Probability cannot be conserved over time, because there is no translation-invariant probability measure on the space of times.
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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If there is an ontologically real probability measure on configurations of the universe, one might expect it not to only be defined conditioned on time. And if so, how total probability changes over time is something one could reasonably ask of a time evolution operator.
End of conversation
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