Dirac operator normality and chiral properties

Normality and $\ga$-hermiticity are what gives rise to chiral properties and rules. The Ginsparg-Wilson (GW) relation is only one of the possible spectral constraints. The sum rule for chiral differences of real modes has important consequences. The alternative transformation of L\"uscher gives the same Ward identity as the usual chiral one (if zero modes are properly treated). Imposing normality on a general function of the hermitean Wilson-Dirac operator $H$ leads at the same time to the GW relation and to the Neuberger operator.