Implications of $Tr{\gamma }_5=0$ in Lattice Gauge Theory

Abstract

We analyze the implications of the relation $Tr{\gamma }_5=0$, 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 $Tr{\gamma }_5$ without any suppression. In this sense, the relation $Tr{\gamma }_5=0$ 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 unphysical states, the trace $Tr{\gamma }_5$ is replaced by $Tr{\Gamma }_5\equiv Tr{\gamma }_5(1-1/2aD)$ which gives rise to the Pontryagin index, to be consistent with the continuum analysis.

Comments: 12 pages, Talk given at Chiral'99, September 13-18,1999, Taipei, Taiwan(To be published in the Proceedings)