is quadratic around the origin and so does not have such problem. Doubling can be avoided if a discontinuous propagator is used, but this results in a non-local theory. The presence of doublers is also reflected in the fermion dispersion relation. Since this is a relation between the energy of the fermion and its momentum, it requires performing an inverse Wick transformation , with the dispersion relation arising from the pole of the propagator The zeros of this dispersion relation are local energy minima around which excitations correspond to different particle species. The above has eight different species arising due to doubling in the three spatial directions. The remaining eight doublers occur due to another doubling in the Euclidean temporal direction, which seems to have been lost. But this is due to a naive application of the inverse Wick transformation. The theory has an obstruction that does not allow for the simple replacement of and instead requires performing the full contour integration. Doing this for the position space propagator results in two separate terms, each of which has the same dispersion relation of eight fermion species, giving a total of sixteen. The obstruction between the Minkowski and Euclidean naive fermion lattice theories occurs because doubling does not occur in the Minkowski temporal direction, so the two theories differ in their particle content. Resolutions to fermion doubling Fermion doubling is a consequence of a no-go theorem in lattice field theory known as the Nielsen–Ninomiya theorem. It states that any even dimensional local, hermitian, translationally invariant fermionic theory always has the same number of left-handed and right-handed Weyl fermions, generating the additional fermions when they are lacking. The theorem does not say how many doublers will arise, but without breaking the assumptions of the theorem, there will always be at least one doubler, with the naive discretization having fifteen. A consequence of the theorem is that the chiral anomaly cannot be simulated with chirally invariant theories as it trivially vanishes. Simulating lattice field theories with fermion doubling leads to incorrect results due to the doublers, so many strategies to overcome this problem have been developed. While doublers can be ignored in a free theory as there the different tastes decouple, they cannot be ignored in an interacting theory where interactions mix different tastes, since momentum is conserved only up to modulo . For example, two taste fermions can scatter by the exchange