Quantum anomaly and geometric phase; their basic differences

TL;DR

This paper clarifies the fundamental differences between quantum anomalies and geometric phases, demonstrating that they are distinct phenomena with different topological and dynamical origins, despite superficial similarities.

Contribution

The paper provides a detailed analysis showing that quantum anomalies and geometric phases are fundamentally different, especially in their topological and dynamical aspects, using a specific quantum model as an example.

Findings

The geometric term in the model is topologically trivial for finite time intervals.

The anomaly-related Wess-Zumino term is unrelated to the geometric phase.

Differences are rooted in the role of adiabatic approximation and topology in each phenomenon.

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.