# Differences between perturbation theory in quantum mechanics, quantum field theory and condensed matter physics?

For perturbation theory method, are there any difference or links in basic quantum mechanics, quantum field theory and in condensed matter physics?

I know the meaning in basic quantum mechanics (focus on single particle problem, especially), for example, considering the following partition for total Hamiltonian $H$ $$H= H_0 + \mu H_1$$ if one knows the solution for $H_0$ and then one can derive the approximate solution (to evaluate the energy correction, wavefunction correction) of total Hamiltonian $H$ based on the solution of $H_0$ when $\mu$ is a very small parameter.I think this must be the original idea developed to attack the real physical problems in quantum mechanics.

But how this method can be applied in quantum field theory (considering relativistic effect) and also survives in condensed matter physics (many-body), both based on Green's function rather than wavefunction? What's the missing history? How to bridge this gap?

Last but not least, the equations to solve are usually non-linear. That means, in general, they cannot be solved. Actually, this is nothing really new: also in QM, this happens (but less often). And as you already mentioned, in order to progress perturbation theory is used. The Hamiltonian is split into a non-interacting part and an interacting part. This is crucially based on the existence of a small parameter (call it $\mu$ or $\alpha$ or $\lambda$ etc.). The principle is the same in QM as well as in QFT. The perturbation theory is then based on the development of the evolution operator $|\Psi(t)> = U(t,t_0)|\Psi(t_0)>$. The evolution operator can be written in a series called Dyson-series with the solution (see for instance the Wikipedia page on the subject "Dyson-series")
$$U(t,t_0) = T \exp(-i \int_{t_0}^{t} d\tau V(\tau))$$
where $V$ is the small interaction part of the Hamilton operator $H=H_0+V$ and $T$ is the time-ordering operator (for more see Wikipedia or QFT textbooks). Each term of this series can be represented by a Feynman-diagram. I think the Dyson-series can already be used in non-relativistic QM. However, the interaction operator $V(\tau)$ is rather different from the one of familiar QM. In QM it describes in most cases an external field or some "wave-function-different" operator; in QFT the mutual interaction between the field in consideration and the interaction, typically another field, is taken into account. In this way you get the Green's functions in, in particular, if self-interaction $V=\frac{\lambda}{4!}\phi^4$ of the field $\phi$ is considered.