born-rulequantum mechanicsquantum-interpretations
The Many Worlds interpretation suffer from at least 2 "wounds", the preferred basis issue and perhaps the most notorious probability issue.
How do you make sense of probability in a model where everything happens?
There are all these elaborate attempts at deriving Born Rule by Wallace et al. and at least 20 different people at arXiv.
And there are some very good papers that provide criticism for these attempts, but my question is why isn't Gleason's Theorem enough?
A simpler example than putting humans inside your equations might be more clear. Imagine a single electron with some spin.
It is going to enter a Stern-Gerlach device as a beam going in the positive y direction and the Stern-Gerlach device will deflect a spin up beam entirely left. And the Stern-Gerlach device will deflect a spin down beam entirely right. These directions, y , left, and right were all details that were determined by the initial setup of the beam and the device.
But the device does this not because the Stern-Gerlach device is magical, it does so because the Stern-Gerlach device has an inhomogeneous magnetic field and there is a particular Hamiltonian for magnetic fields and spin and the Hamiltonian dictates how things evolve. So anyone that agrees about how the Schrödinger equation evolves states agrees this happens. Copenhagen, Many-Worlds Interpretation (which was not called Many Worlds by its creator and I think its creator did not mention the words many worlds), Decoherence, Ithaca, dBB, etcetera.
So the Schrödinger equation has a wave that is a beam spread in the x and z direction and with an initial probability current pointing in the y direction for an initial condition of the wave function. And it evolves according to the Schrödinger equation. Now, what if the spin of the particle is a superposition of spin up and spin down? Again, we can just mathematically change the initial wavefunction and then mathematically evolve it according to the Schrödinger equation for the particular Hamiltonian that accurately describes the actual inhomogeneous magnetic field in the exact actual Stern-Gerlach device and then again every theory that agrees with the Schrödinger equation can evolve it and they all predict the same thing. They see two beams coming out, one with spin polarized as spin up, one with spin polarized as spin down.
And again, any theory that uses the Schrödinger equation can track the so-called probability current and see streamlines in the initial beam, some of which end up in that left deflected beam, some which end up in that right deflected beam. And they can see the beam, as a whole, mathematically split into two like a stream that forks when there is a hill in the way. They see that there is more beam deflected left or right (in the L2 norm sense) depending on how big a superposition you had of spin up versus spin down in the initial wave.
And the Schrödinger equation when used for the actual experimental setup also predicts that the spin of the particle continuously evolves over time so that eventually the left beam has a spin that is up and the spin of the right beam has a spin that is down.
Nobody disagrees that this is exactly what the Schrödinger equation predicts for the actual experimental setup. So the equation that makes the actual predictions predicts that one beam with spin not aligned purely in the up or down direction becomes (continuously over time) two beams each of which has a spin that is aligned in the up or down direction.
Effectively the Stern-Gerlach device has co-evolved the spin and the position so the two are now entangled you have (wave on the left and spin up) added to (wave on the right and spin down) and they are orthogonal, orthogonal for two reasons now, whereas when it was incoming beam with spin of (superposition of up and down) they were orthogonal for only one reason, the spin, which could have been written as a super position of many different orthogonal states. Now that the interaction has separated the beam into two spatially separate beams, future things that interact based on where the beam is can by surrogacy be correlated with the spin.
Great. And everyone that uses the Schrödinger equation and is also willing to bother to use it on the actual experimental setup (a rarely utilized option) agrees this is what happens.
Now it might help to reveal a big fact that isn't mentioned much, and again it is just about the Schrödinger equation. If you have two particles or more, then the wave function is not a wave like the electric field or the magnetic field, it is not defined in actual space. It is defined on configuration space, a space that has a different x for every particle a different y for every particle and a different z for every particle. So for two particles your function is $\Psi=\Psi(x_1,y_1,z_1,x_2,y_2,z_2,t)$ so for every point in the 6d configuration space you are specifying a full configuration, you are saying where each and every particle is.
So this space is huge. If two beams go off in slightly different directions and bounce off separate things and go into different directions and possibly different and changing speeds when you have $10^{26}$ or more particles the chance that two beams ever cross each other once they get pretty far away is vanishingly small. This is key.
Because the Schrödinger equation says that if two beams don't overlap and beam one by itself evolves its own way and beam two by itself evolves its own way and those evolved versions never overlapped then the evolution of the sum evolves as the sum of those two separately evolved beams, including the probability current at each point being the probability current in the separate beams.
So the Schrödinger equation predicts that in absolutely every way the two beams evolve as if they were the only beam when they get to a point in time where they will never overlap again. And this happens when devices like Stern-Gerlach devices separate them and then further interactions with lots of particles have the lots of other particles evolve differently with the two different beams. So far there is no mention of Many Worlds. And any theory that uses the Schrödinger equation for the actual setup will predict this.
