# I Cardinality of Universes in the Multiverse

1. Sep 8, 2016

### tzimie

Is there a consensus on the cardinality of the infinite number of universes in the Multiverse?
Is it countable or more than countable?
Is it the same in different theories?

2. Sep 8, 2016

### Lucas SV

It is definitely uncountable. Even for the simple case of a free particle in quantum mechanics, travelling from point $a$ to point $b$, there are uncountably many histories (paths that can be taken from $a$ to $b$).

3. Sep 8, 2016

### tzimie

It is not obvious for me.
In QM (contrary to Newton mechanics) finite mass in finite volume can be only in finite number of states. So the cardinality of states of our universe is aleph-null, for that reason (based on Max Tegmark) Hubble spaces necessarily repeat.

4. Sep 8, 2016

### Lucas SV

Ok, if we consider distinct universes as arising from distinct states, then even so we have usually, at least in idealized textbook quantum systems, infinitely many states. An electron in a hydrogen atom has a countably infinite number of states. This is because the electron is bound. For free particles, the number of states are uncountable because the spectrum of the Hamiltonian becomes continuous. Such states where the energy is continuous, are called scattering states.

I do not know, however, in what way that the many-worlds interpretation is mathematically formulated, so that I cannot say how scattering states are related. Moreover, it may be that none of the idealizations of uncountably many states are actual, so that the observable universe contains only a finite amount of information. Again I have not reached the required level in the study of quantum information theory, to be qualified to answer this.

5. Sep 8, 2016

### tzimie

Hmmm, I am puzzled. Because what you are saying violates this: https://en.wikipedia.org/wiki/Bekenstein_bound

This is based on the assumption that particles are "free". While this abstraction could be useful in practice, for me it is just an idealization. Very likely all "particles" have common past and are correlated with each other, so there are no "free" particles in a given Hubble volume.

In any case it violates Bekenstein bound, as 2 "free" particles carry an infinite amount of information.

6. Sep 8, 2016

### PeroK

How do you know that?

7. Sep 8, 2016

### Lucas SV

I don't know how information is defined mathematically in different places, but in the article it is defined as entropy. The entropy is the number of microstates that correspond to a given macrostate. The entropy is not the same quantity as the number of possible quantum states in a system, but rather those quantum states that correspond to given macroscopic quantities such as temperature.

Yes it is an idealization, albeit a very useful one. For a cosmologist, it may be important to talk about the entanglement of particles in the big bang. For someone working on fundamental quantum physics, it may also be important. But for a particle physicist working in the LHC, when looking in a scattering experiment, it often suffices to think of particles in the far past as being free, and particles in the far future as being free. Scattering theory based on QFT relies heavily on these assumptions. Yet the predictions of quantum field theory are incredibly accurate, take QED for example.

In my opinion this is a question of scale. In some scales we may ignore the physics at different scale in our description of the physics of the current scale. These are the ideas in which effective field theory is based upon.

8. Sep 8, 2016

### Lucas SV

Ok now I retract my statement about there being uncountably many multiverses, since I don't even know much about the Many worlds interpretaton of quantum mechanics. However, I am sure that there are uncountably many paths from $a$ to $b$ (assuming we allow any path in $\mathbb{R}^3$), that is just a mathematical statement. Easy to prove: Take a point $c$ different from $a$ and $b$. Now take a line going through $c$, but not $a$ nor $b$. For each point $d$ in the path, there exists a path going through $d$. Since the line has uncountably many points, there are uncountably many paths.

9. Sep 8, 2016

### MathematicalPhysicist

Last time I read/heard in Superstring theory/M theory there should be in the landscape something like 10^500 different universes at least which is still finite.
https://en.wikipedia.org/wiki/String_theory_landscape

I am not well versed with the technicalities, there's a lot of literature to read, vast.

10. Sep 8, 2016

### tzimie

In QM there is no such thing as an exact "path" of a "particle" from point A to point B, unless you are working with Bohmian version

11. Sep 8, 2016

### tzimie

This is a number of potential geometries, so the number of categories of different universes, not the number of universes.

12. Sep 8, 2016

### Lucas SV

In the path integral formulation, you compute the amplitude by summing (integrating) over paths, with a factor $e^{i S}$ assigned to each path, where $S$ is the classical action. Of course, I'm not saying the particle is following a particular path, it is just a method to compute amplitudes.

Also, by using the word 'particle', I'm not implying a localized state, but I use the word in the same way particle physicists around the world would use the word. If you want me to be technical, I can say that by a (non-relativistic) particle travelling from point $a$ to point $b$, I mean I am trying to calculate the amplitude $\langle b | a \rangle$, which is the inner product of position eigenstates. Then any other amplitude can be calculated, since any state can be expressed as a superposition of position eigenstates, whose coefficients are what we call the wavefunction.

Last edited: Sep 8, 2016
13. Sep 8, 2016

### MathematicalPhysicist

So where is it stated the number of universes? Is it mentioned somewhere?

Mind you I don't think LQG necessarily resort to a multiverse solution to their problems.

14. Sep 9, 2016

### JorisL

That's restricted to the case of Type IIB flux vacua though.

RE the cardinality, I'm not too certain, a scan I performed to find de Sitter vacua (in 6D) came up with 24ish geometries upto isomorphisms.
If you include every geometry without "modding out" equivalent geometries the cardinality could be that of $\mathbb{R}$.
The critical points of the vacuum potential for this construction can be found anywhere in $\mathbb{R}^n$ for $n$ parameters which are somewhat constrained (but not very much). In principle that is, it's likely that if such points exist they are sparse in the total set.

I think the best way forward would be to check some totally explicit constructions. For example Minkowski or AdS solutions.
Then you can try and see for yourself.