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Geometric phase of a parallel transport over the surface of a sphere 
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#1
Nov113, 06:37 AM

P: 1

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.
The vector performing parallel transport is defined as ψ=(e+ie')/√2, satisfying the parallel transport law, Imψ*・dψ=0. Then another local basis was defined, n(r)=(u(r)+iv(r))/√2, and ψ=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 between n and ψ here, 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. 


#2
Nov213, 11:09 AM

Sci Advisor
HW Helper
PF Gold
P: 2,602

The phase is local, expressed as ##\alpha=\alpha(\mathbf{t})##, where ##t## is a parameter on the path. You should be able to compute that ## \mathrm{Im}\mathbf{n}^* \cdot d\mathbf{n} = d\alpha##.



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