of a highly virtual gauge boson to produce two taste fermions without violating momentum conservation. Therefore, to overcome the fermion doubling problem, one must violate one or more assumptions of the Nielsen–Ninomiya theorem, giving rise to a multitude of proposed fermions: Domain wall fermion: explicitly violates chiral symmetry, increases spatial dimensionality. Ginsparg–Wilson fermion: explicitly violates chiral symmetry. Overlap fermion: explicitly violates chiral symmetry (type of Ginsparg–Wilson fermion). Perfect lattice fermion: nonlocal formulation. SLAC fermion: nonlocal formulation. Stacey fermion: nonlocal formulation. Staggered fermion (Kogut–Susskind fermion): explicitly violates translational invariance, reduces number of doublers. Twisted mass fermion: explicitly violates chiral symmetry (type of Wilson fermion). Wilson fermion: explicitly violates chiral symmetry. These fermion formulations each have their own advantages and disadvantages. They differ in the speed at which they can be simulated, the easy of their implementation, and the presence or absence of exceptional configurations. Some of them have a residual chiral symmetry allowing one to simulate axial anomalies. They can also differ in how many of the doublers they eliminate, with some consisting of a doublet, or a quartet of fermions. For this reason different fermion formulations are used for different problems. Derivative discretization Another possible although impractical solution to the doubling problem is to adopt a derivative discretization different from the symmetric difference used in the naive fermion action. Instead it is possible to use the forward difference or a backward difference discretizations. The effect of the derivative discretizations on doubling is seen by considering the one-dimensional toy problem of finding the eigensolutions of . In the continuum this differential equation has a single solution. However, implementing the symmetric difference derivative leads to the presence of two distinct eigensolutions, while a forward or backward difference derivative has one eigensolution. This effect carries forward to the fermion action where fermion doubling is absent with forward or backward discretizations. The reason for this particle content disparity is that the symmetric difference derivative maintains the hermiticity property of the continuum operator, while the forward and backward discretizations do not. These latter discretizations lead to non-hermitian actions, breaking the assumptions of the Nielsen–Ninomiya theorem, and so avoid the fermion doubling problem. Developing an interacting theory with a non-hermitian derivative discretization leads to a theory with non-covariant contributions to the fermion self-energy and vertex function, rendering the theory non-renormalizable and difficult to work with. For this reason such a resolution to the fermion doubling problem