Early attempts to derive the Born Rule in the MWI

When Everett wrote his thesis in 1957 on the '"Relative State" Formulation of Quantum Mechanics', he certainly needed to address how the Born Rule probabilities fit into his new interpretation of QM. While the MWI remains provocative even today, it was not taken seriously in 1957 except by a few people, to the extent that Everett had to call it "Relative State" rather than "Many Worlds". So it is perhaps fortunate that he did not realize the true challenges of fitting the Born Rule into the MWI, which could have derailed his paper. Instead, he came up with a short derivation of the Born Rule, using assumptions that he did not realize lacked justification.

Of course, the Born Rule issue has long since returned to haunt the MWI. Historically, what has happened several times was that a derivation of the Born Rule that seemed plausible to MWI supporters was produced, but soon it attracted critics. After a few years it became clear to most physicists that the critics were right, and the MWI fell into disrespect until a new justification for the Born Rule was produced. This cycle continues today, with the decision-theoretic Deutsch-Wallace approach being considered the best by many, and now attracting growing (and deserved) criticism.

When considering claimed derivations of the Born Rule in the MWI, it is often useful to keep in mind an 'alternative rule' that is being ruled out, and to question the justification for doing so. Two useful ones are as follows:

a) The unification rule: All observations that exist have the same measure. In this case, branch amplitudes don't matter, as long as they are nonzero (and they always are, in practice).

b) David Albert's fatness rule: The measure of an observer is proportional to the squared amplitude (of the branch he's on) multiplied by his mass. Here, amplitudes matter, but so does something else. This one is especially interesting because it illustrates that not all observers necessarily have the same measure, even if they are on the same branch of the wavefunction. While it is obviously implausible, it's a useful stand-in for other possibilities that may seem better more justifiable, such as using the number of neurons in the observer's brain instead of his mass, or any other detail of the wavefunction.

Another useful thing to keep in mind is the possibility of a modified counterpart to quantum mechanics, in which squared-amplitude would not be a conserved quantity. We would expect that the Born Rule might no longer hold, but some other Rule should, even in the absense of conserved quantities. Presumably, if the modification is small, so would be any departure from the Born Rule. Thus, one should not think that conserved quantities must have any special a priori importance without which no measure distribution is possible.

Let us examine a few of the early attempts to derive the Born Rule within the MWI:

1) Everett's original recipe

In Everett's 1957 paper, he models an observer in a fairly simple way, considering only a set of memory elements. This is a sort of rough approximation of a computational model, but without the dynamics (which are crucial for a well-defined account of computation). Thus, Everett was a visionary pioneer in applying computationalist thinking to quantum mechanics, but he never confronted the complexity of what would be required to do a satisfactory job of it.

He assumed that the measure of a branch would be a function of its amplitude only, and thus would not depend on the specific nature of that branch. This is a very strong assumption, and arguably contains his next assumption as a special case already. [A more general approach would allow other properties to be considered, such as in Albert's fatness rule.]

[Note: Everett's use of the term 'measure' is not stated to refer specifically to the amount of consciousness, but in this context, the role it plays is essentially the same as if it did. Some authors use 'measure of existance' to specifically mean the squared amplitude by definition; obviously Everett did not, since he wanted to prove that his measure was equal to the squared amplitude. I recommend avoiding overly suggestive terms (like 'weight') for the squared amplitude.]

Next, he assumed that measure is 'additive' in the sense that if two orthogonal branches are in superposition, they can be regarded as a single branch, and the same function of amplitude must give the same total measure in either case.

If the definition of a 'branch' is arbritrary in allowing combinations of orthogonal components, the 'additivity' assumption makes sense, since it means that it does not matter how the branches are considered to be divided up into orthogonal components. [An argument similar to that would be presented years later in Wallace's 2005 paper, in which Wallace defended the assumption of 'equivalence' (branches of equal amplitude must have equal measure) against the idea of sub-branch-counting, based on the impossibility of defining the specific number of sub-branches. Everett did not get into such detail.]

With the previous assumption, 'additivity' would only hold if the measure is proportional to the squared amplitude; thus, he concluded that the Born Rule holds.

