Gauge symmetries in geometric phases

TL;DR

This paper reviews geometric phases highlighting gauge symmetries, introduces a hidden local gauge symmetry in second quantization, and clarifies its role in physical observables and differences from other gauge concepts.

Contribution

It reveals a hidden local gauge symmetry in geometric phases within second quantization, offering a new perspective that replaces traditional notions of parallel transport and holonomy.

Findings

Hidden local gauge symmetry clarifies physical observables.

This symmetry differs from Aharonov-Anandan gauge symmetry.

Provides a unified framework for geometric phases and gauge invariance.

Abstract

The analysis of geometric phases is briefly reviewed by emphasizing various gauge symmetries involved. The analysis of geometric phases associated with level crossing is reduced to the familiar diagonalization of the Hamiltonian in the second quantized formulation. A hidden local gauge symmetry becomes explicit in this formulation and specifies physical observables; the choice of a basis set which specifies the coordinates in the functional space is arbitrary in the second quantization, and a sub-class of coordinate transformations, which keeps the form of the action invariant, is recognized as the gauge symmetry. It is shown that the hidden local symmetry provides a basic concept which replaces the notions of parallel transport and holonomy. We also point out that our hidden local gauge symmetry is quite different from a gauge symmetry used by Aharonov and Anandan in their definition of non-adiabatic phases.