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mfb comments on Causal Reference - Less Wrong
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The Standard Model of particle physics with MWI is time-symmetric (to be precise: CPT symmetric) and conserves information. If you define the precise state at one point in time, you can calculate the unique past which lead to that state and the unique future which will evolve from that state. Note that for general states, "past" and "future" are arbitrary definitions.
(Which is why I specified a different set of laws of physics.)