Geometric phase and chiral anomaly; their basic differences

TL;DR

This paper clarifies the fundamental differences between geometric phases and chiral anomalies using a second quantized framework, revealing that they are distinct phenomena with different underlying mechanisms.

Contribution

The paper introduces a second quantized formulation to analyze geometric phases and demonstrates their topological triviality and differences from chiral anomalies.

Findings

Geometric phases are topologically trivial in this formulation

Hidden local symmetry governs both adiabatic and non-adiabatic phases

Geometric phase and chiral anomaly are fundamentally different phenomena

Abstract

All the geometric phases are shown to be topologically trivial by using the second quantized formulation. The exact hidden local symmetry in the Schr\"{o}dinger equation, which was hitherto unrecognized, controls the holonomy associated with both of the adiabatic and non-adiabatic geometric phases. The second quantized formulation is located in between the first quantized formulation and the field theory, and thus it is convenient to compare the geometric phase with the chiral anomaly in field theory. It is shown that these two notions are completely different.