ar X iv : h ep - l at / 9 91 10 17 v 1 1 5 N ov 1 99 9 A comment on the index of the lattice Dirac operator and the Ginsparg - Wilson relation

We pursue Ginsparg and Wilsons’ block spin approach in the derivation of the Ginsparg-Wilson relation and give a formal argument that the index of the lattice Dirac operator agrees with that of the continuum theory, by identfying unphysical modes of the lattice Dirac operator which do not correspond to any physical modes of the continuum Dirac operator, and by studying the correspondence of the zero modes in the continuum and lattice theories. One of the recent developments in the treatemnt of the chiral symmetry on a lattice based on the Ginsparg-Wilson relation [1] Dγ5 + γ5D lat = aDγ5D lat (1) is the interesting index relation [2, 3] trγ5(1− a 2 D) = n + − n lat − (2) on a lattice and its implications for the understanding of the anomaly [1]-[13]. In these equations, D is a Dirac operator describing a fermion on a lattice, n± are the numbers of the right-handed and left-handed zero modes of D, a is the lattice spacing and tr is the trace in the lattice theory. The role of the factor (1− a 2 D) in Eq. (2), which is absent in the continuum index relation [14, 15] trγ5 = n c + − n c −, (3) is further studied in Refs. [10, 11] based on the representation of the algebra (1), leading to the observation that the mismatch of the chiralities between the zero modes of D should be compensated by the mismatch of the chiralities of its eigenmodes with the eigenvalue a 2 to ensure the relation trγ5 = 0 in the lattice theory. In this short note, we pursue Ginsparg and Wilsons’ block spin approach in the derivation of the Ginsparg-Wilson relation [1] and study the relation between the eigenmodes of a continuum Dirac operator D and those of the lattice Dirac operator D constructed from D following the block spin transformation, in the hope that such analyses will clarify undestandings of the eigenmodes of D from a physical point of view. The eigenmodes of D with the eigenvalue a 2 and the zero modes of D and D are investigated after introducing a suitable cut-off in D to make our procedures free from divergences. Our study gives a physical interpretation, even though at a formal level, of Eq. (2) and the relation between Eq. (2) and (3). We begin with an action S(φ̄x, φx) of the fermionic fields φx and φ̄x defined in the continuum Euclidean space-time. From this action Ginsparg and Wilson constructed a new action S(ψ̄n, ψn) on a lattice by block spin transformation. First we define the block variables ρn and ρ̄n corresponding to the continuum fields φx and φ̄x as ρn = ∑