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 $\psi$ 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, Bibcode:1981PhLB..105..219N, doi:10.1016/0370-2693(81)91026-1
• Nielsen, H.B.; Ninomiya, M. (1981), "Absence of neutrinos on a lattice: (I). Proof by homotopy theory", Nucl. Phys., B185: 20, Bibcode:1981NuPhB.185...20N, doi:10.1016/0550-3213(81)90361-8
• Nielsen, H.B.; Ninomiya, M. (1981), "Absence of neutrinos on a lattice: (II). Intuitive topological proof", Nucl. Phys., B193: 173, Bibcode:1981NuPhB.193..173N, doi:10.1016/0550-3213(81)90524-1
• Friedan, D. (1982), "A Proof of the Nielsen-Ninomiya Theorem", Commun. Math. Phys., 85: 481, Bibcode:1982CMaPh..85..481F, doi:10.1007/BF01403500
• 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