Everett considered the additivity requirement equivalent to saying that measure is conserved; thus, when a branch splits into two branches, the sum of the new measures is equal to the measure of the original branch. He gave no justification for the conservation of measure, perhaps considering it self-evident.

In classical mechanics, conservation of probability is self-evident because the probability just indicates something about what state the single system is likely to be in. If the probabilities summed to 2, for example, a single system couldn't explain it; perhaps there would have to be 2 copies instead of one. Yet the existance of multiple copies is precisely what the MWI of QM describes, and in this case, there is no a priori reason to believe that the total measure can not change over time.

Everett's attempted derivation of the Born Rule is not considered satisfactory even by other supporters of the MWI, because he did not justify his assumptions. Soon, other attempts to explain the probabilities emerged.

2) Gleason's Theorem

Also discovered in 1957, Gleason's theorem shows that if probabilities are non-contextual, meaning that the probability of a term in the superposition does not depend on what other terms are in the superposition, then the only formula which could give the probabilities is based on squared expansion coefficients. It is straighforward to argue that the correct expansion to use is that for the current wavefunction; thus, these coefficients are the amplitudes, which gives Born's Rule.

Unfortunately, there is no known justification for assuming non-contextuality of the probabilities. If measure is not conserved, the probabilities can not generally be noncontextual. Gleason's theorem is sometimes cited in attempts to show that the MWI yields the Born Rule, but it is not a popular approach since usually those attempts make (unjustified) assumptions which are strong enough to select the Born Rule without having to rely on the more complicated math required to prove Gleason's theorem.

3) The infinite-measurements limit and its frequency operator

The frequency operator is the operator associated with the observable that is the number of cases in a series of experiments that a particular result occurs, divided by the total number of experiments. If is assumed that just the frequency itself is measured, and if the limit of the number of experiments is taken to infinity, the eigenvalue of this frequency operator is unique and equal to the Born Rule probability. The quantum system is then left in the eigenstate with that frequency; all other terms have zero amplitude, as shown by Finkelstein (1963) and Hartle (1968).

This scheme is irrelevant for two reasons. First, an infinite number of experiments can never be performed. As a result, terms of all possible frequencies remain in the superposition. Unless the Born Rule is assumed, there is no reason to discard branches of small amplitude. Assuming that they just disappear is equivalent to assuming collapse of the wavefunction.

Second, in real experiments, individual outcomes are recorded as well as the overall frequency. As a result, there are many branches with the same frequency and the amplitude of any one branch tends towards zero as the number of experiments is increased. If one discards branches that approach zero amplitude in the limit of infinite experiments, then all branches should be discarded. Furthermore, prior to taking the infinite limit, the very largest individual branch is the one where the highest amplitude outcome of each individual experiment occurred, if there is one.

A more detailed critique of the frequency operator approach is given here. The same basic approach of using infinite ensembles of measurements has been taken recently by certain Japanese physicists, Tanaka (who seems unaware of Hartle's work) and (seperately) Wada. Their work contains no significant improvements on the old, failed approach.

4) Graham's branch counting

Neil Graham came out with a paper in 1973 that appears in the book "The Many Worlds Interpretation of Quantum Mechanics" along with Everett's papers and others.

Graham claimed that the actual number of fine-grained branches is proportional to the total squared amplitude of a course-grained macroscopically defined branch. Such sub-branches would be produced by splits due to microscopic scattering events and so on which act as natural analogues of measurements.

If it were true, it could also begin to give some insight into why the Born Rule would be true, beyond just a mathematical proof; that is, each fine-grained branch would presumably support the same number of copies of the observer. (That assumption would still need to be explained, of course.)

Unfortunately, and even aside from the lack of precise definition for fine-grained branches, he failed to justify his statistical claims, which stand in contradiction to straightforward counting of outcomes. He simply assumed that fine-grained branches would on average have equal amplitudes regardless of the amplitude of the macroscopic branch that they split from.

In the next post, the more recent attempts (other than my own) to derive the Born Rule within the MWI will be described.