Hi folks, I'm assured that scattering cross-sections in QFT computed at tree level correspond to cross-sections in the classical theory. For example the tree level cross-section for electron-electron scaterring in QED corresponds to scattering of classical point charges. But I'm not sure I understand how this can be. In particular, I think I have two big misunderstandings and if anyone could set me straight I would be grateful! Suppose we have an amplitude for 2-> 2 scattering, represented by A(p1,p2,q1,q2) where p1...qn encode the state-dependence and state-independent properties of the particles involved. (1) Since inelastic processes -- by which I mean processes that change the kinds of particles involved -- are not allowed classically, I take it that the contribution from tree diagrams must be zero in these cases. But it doesn't seem to be the case that tree diagrams give zero in inelastic processes in QFT! Could someone tell me what the error is in my thought here? (2) Since classical field theory is deterministic, it strikes me that the classical amplitude for any one process must always be 1 or 0. But of course the sum of tree-level terms in a QFT is not always going to sum to either of these! Can anyone help me interpret this supposed classical limit, because I'm clearly not getting this gist here at all? Thanks a lot!
Quantum theories correspond to classical theories in the so-called classical limit. Only in the cassical limit does (and should!) a quantum calculation agree with a classical calculation for some given process.
For a quantum theory to be correct, any result obtained using classical theory must somehow be obtainable using the quantum description. In the case of scattering in QFT, classical results are produced at tree-level. However this doesn’t mean that all tree-level results can be obtained or understood classically. The classical and quantum concepts involved in the description of the same (or any) system are profoundly different with quantum theory giving a way to understand physics that is deeper and more general than the corresponding classical description of a given system. What you need to investigate is the precise correspondence between classical and quantum descriptions and in particular how a classical limit, when one exists, emerges from a quantum theory (sources for which abound).