A lattice fermion without doubling
  A free fermion without doubler is formulated on 1+D dimensional discrete Minkowski space-time. The action is not hermitian but causes no harm. In 1+3 dimensional massless case the equation describes a single species of Dirac particle in the continuous space-time limit. In 1+1 dimensional massless case the equation is the same as the automaton equation by 't Hooft and describes a chiral fermion. The time evolution operator is unitary and the norm is conserved. For interacting fermions with gauge fields the evolution operator is not unitary. If it is considered as an approximation for the theory on continuous space-time, the path integral formalism can be applied, where the fermion without doubling is used. Consequences of loosening the unitarity condition on the time evolution operator is discussed. 
A. Hayashi, T. Hashimoto, M. Horibe and H. Yamamoto