. Therefore, a fermion with momentum near the center of the Brillouin zone is mapped to one of its corners while one of the corner fermions comes in to replace the center fermion, showing that the transformation acts to exchange the tastes of the fermions. Since this is a symmetry of the action, the different tastes must be physically indistinguishable from each other. Here the Brillouin momentum for small is not the physical momentum of the particle, rather that is . Instead acts more as an additional quantum number specifying the taste of a fermion. The term is responsible for changing the representation of the -matrices of the doublers to , which has the effect of changing the signs of the matrices as . Since any such sign change results in a set of matrices still satisfying the Dirac algebra, the resulting matrices form a valid representation. It is also the term that enters the wave function of the doublers given by and , where and are the usual Dirac equation solutions with momentum . Propagator and dispersion relation In the continuum theory, the Dirac propagator has a single pole as the theory describes only a single particle. However, calculating the propagator from the naive action yields for a fermion with momentum . For low momenta this still has the expected pole at , but there are fifteen additional poles when . Each of these is a new fermion species with doubling arising because the function has two poles over the range . This is in contrast to what happens when particles of different spins are discretized. 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. The necessity of fermion doubling can be deduced from the fact that the massless fermion propagator is odd around the origin. That is, in the continuum limit it is proportional to , which must still be the case on the lattice in the small momentum limit. 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 for the naive fermion propagator. This is in contrast to the bosonic propagator which