I agree that the CS might be the most physically relevant thing, but at the end of the day the integration over LIPS volume is essentially classical physics (including SR). All the quantum stuff is in the matrix elements.
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cross sections are cool. and as a structural realist, i think they are real patterns. but i don't think they are nearly fundamental. they are a statistic that summarizes some facts about effective field theory in a limiting case.
I feel like we could make that same statement with matrix elements though of course those aren’t “purely” measurable but perhaps they are in principle