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?

  • $\begingroup$ Suggestion: You could add a few links to make the question easy accessible for the reader. $\endgroup$
    – Qmechanic
    Commented Feb 10, 2012 at 13:11
  • $\begingroup$ Don't really know what else I can add other than a wiki link $\endgroup$ Commented Feb 10, 2012 at 13:23
  • 1
    $\begingroup$ You could, e.g., link to Wallace et al. and the most relevant papers from the arXiv. $\endgroup$
    – Qmechanic
    Commented Feb 10, 2012 at 13:28
  • $\begingroup$ But Wallace et al employ decision-theoretic apporoach, there exist nearly countless amounts of criticism towards that approach. But I can't find the same slamdunks against Gleason $\endgroup$ Commented Feb 10, 2012 at 13:47
  • $\begingroup$ The link to wiki was broken, I submitted an edit that must be approved... $\endgroup$
    – FrankH
    Commented Feb 10, 2012 at 14:53

4 Answers 4


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.

  • $\begingroup$ I think it is fair to say you have studied this area. Great answer, enjoyed it... $\endgroup$
    – MoonKnight
    Commented Aug 7, 2013 at 15:22
  • $\begingroup$ When you say that the splitting of the worlds is observer dependent, does that mean that it violates Lorentz invariance? $\endgroup$
    – user86072
    Commented Feb 11, 2017 at 6:29
  • $\begingroup$ @2117 No, this is about something else. The problem is, given a wavefunction, what are the worlds, how do you divide it into worlds? A layperson imagines, naturally enough, that in the many worlds interpretation, you have many worlds objectively existing alongside each other, but it's not like that at all, except in the "many interacting worlds" interpretation, which is a modern minority view. In the standard approach to many worlds, what exists is just a wavefunction, and then you can slice and dice it into parts in many different ways (called choice of basis in ordinary QM). $\endgroup$ Commented Feb 11, 2017 at 9:53
  • $\begingroup$ @MitchellPorter Some say that its just that each branch of the universe splits into more branches, but the universe as a whole still has a single wave function that never splits. Would that solve the problem? What interpretation of QM do you take? $\endgroup$
    – user86072
    Commented Feb 20, 2017 at 18:34
  • $\begingroup$ @MitchellPorter and what does it mean when people says that the splitting is observer dependent? $\endgroup$
    – user86072
    Commented Feb 20, 2017 at 23:24

The main issue is that there are people who will not accept that a deterministic theory has any probability at all. I happened to meet David Albert at a coffee shop today, and he presented the argument to me in person as follows:

  • Captain Kirk is on the Enterprise, and he will be beamed down to the surface of a planet. Only the transporter is malfunctioning, and a million Kirks appear in different places. One of the Kirks is standing on a rock. The rest are not. Kirk knows about the malfunction, and is ready to be duplicated a million-fold. Is it reasonable for the Kirk who is standing on the rock to be surprised at the outcome?

The argument for yes is that the probability is small that Kirk would end up on a rock, since most Kirks in a counting sense are not on a rock. But which Kirk should be surprised exactly? The Kirk that ended up on a rock is by definition the exact one that ended up on a rock, and all the rest are different Kirks which didn't end up on a rock. It seems you need a philosophically unambiguous object over which you are making a probability distribution, and this data seems to come from nowhere.

Below is my response to David Albert, which also answers this question. I believe the data comes from nowhere. If you accept the stuff below, Gleason's theorem is sufficient. But the stuff in this answer is pure philosophy, since the whole issue is pure philosophy, unfortunately. The main point is that there is always additional data in a physical theory beyond the state variables, namely the map from state variables to abstract experience, and this map is part of the physical description and contains a growing amount of data which is irreducible data associated with the experience of observers.

Apologies to David Albert if I misrepresented his position.

