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Why are amplitudes complex?

[By prior agreement, this post will be cross-posted on Microsoft’s Q# blog, even though it has nothing to do with the Q# programming language.  It does, however, contain many examples that might be fun to implement in Q#!] Why should Nature have been quantum-mechanical?  It’s totally unclear what would count as an answer to such a question, and also totally clear that people will never stop asking it. Short of an ultimate answer, we can at least try to explain why, if you want this or that piece...

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