Quantum anomaly and geometric phase; their basic differences

Abstract

It is sometimes stated in the literature that the quantum anomaly is regarded as an example of the geometric phase. Though there is some superficial similarity between these two notions, we here show that the differences bewteen these two notions are more profound and fundamental. As an explicit example, we analyze in detail a quantum mechanical model proposed by M. Stone, which is supposed to show the above connection. We show that the geometric term in the model, which is topologically trivial for any finite time interval $T$, corresponds to the so-called ``normal naive term'' in field theory and has nothing to do with the anomaly-induced Wess-Zumino term. In the fundamental level, the difference between the two notions is stated as follows: The topology of gauge fields leads to level crossing in the fermionic sector in the case of chiral anomaly and the {\em failure} of the adiabatic approximation is essential in the analysis, whereas the (potential) level crossing in the matter sector leads to the topology of the Berry phase only when the precise adiabatic approximation holds.

28 pages. The last sentence in Abstract has been changed, the last paragraph in Section 1 has been re-written, and the latter half of Discussion has been replaced by new materials. New Conclusion to summarize the analysis has been added. This new version is to be published in Phys. Rev. D

References in corpus (7)

• Anomalous Hall Effect and Magnetic Monopoles in Momentum-Space
• Anomalous Hall effect in ferromagnetic semiconductors
• Spin Anisotropy and Quantum Hall Effect in the Kagome Lattice - Chiral Spin State based on a Ferromagnet-
• Topological Nature of Anomalous Hall Effect in Ferromagnet
• Second quantized formulation of geometric phases
• Geometric phases and hidden local gauge symmetry
• Topological properties of Berry's phase

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