Probably-incomprehensible decision theory exposition and speculation

you meant what's the effect on entangled particles at different locations?

No, I just meant that your Hilbert space is associated with a preferred foliation. The states in the Hilbert space are superpositions of configurations on the slices of that foliation. If you follow Copenhagen, observables are real, wavefunctions are not, and this foliation-dependence of the wavefunctions doesn't matter. It's like fixing a gauge, doing your calculation, and then getting gauge-invariant results back for the observables. These results - expectation values, correlation functions... - don't require any preferred foliation for their definition. The wavefunctions do, but they are just regarded as constructs.

So Copenhagen gets to be consistent with special relativity at the price of being incomplete. Now according to Many Worlds, we can obtain a complete description of physical reality by saying that wavefunctions are real. What I am pointing out is that wavefunctions are defined with respect to a reference frame. Time is not an operator and you need surfaces of simultaneity for Schrodinger evolution. The surface of simultaneity that it lives on is one of the necessary ingredients for defining a wavefunction. If the wavefunction is real, then so is the surface of simultaneity, but the whole point of special relativity is that there is no absolute simultaneity. So how do you, a wavefunction realist, get around this?

If you want an explanation of how you get a probabilistic state from an entangled state ("how the theory predicts what we observe"), check out partial traces.

So, let's return to your example. The wavefunction of the universe is "|U> x ( 3/5 |1> + 4/5 |0> )". Well, this isn't a great example because the wavefunction factorizes. But anyway, let's suppose that the reduced density matrix of your two-state system is c_00 |0><1|. You still need to explain how the Born rule makes sense in terms of a multiverse.

Perhaps an analogy will make this clearer. Suppose I'm a car dealer, and you place an order with me for 9 BMWs and 16 Rolls-Royces. Then you come to collect your order, and what you find is one BMW with a "3" painted on it, and one Rolls-Royce with a "4" painted on it. You complain that I haven't filled the order, and I say, just square the number painted on each car, and you'll get what you want. So far as I can see, that's how MWI works. You work with the same wavefunctions that Copenhagen uses, but you want to do without the Born rule. So instead, you pull out a reduced density matrix, point at the coefficients, and say "you can get your probabilities from those".

That's not good enough. If quantum mechanics is to be explained by Many Worlds, I need to get the Born rule frequencies of events from the frequencies with which those events occur in the multiverse. Otherwise I'm just painting a number on a state vector and saying "square it". If you don't have some way to decompose that density matrix into parts, so that I actually have 9 instances of |1> and 16 instances of |0>, or some other way to obtain Born frequencies by counting branches, then how can you say that Many Worlds makes the right predictions?

By the way, you're doing an excellent job of explanation, but I hope you see by now what I meant by "playing Whack-A-Mole". Every time you make a point, rather than acknowledge it, he'll just restate his vague objection in more elevated jargon.

how we should regard the size of some part of the configuration space compared to some other part, the L^2 norm is the blindingly obvious mathematical answer because of the properties of the wavefunction.

Does "part of the configuration space" refer to a single state vector, or a whole region that a state vector might belong to? My impression is that measuring the latter sort of thing is problematic from a rigorous mathematical standpoint. Is this correct, and does it have consequences for your discussion?

Okay, I have tried to understand what sort of ontology could answer to your description. A key consideration: you say we should judge "the size of some part of the configuration space compared to some other part" according to "the L^2 norm of the wavefunction". You also talk about "mind-configurations similar to me".

A wavefunction may evolve over time, but configuration space does not. Configuration space is a static arena, and the amplitudes associated with configurations change (unless we're talking about a timeless wavefunction of the universe; I'll come to that later). In general, I infer from your discussion that configurations are real - they are the worlds or branches - and the wavefunction determines a measure on configuration space. The measure can't be identified with the wavefunction - the phase information is lost - so, if we are to treat the wavefunction as also real, we seem to have a dualism remotely similar to Bohmian mechanics: The wavefunction is real, and evolves over time, and there is also a population of configurations - the worlds - whose relative multiplicity changes according to the changing measure.

I want to note one of the peculiarities of this perspective. Since configuration space does not change, and since the different configurations are the worlds, then at every moment in the history of the universe, every possible configuration exists (presumably except for those isolated configurations which have an individual measure of exactly zero). What distinguishes one moment from the next is that there is "more" or "less" of each individual configuration. If we take the use of the mathematical continuum seriously, then it seems that there must be an uncountable number of copies of each configuration at each moment, and the measure is telling us the relative sizes of these uncountable sets.

This scenario might be simplified a little if you had a timeless wavefunction of the universe, if the basic configurations were combinatorial (discrete degrees of freedom rather than continuous), and if amplitudes / probabilities were rational numbers. This would allow your multiverse to consist of a countable number of configurations, each duplicated only finitely often, and without the peculiar phenomenon of all configurations having duplicates at every moment in the history of the universe. This would then land us in a version of Julian Barbour's Platonia.

There are three features of this analysis that I would emphasize. First, relativity in any space-time sense has disappeared. The worlds are strictly spatial configurations. Second, configurations must be duplicated (whether only finitely often, or uncountably infinitely often), in order for the Born frequencies to be realized. Otherwise, it's like the parable of the car dealer. Just associating a number with a configuration does not by itself make the events in that configuration occur more frequently. Third, the configurations are distinct from the wavefunction. The wavefunction contains information not contained in the measure, namely the phase relations. So we have a Bohm-like dualism, except, instead of histories guided by a pilot wave, we have disconnected universe-moments whose multiplicities are determined by the Born rule.

There are various ways you could adjust the details of this ontology - which, I emphasize, is an attempt to spell out the ontological commitments implied by what you said. For example, your argument hinged on typicality - being a typical mind-configuration. So maybe, instead of saying that configurations are duplicated, you could simply say that configurations only get to exist if their amplitude is above some nonzero threshold, and then you could argue that Born frequencies are realized inside the individual universe-configuration. This would be a version of Everett's original idea, I believe. I thought it had largely been abandoned by modern Many Worlds advocates - for example, Robin Hanson dismisses it on the way to introducing his idea of mangled worlds - but I would need to refresh my knowledge of the counterarguments to personally dismiss it.

In any case, you may wish to comment on (1) my assertion that this approach requires dualism of wavefunction and worlds (because the wavefunction can't be identified with the ensemble of worlds, on account of containing phase information), (2) my assertion that this approach requires world duplication (in order to get the frequencies right), and (3) the way that configuration has supplied a definitely preferred basis in my account. Most Many Worlds people like to avoid a preferred basis, but I don't see how you can identify the world we actually experience with a wavefunction-part unless you explicitly say that yes, that wavefunction-part has a special status compared to other possible local basis-decompositions. Alternatively, you could assert that several or even all possible basis-decompositions define a "valid" set of worlds, but validity here has to mean existing - so along with the ensemble of spatial configurations, distinct from the wavefunction, you will end up with other ensembles of worlds, corresponding to the basis wavefunctions in other choices of basis, which will also have to be duplicated, etc., in order to produce the right frequencies.

To sum up, my position is that if you do try to deliver on the claims regarding how Many Worlds works, you have to throw out relativity as anything more than a phenomenological fact; you have to have duplication of worlds in order to get the Born frequencies; and the resulting set of worlds can't be identified with the wavefunction itself, so you end up with a Bohm-like dualism.