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.

Since the Standard Model is chiral (left- and right-handed fermions are treated differently by weak interactions, for example), the Nielsen–Ninomiya theorem implies that for simulating some Standard Model phenomena at least one of the assumptions of the theorem needs to be violated, otherwise there would be fermion doubling to the quantum system.