So now let's talk about Many Worlds. These beams that never again overlap act like they are the only beam, they act like they are a world unto their own. The math says so. The math everyone uses that uses the Schrödinger equation. So why not just let them them be a World unto their own? Your personal experience is generated by the state of your neurons and such, so the wave function of every particle includes the configurations of your neurons and such. So there is a beam with your neurons in some configurations (the wave is nonzero there) and there is a completely separate non overlapping beam (the beam is defined in configuration space so if just one particle has the beam separated to be non overlapping in the direction for that particle it is like separating a beam in the x direction the whole beam is separated). A separate non overlapping beam for your neurons is a possibly different collection of configurations (or possibly the same, maybe the beam has separated in other places but the dynamics of your neurons haven't changed yet, but the beams as a whole are separate so the math that every single person that uses the Schrödinger equation trusts says the beams won't affect the so called probability-current or the evolution of the other if they never overlap again).
OK, so in the future devices make beams become non overlapping. And if they interact later with large numbers of particles the huge number of particles make it incredibly unlikely they will ever overlap again, so each can and does evolve as if it were the only beam. In Many Worlds you call those worlds (and the creator didn't but the first major popularizer did). But that is just because they then evolve as if they were the only beam in the world. The key is their separateness, a separateness that is already and solely predicted by the Schrödinger equation. That same Schrödinger equation predicts that each of those beams exists and each acts as if it were the only one.
So there is one configuration space, and one wavefunction, the one predicted by the one Schrödinger equation. It's just that it naturally evolves into separated beams that then act independently as if they were the only beam in the world. In Many Worlds you realize that each acts on its own after it separates and realize that any subjective experience is based on your body's configuration so each separately acting beam has a separately acting arrangement of your body, so the different beams include possibly differently arranged yous and each are just as important and valid because we predict them all and each acts like it is the only one that survived the beam splitting and interaction with a large number of particles.
Instead of calling it the Copenhagen interpretation you could call it the Solipsism interpretation (after all Many Worlds wasn't called Many Worlds by its creator) because with a Solipsist's Interpretation, only one of those beams can be real the one with "you" in it.
And a die hard solipsists will sometimes go so far as to throw the baby of science out the window with the bathwater of non-solipsism.
Why? Because the Schrödinger equation predicted all these separate beams. So how can you end up with just one? You'd have to do something other than the Schrödinger equation. And so you'd have to draw a line somewhere and say that some magic happens somewhere and the Schrödinger equation doesn't hold there or then. But if we do an experiment to test that, you are always wrong.
Why? Because by carefully arranging beam reflectors we can get those beams to bounce back and overlap and show that every beam was still there all along, every single one, for any length of time, so was there all along. Always, every single time. So you can just find out how many beam reflectors we can make and how precisely we can arrange them (find out our current technical expertise) and then postulate that raw unobservable (because you specifically designed it to happen where we can't test it) magic happens and that the Schrödinger equation doesn't hold there because it interferes with your self centered egotistical solipsism. Which is really just a kind of sexism and racism and hating of other people and things that is so profound that you just refuse to believe that anything other than you exists (even a body just like yours that was entirely the same up until one day it dynamically evolved the way you would if you interacted with a world where a Stern-Gerlach device deflected a beam differently than the one you personally subjectively experienced) to the point were you make unscience to defend your solipsism.
Why unscience? Because drawing the lines in different places means when our technical expertise advances we get an infinite number of different predictions. So Copenhagen can't even make predictions. It, Copenhagen, isn't even science when it wants to have magical collapses to defend solipsism. And there is no reason to hold one of the splittings of the beam as special just because you experienced it.
Let's get this straight, there is nothing wrong about treating those other worlds as if they don't exist because they don't affect your world. But to claim that something happened to them just because you want to feel special and to destroy science itself and the ability to make predictions just because you want to have your individual and personal subjective experience be the center of the universe is misguided.
Ignore the other worlds because you can. And be aware of when you can and don't do it too early or in situations where it doesn't apply. Pretending they aren't there is a bit dishonest and can lead to wrongness.
I think I have understood your question now (and I deleted my previous answer since it actually referred to the wrong question). Let me try to summarize.
On the one hand we have a wavefunction $\psi$ in the Hilbert space $L^2(\mathbb R)$ for a given quantum system $S$ and we know that $\psi$ determines the state of $S$ in some (unspecified) sense: it can be used to extract transition probabilities and probabilities of outcomes when measuring observables.
($\psi$ could arise from some analogy optics - mechanics and can have some meaning different from that in Copenaghen intepretation, e.g. a Bohmian wave.)
On the other hand we know, from Gleason's theorem, that an (extremal to stick to the simplest case) probability measure associated to $S$ can be viewed as a a wavefunction $\phi \in L^2(\mathbb R)$ and Born's rule can be now safely used to compute the various probabilities of outcomes.
