What kind of background do I need to study many-body quantum field theory?
This depends on what you understand under "many-body" and under "quantum field theory". Could be anything from solid state or nuclear physics, over statistical mechanics, to quantum chemistry. Oh, and some quantum field people might also be doing that.
As I understand it, "many-body" and "quantum field theory" are redundant, in the sense that quantum field theory is already inherently a many-body theory. (It's not quite true the other way around--you can do many-body quantum mechanics without field theory.)
Oh... right. That is is even another vector of confusion. Many-body theories can still be interacting (primarily dealing with particles interacting with each other via some two- or three, or more-particle forces) and non-interacting (primarily dealing with free particles, possibly interacting via symmetries (Fermionic/bosinic) and maybe a common one-body external potential).
In some areas of physics, "many-body treatment" would suggest the former, and all the areas I noted (except for maybe for QFT) are more concerned with interacting systems and ways of turning interacting systems (for which the Hamiltonian is known) into effective non-interacting systems (which are easier to interpret... e.g., Fermi liquids, Orbitals, shell models of nuclei). However, even for free particles many-body theory can be highly non-trivial, so large parts of QFT are highly concerned with those, although this kind of research would not be considered "many-body treatment" in quantum chemistry, for example.
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