Exact factorization of a many-body wavefunction beyond the electron–nuclear problem

This review is devoted to the presentation of the exact factorization as a framework employed to study a variety of quantum-mechanical many-body problems. Since its original formulation in the 70s, the main applications of the exact factorization were directed toward understanding the properties of multi-component systems, e.g., electron–nuclear systems. Especially in the electron–nuclear case, the exact factorization is viewed as an exactification of the Born–Oppenheimer approximation, thus it was, and still is, largely employed in nonadiabatic dynamics. Nonetheless, as early as the 80s, the formalism was employed to study many-electron interacting systems and, quite recently, i.e., less than a decade ago, it was extended to study the behavior of molecules in the context of cavity quantum electrodynamics. These formulations, perhaps less popular than the electron–nuclear formulation, have attracted a great deal of attention over the years. Therefore, we review here the exact electron-only factorization and the exact photon–electron–nuclear factorization.