You would like to understand if necessarily $\psi=\phi$ up to phases as a consequence of Gleason's theorem.
Without further requirements on the procedure to extract transition probabilities (you only say that transitions probabilities can be extracted from $\psi$ with some unspecified procedure) it is not possible to conclude that $\psi=\phi$ up to phases in spite of Gleason's theorem.
We can only conclude that there must be an injective map $$F : L^2(\mathbb R) \ni \psi \to [\phi_\psi] \in L^2(\mathbb R)/\sim$$ where $[\cdot]$ denotes the equivalence class of unit vectors up to phases.
A trivial example of $F$ is $$\phi_\psi := \frac{1}{||\psi+ \chi||}(\psi + \chi)\quad\mbox{and} \quad \phi_{-\chi} := -\chi$$ where $\chi$ is a given (universal) unit vector.
This map is evidently non-physical since there is no reasonable way to fix $\chi$ and, assuming this form of $F$, some argument based on homogeneity of physical space would rule out $\chi$. However much more complicated functions $F$ (not affine nor linear) can be proposed and in the absence of further physical requirements on the correspondence $\psi$-$\phi_\psi$ (e.g. one may assume that some superposition principle is preserved by this correspondence) or on the way to extract probabilities from $\psi$, Gleason's theorem alone cannot establish the form of $F$.
Best Answer
The immediate problem with obtaining the Born rule in the many-worlds interpretation is quite elementary: you can't even begin to attach probabilities to "worlds" (or to events within worlds), in your theory of many worlds, if the theory isn't even clear on what a world is.
Physical states according to various interpretations
In classical physics, a physical state is a configuration of particles and/or fields.
In quantum physics according to the Copenhagen interpretation, a "quantum state" (for the present discussion, let's say it's a vector in a Hilbert space) is an abstract, second-order "state" which provides probabilities regarding the actual physical state. The actual physical state is like a classical physical state (configuration of particles and/or fields) except that the uncertainty principle prevents a complete specification.
In quantum physics according to the many-worlds interpretation, the quantum state is the physical state. But then we need to understand how the physical reality we observe relates to this quantum state. It should be a particular "part" of the quantum state, with other, similar parts being the other worlds parallel to our own.
However, there is no consensus among many-worlds advocates on how to answer that question. One might suppose that superposition has something to do with the answer, because it is about putting two quantum states together to get a combined quantum state. But given a quantum state, there is no unique decomposition of it into a set of superposed states. There are infinitely many sets of basis vectors available, and even if we restrict ourselves to states which are eigenstates of classical observables like position or momentum, you still have the choice forced on you by the uncertainty principle.
What is an Everett world?
If you look into the many-worlds literature, formal and informal, you will find people advocating the position basis, people advocating a basis determined by decoherence, and people saying that all bases are equally valid. There was at one time a hope that Gell-Mann and Hartle's consistent histories formalism would lead to the discovery of a unique basis that is quasiclassical and maximally fine-grained, but I don't see people talking about that any more.
Conversations with ordinary physicists who believe in many worlds have left me with the impression that most of them don't have a logically coherent concept of what an Everett world is. The worldview seems to be operationally the same as Copenhagen - use state vectors to obtain probabilities - but this is then overlaid with a belief that "the wavefunction exists", and some vague significance attached to decoherence.
If you think imprecisely like that, you are in danger of never even noticing the real problems that a many-worlds interpretation faces. Einstein once described the Copenhagen interpretation as a "tranquilizing philosophy", and it seems that this informal version of many-worlds, in which one goes on using quantum mechanics exactly as before, but one now proclaims that the quantum state is reality, similarly provides many contemporary physicists with mental peace, without actually providing answers.
An example of a many-worlds theory which does specify exactly what the worlds are
So we can't even begin to have this discussion unless we settle on a particular version of many-worlds; and some versions of many-worlds are just logically incoherent - for example, one according to which the "splitting into worlds" is observer-dependent. You the observer are supposed to be inhabiting just one world of many, so this would make your own individual existence "observer-dependent". A lot of the prose written about many-worlds eventually lapses into incoherence, by talking about observer-dependent observers, worlds that differ in their degree of realness, and other conceptual misadventures - though the authors of these concepts no doubt regard them as daring insights that need to be accepted or contemplated.
Ideas like that can't be analyzed in the way you would normally evaluate a proposition about physical reality, e.g. by checking it against the evidence. All you can do is try to bring out the conceptual incoherence and make it obvious, which is a thankless task. So I won't further try to address that sort of many-world theory. Instead, for the purposes of discussion, I will focus on Julian Barbour's "Platonia" theory.
Barbour is at least very clear about what he thinks exists. He is a quantum cosmologist, and he proposes that what exists are all possible spatial configurations of the universe. He calls them "time capsules": time is not real, nothing is actually happening anywhere, but some of these static configurations contain what looks like evidence of a past - memories or other physical traces.
