It is increasingly acknowledged that the embarrassment of modern theoretical physics is the complete failure to describe strongly interacting finite density *fermion* matter using general- and controlled mathematical methods. This is a very old problem that already bothered for instance Feynman but for largely historical reasons the physics community managed to stare away from it. However, the lack of success of the theorists trying to make sense of modern experiments on strongly interacting electron systems as found in e.g. high Tc superconductors has changed this.

One way of stating the problem is: all of the great successes of modern quantum field theory eventually rest on the mapping via the euclidean path integral on a Bolzmannian statistical physics problem. The great succes story is in fact equilibrium stat-phys: our fathers have managed to enumarate this completely. However, a generical quantum many particle system produces a path integral with a non-probabilistic structure which cannot be handled: the “fermion signs” are the case in point. In the Leiden group there is a sustained effort in trying to make progress in this “sign problem”. In close collaboration with the Weng group at Tsinghua, Beijing, it is demonstrated that the quantum statistics of a doped fermionic Mott insulator is of a completely different nature as the Fermi-Dirac statistics of free fermions [143,183]. It was speculated that the strange metals are associated with a “Mott-collapse” [166] where this strange statistics “collides” with the usual free fermion variety.

Some serious progress was made [142] departing from a mathematical reformulation of the Fermion path integral: the “constrained” path integral discovered by Ceperley. The full density matrix specifying the world history of all fermion wordlines together is at the center. The statement is that the path integral becomes probabilistic under the condition that only worldhistories are summed over that do not change the overall sign of the associated full density matrix. This geometrizes the sign problem in the sense that one only needs to know the “nodal surface” of the density matrix, defined as its hypersurface of zero’s, in order to completely enumarate the influence of the fermion statistics. One can now argue that in order to realize a quantum critical state of finite density fermions, this nodal surface has to turn into a geometrical fractal. It was demonstrated that actually the Feynman fermionic backflow system can be tuned into a non-Fermi-liquid critical state which is characterized by such a fractal nodal surface [145, also picture(s)].

This has also tight connections to the use of entanglement entropy to characterize non-Fermi liquid states of matter. A research project financed by the Templeton foundation is at present going on, aimed at using holography to address the nodal structure information in the string theoretical strange metals, and to specify how it relates to the entropy measures of quantum information.

Jan Zaanen