Algebraic generalization of the Ginsparg–Wilson relation Original Research Article

A specific algebraic realization of the Ginsparg–Wilson relation in the form γ5(γ5D)+(γ5D)γ5=2a2k+1(γ5D)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 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 Neubergerʹs 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.