You might think this is a trivial question. I have read some articles on this topic, and I know there is a surging interest on this topic recently. A lot of researchers, particularly scientists and engineers in optics are actively looking for new possible applications of Berry's phase. What I really mean is, people may have mixed up something else with Berry's phase. Let me say a few things. I might be totally wrong. Please point out any silly remarks or mistakes, misunderstandings in case what I say in the following
1 How is Berry's phase is incorporated in Feynman's path integral approach? I haven't found a good paper on that.
2. Is Hyungen's principle, which has fundamental importance in optics, compatible with Berry's phase?
3 Let's just talk about electromagnetic waves. Berry's phase is related to the enclosed area of a closed loop that light travels. It seems to me that at least in certain cases, there is a fundamental difference between different close contours of different topological nature. It seems to me that certain experimental verification of Berry's phase in optics might just be the effect of singularities.
4 What's the connection between topological singularities in the electromagnetic field and Berry's phase? I have a feeling that people are talking indiscriminately about a lot of things all labeled "Berry's phase". I'm not mathematically good enough to sort this all out. For example, as far as light polarization is concerned, I feel that there should be some connections between the group structure of SL(C,2) and Berry's phase. I might be wrong, or it might be obvious.
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