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?
Thanks for the comments!
I agree with all of these points. In fact, with respect to (4) it is even plausible that some “once infected” people never go on to develop the kind of antibodies that are being tested for. Point (3) is why I control for age/sex, but of course there are a number of further complexities.
These further complexities, along with (1) and (2) are currently “un-modelable complexities” for me. There are just so many selection effects in play that it isn’t clear if you gain anything from trying to take them into account. Given that there are a number of papers that try to calculate some kind of IFR making a number of basic mistakes, I wanted to set out to see what the data + simple models gets if you do the maths correctly, as opposed to doing ridiculous things like caring about the median study like the Ioannadis meta-analysis does. After all, it seems like the IFR is what everyone cares about, so it would be nice if we were doing things “less wrong” here.