Implications of Trγ 5 = 0 in Lattice Gauge Theory

We analyze the implications of the relation Trγ5 = 0, which is customarily assumed in practical lattice calculations. On the basis of the finite dimensional representations of the Ginsparg-Wilson algebra, it is shown that this relation reflects the species doubling in lattice theory; topological excitations associated with species doublers, which have eigenvalue 2/a, contribute to Trγ5 without any suppression. In this sense, the relation Trγ5 = 0 is valid only when we allow the presence of unphysical states in the Hilbert space; this statement is also valid in the Pauli-Villars regularization. If one eliminates the contributions of the unpysical states, the trace Trγ5 is replaced by TrΓ5 ≡ Trγ5(1− 12aD) which gives rise to the Pontryagin index, to be consistent with the continuum analysis.