Dirac operator normality and chiral fermions

Normality of the Dirac operator is shown to be necessary for chiral properties. From the global chiral Ward identity, which in the continuum limit gives the index theorem, a sum rule results which constrains the spectrum. The Ginsparg-Wilson relation is to be restricted to its simple form and is a member of a set of spectral constraints. A family of alternative chiral transformations is introduced. The one of L"uscher is a special case which transports only the anomaly term to the measure. An alternative transformation would also be needed to correct Fujikawa’s path-integral approach. From a general function of the hermitean Wilson-Dirac operator the one of Neuberger follows.