Some remarks on the Ginsparg-Wilson fermion

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We note that Fujikawa's proposal of generalization of the Ginsparg-Wilson relation is equivalent to setting R = (a γ₅ D)²ᵏ in the original Ginsparg-Wilson relation D γ₅ + γ₅ D = 2 a D R γ₅ 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.