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To my knowledge weak localization is a result of constructive interference between scattering processes and their time reversed.

Traveling a given path in a vector potential leads to the phase $q\int d\vec{x}\ \vec{A}(x)$. A particle traveling the time reversed path gets an additional "$-$"sign as the path is inverted.

Somehow the spin flip seems to give an additional sign resulting in constructive interference. How is that? How is the phase of a particle in an vector potential connected to the spin?

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The spin of a particle flips under time reversal. It has nothing to do with the phase of the particle! You may go through time reversal symmetry for more information about spin flipping.

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