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Cardinality of class of worlds in quantum MWT

by nomadreid
Tags: cardinality, class, quantum, worlds
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Jan15-13, 03:35 AM
P: 562
Two (related) questions:
(A) If I understand correctly (no guarantee to that), in an Everett-type Many-Worlds-Theory of Quantum Mechanics, every probability amplitude is associated with a world. This would mean, for a single particle, that there would be as many worlds ("be" in the sense of a model similar to a Kripke Frame, without taking a stand on any other type of existence) as the continuum. In fact, not only for a particle, but for a point (or quantum grain) in spacetime. Given that the largest proposed cardinality for the number of points or grains seems to also be that of the continuum, the product still ends up giving the number of worlds as the continuum. Is this a valid conclusion?
(B) However, (and here is where things get really shaky), given 2 particles, then are there also various possible relations between the two particles in the different worlds (e.g., different strengths of gravity,etc.), or are these considered to be the same (that is, if two particles are the same in two different worlds, can one assume that the interactions between them are also the same?). In the former case, then given c (continuum) particles (or grains, or points), the number of relations would be 2c, thus giving 2c number of worlds, whereas in the latter case, the number of worlds would remain at c. Which one, or neither, is applicable?
(I am not sure whether physicists care about cardinality, but a mathematician definitely would.)
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Jan16-13, 01:05 AM
Sci Advisor
P: 5,442
You should not confuse diffent meanings of 'Many Worlds'.

1) The MWI according to QM refers to one theory where the evolution of the wave function is interpreted as branching into many worlds.
2) String theory indicates that there may exist different solutions in a landscape of theories described by different interactions.
3) According to inflationary scenarios there may exist universes in a larger multiverse where there may.

The fundamental difference is where these worlds exist and in which sense one has to count them, or in which sense one can define a probablity measure or whatever

Worlds in the sense of 1) exist in a Hilbert space
Worlds in the sense of 2) live in 'theory space'
And worlds in the sense of 3) live in a spacetime manifold (or some generalization)
Jan17-13, 08:59 AM
P: 562
thanks, tom.stoer; as I was interested in the MWT#1 in your list, I guess the answer is that the cardinality is that of the continuum 2[itex]\aleph[/itex]0, since Hilbert space is defined over the complex numbers.

I am interested in counting them to know how the "worlds" would look as "worlds" in a Kripke frame or something similar.

Jan17-13, 10:55 AM
P: 11,869
Cardinality of class of worlds in quantum MWT

To be considered as different worlds, those regions in Hilbert space should be separated enough to have a negligible interaction - in other words, they should be decoherent. If your initial state has some similarity to a classical state ("one world") or a superposition of some classical states, you get a finite (but extremely large) number of worlds after a finite time.
Jan18-13, 01:08 AM
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P: 5,442
How should one define the relevant 'distance' in the Hilbert space? How can one define 'sufficient decoherent'? Is there a geometrical definition using rays or density matrices? Is there a way to 'partition the unit sphere in different worlds

And based on that - is there a way to define the counting?
Jan18-13, 02:07 AM
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PF Gold
Hurkyl's Avatar
P: 16,091
If the density matrix A of the state can be approximately written as a convex combination of the density matrices of pure states:

[tex]A \approx \sum_{i=1}^n c_i | \psi_i \rangle \langle \psi_i |[/tex]

then this summation remains valid after unitary evolution too. So, it's fair to describe the state described by A comprising multiple worlds each described by the [itex]| \psi_i \rangle[/itex], weighted by the values [itex]c_i[/itex].

"Counting" really isn't the right word, except in the very special case where you write a sum with all of the [itex]c_i[/itex] the same value.

Note that the decomposition described above is not unique; e.g. for a qubit, the state comprising spin up around Z and spin down around Z in equal weights is the same as the state comprising spin up around X and spin down around X in equal weights.
Jan18-13, 09:32 AM
P: 11,869
Quote Quote by tom.stoer View Post
How should one define the relevant 'distance' in the Hilbert space?
How can one define 'sufficient decoherent'?
Anything you like, the "number of worlds" will depend on that, but I think this number is not really interesting anyway.
Jan18-13, 10:18 AM
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PF Gold
bcrowell's Avatar
P: 5,597

In any case, the cardinality of an infinite set is never something that can be physically tested. If a theory describes nature using an infinite set of this cardinality or that cardinality, that's an incidental feature of that particular presentation of the theory. The set of possible experiments and experimental outcomes is countable. We could construct all our physical theories using the rationals rather than the reals. The reals are just more convenient.

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