In lattice field theory, fermion doubling occurs when naively putting fermionic fields on a lattice, resulting in more fermionic states than expected. For the naively discretized Dirac fermions in Euclidean dimensions, each fermionic field results in identical fermion species, referred to as different tastes of the fermion. The fermion doubling problem is intractably linked to chiral invariance by the Nielsen–Ninomiya theorem. Most strategies used to solve the problem require using modified fermions which reduce to the Dirac fermion only in the continuum limit. Naive fermion discretization For simplicity we will consider a four-dimensional theory of a free fermion, although the fermion doubling problem remains in arbitrary dimensions and even if interactions are included. Lattice field theory is usually carried out in Euclidean spacetime arrived at from Minkowski spacetime after a Wick rotation, where the continuum Dirac action takes the form This is discretized by introducing a lattice with lattice spacing and points indexed by a vector of integers . The integral becomes a sum over all lattice points, while the fermionic fields are replaced by four-component Grassmann variables at each lattice site denoted by and . The derivative discretization used is the symmetric derivative discretization, with the vectors being unit vectors in the direction. These steps give the naive free fermion action This action reduces down to the continuum Dirac action in the continuum limit, so is expect to be a theory of a single fermion. However, it instead describes sixteen identical fermions, with each fermion said to have a different taste, analogously to how particles have different flavours in particle physics. The fifteen additional fermions are often referred to as doublers. This extended particle content can be seen by analyzing the symmetries or the correlation functions of the lattice theory. Doubling symmetry The naive fermion action possesses a new taste-exchange symmetry not found in the continuum theory acting on the fermion fields as where the vectors are the sixteen vectors with non-zero entries of specified by . For example, , , , and . The Dirac structure in the symmetry is similarly defined by the indices of as where and ; for example with . The presence of these sixteen symmetry transformations implies the existence of sixteen identical fermion states rather than just one. Starting with a fermion field , the symmetry maps it to another field . Fourier transforming this shows that its momentum has been shifted as