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Background and Question

"Measurements are modelled as interactions between subsystems of the universe and manifest themselves as a branching of the universal state." - Wikipedia

What is the cardinality of the set of universes that branch out from a particular universe?

Essentially I am under the impression that while everyone within the many world interpretation of quantum mechanics agree on branching I feel they are vague on how many worlds exist due to this branching.

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  • $\begingroup$ To whomever answered the question and then deleted it: I found the answer interesting and was going to ask since measure theory does make sense in QM if there was any way that could be used to divide the area? (sorry if this is irrelevant but I didnt have too much time to read the answer let alone ponder it's implications) $\endgroup$ – More Anonymous Sep 11 at 2:17
  • $\begingroup$ To be an "interpretation" the theory must use the same mathematics as the other "interpretations". In the path integral mathematical formulation of quantum mechanics it is obvious that there are infinite possible paths. Interpreting them as different worlds does not reduce the infinities. imo in any mathematical formulation of many worlds the same will be true. Otherwise it is not an "interpretation" but a new theory replacing quantum mechanics! $\endgroup$ – anna v Sep 11 at 2:50
  • $\begingroup$ Surely in the case of quantum mechanics where measure theory is well defined .. One can answer this kind of question? Infact I suspect the many worlds would have the same cardinality as the infinite possible paths (?). I feel the essense of what your saying is in many worlds they use branching but have no precise idea of what their talking about? $\endgroup$ – More Anonymous Sep 11 at 2:53
  • $\begingroup$ Yes, I think they hand wave their "interpretation" but are not in a position mathematically to support it, imo. $\endgroup$ – anna v Sep 11 at 2:59
  • $\begingroup$ While I have similar fears I hope to be proven wrong by asking this question. $\endgroup$ – More Anonymous Sep 11 at 3:01
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The many worlds interpretation (MWI) takes quantum mechanics (QM) seriously as a description of how the world works.

A measurement is an interaction that takes information from a system that can be copied into other systems. This copyable information for a particular interaction consists of eigenvalues of some observable:

https://arxiv.org/abs/1212.3245

If information about some observable is copied widely then that prevents interference between versions of the system with different values of that observable. The suppression of interference makes these different versions act like a collection of parallel universes to a good approximation:

https://arxiv.org/abs/quant-ph/0104033

The set of values of an observable form a discrete and finite set for any finite system. So the number of distinguishable versions of a system is some finite but large number. To calculate the number of universes that could be produced you would do something like the calculations of entropy in "Universal upper bound on the entropy-to-energy ratio for bounded systems":

http://old.phys.huji.ac.il/~bekenste/PRD23-287-1981.pdf

This number isn't particularly interesting or fundamental and it doesn't really matter for making predictions, so there's not much reason to calculate it.

Now, if we look at some particular member of the set of measurable values, it has a real valued number that represents the probability of getting that result. So each branch consists of a set of instances of a system that aren't distinguishable by any measurement. The "number" of instances is either a real number or a continuous infinity depending on how you want to think about it.

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  • $\begingroup$ Thank you for your answer. You seem to be saying the number of the branches depend on experiment outcomes? Also In the case of infinity is it $2^{\aleph_0}$ or greater? $\endgroup$ – More Anonymous Sep 12 at 10:15
  • $\begingroup$ The number of instances in each branch has the same cardinality as the set of real numbers, which is $2^{\aleph_0}$. The number of branches depends on the details of the interactions of the systems involved, and since those interactions are used in experiments the number of possible experimental outcomes is related to the number of branches. $\endgroup$ – alanf Sep 12 at 10:22
  • $\begingroup$ Ah ... I was under the impression that it always the same infinity however when there are a finite number of outcomes (say $2$) you have half outcomes A and the other half B. $\endgroup$ – More Anonymous Sep 12 at 10:29
  • $\begingroup$ No. The probabilities have to add up to 1 but they don't always have to be 0.5 and 0.5. $\endgroup$ – alanf Sep 12 at 11:46
  • $\begingroup$ Yes, I am aware of that and was providing an example when I said "(say $2$)" ... Yes there can be systems where there are $2$ outcomes of say $ .7$ and $.3$ $\endgroup$ – More Anonymous Sep 12 at 11:47

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