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A version of many worlds that works

And why and how you should erase all the other worlds

The many worlds interpretation vaguely envisions some splitting of the world, at special moments that cannot be determined – because there are no special moments in quantum mechanics; according to observables that cannot be determined – because there are no special observables in quantum mechanics; to an unknown number of worlds – because probabilities in quantum mechanics aren't rational in general. And there are other reasons that guarantee that no meaningful many worlds interpretation can exist.

But one may design a many world interpretation that works. However, it's useless: the interpretation may be described as an overly redundant "visualization of subjective probabilities". How does it work?

At every moment \(t\), for every state \(\ket\psi,\) for every linear Hermitian operator \(L,\) for every \(\mu\in(0,1),\) there exists the world \(W(\psi,L,\mu)\).
What do the arguments mean? The time \(t\) is just some time. The wave function \(\psi\) is a state vector associated to the world. Most nontrivially, \(L\) is the first observable that may be measured. When it's measured, the world splits according to the eigenvalues of \(L\) to a continuum of new extra universes labeled by \(\mu\). And in the fractions of the \(\mu\) interval \((0,1)\) corresponding to the measured eigenvalue of \(L\), one collapses the wave function \(\ket\psi\) to \(P(L=\lambda_i)\ket\psi\) at the following moment.

From each world \(W(\psi,L,\mu)\), there continues some regular unitary evolution in which \(\ket\psi\) and \(L\) are evolving according to the particular picture you prefer – Schrödinger, Heisenberg, Dirac, it's your choice. However, from each world, there are also special arrows pointing to all the possible "collapsed worlds" \(W(P(L=\lambda_{i(\mu)})\ket\psi,L_{\rm new},\mu_{\rm new})\).

Here, the subscripts "new" indicate that the arrow leads to all possible values of the new \(L\) and \(\mu\) that is independent of the starting one. However, the new state vector is determined from the old one as \(P(L=\lambda_{i(\mu)})\ket\psi\) where \(i(\mu)\) is the label that determines the possible eigenvalues of the old \(L\) so that \(i(\mu)\) is a non-decreasing function of \(0\lt \mu\lt 1\), all \(i(\mu)\) are eigenvalues of the old \(L\), and the lengths of the intervals at which a given eigenvalue is realized are calculated by Born's rule from the corresponding eigenvalue and the old \(\ket\psi\).

How will you use these many worlds? You just imagine that you live in one of the worlds \(W(\psi,L,\mu)\). You have determined \(\ket\psi\) so you know what it is. However, you're not decided what is \(L\) in your world – you haven't decided what to measure yet – and you don't know the value of \(\mu\) which is randomly distributed in the unit interval. This value of \(\mu\) was generated randomly during the last observation of yours and it's fixed while \(\ket\psi,L\) are evolving according to the equations of the picture.

At any moment, you may decide that you measure something, \(L\). If you decide to measure a particular operator \(L\), it proves that you were in one of the worlds \(W(\psi,L,\mu)\) where both \(\ket\psi\) and \(L\) are known – the state vector is given because Nature imposed it on you during the previous recent measurement; and \(L\) was picked by you because you chose what to measure. Once you measure \(L\), you may be sent to one of the new worlds where \(L_{\rm new}\) is an arbitrary new Hermitian operator that you don't know yet and where the new \(\ket\psi\) is completely determined as \(P(L=\lambda_{i(\mu)})\ket\psi\) – perhaps normalized so that its norm is one.

So these many worlds are always completely ready for any measurement you may want to do as the first one in the coming future (they include all possible "Heisenberg choices"); and they're completely ready to produce all the possible results of the measurement (the "Dirac choice") with the right probabilities. The probabilities are equal to what they should be because the probability is interpreted as the relative fraction of the worlds with a known value of the state vector and the operator which is measured by the length on the unit interval for \(\mu\) – the probability distribution for \(\mu\) is always assumed to be uniform. The randomly generated numbers always arise from \(\mu\) that is always randomly created during every measurement (but the observer, you, can't know what the value is) and it's waiting to affect the projection associated with the new measurement.

Now, what have we gained? We have gained absolutely nothing relatively to proper, "Copenhagen school" quantum mechanics. We still work with a particular world with a particular \(\ket\psi\), we choose a particular \(L\) that we want to measure and it's still up to our free will (but the observable should be sufficiently slowly changing with time for the question to be stable enough; and decohered enough to admit mutually exclusive perceptions), and the only thing we can predict are the probabilities. So all the remaining worlds are absolutely unphysical.

