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entropy1
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Because I understand that for unitary evolution, MWI is required, which suggests that for different interpretations, there may not be unitary evolution?
Is collapse a subset of MWI?If MWI is correct, then the apparent non-unitary evolution of the standard interpretation is can be derived, rather than needing to be postulated.
No.Is collapse a subset of MWI?
Is the math different in either case?They are two different ways of explaining the same thing
The consequence of collapse is selection of a single eigenstate, where the consequence of unitary evolution would be coexistence of all eigenstates, right? (Well, not really 'coexistence' I guess )The maths is the same. Otherwise the consequences would be different.
for unitary evolution, MWI is required
Is that not so?Why do you think that?
Is that not so?
the maths is different for standard quantum mechanics and for the MWI
I think this needs to be clarified. The math that actually makes predictions that are compared with measurements is the same for all interpretations of QM. MWI, as you say, derives this math from the assumption of unitary evolution at all times, even through measurements; but that assumption does not lead to any different experimental predictions, it's just a different underlying set of assumptions.
To clarify even more, I would say "tries to derive" since there is no consensus that the derivation is possible.
They are the same for all practical purposes because of the practical impossibility of observing interference between macroscopically different possibilities.
I would view this somewhat differently. As I view it, in the standard math of QM, the definition of "macroscopic" is "observing interference is impossible". Or, to put it another way, the standard math of QM only says that a measurement has occurred when observing interference is impossible. (The decoherence program fleshes all this out with a lot more detail about how the point when observing interference is impossible is reached in practice.) If we had two alternatives that seemed "macroscopic" intuitively but between which interference was possible, a collapse interpretation would say no measurement had yet occurred.
If there were differences then it should be possible to point them out explicitly, say this equation or expression is different in this interpretation .
Yes, but this is a difference in the interpretation not the mathematics. In a collapse interpretation one says that after a measurement the wave function collapses, say from ##\alpha|a\rangle+\beta|b\rangle## to ##|a\rangle##, and from then on you use ##|a\rangle##. In a different interpretation one says that after the measurement the universe splits (or some such thing), and since you observed ##\alpha## and things have decohered from then on you are in that branch and you use ##|a\rangle##.In a collapse interpretation, the wave function changes following a measurement. So the probabilities for the next measurement are different, depending on whether there was a previous collapse, or not. In principle, that's a difference, but in practice, it's not observable.
So is 'outcome ##|a\rangle## given measurement outcome ##\alpha##' the same as ##|U_a\rangle|a\rangle##?Yes, but this is a difference in the interpretation not the mathematics. In a collapse interpretation one says that after a measurement the wave function collapses, say from ##\alpha|a\rangle+\beta|b\rangle## to ##|a\rangle##, and from then on you use ##|a\rangle##. In a different interpretation one says that after the measurement the universe splits (or some such thing), and since you observed ##\alpha## and things have decohered from then on you are in that branch and you use ##|a\rangle##.
Yes, but this is a difference in the interpretation not the mathematics. In a collapse interpretation one says that after a measurement the wave function collapses, say from ##\alpha|a\rangle+\beta|b\rangle## to ##|a\rangle##, and from then on you use ##|a\rangle##. In a different interpretation one says that after the measurement the universe splits (or some such thing), and since you observed ##\alpha## and things have decohered from then on you are in that branch and you use ##|a\rangle##.
Yes, but whether you include it or not is a matter of interpretation, not mathematics. For instance you can sayNow, for practical purposes, if ##C## and ##D## are macroscopically distinguishable, then either ##P_{CB}## or ##P_{DB}## will be completely negligible. So the interference term will be effectively zero. But mathematically, it's not exactly zero.
If the difference between ##C## and ##D## is that it represents two alternative intermediate measurement results, then a split of the world means that the interference term should be left out of the probability. No split means that the interference term should be included. That's what I mean by saying that there is no a mathematical difference between interpretations.If the difference between ##C## and ##D## is that it represents two alternative intermediate measurement results, then a collapse means that the interference term should be left out of the probability. No collapse means that the interference term should be included. That's what I mean by saying that there is a mathematical difference between the collapse and no-collapse interpretations.
Yes, but whether you include it or not is a matter of interpretation, not mathematics.
If the difference between ##C## and ##D## is that it represents two alternative intermediate measurement results, then a split of the world means that the interference term should be left out of the probability. No split means that the interference term should be included. That's what I mean by saying that there is no a mathematical difference between interpretations.