Algebraic Generalization of the Ginsparg-Wilson Relation

A specific algebraic realization of the Ginsparg-Wilson relation in the form γ5(γ5D) + (γ5D)γ5 = 2a (γ5D) 2k+2 is discussed, where k stands for a nonnegative integer and k = 0 corresponds to the commonly discussed Ginsparg-Wilson relation. From a view point of 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, γ5D + Dγ5 = 2aDγ5αD, by choosing α as an operator containing D ( and 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 irrelevent 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 accomodate a Dirac operator with a large value of k.