Nielsen–Ninomiya theorem

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

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

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

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

${\displaystyle S=\sum _{x,y,\mu }\overline{\psi }(x)i{\gamma }_{\mu }{F}_{\mu }(x,y)P_R\psi (y),}$

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 ${\displaystyle F_{\mu }(-x)=F_{\mu }(x{)}^{\ast }}$.
• 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.

Proof summary

The simplified Euclidean version of the theorem has a much shorter proof, relying on a key theorem from differential topology known as the Poincaré–Hopf theorem. It can be summarized as follows. From the locality assumption, the Fourier transform of the inverse propagator ${\displaystyle F_{\mu }(k)}$ must be a continuous vector field on the Brillouin zone whose isolated zeros correspond to different species of particles of the theory. Around each zero the vector field behavior is either a saddle singularity or a sink/source singularity. This is captured by the index of the vector field at the zero which takes the values ${\displaystyle \pm 1}$ for the two cases. It can be shown that the two cases determine whether the particle is left-handed or right-handed. The Poincaré–Hopf theorem states that the sum of the indices of a vector field on a manifold is equal the Euler characteristic of that manifold. In this case, the vector field lives on the Brillouin zone which is topologically a 4-torus which has Euler characteristic zero. Therefore, there must be an equal number of left-handed and right-handed particles.