Geometric phases and hidden local gauge symmetry

TL;DR

This paper explores geometric phases in quantum systems through a second quantized approach, revealing a hidden local gauge symmetry that clarifies physical observables and offers an alternative perspective to traditional holonomy analysis.

Contribution

It introduces a hidden local gauge symmetry in the analysis of geometric phases, providing a new framework that aligns with Pancharatnam's prescription and distinguishes cyclic from noncyclic evolutions.

Findings

Hidden local gauge symmetry clarifies physical observables.

Analysis aligns with Pancharatnam's geometric phase.

Differences identified between cyclic and noncyclic phases.

Abstract

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, which is associated with the arbitrariness of the phase choice of a complete orthonormal basis set, becomes explicit in this formulation (in particular, in the adiabatic approximation) and specifies physical observables. The choice of a basis set which specifies the coordinate 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. We discuss the implications of this hidden local gauge symmetry in detail by analyzing geometric phases for cyclic and noncyclic evolutions. It is shown that the hidden local symmetry provides a basic concept alternative to the notion of holonomy to analyze geometric phases and that the analysis based on the hidden local gauge symmetry leads to results consistent with the general prescription of Pancharatnam. We however note an important difference between the geometric phases for cyclic and noncyclic evolutions. We also explain a basic difference between our hidden local gauge symmetry and a gauge symmetry (or equivalence class) used by Aharonov and Anandan in their definition of generalized geometric phases.