Can QFT not be done with only tree diagrams?

Question

Here's the methodology: We first write out the equation of motion for $\phi(x,t)$ in the Heisenberg picture. This is identical to classical field equation. Now, we solve the equations perturbatively, pretending that we're perturbatively solving the interacting Euler Lagrange equations from classical field theory. This perturbation series will involve only tree diagrams.

After we have the perturbation series, we can re-interpret $a_p$ and $a^{\dagger}_p$ appearing in the series as operators satisfying $[a, a^{\dagger}]=\delta$. The correspondence between Poisson brackets and commutators guarantees that we can re-interpret the variables as operators.

After we have the full $\phi(x,t)$ solution in the interacting theory, we can plug it into the LSZ formula to get interaction amplitudes.

This method is computationally complicated, which is probably why we use the full interaction picture for QFT. But nevertheless, doesn't it show that QFT involves only tree diagrams identical to CFT, provided that we work in the Heisenberg picture? Or is it possible that this method is wrong?

Answer 1

1. In the path integral formalism, the loop-expansion is the same as the $\hbar$-expansion if the action $S$ does not depend explicitly on $\hbar$, cf. e.g. my Phys.SE answer here. In other words, the classical theory ($\hbar=0$) corresponds to tree-level in the path integral.

2. The operator formalism (and in particular the interaction picture) is formally equivalent to the path integral formalism. The consistent co-existence of the Heisenberg equations of motion and the contact-terms in the Schwinger-Dyson (SD) equations follows from different time-orderings $T$ and $T_{\rm cov}$, cf. e.g. my Phys.SE answers here & here. Therefore the operator formalism contains loop-diagram contributions of the path integral formalism.

3. Now the notion of tree-level depends on the formalism and context. Examples:

   • (A sum of all possible) connected diagrams is (a sum of all possible) trees of connected propagators and (amputated) 1PI vertices, cf. e.g my Phys.SE answer here.

   • One can argue that the perturbation theory with the Lippmann-Schwinger equation/Born series can be viewed as series of diagrams with the topology of trees. The first term, the Born-approximation, usually represents the classical theory.

   In the above 2 examples, the trees contain the full QFT, cf. OP's title question.

4. Note that the $\hbar$-dependence in itself is subtle. Example: It makes a difference if in the free Klein-Gordon equation $$ (\pm\Box-\mu^2)\phi(x)~=~0, $$ we treat $\mu\equiv\frac{mc}{\hbar}$ or $m$ as $\hbar$-independent. There are similar issues with the interaction terms.

5. Related post by OP: "Tree diagrams appear in classical field theory and non-relativistic QM because of the absence of the delta in the Schwinger-Dyson equation"