Algebraic generalization of the Ginsparg-Wilson relation

A specific algebraic realization of the Ginsparg-Wilson relation in the form gamma (5)(gamma D-5) + (gamma D-5)gamma (5) = 2a(2k+1) (gamma D-5)(2k+2) is discussed, where k stands for a non-negative integer and k = 0 corresponds to the commonly discussed Ginsparg-Wilson relation. From a view point or algebra, a characteristic property of our proposal is that we have a closed algebraic relation for one unknown operator D, although this relation itself is obtained from the original proposal of Ginsparg and Wilson, gamma D-5 + D gamma (5) = 2aD gamma (5)alphaD, by choosing alpha as an operator containing D land thus Dirac matrices). In this paper, it is shown that we can construct the operator D explicitly for any value of k. We first show that the instanton-related index of all these operators is identical. We then illustrate in detail a generalization of Neuberger's overlap Dirac operator to the case k = 1. On the basis of explicit construction, it is shown that the chiral symmetry breaking term becomes more irrelevant for larger k in the sense of Wilsonian renormalization group. We thus have an infinite tower of new lattice Dirac operators which are topologically proper, but a large enough lattice is required to accommodate a Dirac operator with a large value of k. (C) 2000 Elsevier Science B.V. All rights reserved.