Nielsen–Ninomiya theorem

The theorem was originally formulated in the Hamiltonian formulation of lattice field theory where time is continuous but space has been discretized.[1][2] Consider a theory with a Hamiltonian of the form

${\displaystyle H=\sum _{{\boldsymbol {x}},{\boldsymbol {y}}}\psi ^{\dagger }({\boldsymbol {x}})F({\boldsymbol {x}},{\boldsymbol {y}})\psi ({\boldsymbol {y}})}$

together with a charge ${\displaystyle Q}$. 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[6]

• Translational invariance: Implies that ${\displaystyle F({\boldsymbol {x}},{\boldsymbol {y}})=F({\boldsymbol {x}}-{\boldsymbol {y}})}$.
• Locality: ${\displaystyle F({\boldsymbol {x}}-{\boldsymbol {y}})}$ must vanish fast enough to have a Fourier transform with continuous derivatives.
• Hermiticity: For the Hamiltonian to be Hermitian, ${\displaystyle F({\boldsymbol {x}})}$ 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[7]

${\displaystyle S=\sum _{x,y,\mu }{\bar {\psi }}(x)i\gamma _{\mu }F_{\mu }(x,y)P_{R}\psi (y),}$

where ${\displaystyle P_{R}}$ is the right-handed projection operator, together with three assumptions

• Translational invariance: ${\displaystyle F_{\mu }(x,y)=F_{\mu }(x-y)}$.
• Hermiticity: For the action to be hermitian, it must hold that ${\displaystyle F_{\mu }(-x)=F_{\mu }(x)^{*}}$.
• 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.[8]

The Nielsen–Ninomiya theorem can be generalized to all possible regularization schemes, not just lattice regularization.[4] This general no-go theorem states that no regularized chiral fermion theory can satisfy all the following conditions

• Invariance under at least the global part of the gauge group.
• Different number of left-handed and right-handed Weyl species for a given combination of generators.
• The correct chiral anomaly.
• An action bilinear in Weyl fields.

A short proof by contradiction points out that the Noether current acquired from some of assumptions is conserved, while other assumptions imply that it is not.

Every regularization scheme must violate one or more of the conditions. For lattice regularization the Nielsen–Ninomiya theorem leads to the same result under even weaker assumptions where the requirement for the correct chiral anomaly is replaced by an assumption of locality of interactions. Dimensional regularization depends on the particular implementation of chirality. If the ${\displaystyle \gamma _{5}}$ matrix is defined as ${\displaystyle \gamma _{[1}\gamma _{2}\cdots \gamma _{4+\epsilon ]}}$ for infinitesimal ${\displaystyle \epsilon }$ then this leads to a vanishing chiral anomaly, while using ${\displaystyle \gamma _{[1}\gamma _{2}\gamma _{3}\gamma _{4]}}$ breaks global invariance. Meanwhile, Pauli–Villars regularization breaks global invariance since it introduces a regulator mass.

• Lattice gauge theory