Relation Trγ 5 = 0 and the index theorem in lattice gauge theory

نویسنده

  • Kazuo Fujikawa
چکیده

The relation Trγ5 = 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 Trγ5 = 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 T̃ rγ5 = n+ − n− which is equal to the Pontryagin index. A smooth continuum limit of T̃ rγ5 = Trγ5(1 − (a/2)D) = n+ − n− for the Dirac operator D satisfying the GinspargWilson relation leads to the natural treatment of chiral anomaly in continuum path integral. In contrast, the continuum limit of Trγ5 = 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 Trγ5 = 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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تاریخ انتشار 1999