Axial vector current of exact chiral symmetry on the lattice

We discuss the exact chiral symmetry and its spontaneous breakdown in lattice QCD with the Dirac operators satisfying the GinspargWilson relation. The axial vector current, which turns out to be related to the vector current simply by the insertion of the operator γ5(1−aD), is explicitly constructed in the case of the Neuberger-Dirac operator. We also consider an Euclidean proof of the Nambu-Goldstone theorem using the Ward-Takahashi identity of this symmetry. e-mail address: [email protected] e-mail address: [email protected] 1 Recently, two Dirac operators which describe exactly massless fermions on a lattice have been proposed. The first one is the fixed point Dirac operator considered by Hasenfratz et. al. [1, 2, 3, 4, 5] based on the renormalization group method [6, 7]. The second one is the Dirac operator derived by Neuberger [8, 9] based on the overlap formalism of chiral determinant [10, 11] and its explicit form is known as aDnm = (