Nielsen-Ninomiya No-go Theorem

 An interesting paper which I have been fascinated by recently is the paper by Holger Nielsen and Masao Ninomiya, which shows that there exists a no-go theorem concerning fermion doubling with chiral fermions on a lattice. Naively, the idea is that the Hamiltonian for fermion field has fermion doubling if the following conditions are satisfied: (1) translational invariance is satisfied, (2) the charges are conserved and quantized, and (3) the interaction Hamiltonian is Hermitian and local. In this sense, there is a no-go theorem, the original arguments of which concern homotopy theory. There is also a very good paper by Friedan on a proof of this theorem. There were some arguments concerning if one could be a little more lenient with the hermiticity condition, and impose something like the PT symmetry, see for instance The Nielsen-Ninomiya theorem, PT-invariant non-Hermiticity and single 8-shaped Dirac cone. There is also a very good paper on the approach from loop quantum gravity by Jake Barnett and Lee Smolin, which seems to have some interesting features. Seemingly, there seems to be emphasis on circumventing the Nielsen-Ninomiya theorem in LQG, which seems to be an interesting remark. A part of the theorem's popularity also lies with the motivation it provides to other things with fermions, most famously massive fermions, which is something that David Tong has worked on.