On geometric interpretation of the Berry phase

TL;DR

This paper provides a geometric perspective on the Berry phase and its non-Abelian generalization, emphasizing the importance of connections on fiber bundles over topological properties, with implications for quantum physics.

Contribution

It offers a detailed geometric interpretation of the Berry phase and shows that nontrivial effects can arise from connections even when the underlying fiber bundle is trivial.

Findings

Principal fiber bundles can be trivial while connections are nontrivial.

The Berry phase is primarily a geometric effect, not a topological one.

Non-Abelian generalizations are also interpreted geometrically.

Abstract

A geometric interpretation of the Berry phase and its Wilczek--Zee non-Abelian generalization are given in terms of connections on principal fiber bundles. It is demonstrated that a principal fiber bundle can be trivial in all cases, while the connection and its holonomy group are nontrivial. Therefore, the main role is played by geometric rather than topological effects.