I'm currently reviewing some of my notes on Quantum Field Theory (the version of Greiner) and I was wondering if QFT always works in the Hartree-Fock approximation ? Or at least that's what it seems to me!
We have our field-operators $\hat{\psi}(\vec{r},t)$ and $\hat{\psi}^\dagger(\vec{r},t)$ which annihilate or create a particle in $(\vec{r},t)$. By using the appropriate commutation-relations we get fermions or bosons. But these are ONE-PARTICLE operators which obey the correct commutation-relations, or which give the right symmetry (using Fock-space structure).
Now intuïtively I can see this for the free-particle Hamiltonians that this will give an exact result since we'll be able to rewrite them as:
$\hat{H}_0=\sum\limits_nE_n\hat{a}^\dagger_n\hat{a}_n,$
which indeed yields a result in the sense of product functions (since every eigenfunction of $\hat{a}^\dagger_n\hat{a}_n$ is also one of $\hat{H}_0$.
Now the problem starts when we get two-particle (of many-particle) interactions since the Hamiltonian isn't diagonizable in any easy way. This forces us to use perturbation theory and hence the scattering matrix. Upon applying Wick's theorem we can chop up the n-th order term of the scatering matrix into operators of the form $\hat{a}^\dagger_n\hat{a}_n$ which we can calculate in terms of our product basis. Which can also be expressed in terms of a product basis set.
Now long question short: Do we always work in the Hartree-Approximation when we're doing QFT, or am I mistaken?
