Exact Abelian and Non-Abelian Geometric Phases

TL;DR

This paper explores the geometric phases in quantum systems using the framework of Hopf fibrations, revealing deep connections between quantum states, their evolution, and complex or quaternionic geometrical structures.

Contribution

It introduces an exact formalism for Abelian and non-Abelian geometric phases based on fiber bundle structures and geometric connections, providing new insights into quantum state evolution.

Findings

Explicit expressions for geometric phases derived from fiber bundle formalism

Verification of the formalism through physical examples

Connection established between geometric phases and Kahler/hyper-Kahler geometries

Abstract

The existence of Hopf fibrations S^{2N+1}/S^1 = CP^N and S^{4K+3}/S^3 = HP^K allows us to treat the Hilbert space of generic finite-dimensional quantum systems as the total bundle space with respectively $U(1)$ and $SU(2)$ fibers and complex and quaternionic projective spaces as base manifolds. This alternative method of studying quantum states and their evolution reveals the intimate connection between generic quantum mechanical systems and geometrical objects. The exact Abelian and non-Abelian geometric phases, and more generally the geometrical factors for open paths, and their precise correspondence with geometric Kahler and hyper-Kahler connections will be discussed. Explicit physical examples are used to verify and exemplify the formalism.