In lattice field theory, the Nielsen–Ninomiya theorem is a no-go theorem about placing chiral fermions on a lattice. In particular, under very general assumptions such as locality, hermiticity, and translational symmetry, any lattice formulation of chiral fermions necessarily leads to fermion doubling, where there are the same number of left-handed and right-handed fermions. It was first proved by Holger Bech Nielsen and Masao Ninomiya in 1981 using two methods, one that relied on homotopy theory and another that relied on differential topology. Another proof provided by Daniel Friedan uses differential geometry. The theorem was also generalized to any regularization scheme of chiral theories. One consequence of the theorem is that the Standard Model cannot be put on a lattice. Common methods for overcoming the fermion doubling problem is to use modified fermion formulations such as staggered fermions, Wilson fermions, or Ginsparg–Wilson fermions, among others.
Lattice regularization The theorem was originally formulated in the Hamiltonian formulation of lattice field theory where time is continuous but space has been discretized. Consider a theory with a Hamiltonian of the form
H = ∑ x , y ψ † ( x ) F ( x , y ) ψ ( y ) {\displaystyle H=\sum _{{\boldsymbol {x}},{\boldsymbol {y}}}\psi ^{\dagger }({\boldsymbol {x}})F({\boldsymbol {x}},{\boldsymbol {y}})\psi ({\boldsymbol {y}})}
together with a charge Q {\displaystyle Q} . The Nielsen–Ninomiya theorem states that there is an equal number of left-handed and right-handed fermions for every set of charges if the following assumptions are met
Translational invariance: Implies that F ( x , y ) = F ( x − y ) {\displaystyle F({\boldsymbol {x}},{\boldsymbol {y}})=F({\boldsymbol {x}}-{\boldsymbol {y}})} . Locality: F ( x − y ) {\displaystyle F({\boldsymbol {x}}-{\boldsymbol {y}})} must vanish fast enough to have a Fourier transform with continuous derivatives. Hermiticity: For the Hamiltonian to be Hermitian, F ( x ) {\displaystyle F({\boldsymbol {x}})} must also be Hermitian. The charge is defined locally through some local charge density. The charge is quantized. The charge is exactly conserved. This theorem trivially holds in odd dimensions since odd dimensional theories do not admit chiral fermions due to the absence of a valid chirality operator, that is an operator that anticommutes with all gamma matrices. This follows from the properties of Dirac algebras in odd dimensions. The Nielsen–Ninomiya theorem has also been proven in the Euclidean formulation. For example, consider a weaker version of the theorem which assumes a less generic action of the form
S = ∑ x , y , μ ψ ¯ ( x ) i γ μ F μ ( x , y ) P R ψ ( y ) , {\displaystyle S=\sum _{x,y,\mu }{\bar {\psi }}(x)i\gamma _{\mu }F_{\mu }(x,y)P_{R}\psi (y),}
where P R {\displaystyle P_{R}} is the right-handed projection operator, together with three assumptions
Translational invariance: F μ ( x , y ) = F μ ( x − y ) {\displaystyle F_{\mu }(x,y)=F_{\mu }(x-y)} . Hermiticity: For the action to be hermitian, it must hold that F μ ( − x ) = F μ ( x ) ∗ {\displaystyle F_{\mu }(-x)=F_{\mu }(x)^{*}} . Locality: The inverse propagator decreases fast enough so that its Fourier transform exists and all its derivatives are continuous. If all these conditions are met then there is once again an equal number of left-handed and right-handed fermions.
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