Quartic terms as two particle interaction? In some lecture notes I found the sentence:

[...] considering two toy models: the classical and quantum harmonic oscillator with quartic perturbation. The quartic terms $\sim x^4$ and $\sim \hat x ^4$, respectively, correspond to the two-particle interaction in fermionic many-body models.

Does the author only refer to a formal similarity (as in second quantization interaction terms are quartic) or can a perturbated harmonic oscillator really represent an interacting system? If so, how can one see this?
 A: Quantum mechanics, e.g. the harmonic oscillator with
$$ H = \frac{p^2}{2m}+ \frac{kx^2}{2}$$
literally is a quantum field theory – Klein-Gordon theory. However, the observables, $x(t)$ and $p(t)$, only depend on one spacetime dimension, time, so it is a quantum field theory in $0+1$ dimensions. That's true even for a 3-dimensional harmonic oscillator with
$$ H = \frac {|\vec p|^2}{2m} + \frac{k|\vec x|^2}{2} $$
which still depend on $t$ only. If one doesn't see it, he should rename $x(t)$ as $\phi(t)$. It is nothing else than a spacetime field in a $0+1$-dimensional spacetime.
Because anharmonic (harmonic plus quartic perturbation) oscillator is literally a quantum field theory, we may also use pretty much all the tools of quantum field theory, including the Feynman diagrams. The quartic term still adds a vertex with 4 external legs.
So the claim that the anharmonic oscillator is a model of the "quartic-interacting Klein-Gordon" or another quantum field theory is not just a remote analogy. The quantum harmonic oscillator with a polynomial is exactly and rigorously a special case of an interacting quantum field theory. 
Quantum field theories in $3+1$ dimensions (or other higher dimensionalities) differ from  the anharmonic quantum oscillator only because the fields depend on several spacetime coordinates such as $x,y,z,t$, and that's true even after the Fourier transform – in the loops, we integrate over a higher-dimensional momentum space, for example. Quantum mechanics, due to its low spacetime dimensionality, also avoids ultraviolet divergences while it makes various infrared divergences in the Feynman diagrams etc. omnipresent. But the general mathematical laws of quantum field theory work in all these examples.
