Abstract
We note that Fujikawa's proposal of generalization of the Ginsparg-Wilson relation is equivalent to setting
R=(a
γ
5
D
)
2k
in the original Ginsparg-Wilson relation
D
γ
5
+
γ
5
D=2aDR
γ
5
D
. An explicit realization of D follows from the Overlap construction. The general properties of D are derived. The chiral properties of these higher-order (k > 0) realizations of Overlap Dirac operator are compared to those of the Neuberger-Dirac operator (k = 0), in terms of the fermion propagator, the axial anomaly and the fermion determinant in a background gauge field. Our present results (up to lattice size 16 x 16) indicate that the chiral properties of the Neuberger-Dirac operator are better than those of higher-order ones.