Is this in conflict with the fact that time evolution is unitary?
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This is because events become more certain as time passes? A coin flip with a 50% probability of heads becomes closer to 100% or 0% as time progresses.
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No, I mean the probability of it currently being between times t and t+𝛿 cannot be independent of t.
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Of course, what people really mean isn't that the total of P(X = x and Time = t) over all x is preserved/invariant over all times t, but rather, that the total of P(X = x | Time = t) over all x is invariant (indeed, invariantly 1) over all times t, no?
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Sure, but if probability is squared norm in QM, and a time evolution operator changes total probability, wouldn't that be non-unitary? We could still scale the result back to norm 1, and the composition of the operator and the rescaling would be non-linear.
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The unitary time evolution operator is the one which evolves/yields information about P(X = x | T = t). You shouldn't think of it as giving you any information whatsoever about P(T = t); the information it encodes and acts upon is entirely separate from that.
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Yes, but conditioning everything on what time it is strikes me as an odd thing to be happening in fundamental physics.
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It hardly seems like an odd thing for the "time evolution" operator to be doing, though...
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That is, it's precisely what it means to be the "time evolution" operator, to be the operator which evolves the probability distribution P_t into the probability distribution P_{t'}.
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