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 ) but I fail to link it to my intuitive model of waves superimposing nonlinearly. Let's calculate the 2-point function $\left&lt;\phi(x_1)\phi(x_2)\right&gt;$ 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 &quot;point&quot; 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?
We always say that tree levels are classical but loop diagrams are quantum.  Let's talk about a concrete example： $$\mathcal{L}=\partial_a \phi\partial^a \phi-\frac{g}{4}\phi^4+\phi J$$ where $J$ is source.   The equation of motion is $$\Box \phi=-g \phi^3+J$$  Let's do perturbation, $\phi=\sum \phi_{n}$ and $\phi_n \sim \mathcal{O}(g^n) $. And define Green function $G(x)$ as $$\Box G(x) =\delta^4(x)$$  Then   Zero order:  $\Box \phi_0 = J$   $\phi_0(x)=\int d^4y G(x-y) J(y) $  This solution corresponds to the following diagram:  First order:  $\Box \phi_1 = -g \phi_0^3   $  $\phi_1(x)=-g \int d^4x_1 d^4x_2 d^4x_3 d^4x_4 G(x-x_1)G(x_1-x_2)G(x_1-x_3)G(x_1-x_4)J(x_2)J(x_3)J(x_4) $  This solution corresponds to the following diagram:  Second order:  $\Box \phi_2 = -3g \phi_0^2\phi_1  $  $\phi_2(x)= 3g^2 \int d^4x_1 d^4x_2 d^4x_3 d^4x_4 d^4x_5 d^4yd^4z  G(x-y)G(y-x_1)G(y-x_2)G(y-z)G(z-x_3)G(z-x_4)G(z-x_5) J(x_1)J(x_2)J(x_3)J(x_4)J(x_5) $ This solution corresponds to the following diagram:  Therefore, we've proved in brute force that up to 2nd order, only tree level diagram make contribution.  However in principle the first order can have the loop diagram, such as  but it really does not occur in above classical calculation.  My question is:   What's the crucial point in classical calculation, which forbids the loop diagram to occur? Because the classical calculation seems similiar with quantum calculation. How to prove the general claim rigorously that loop diagram will not occur in above classical perturbative calculation.
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