Tree level QFT and classical fields/particles

It is well known that scattering cross-sections computed at tree level correspond to cross-sections in the classical theory. For example the tree level cross-section for electron-electron scattering in QED corresponds to scattering of classical point charges. The naive explanation for this is that the power of $\hbar$ in a term of the perturbative expansion is the number of loops in the diagram.

However, it is not clear to me how to state this correspondence in general. In the above example the classical theory regards electrons as particles and photons as a field. This seems arbitrary. Moreover, if we consider for example $\phi^4$ theory than the interaction of the $phi$-quanta is mediated by nothing except the $phi$-field itself. What is the corresponding classical theory? Does it contain both $phi$-particles and a $phi$-field?

Also, does this correspondence extend to anything besides scattering theory?

Summing up, my question is:

What is the precise statement of the correspondence between tree-level QFT and behavior of classical fields and particles?

Answer

This was something that confused me for awhile as well until I found this great set of notes: homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf

Let me just briefly summarize what's in there.

The free Klein-Gordon field satisfies the field equation

$$(\partial_{\mu} \partial^{\mu} +m^2) \phi(x) = 0$$ the most general solution to this equation is $$\phi(t, \vec{x}) = \int_{-\infty}^{\infty} \frac{d^3k}{(2\pi)^3} \; \frac{1}{2E_{\vec{k}}} \left( a(\vec{k}) e^{- i( E_{\vec{k}} t -\vec{k} \cdot \vec{x})} + a^{*}(\vec{k}) e^{ i (E_{\vec{k}} t- \vec{k} \cdot \vec{x})} \right)$$ where $$\frac{a(\vec{k}) + a^{*}(-\vec{k})}{2E_{\vec{k}}} = \int_{-\infty}^{\infty} d^3x \; \phi(0,\vec{x}) e^{-i \vec{k} \cdot \vec{x}}$$ and $$\frac{a(\vec{k}) - a^{*}(-\vec{k})}{2i} = \int_{-\infty}^{\infty} d^3x \; \dot{\phi}(0,\vec{x}) e^{-i \vec{k} \cdot \vec{x}}$$

Introducing an interaction potential into the Lagrangian results in the field equation

$$(\partial^{\mu} \partial_{\mu} + m^2) \phi = -V'(\phi)$$

choosing a phi-4 theory $V(\phi) = \frac{g}{4} \phi^4$ this results in

$$(\partial^{\mu} \partial_{\mu} + m^2) \phi = -g \phi^3$$

Introduce a Green's function for the operator

$$(\partial^{\mu} \partial_{\mu} + m^2) G(x) = -\delta(x)$$

which is given by

$$G(x) = \int \frac{d^4k}{(2\pi)^4} \; \frac{-e^{-i k \cdot x}}{-k^2 + m^2}$$

now solve the full theory perturabtively by substituting

$$\phi(x) = \sum_{n} g^n \phi_{n}(x)$$

into the differential equation and identifying powers of $g$ to get the following equations

$$(\partial^{\mu} \partial_{\mu} + m^2) \phi_0 (x) = 0$$

$$(\partial^{\mu} \partial_{\mu} + m^2) \phi_1(x) = -\phi_0(x)^3$$

$$(\partial^{\mu} \partial_{\mu} + m^2) \phi_2 (x) = -3 \phi_0(x)^2 \phi_1(x)$$

the first equation is just the free field equation which has the general solution above. The rest are then solved recursively using $\phi_0(x)$. So the solution for $\phi_1$ is

$$\phi_1(x) = \int d^4y\; \phi_0(y)^3 \, G(x-y)$$

and so on. As is shown in the notes this perturbative expansion generates all no-loop Feynman diagrams and this is the origin of the claim that the tree level diagrams are the classical contributions...