Berry’s geometric phase is a correction to the wave function arising in the study of adiabatic quantum systems; it has been discovered by M. V. Berry. There are analogous effects for other wave phenomena; there has been also much earlier work of Pancharatnam on the related phenomenon in optics. The origin of the Berry’s phase is in nonflatness of a parallel transport which appears in the corresponding phase factors.
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<http://www.mi.infm.it/manini/berryphase.html>
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