Exact Abelian and Non-Abelian Geometric Phases

2013/05/17 by Chopin Soo, Soo, Chopin, Huei-Chen Lin +1
Medicine · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Biofield Effects and Biophysics #FOS: Physical sciences #Quantum Mechanics and Applications #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.1305.3992

openalex publication_date 2013/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The existence of Hopf fibrations S^2N+1/S¹ = CP^N and S^4K+3/S³ = 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.

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