Possible Consequences of Conjectural Periodicity of Spectrum of Lattice Dirac Operator

Abstract

Some consequences which follow from the periodicity assumption for spectral density of Wilson--Dirac operator are studied. Such an assumption allows to obtain simple representations for quark propagator, which reveals an important role of $m\leftrightarrow -m$ symmetry. It is argued that this symmetry is restored when the mirror fermion mass $m_r$ tended to infinity. The constrains on zero modes of Wilson--Dirac operator in a toy model approximation are also discussed.