The theory is therefore quite crazy - he's saying that time isn't real, that despite appearances one moment does not flow into another. It also has the feature that it doesn't ontologically satisfy special relativity - for that you have to have space-time, and here you only have space. This is a problem that will plague many attempts to be precise about what the Everett worlds are. Copenhagen quantum mechanics is relativistic because reality is events in space-time, a change of coordinate systems is just a relabeling of events, and state vectors are just calculating devices. But the many-worlds interpretation reifies state vectors (it stipulates that they are "elements of reality"), and it's really hard to see how you can do that without also reifying the reference frame in which they are defined.
However, your question was about the Born rule, and not relativity, so let us leave these other problems and return to Barbour's theory. Barbour interprets the wavefunction of the universe by saying that the various configurations making up "configuration space" are what's real, and the Born rule supplies the "measure" which tells us how to "count" them. Normally we would say it's a probability measure, but here, by hypothesis, all these worlds are equally real, so perhaps we should say it's a "reality measure".
Even though we have here arrived at a precise statement from Barbour about what it is that exists (at the level of "worlds"), there are still formidable problems in making sense of it (beyond the already stated problems, the problem of time being unreal, and the problem of relativity not applying ontologically). It seems that, in order to explain the observation of Born-rule frequencies in reality, we have to regard the measure on configuration space as a prior (in the Bayesian sense), which we can then combine with intra-world relative frequencies in order to obtain conditional probabilities for the outcome of experiments. That is, if physical occurrence A is accompanied by physical occurrence B1 3/4 of the time, and by the alternative B2 just 1/4 of the time, that is because the combined measure of the (A & B1) worlds is three times the combined measure of the (A & B2) worlds.
But it seems a little strange to be using a nonuniform measure at all. When you do calculus, you start with a uniform measure like Lebesgue measure, and then the "nonuniformity" of the integral comes about because the function you're integrating over is not constant. Here we are asked to introduce the nonuniformity at the level of the measure itself. This is mathematically possible, but does it make sense as a statement about reality? In my opinion, the sensible interpretation of a nonuniform measure in a multiverse theory (insofar as one can ever be "sensible" about such matters) is that it means that the worlds are duplicated, in proportion to the deviation from uniformity. The true measure will be the natural, uniform one, and the Born frequencies have to come about from the duplication of worlds.
So what about Gleason's Theorem?
So far I haven't said a thing about Gleason's Theorem. But I consider it essential to first spell out what a real discussion of a many-worlds ontology would look like. Either your theory has to say exactly what the worlds are, so we can then have the discussion about how the Born rule could work in that model, or we are stuck in the mystical realm of hugging the wavefunction and loving its many-in-oneness. Hopefully it's obvious why Gleason's Theorem is not enough to obtain the Born rule in the latter type of many-worlds interpretation: there isn't actually a theory there. But the resistance to taking the other path is immense, because all this ugliness like having a preferred basis and even an ontologically preferred reference frame tends to appear. Perhaps it's a point in favor of the physical intuition of "mystical" many-worlders that they don't want to take that path - they sense the ugliness of the consequences - but remaining content with a studiously vague concept of Everett world is a point against their intellectual rigor.
As for the ontological implications of Gleason's Theorem - whether for a genuinely rigorous many-worlds theory, or for some other interpretation of quantum mechanics - I'm really not sure. It seems hard to escape the conclusion that a many-worlds theory in which the worlds are defined has something like a preferred basis. In that case, applying the Born rule is certainly consistent with the theorem (though there would still remain the question of what a nonuniform measure on the worlds means ontologically - are the worlds duplicated? are the actual worlds just an appropriately weighted discrete sampling from a continuum of possible worlds?).
But it would be a somewhat trivial consistency, because of the preferred basis. The interesting thing about Gleason's measure is that it is defined for subspaces in a basis-independent way. This is one reason why it's appealing for mystical many-worlders who don't want to have an ontologically preferred basis; it seems to promise a perspective in which the quantum state is primary, and a division into individual worlds is just a matter of perspective. But this leads to the paradox of observer-dependent observers, or the problem of oneself being something less than absolutely real.
I note that Gleason's theorem has played a small role in the reception accorded to a completely different interpretation, Bohmian mechanics. Gleason's theorem was at one time taken as a proof of the impossibility of hidden variables, but John Bell pointed out that it's only inconsistent with noncontextual hidden-variable theories, in which all observables simultaneously have sharp values. Bohmian mechanics is a contextual theory in which position has a preferred status, and in which other observables take on their measured values because of the measurement interaction. This runs against the belief in ontological equality of all observables; but perhaps reflecting on the status of Gleason's theorem within the Bohmian ontology will tell us something about its meaning for the real world.