Basically all of it. No one has rigorously constructed a non-trivial quantum field theory in four dimensions (the physically realistic case; 3 spatial + 1 time)

Basically all of it. No one has rigorously constructed a non-trivial quantum field theory in four dimensions (the physically realistic case; 3 spatial + 1 time). I am also much more pessimistic than Seiberg about the possibility that a priori mathematical investigation is going to say something useful about quantum gravity. The following essay gives a pessimistic point of view on a particular program to rigorize QFT, but the arguments generalize: http://philsci-archive.pitt.edu/8890/ http://philsci-archive.pitt.edu/8890/. But, that's a long paper. For a super-condensed take on why the project might be hopeless, read section two of this paper: http://philsci-archive.pitt.edu/15296/1/qm-continuum%20revised.pdf http://philsci-archive.pitt.edu/15296/1/qm-continuum%20revis....