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 study the correspondence of the eigenmodes of the Dirac operators in the continuum and lattice theories. After introducing a suitable cut-off in the continuum theory, we identify unphysical modes of the lattice Dirac operator which do not correspond to any physical modes of the regulated continuum Dirac operator. We also consider zero modes in the continuum and lattice theories. Our studies give a physical interpretation of the expression of the index defined on a lattice and a formal argument on the relation of the indices between the continuum and lattice theories. One of the recent developments in the treatment 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 2 a 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 correspondence 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 understandings of the eigenmodes of D from a physical point of view. The eigenmodes of D with the eigenvalue 2 a and the zero modes of D and D are investigated after introducing a suitable cut-off in D to make our analysis free from divergences. This cut-off procedure, which was not considered in Ref. [1], is an important step to derive a clear correspondence of the eigenmodes in our study. We will see that the eigenmodes of D with the eigenvalue 2 a do not correspond to any physical modes of D, thus they are considered to be unphysical [16]. Based on this criterion of the unphysical modes, we interpret that the role of the factor (1− a 2 D) in the trγ5(1 − a 2 D) is to ensure that the unphysical modes λn satisfying D λ = 2 a λ are omitted in the evaluation of the trace. Also we will show that the zero modes of D are transformed to the zero modes of D preserving the chirality so that n± = n lat ± . These two observations provide us a physical interpretation of the index expression (2) and the identity trγ5 = tr γ5(1− a 2 D) (4) at a formal level. 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 1 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 = ∑