Cardinality of class of worlds in quantum MWT

In summary: I guess.In summary, the MWI suggests that there are an infinite number of worlds, each with its own set of probabilities. String theory indicates that there may exist different solutions in a landscape of theories described by different interactions. According to inflationary scenarios there may exist universes in a larger multiverse where there may be an infinite number of worlds. The cardinality of an infinite set is never something that can be physically tested.
  • #1
nomadreid
Gold Member
1,665
203
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.)
 
Physics news on Phys.org
  • #2
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
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.
 
  • #4
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
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:
  • #6
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.
 
  • #7
tom.stoer said:
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.
 
  • #8
https://www.physicsforums.com/showthread.php?t=632734

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.
 
Last edited:

1. What is the cardinality of the class of worlds in quantum Many-Worlds Theory (MWT)?

The cardinality of the class of worlds in quantum MWT is infinite. This means that there are an infinite number of parallel universes or "worlds" in which every possible outcome of a quantum event exists simultaneously.

2. How does the concept of cardinality apply to quantum MWT?

In quantum MWT, the concept of cardinality refers to the number of parallel universes or worlds that exist. Each world represents a different outcome of a quantum event, and the cardinality represents the infinite possibilities that exist.

3. Is the concept of cardinality in quantum MWT supported by scientific evidence?

While the concept of cardinality in quantum MWT is a popular interpretation of quantum mechanics, it is still a subject of debate and does not have conclusive scientific evidence to support it. However, recent experiments have shown that quantum superposition, which is a key component of MWT, can occur on a macroscopic scale.

4. How does the cardinality of worlds in quantum MWT differ from other interpretations of quantum mechanics?

The concept of cardinality in quantum MWT differs from other interpretations in that it suggests the existence of an infinite number of parallel universes, while other interpretations may propose a single universe or a limited number of universes. Additionally, MWT suggests that all possible outcomes of a quantum event occur, while other interpretations may propose that only one outcome is realized.

5. Can we ever prove or disprove the existence of an infinite number of worlds in quantum MWT?

It is currently not possible to prove or disprove the existence of an infinite number of worlds in quantum MWT. As it is a theoretical interpretation of quantum mechanics, it falls outside the realm of empirical science and is thus not testable. Some scientists argue that the concept of an infinite number of worlds is not falsifiable, making it unscientific. However, others believe that future advancements in technology and understanding of quantum mechanics may provide evidence for or against the concept.

Similar threads

Replies
2
Views
2K
  • Beyond the Standard Models
Replies
4
Views
2K
  • Beyond the Standard Models
Replies
13
Views
1K
  • Beyond the Standard Models
Replies
11
Views
2K
  • Beyond the Standard Models
Replies
0
Views
368
  • Set Theory, Logic, Probability, Statistics
Replies
16
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
966
  • Quantum Interpretations and Foundations
Replies
3
Views
2K
  • Beyond the Standard Models
Replies
6
Views
2K
  • Quantum Interpretations and Foundations
Replies
7
Views
575
Back
Top