Nielsen-Ninomiya theorem

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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 $S[\psi ]$ be the (Euclidean) action describing fermions $\psi$ 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.

References

Nielsen, H.B.; Ninomiya, M. (1981), "A no-go theorem for regularizing chiral fermions", Phys. Lett. B105: 219
Nielsen, H.B.; Ninomiya, M. (1981), "Absence of neutrinos on a lattice: (I). Proof by homotopy theory", Nucl. Phys. B185: 20
Nielsen, H.B.; Ninomiya, M. (1981), "Absence of neutrinos on a lattice: (II). Intuitive topological proof", Nucl. Phys. B193: 173
I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge University Press 1997
Itzykson, Claude; Drouffe, Jean-Michel (1989), Statistical field theory. Vol. 1, Cambridge Monographs on Mathematical Physics, Cambridge University Press, ISBN 978-0-521-34058-8, MR 1175176

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