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When I read descriptions of the many-worlds interpretation of quantum mechanics, they say things like "every possible outcome of every event defines or exists in its own history or world", but is this really accurate? This seems to imply that the universe only split at particular moments when "events" happen. This also seems to imply that the universe only splits into a finite number of "every possible outcome".

I imagine things differently. Rather than splitting into a finite number of universes at discrete times, I imagine that at every moment the universe splits into an uncountably infinite number of universes, perhaps as described by the Schrödinger equation.

Which interpretation is right? (Or otherwise, what is the right interpretation?) If I'm right, how does one describe such a vast space mathematically? Is this a Hilbert space? If so, is it a particular subset of Hilbert space?

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No one has any justifiable unique answers to such questions. The many-worlds interpretation isn't an actual theory of physics, an actual set of rules, ideas, or equations. It's just a vague and, when looked with any precision, meaningless and vacuous philosophical paradigm. Obviously, proper quantum mechanics doesn't imply any splitting whatsoever. Any rule when a splitting occurs is bound to be unnatural. The only "splitting" that proper QM allows is an approximate one, given by decoherence: the moment when the chances of parts of $\psi$ to "re-interfere" in the future are negligible. – Luboš Motl Jul 21 '12 at 7:11
@LubošMotl your statement that "Obviously, proper quantum mechanics doesn't imply any splitting whatsoever." I don't really understand in this contex. They are not explaining splitting, but the state vector reduction/collapse of the wavefunction. I agree that the many-world interpretation is pysically flawed and has no mathematical basis as a theory. However, interpretations like the Many-Minds/multi-consciousness interpretation do. Moreover, this particular theory is complete, well defined and cannot be disprooved from a physical stand-point. Of course, this does not make it correct! – Killercam Jul 21 '12 at 10:43

The Many Worlds interpretation is popularly misunderstood. The wave function itself contains a spectrum of universes, one corresponding to each eigenvalue for a given operator. The "splitting" of the "many worlds" is represented by the time evolution of the wave function described by the Schrodinger equation. As Lubos mentions above, these "universes" only become separate through decoherence.

Consider, for example, a wave function in the position-basis given by a delta-function at x=0. This represents one universe. Now time-evolve the wave function using the schrodinger equation. The delta-function has now spread-out a bit. It is peaked at x=0, but has non-zero values at x=+1 and x=-1. This represents the existence of universes in which the position of the particle is at x=0, x=+1, and x=-1. In some sense there are "more" universes at x=0 than at x=+-1, because the wave function is more highly peaked at x=0. This is where some of the difficulty in the Many Worlds interpretation comes in: what ontology to use to describe the "splitting", "how many universes" are at x=0 vs x=+-1, and so on. The main point I want to make is that the "splitting" is just an interpretation of what is happening with the evolution of the wave function according to the schrodinger equation. Nothing "more" is actually happening. You model the "splitting" using the tried-and-true schrodinger evolution of the wave function.

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You imply that there is a spectrum (a countable infinity) of "possible universes". But is it actually a continuum (an uncountable infinity) of "possible universes"? Can the delta-function have non-zero values at locations everywhere between 0 and +/-1? Or maybe a better example (since I don't understand the delta-function), in the double-slit experiment, can't a particular photon hit the detector plane at any point on the plane? (<- thus uncountable infinite possible universes) – John Berryman Jul 21 '12 at 13:04
@John Berryman The word 'spectrum' does not imply a countable infinity. It is a continuum representing an uncountably infinite number of universes in the example I gave. You can think of a delta function like a very narrow spike. The schrodinger equation time-evolves a narrow spike into a wider and wider gaussian shape. In the example, in order to keep things simple, I approximated this as {-1,0,1) (a very rough approximation, but serves to illustrate the point). – user1247 Jul 21 '12 at 19:05

Many worlders won't tell you this dirty little secret but how often splitting happens, and how many worlds there are, depends upon the choice of coarse graining, and the coarse graining resolution. No, it's not possible to ramp up the coarse graining all the way to the finest levels because a decoherence/coherence threshold would be crossed. And no, there is no canonical coarse graining either.

The preferred basis depends upon the environment. Always. What is the preferred basis for a closed self-contained universe?

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A more accurate answer than "The preferred basis depends upon the environment. Always." would be that the supporters of the MWI haven't yet described any other mechanism by which it could arise, just as they haven't yet shown how the Born rule would emerge even for a finite system. – Niel de Beaudrap Jul 21 '12 at 11:52
Nice answer. I find that many worlders do actively sweep this under the rug as often as possible. You could improve it by attempting some calculation at some level - for instance merely assuming that there is only one observer - you - in the Universe, how many universes split per second? I can see the coarse graining being undefined even for that! – Tom Andersen Jan 2 '15 at 16:02

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