Malaysian Journal of Mathematical Sciences, June 2014, Vol. 8(S)
Special Issue: The 6th Asia-Pacific Conference and Workshop in Quantum Information Science 2012 (APCWQIS2012)
Exact Abelian and Non-Abelian Geometric Phases
Chopin Soo and Huei-Chen Lin
Corresponding Email: [email protected]
Received date: -
Accepted date: -
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 Kähler and hyper-Kähler connections will be discussed. Explicit physical examples are used to verify and exemplify the formalism.
Keywords: -
