Count unwanted light modes before interpreting a fermion field

For a free fermion on a d-dimensional Euclidean hypercubic lattice, use Hermitian gamma matrices and the symmetric nearest-neighbor derivative. In momentum space each derivative becomes i sin(apμ)/a. Restrict p to a half-open Brillouin interval of width 2π/a in every direction, so opposite endpoints are identified rather than separately counted. At zero mass, all sine components vanish when each momentum component is either zero or the zone-edge value π/a modulo reciprocal periods.

There are therefore 2 to the d distinct zero neighborhoods. In four Euclidean spacetime dimensions this gives sixteen species for a naive Dirac field. A Hamiltonian treatment with only three spatial directions counts eight momentum corners instead, before its own spin/chiral interpretation. Stating the dimension avoids confusing these two familiar numbers. A small nonzero mass gaps all the naive modes together rather than preferentially lifting only the unwanted corners.

Near a corner, write pμ as its corner value plus a small displacement. The sine’s linear coefficient is positive at zero and negative at a π corner. These sign changes can be absorbed into redefined gamma matrices, revealing an independent relativistic-like light field around each zero. The orientation of those velocity signs determines the chirality assignment in the corresponding chiral-band picture. Ignoring a corner because it sits at large lattice momentum misses that its excitation energy can still be small.

No-go statements rely on assumptions such as locality, translational structure, appropriate Hermiticity, and the conventional chiral anticommutation relation, together with a regular compact momentum-space description. There are several related formulations, so a theorem’s exact hypotheses should accompany its use. Wilson fermions relax naive chiral symmetry, while Ginsparg–Wilson formulations replace its lattice form. Neither strategy disproves the theorem; each changes the mathematical setup to retain a desired continuum limit.

At a finite box size, periodic or antiperiodic boundary conditions choose different discrete momenta. Antiperiodic temporal boundaries can remove an exact zero at one sampled momentum while leaving modes arbitrarily close to each corner as the box is enlarged. Thus absence of an exact zero on a particular finite grid does not by itself remove a species in the continuum theory. Both the spacing limit and the physical-volume limit need an analysis consistent with the intended spectrum.

The modular lab illustrates an indexing distinction. Reset and set modulus to 8 and additive step to 1; predict period 8. Reset and set modulus to 8 and step to 2; predict period 4. A stride can skip half the indices, but that does not prove that a local fermion operator lost its extra light modes. This additive orbit has no gamma matrices, sine derivative, dispersion spectrum, or fermion determinant; the physical zero count must be calculated separately.