Which is the state of the art of relativistic many-body QFT?
We have a class of relativistic quantum field theories, typically used to calculate particle interactions (scattering) or to extend the Standard Model. Typically one starts with a "free" theory, then adds self-interactions of the field with itself, or interactions with other fields: perturbation theory is essential and the concept of "particle" is that of a "packet" that moves with a dispersion relation in vacuum (namely, in such theories we must have few particles in vacuum, otherwise the "particle concept" is lost). More precisely, the ground state of relativistic QFT is the vacuum, and it is an invariant state under <PERSON> transformations. This is not the case when we are in a medium (finite density): the density is not a <PERSON> invariant, plus we have no currents only in the rest-frame of the medium (i.e.: from the point of view of <PERSON> transformations, the "finite density" ground state is different from the vacuum, see the "Note" below).
On the other hand, we have non-relativistic Quantum Mechanics that is used in condensed matter for finite density systems. This can be expressed in a "second quantization" fashion, becoming a non-relativistic QFT. 651
Moreover, we also have a sort of "thermal QFT" that allows us to calculate the thermodynamic equilibrium properties of many-body quantum systems (i.e., QFT but after Wick rotation).
How about a "relativistic QFT of finite-density systems"?
Namely, a QFT that aims at describing a dense environment of relativistic particles (not just scattering in vacuum). Do we have such a theory or the only thing we have are some "approximations"? (by "approximation" I mean a theoretically problematic situation resembling that of the "early relativistic QM", which was then replaced by relativistic quantum field theories).
Note: exercise V.2.3 of Zee "QFT in a nutshell" (2nd edition) asks us to develop "QFT at finite density", which would be exactly the "relativistic many-body QFT" theory I am referring to. It seems that we have to add a chemical potential term to the Lagrangian. Moreover, "finite density, as well as finite temperature, breaks Lorentz invariance". See also this article.
Useful questions and answers: https://physics.stackexchange.com/q/363666/226902, https://physics.stackexchange.com/q/209291/226902, https://physics.stackexchange.com/a/34633/226902, https://physics.stackexchange.com/a/434439/226902. Why do we use Lagrange multipliers to impose a certain density? See: https://richardmmyers.com/lagrange-multipliers-and-path-integrals/ 298