How probability works in many-worlds

The main issue here is the emergence of probability in a deterministic setting from the subjective point of view. The philosophical problem is the Captain Kirk problem--- N Kirks, one on a rock. Kirk knows the score, after the beaming, the question is whether the Kirk is allowed to have a probabilistic description of the situation..

The issue you seemed to raise is that it is a "probability of what, exactly?" The solution to this problem, I am sure, is that when you have a conscious experience, this must be identified with an abstract computer program, whose running state is encoded in some integer, and whose algorithm is encoded in a recursive function on this integer. The inputs are some external variables, and the whole thing lives in some Platonic realm. You can take all of this to be a figure of speech--- the notion of matching a program to a physical instantiation is well defined from a logical positivist point of view--- you can run the program, map it to the physical instantiation and see if the two agree arbitrarily far into the future.

If you are living in the world, like Kirk, you need this map from the Platonic "Kirk" program (by Platonic, I mean the Kirk program, the space of computer program I take as the definition of the Platonic realm), which is a model of the contents of Kirk's consciousness, to some physical variables in the system. This map is such that the image of each possible memory state of Kirk is some collection of assemblage of atoms in the physical system, with some well defined relative positions and momenta (Kirk's world is classical). And the map of the Platonic Kirk to the physical Kirk identifies which part of the universe is instantiating the Platonic Kirk mind at any one time.

After the beaming, the Kirk system is split, so there is an extra counter with log_2(N) bits of data specifying how the platonic Kirk maps to the physical Kirks, and it is legitimate to place a probability distribution on this data, since it is extra data unknown to the post-split Kirk, defined by the map between Platonic mind (meaning computer program) and matter (meaning positions and momenta of atoms).

This is in effect a non-material variable which is necessary due to the requirement that to match sense experience to physical data, there must be a map between the abstract computation encoding said sense experience and the physical data which gives this abstract computation a physical instantiation. This map contains additional information, beyond the information in the classical atoms positions and velocities--- it is telling you which subset of the atoms' state variables carries the computation.

This extra data in the map is "mystical" in the sense that it is not contained in the atoms positions and velocities, but it is minimally mystical, since it preserves all essential features of physicalism---- there is no behavior of the "soul" which is not reflected in motions of the atoms, there is no causal agency in the abstract computation which is not instantiated in the atoms, the map is just there to tell you which atoms the computation is instantiated in.

This map is required from the point of view of classical mechanics whenever you have splitting observers. So that a conscious AI that you can duplicate is an example.

In order for the map to be sensible the motion of the atoms (say after 1s) must reproduce the same primitive recursive function which produces the next abstract Kirk state from the previous one. You might then say that the next Kirk state is defined by the motion of the atoms just as well, but this is no good when the Kirk's split, since it is exactly the question "which atoms" which needs an aswer. But when the abstract computation matches to the physical computation, everything is defined precisely, and there is no further ambiguity. This is the mind-body parallelism. But there is no requirement that this map for each individual Kirk does not contain extra data beyond the physical description, and for the case of duplicated Kirk's, in fact, it contains extra data.

The map in many worlds between sense experience and physical state identifies the abstract computation in the mind with the physical instantiation in a decohered branch of the wavefunction, with the property that with a suitably large probability the time evolution of the branch will reproduce the correct primitive recursive (or primitive recursive stochastic) computation in the mind, so that the concrete and abstract things match up. There is extra data in the map between the abstract computation of the mind and the concrete instantiation of the computation in the computer program, but this is not a big deal--- it is no worse philosophically than the extra data in the map in the classical Kirk case. So the emergence of probability many worlds interpretation does not in any way involve any worse philosophical twisting and turning than the classical splitting-observer scenario.

But the more pressing problem is the problem of measure. If the Kirks appear once a year, and the first Kirk is on a rock, and you wait one year, then another Kirk appears, not on a rock, and so on for a million years. Or worse, if the Kirks appear on a rock or not according to a switch which is set by hand in the Enterprise once a year. Then there is the question of the subjective probability of the events to Kirk. Does it change with time? What if the Kirk's keep coming forever?

