Probability cannot be conserved over time, because there is no translation-invariant probability measure on the space of times.
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Replying to @ModelOfTheory
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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Replying to @TheMichaelBurge
No, I mean the probability of it currently being between times t and t+𝛿 cannot be independent of t.
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Replying to @ModelOfTheory @TheMichaelBurge
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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Replying to @RadishHarmers @TheMichaelBurge
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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Replying to @ModelOfTheory @TheMichaelBurge
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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Replying to @RadishHarmers @TheMichaelBurge
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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Replying to @ModelOfTheory @TheMichaelBurge
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.
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