# Cardinality of class of worlds in quantum MWT

1. Jan 15, 2013

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.)

2. Jan 16, 2013

### tom.stoer

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)

3. Jan 17, 2013

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$\aleph$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.

4. Jan 17, 2013

### Staff: Mentor

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.

5. Jan 18, 2013

### tom.stoer

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?

Last edited: Jan 18, 2013
6. Jan 18, 2013

### Hurkyl

Staff Emeritus
If the density matrix A of the state can be approximately written as a convex combination of the density matrices of pure states:

$$A \approx \sum_{i=1}^n c_i | \psi_i \rangle \langle \psi_i |$$

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 $| \psi_i \rangle$, weighted by the values $c_i$.

"Counting" really isn't the right word, except in the very special case where you write a sum with all of the $c_i$ 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.

7. Jan 18, 2013

### Staff: Mentor

Anything you like, the "number of worlds" will depend on that, but I think this number is not really interesting anyway.

8. Jan 18, 2013

### bcrowell

Staff Emeritus