Fermion doubling

In lattice theories, fermion fields experience (at least) a doubling of the number of particle types in a lattice.

A lattice is a periodic arrangement of vertices. If you Fourier transform a lattice, the space of momenta is a torus with the shape of the fundamental domain of the reciprocal lattice.

This means if we look at the wave solutions over a lattice, the energy (aka frequency) as a function of momentum (aka wave vector) has to be periodic.

For a bosonic field, the action is quadratic and so, the energy tends to have the form

$E=\sqrt{4\sin^2(kL/2)/L^2+m^2}$

or something like that where m<<1/L. At scales much larger than the lattice spacing (i.e. at low energies) only the momenta around k=0 dominate and we have a single species of boson.

Fermions, on the other hand, are described by first order equations. So, we might have something which goes like

$E={sin(kL)\over L}$

at least with one spatial dimension, but the higher dimensional cases are analogous. If we look at the low energy limit, we see two different regions; one about k=0 and the other about k=π/L. They behave like two different kinds of particles. This is called fermion doubling.

Fermion doubling is a generic consequence of local actions and Hamiltonians. However, the Ginsparg-Wilson action , which is nonlocal solves it (partially, by reducing the number of species).

Categories: Quantum chromodynamics

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