonlinear superposition, i.e. interaction. Now when these different branches of the wave come together again, that is exactly self-interaction, no? and seems to produce exactly the processes that are depicted by loop diagrams?

The integral $$-\lambda^2 \int\mathrm{d}^4\!y_1 \mathrm{d}^4\!y_2 \; G(x_1, y_1) G^3(y_1, y_2) G(y_2,x_2)$$ reads just like two elemental nonlinear superpositions, to me.

How does nonlinear superposition (not) relate to self-interaction?

Speaking in words of perturbation theory, how does nonlinear superposition of the different branches of a classical wave not lead to loop diagrams? What makes them appear in QFT then?

How does the full propagator of nonlinear field theory look like? If it does not differ from the free one, why not?

• Commented Jun 6, 2021 at 16:26
• Hey, thanks for the quick reply. I've read the former and the construction makes very much sense to me. Still, it doesn't explicitly explain why loop diagrams do not occur. It rather seems like the construction they're doing, is different to my one. It explains how the full solution can be recovered from all n-point correlation functions, but not why e.g. the full two-point function wouldn't differ from the free propagator. I'll definitely look into the other paper :) Commented Jun 6, 2021 at 16:36
• An $n$-loop diagram is proportional to $\hbar^n$, i.e. it only appears in a quantum theory. Possible duplicate: Use my example to explain why loop diagram will not occur in classical equation of motion? Commented Jun 6, 2021 at 17:03
• Just like the above mentioned lecture notes, your answer to the SE question you posted shows how tree-level diagrams appear (and are enough) in that specific construction (which is just what perturbation theory is, ok). That does not help me understand how nonlinear superposition does not correspond to loop-diagrams. Commented Jun 6, 2021 at 17:57
• To reopen this post (v5) consider to make your mental/intuitive model more concrete so it can be assessed how it differs from the duplicate. Commented Jun 14, 2021 at 11:50