I’m very confused by the mathematical setup. Probably it’s because I’m a mathematician and not a physicist, so I don’t see things that would be clear for a physicists. My knowledge of quantum mechanics is very very basic, but nonzero. Here’s how I rewrote the setup part of your paper as I was going along, I hope I got everything right.
You have a system which is some (seperable, complex, etc..) Hilbert space. You also have an observer system O (which is also a Hilbert space). Elements of various Hilbert spaces are called “states”. Then you have the joint system of which is an element of, which comes with a (unitary) time-evolution . Now if were not being observed, it would evolve by some (unitary) time-evolution . We assume (though I think functional analysis gives this to use for free) that is an orthonormal basis of eigenfunctions of , with eigenvalues .
Ok, now comes the trick: we assume that observation doesn’t change the system, i.e. that the -component of is . Wait, that doesn’t make sense! doesn’t have an “-component”, something like an -component makes sense only for pure states, if you have mixed states then the idea breaks down. Ok, so we assume that , when acting on pure states, is equal to . So this would give , where is defined so that this holds. Presumably something goes wrong if we do this, so we instead require the weaker . And bingo! Since the are eigenfunctions, we get that , and let’s redefine to include the term because why not. Now, if we extend by linearity we get that . Applying again gives , and the same for further powers.
Ok, let’s interpret that last part in terms of “observations”. If we take states of the combined system , then time-evolution maps pure states with only a component to pure states with only a component. Wait, that’s exactly what we assumed, why should we be surprised? Well yeah, but if you started out with some linear combination of eigenfunctions, these will be mapped to a linear combination of pure states, and each pure state in this linear combination evolves as assumed, which may or may not be abig deal to you. In a mixed state that is a linear combination of pure states, we call each pure state a “separate observer” or something like this. Of course, mixed states in a tensor product state cannot be uniquely be written as a sum of pure states. However, if we take our preferred basis and express our mixed states as pure states with respect to that basis in the -component, this again makes sense.
So it’s super important that we have already distinguished the eigenfunctions of at the start, we unfortunately don’t get them out “naturally”. But I guess we learn something about consistency, in the sense that “if eigenfunctions are important, then eigenfunctions are important”.
Ok, now assume our system is itself a tensor-product of subsystems , which we think of as “repeating a measurement”. Now what we get if we start with some pure-state is (in general) a mixed state which can be written as a linear combination of pure states of eigenfunctions. As the eigenfunctions of the different systems are different (they are elements of different spaces), if you start out with some non-eigenfunction in each subsystem, you’ll end up with some mixed state that contains different eigenfunctions for the different systems. The “derivation of the Born rule” doesn’t need this step with multiple systems. Basically, we can see this already with just one system. If we start with a non-eigenfunction , then this gets mapped to some linear combination of pure states via the time-evolution. As the time-evolution is unitary, and the |a_i|^2 sum to 1, we can see that each pure state has “length” |a_i|^2.
Thanks for the great paper! I think I’ve finally understood the Everett interpretation.I think the basic point is that if you start by distinguishing your eigenfunctions, then you naturally get out distinguished eigenfunctions. Which is kind of disappointing, because the fact that eigenfunctions are so important is what I find weirdest about QM. I mean I could accept that the Schrödinger equation gives the evolution of the wave-function, but why care about its eigenfunctions so much?
While I agree with the conclusion, I really do vehemently object to the framing. Who is “we”, and what on earth is “global democratic culture”?
I mean these are two separate things, and depending on who you ask rewriting history is what the West is pretty good at. But again, what “Western consensus”? Depending on who you ask, either Russia or the West is massively “rewriting history” regarding Ukraine.
If I’m not mistaken, Bandera was interned at a German concentration camp when the atrocities took place. Which is quite a major detail, “our Western consensus” generally considers people innocent by default.
Ah, a German perspective, this somehow fits in very well with the extreme focus on nazis and/or nazi symbols as a kind of axiom from which everything else is derived. Here’s an anecdote: I was friends with an Ukranian, who once wore a stylized swaztika necklace. Was she a Nazi? She was adamant that she was not. Turns out she just liked the pattern, had seen it being used in India, and didn’t treat it as anathema the way a German would. The take that symbols that were used by nazis aren’t proof of evil is an underrated one (especially in Germany). It’s your actions that define you, not which symbols or Russell conjugation you use (or don’t use) when talking about them (“I am a patriot, he is a nationalist, they are a government with a democratic mandate to put their population first”).
That said: nationalism really is a problem in Ukraine at the moment. But I’m more worried about the kind of nationalism the West tends to support (making sure everyone says “Kharkiv” instead of “Kharkov”, calling Russian soldiers “orcs”, insisting that Crimea “is” Ukranian), rather than the “using Nazi symbols sometimes” form.
Unfortunately I don’t think this harms their European support much—these are all things that are far in the past, and haven’t stopped the EU waxing poetic about how Ukraine is fundamentally one of them. In fact, since in Ukraine, nationalism = against Russia, and “against Russia” is popular in Eastern Europe (for obvious reasons), the nationalist aspects may even find them support from the West. That said, it’s just morally a bad policy, and absolutely affects how Ukraine is seen by Russia.
Unfortunately not, people who don’t want to be in Crimea will have left, and polls aren’t always reliable.
The non-Crimeans Tatars are somewhat irrelevant here given that most are Russian.
There’s a strong case that punishing Syrians for being ruled by Assad might just feel morally wrong too, but who am I to argue with a moral authority like the “Western consensus”.
Thank you for the great post on the clusterf*ck that is Ukraine and Ukrainian politics. Really this is a great case of a situation where it isn’t clear what is actually going on given the rather one-sided reporting on both sides.
No they don’t. The “referendum” of 2014 was a joke. Nobody needs to respect this joke.