# 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?

Any relevant comments and references are beneficial. Thanks in advance.

## 1 Answer

Quantum field theory is a very abstract and complicated theory; one needs in general time to assimilate it, in particular as the degree of abstraction is much higher than that of 1-particle quantum mechanics. What is the bridge between QM and QFT? The basis of QFT is still a Hilbert space (called Fock space), but its states are no longer one-particle wave functions, but multi-particle states. That's a large leap in abstraction; however, even if the mathematical formalism in principle is the same, it does not look the same, though. Operators are defined on the Fock space as well as on the 1-particle Hilbert space. The Hamiltonian operator can be defined (for a non-interacting theory, for an interacting theory it's more complicated in general), as well a momentum operator etc. These operators can be expressed by the (most) basic operators of the Fock space: the creation and annihilation operators. If you don't feel familiar with these operators, study the 1-particle harmonic oscillator. In that context (1-particle Hilbert space) these operators are already defined, certainly with the "nasty" property of being non-diagonal in the usual Fock-space representation. BTW in case of the 1-particle harmonic oscillator, they are also non-diagonal. Once you get used to it, this turns out to be a rather nice property in fact.

Further abstraction provides the Heisenberg-picture in QFT, in contrast, to commonly used Schrodinger picture. But as you certainly know you can also swap from one picture to the other by a unitary transformation, and the physics does not change. This rule is valid in QM as well as in QFT. Probably you can also formulate QFT in the Schrodinger picture, but the formalism will become even more complicated than it is already, so better not to try it. So time-dependence is transferred to the operators. This is necessary to make the theory explicitly relativistic invariant: time and space should be treated on the same footing. This is certainly one of the biggest difference between QM and QFT. But time does not lose completely the special role it has in QM. Time also remains a parameter in QFT as well it is in QM. It is more the other way around: Whereas position is an operator in QM, in QFT it is downgraded to a parameter like time, as it should be. This is BTW well described in the QFT book of Srednicki.

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.

I think these are the main characteristics of how QFT is different from QM. Certainly, there is much more to say, but this can be read in QFT-books. I admit, most books advance in big steps towards Feynman diagrams and renormalization, so the basics are often treated quickly. One last advice: forget about wave-functions; they no longer exist in QFT. Field operators, often designated by the same symbol have nothing to do with wave-functions. The multi-particle states take over the role of the wave-functions. Read books or/and ask your professor. Most well-known sources are Peskin-Schroeder, Srednicki, (both are not so easy to read for a beginner), Zee's book QFT in a nutshell, L.H.Ryder QFT etc.