I have this question on the calculation of the geometric phase (Berry phase) of a parallel transporting vector over the surface of a sphere, illustrated by Prof. Berry for example in the attached file starting on page 2.(adsbygoogle = window.adsbygoogle || []).push({});

The vector performing parallel transport is defined asψ=(e+ie')/√2,

satisfying the parallel transport law, Imψ*・dψ=0.

Then another local basis was defined,n(r)=(u(r)+iv(r))/√2,

andψ=n(r)exp(-iα).

Together the geometric phase (or so called anholonomy) is given as

α(C)=Im∫_{C}n*・dn.

I can't see the difference betweennandψhere, except for a phase factor α. I think both of them performing the same parallel transport with α being constant. But why

Imψ*・dψ=0 while Imn*・dn≠0, even with the latter being a gauge of the geometric phase?

Thanks in advance.

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# Geometric phase of a parallel transport over the surface of a sphere

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