Nielsen–Ninomiya theorem

(Redirected from Nielsen-Ninomiya theorem)
The Nielsen–Ninomiya theorem is a no-go theorem in physics, in particular in lattice gauge theory, concerning the possibility of defining a theory of chiral fermions on a lattice in even dimensions. The theorem can be stated as follows: let ${\displaystyle S[\psi ]}$ be the (Euclidean) action describing fermions ${\displaystyle \psi }$ on a regular lattice of even dimensions with periodic boundary conditions, and suppose that S is local, hermitian and translation invariant; then the theory describes as many left-handed as right-handed states. Equivalently, the theorem implies that there are as many states of chirality +1 as of chirality -1. The proof of the theorem relies on the Poincaré-Hopf theorem or on similar results in algebraic topology.