Ginsparg–Wilson equation

Ginsparg–Wilson fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry. More precisely, the continuum chiral symmetry relation ${\displaystyle D{\gamma }_5+{\gamma }_{5}D=0}$ (where ${\displaystyle D}$ is the massless Dirac operator) is replaced by the Ginsparg–Wilson equation

${\displaystyle D{\gamma }_5+{\gamma }_{5}D=aD{\gamma }_{5}D}$

which recovers the correct continuum expression as the lattice spacing ${\displaystyle a}$ goes to zero.

In contrast to Wilson fermions, Ginsparg–Wilson fermions do not modify the inverse fermion propagator additively but multiplicatively, thus lifting the unphysical poles at ${\displaystyle p_{\mu }=\pi /a}$. The exact form of this modification depends on the individual realisation.