Geometric Phase and Chiral Anomaly in Path Integral Formulation

TL;DR

This paper unifies the description of geometric phases in quantum mechanics using path integral formulation, revealing their topological triviality and relating them to chiral anomaly.

Contribution

It introduces a second quantized path integral approach that unifies adiabatic and non-adiabatic geometric phases and clarifies their topological nature.

Findings

All geometric phases are topologically trivial.

The formulation reveals a hidden local symmetry in the Schrödinger equation.

Comparison with chiral anomaly highlights their natural formulation in path integrals.

Abstract

All the geometric phases, adiabatic and non-adiabatic, are formulated in a unified manner in the second quantized path integral formulation. The exact hidden local symmetry inherent in the Schr\"{o}dinger equation defines the holonomy. All the geometric phases are shown to be topologically trivial. The geometric phases are briefly compared to the chiral anomaly which is naturally formulated in the path integral.