In particular, we don't change anything about the fact that there is no preferred rule that would dictate us "whether we want to measure something at all and when", "what observable we should exactly measure", or "how much decoherence there should have been to allow us to measure it at all". And concerning the Dirac choice – Nature's random generator for the results – we don't gain anything at all, either. We're just "drawing" the other possibilities that were possible before the measurement, but didn't materialize, as "real worlds" somewhere. But we can never restore any physical access to these other worlds.

If someone is driven towards the many worlds faith by the idea that there is some "objective cosmic directive" that tells everyone what should be measured and when it should be measured, he must be disappointed. Nothing like that exists. Also, the random generator is a real one, not a pseudorandom generator. So if someone wants to "predict" all the random numbers from the Dirac choice, he must be disappointed, too.

There's one more thing that unavoidably remains Copenhagen-like. The determination of the "world where I live" remains subjective i.e. dependent on the observer. Different observers learn different things from their measurements – which are intrinsically subjective – so they will place themselves into different worlds. The different observers won't exist in the same world. So nothing is changed – and nothing can be changed – about the subjective character of the observations in quantum mechanics. In this sense, my "many worlds" picture should also be called a "many minds" picture.

Again, to summarize, once you understand that the "beef" must remain that of the Copenhagen school, you may design a many worlds interpretation that adds all these extra new worlds. But they're nothing else than the visualization of possible questions that an observer may ask; and possible answers that Nature could have given even though she only gave a particular one. All these other worlds are completely useless and redundant because they won't ever affect anything in a given world again. That's why a scientist – someone who builds on the actual evidence – should consider them immaterial and erase them from his "picture" of the world.

I mentioned the beef of the "Copenhagen school" that my usable version of the many worlds couldn't change. They are:
  1. Heisenberg choice is up to free will: there must exist free will of the observer who must know – without being "told" by anybody or some laws – what operator he wants to measure and when. In principle, all things that "may be measured" are equally allowed. Quantum mechanics only produces answers (in terms of probabilities) once a question is well-defined, and a question requires \(L\) to be determined by the observer.
  2. Dirac choice is generated by a pure random generator: the random results are really probabilistic so there can't be any hidden variables or "pseudorandom generators" that would "explain" the outcomes.
For these two reasons, you can't possibly gain any knowledge or understanding if you try to "go beyond Copenhagen" which simply requires questions we want to predict to be well-defined – the choice of the observable \(L\) must be inserted as a part of the question; and if you try to "go beyond Copenhagen" when it comes to the true randomness of the outcomes. At most, a truly neutral version of the many worlds of my type is allowed. But it must be considered physically equivalent to the Copenhagen school quantum mechanics.

It's just a different "visualization" what's going on – that draws some "no longer relevant" choices or outcomes as "real" although they're inconsequential for the subsequent evolution of the chosen observer's life – and even before quantum mechanics was born, the probability calculus was invented exactly to liberate us from the duty to draw all these no longer relevant options.

My version of the many worlds contains infinitely many worlds – they form a continuum with many continuous coordinates included in \(\ket\psi,L,\mu\). I would claim that the splitting of the many worlds according to \(\ket\psi\) as well as \(L\) as well as some \(\mu\) is necessary (my number is the minimum one) – you need to multiply the number of worlds both to deal with the Heisenberg choice and the Dirac choice. But you could increase the number of the worlds. For example, instead of depending "just" on \(\ket\psi\), the world could depend on a whole pre-history of measurements which remembers "how we got to \(\ket\psi\)". In other words, it would remember a possible path how we got to \(\ket\psi\) through a sequence of projections. Again, this would be just a redundant addition because all the future predictions for \(L\) only depend on \(\ket\psi\), not on how we got to \(\ket\psi\).

Copenhagen school has been using "Occam's razor", if you wish. "Entities [e.g. worlds] are not to be multiplied without necessity" (Non sunt multiplicanda entia sine necessitate). Bohr and his guys avoided imagining e.g. the path above to be "real" because this path isn't relevant for any further predictions. So it's not "real". Only the results of measurements are "real" – but this reality is subjective – and only the probabilities may be predicted. It's important that the predictions don't depend on the path in the tree, on the way how you could write \(\ket\psi\) as a sum of many pieces, and on anything else. That's why a sensible scientist erases these distracting features – he labels them unphysical because they're unobservable, even in principle.

The only physical things are those that the Copenhagen school talks about. That's why my minimal worlds were only labeled by \(\ket\psi\), \(L\), and \(\mu\) – by the knowledge about the physical system, by the choice what he wants to know next, and by something that produces the random numbers. You're advised to reduce this minimal system further – to agree with your perceptions that the "worlds" with other values of \(\ket\psi,L,\mu\) than the relevant ones "don't exist". The usage of this non-existence assumption is nothing else than the standard usage of the probability calculus.

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