Fermion doubling
The fermion doubling problem is intractably linked to chiral invariance by the Nielsen–Ninomiya theorem.
This extended particle content can be seen by analyzing the symmetries or the correlation functions of the lattice theory.
The Dirac structure in the symmetry is similarly defined by the indices of
Since this is a symmetry of the action, the different tastes must be physically indistinguishable from each other.
acts more as an additional quantum number specifying the taste of a fermion.
However, calculating the propagator from the naive action yields for a fermion with momentum
For example, scalars acquire propagators taking a similar form except with
, which only has a single pole over the momentum range and so the theory does not suffer from a doubling problem.
But since any local lattice theory that can be constructed must have a propagator that is continuous and periodic, it must cross the zero axis at least once more, which is exactly what occurs on the Brillouin zone corners where
This is in contrast to the bosonic propagator which is quadratic around the origin and so does not have such problem.
Doubling can be avoided if a discontinuous propagator is used, but this results in a non-local theory.
The presence of doublers is also reflected in the fermion dispersion relation.
of the fermion and its momentum, it requires performing an inverse Wick transformation
, with the dispersion relation arising from the pole of the propagator[7] The zeros of this dispersion relation are local energy minima around which excitations correspond to different particle species.
The above has eight different species arising due to doubling in the three spatial directions.
The remaining eight doublers occur due to another doubling in the Euclidean temporal direction, which seems to have been lost.
But this is due to a naive application of the inverse Wick transformation.
Doing this for the position space propagator results in two separate terms, each of which has the same dispersion relation of eight fermion species, giving a total of sixteen.
[8] The obstruction between the Minkowski and Euclidean naive fermion lattice theories occurs because doubling does not occur in the Minkowski temporal direction, so the two theories differ in their particle content.
Simulating lattice field theories with fermion doubling leads to incorrect results due to the doublers, so many strategies to overcome this problem have been developed.
While doublers can be ignored in a free theory as there the different tastes decouple, they cannot be ignored in an interacting theory where interactions mix different tastes, since momentum is conserved only up to modulo
taste fermions can scatter by the exchange of a highly virtual gauge boson to produce two
Therefore, to overcome the fermion doubling problem, one must violate one or more assumptions of the Nielsen–Ninomiya theorem, giving rise to a multitude of proposed resolutions: These fermion formulations each have their own advantages and disadvantages.
[26] They differ in the speed at which they can be simulated, the easy of their implementation, and the presence or absence of exceptional configurations.
Some of them have a residual chiral symmetry allowing one to simulate axial anomalies.
They can also differ in how many of the doublers they eliminate, with some consisting of a doublet, or a quartet of fermions.
The effect of the derivative discretizations on doubling is seen by considering the one-dimensional toy problem of finding the eigensolutions of
The reason for this particle content disparity is that the symmetric difference derivative maintains the hermiticity property of the continuum
These latter discretizations lead to non-hermitian actions, breaking the assumptions of the Nielsen–Ninomiya theorem, and so avoid the fermion doubling problem.
[28] For this reason such a resolution to the fermion doubling problem is generally not implemented.
