Relation $Tr \gamma_{5}= 0$ and the index theorem in lattice gauge theory

The relation $Tr \gamma_{5}= 0$ implies the contribution to the trace from unphysical (would-be) species doublers in lattice gauge theory. This statement is also true for the Pauli-Villars regularization in continuum theory. If one insists on $Tr \gamma_{5}= 0$, one thus inevitably includes unphysical states in the Hilbert space. If one truncates the trace to the contribution from physical species only, one obtains $\tilde{T}r \gamma_{5} = n_{+} - n_{-}$ which is equal to the Pontryagin index. A smooth continuum limit of $\tilde{T}r \gamma_{5} = Tr \gamma_{5}(1-(a/2)D) = n_{+} - n_{-}$ for the Dirac operator $D$ satisfying the Ginsparg-Wilson relation leads to the natural treatment of chiral anomaly in continuum path integral. In contrast, the continuum limit of $Tr \gamma_{5}= 0$ is not defined consistently. It is shown that the non-decoupling of heavy fermions in the anomaly calculation is crucial to understand the consistency of the customary lattice calculation of anomaly where $Tr \gamma_{5}= 0$ is used. We also comment on a closely related phenomenon in the analysis of the photon phase operator where the notion of index and the modification of index by a finite cut-off play a crucial role.