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

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

1999-04-23

arxiv.org/abs/hep-lat/9904007v2

Phys.Rev. D60 (1999) 074505

Physics

High Energy Physics

High Energy Physics - Lattice

20 pages. Some clarifying comments and additional references are added. Phys. Rev. D (in press)

Scientific paper

10.1103/PhysRevD.60.074505

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.

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