Algebraic Generalization of the Ginsparg-Wilson Relation
A specific algebraic realization of the Ginsparg-Wilson relation in the form $\gamma_{5}(\gamma_{5}D)+(\gamma_{5}D)\gamma_{5} = 2a^{2k+1}(\gamma_{5}D)^{2k+2}$ 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, $\gamma_{5}D+D\gamma_{5}=2aD\gamma_{5} \alpha D$, by choosing $\alpha$ 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 $k$.