You can obviously make the Kirk rock/non-rock number distribution whatever you like by adjusting the beaming device over time. The resulting distribution depends on the procedure.

It is here that one must be strict in the logical positivist point of view. The only probability that one can speak about is one that is observered after Kirk comes and talks to you. In this case, depending on the ensemble of Kirks, the reported internal view of probability will be different depending on how you interact with the ensemble. In other words, it is correct to use the probability distribution which is different at different times, depending on the number of Kirk's present, and it is correct to use any probability distribution which is consistent with the external observer's estimation of how many Kirk's of different kinds the external observer is likely to encounter, from the purely external point of view, of which ones he chooses to talk to.

The point is that the internal experience of any conscious computation is only well defined in the probabilistic sense relative to the interaction of this consciousness with an external world, and with other conscious computations. By talking together, these conscious abstract computations link up to greater wholes, and ultimately the internal experience of each one can be matched up to the point of view of the complete computation. This is the "God's eye" point of view. You can think of it as Everett's solipsistic Copenhagen, where God does the collapse of the wavefunction. Although, I like to think of it as a teleologically consistent many-worlds where all the probability comes from the unique teleological measure defined by which observers any one observer can see.


The problem is that the Gleason's theorem gives little insight in the physical.

For example, in W. H. Żurek, Probabilities from Entanglement, Born's Rule from Envariance (2004) there is stated:

Indeed, Gleason’s theorem [30] is now an accepted and rightly famous part of quantum foundations. It is rigorous – it is after all a theorem about measures on Hilbert spaces. However, regarded as a result in physics it is deeply unsatisfying: it provides no insight into physical significance of quantum probabilities – it is not clear why the observer should assign probabilities in accord with the measure indicated by Gleason’s approach.

[30] A. M. Gleason, J. Math. Mech. bf 6, 885 (1957)

The same paper actually derives the probabilities (i.e. $p_i = |\psi_i|^2$) de facto making use of the Special Relativity - as it bases on subsystems which cannot communicate instantly. It also puts emphasis on distinction when probability is based on ignorance (lack of knowledge) or necessity (i.e. some measurements need to be probabilistic otherwise we could have superluminal communication).


As far as I understand, the Gleason's theorem only states that "the only possible measure of the probability" is such and such (http://en.wikipedia.org/wiki/Gleason%27s_theorem ), but it does not state that this "possible measure of the probability" is indead the actual measure of the probability.

  • $\begingroup$ Hmm, I think the wikipedia for Gleason's Thm, and John Preskill's definition of the theorem in his Quantum Information notes are in disagreement with your comment. Gleason's Theorem is a representation theorem. It says that $tr(\rho P)$ can always represent the probability map from projectors to reals (given a few assumptions). This does not mean it is the only possible representation of such a map. Correct me if I'm wrong. $\endgroup$ Commented May 7, 2017 at 11:19
  • $\begingroup$ @doublefelix: I used the formulation in the Wikipedia that was current at the time I wrote my answer. Do you think that formulation was incorrect? I found it at books.google.com/… as well. $\endgroup$
    – akhmeteli
    Commented May 7, 2017 at 12:34
  • $\begingroup$ Hmm, I checked the wording of the result (on the last page of Gleason's paper). I'd paraphrase it as "for any probability measure under some assumptions, there is always a density matrix $\rho$ so that the probability is equal to $tr(\rho P)$". This doesn't exclude other, very different expressions which also equal the probability. It does guarantee that this type of expression will always work, though. $\endgroup$ Commented May 7, 2017 at 13:18
  • $\begingroup$ @doublefelix: I don't see a significant difference between the two different formulations we use, but "that this type of expression will always work" does not mean it is indeed a probability measure, say, for many-worlds. $\endgroup$
    – akhmeteli
    Commented May 7, 2017 at 18:38

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