[2] Another proof provided by Daniel Friedan uses differential geometry.
[3] The theorem was also generalized to any regularization scheme of chiral theories.
[4] One consequence of the theorem is that the Standard Model cannot be put on a lattice.
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] 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.
For example, consider a weaker version of the theorem which assumes a less generic action of the form[7] where
From the locality assumption, the Fourier transform of the inverse propagator
must be a continuous vector field on the Brillouin zone whose isolated zeros correspond to different species of particles of the theory.
This is captured by the index of the vector field at the zero which takes the values
It can be shown that the two cases determine whether the particle is left-handed or right-handed.
In this case, the vector field lives on the Brillouin zone which is topologically a 4-torus which has Euler characteristic zero.
[4] This general no-go theorem states that no regularized chiral fermion theory can satisfy all the following conditions 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.
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.
Meanwhile, Pauli–Villars regularization breaks global invariance since it introduces a regulator mass.