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The greatest appeal of the Everett-style Many-Worlds Interpretation of QM - that is, the wave equation alone, or standard MWI - is its simplicity in terms of mathematics and physics. While all other interpretations need to add extra things to the wave equation - adding in hidden variables with dynamics of their own, or modifying the wave equation itself to include random wave collapse processes - the Everett MWI states that the standard wave equation alone can explain everything we observe.
Yet, despite numerous attempts and claims to the contrary (and putting aside the possibilities for my own approach for now), the Born Rule probabilities have not been derived from the Everett picture. Thus, it may prove necessary to add new physics to our description of QM after all.
However, some approaches attempt to retain much of the advantage in simplicity of the MWI, as well as its multiple-worlds character, while making such modifications. It's a promising idea, and these MWIs certainly take inspiration from the Everett-style MWI, but they add too much to the pure wave picture to satisfy true Everett-style MWI partisans.
a) Hidden variables introduce complexity not only because of the extra dynamic equations, but because they require some choice of initial conditions. The wavefunction of QM also requires initial conditions, of course, but there is reason to hope that some simple equation could govern those initial conditions; the Hartle-Hawking 'no boundary' condition for the wavefunction of the universe is a well-known example of such a proposal (though it has problems of its own). Particle-like hidden variables do not seem amenable to such simple specification of initial conditions. However, if all possible sets of hidden variable initial conditions are equally real, then the overall simplicity of initial conditions for the multiverse is restored.
'Continuum Bohmian Mechanics' (CBM) is the best-known example of this approach. Like the Pilot Wave Interpretation, it has particle-like hidden variables; but instead of just one set of them, it has a continuous distribution of such sets, which act much like a continuous fluid. In addition to the possibility of simpler initial conditions, CBM may be immune to the fatal flaw of the PWI, which is being 'many-worlds in denial'. In other words, in the 'single-world' PWI, with one set of hidden variables, most of the observers will end up being implemented by the many worlds of the wavefunction, so the hidden variables won't matter. With CBM, the number of hidden variable sets is also infinite, so a typical observer could depend on the hidden variables after all. (This latter claim still needs to be proven compatible with computationalist considerations, but it is plausible.)
The hidden variables in the PWI follow the Born Rule, so CBM should be OK in that regard. But CBM retains the other features of the PWI that many physicists dislike, namely non-locality and a preferred reference frame. It is also not clear how well a relativistic version of the PWI works, and CBM inherits such problems. (Granted, no physics that works for quantum general relativity is known yet.) Also, even CBM is not as simple as the standard MWI.
I regard CBM (and more generally, MWI's with hidden variables) as something useful to keep in mind, as it is one of the few interpretations of nonrelativistic QM that seems to actually work in terms of being compatible with the Born Rule and not having an 'MWI in denial' problem. But other possiblities must be thoroughly explored before I would consider endorsing CBM as being likely to be true.
b) Another approach is to retain a pure wavefunction picture as in Everett's MWI, but to make the wavefunction be discrete instead of continuous. Discrete space is not what is meant here, but rather a discrete nature of the wavefunction itself. Buniy et al advocate such an approach.
The most obvious way to do that might be to assume that the wavefunction is represented by an integer function on configuration space rather than a continuous function. (If configuration space is also discrete, that is one way an approximate discrete numerical representation of a continuous wave function might be done on a digital computer.)
Buniy et al propse a somewhat different different assumption, in which wavefunctions seperated by a term of some mimimum squared amplitude are condidered to be the same.
Because the wave function (or perhaps I should say its 'populated' region) is constantly and rapidly expanding into new areas of configuration space (e.g. as entropy increases), its numerical value is constantly imploding. I will call this the Wavefunction Value Implosion (WVI). If the universe is finite, then this effect will be finite, but exponentially large as a function of the number of particles in the universe. Thus, a discrete wavefunction could not be detected experimentally if its discrete nature is small enough, until such time as the WVI brings the populated part of the wavefunction below that scale, and then presumably time evolution will radically change or effectively stop; I will call this the Crash.
Discrete physics has a certain appeal to some people, independent of any possible role in quantum mechanics. Wolfram's book "A New Kind of Science" discusses such views. Also, the idea that all possible mathematical universes physically exist (the Everything Hypothesis, which will be discussed in a later post) may be somewhat more tractable if it is restricted to digital systems (though, despite its undeniable appeal, it still has problems even then).
If the wavefunction is discrete, would that help explain the Born probabilities? Buniy et al argue that it would, by shoring up the old "frequency operator" attempted derivation by extending it to finite numbers of measurements rather than infinite. This argument notes that, after repeated measurements, terms in the wavefunction which don't have the Born frequencies have much smaller amplitudes than the terms with the 'right' frequencies. With a minimum amplitude cutoff, most of the un-Born terms would be eliminated. This argument does not seem very satisfactory, as we are interested in situations with small numbers of meaurements, and thus small factors of difference in amplitude, while the digital cutoff would have to be very far from a significant fraction of the total amplitude if the Crash is not yet upon us. In practical situations, other factors would affect amplitudes much more. For example, entropy production is not associated with low probability, but it results in numerous sub-branches each of which shares a fraction of the original squared amplitude.
Shoring up the 'Mangled Worlds' argument would seem a more promising approach. There are many sub-branches comprising each macroscopically distinguishable world, and they tend to have a log-normal distribution in squared amplitude. As Hanson showed, a cutoff in the right range of squared amplitudes would lead to Born Rule probabilities. This cutoff must be uniform across branches, which Hanson's 'mangling' mechanism by larger branches actually fails to provide, but a digital cutoff could provide it. I will tentatively say that this is a possible mechanism for the Born Rule, though I need to study it more before I can say for sure that there are no problems that would ruin it. In particular, if the number of worlds changes too much over time or the era in which the Born Rule holds is too short, that would indicate a problem.
c) Another mechanism that improves on 'Mangled Worlds' is my own idea in which random noise in the initial wavefunction means that larger volumes in configuration space per implemented computation are required for low-amplitude sub-branches, which can lead to the Born Rule. This requires new physics in the form of special initial conditions, but hopefully not in terms of time evolution. It is possible that this leads to a Boltzmann Brains problem. I will discuss this, as well as an alternative in which the Born Rule is due to a special way to count computations (which avoids new physics - if it can be justified) in later posts.
d) The Everything Hypothesis (that all possible mathematical stuctures exist) can be used directly in an attempt to predict what a typical observer would observe. Some have argued that this explains what we observe, including the Born Rule. The Everything Hypothesis will be discussed in a post of its own.
e) Other MW schemes for modifying physics have been proposed.
One example is Michael Weismann's idea involving sudden splitting of existing worlds into proposed new degrees of freedom, with a higher rate of such splitting events for higher amplitude worlds. The problem with it is that if new worlds are constantly being produced, then the number of observers would be growing exponentially. The probability of future observations, as far into the future as possible, would be much greater than that of our current observations. Thus, the scheme must be false unless we are highly atypical observers, which is highly unlikely.
David Strayhorn had an idea based on general relativity, in which different topologies correspond to different sub-branches of the wavefunction. This approach is not well-developed as of yet and it has problems that I think will prevent it from working. I discussed it in various post on the OCQM yahoo group.
