  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.
