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If I understand correctly the many worlds interpretation says the universe is continously splitting into multiple branches and quantum measurements occur when decoherence causes a quantum state to select one of its possible outcomes. Each branch gets its own possible outcome and likely outcomes occur in more branches. I added this paragraph because I'm still a bit unsure about this so you can correct me if I'm wrong.

The problem is I don't understand how this would translate to quantum states this so I can't really appreciate the full glory of this theory which is sad because I think it sounds really compelling. Do multiple branches live in superposition of each other? So do you have to add the different states together? Or do you have to tensor product them together somehow?

As an example consider Schrödinger's cat. Let $U(t_2,t_1)$ be an unitary time evolution operator that takes a state from time $t_1$ to $t_2$. The poison to kill Schrödinger's cat is triggered by the decay of an atom. At time $t_1$ the atom and cat are in the state $|\text{not decayed}\rangle\otimes|\text{cat alive}\rangle$. At $t_2$ the atom that would have a 50% chance of being decayed. At $t_3$ the poison has had enough time to do its things (poor cat). Would then the following equation be an example of many worlds theory? \begin{align} U(t_3,t_2)\Big(U(t_2,t_1)\big(\,|\text{not decayed}\rangle\otimes|\text{cat alive}\rangle\,\big)\Big)&=\\ U(t_3,t_2)\big(\,\frac{1}{\sqrt 2}(|\text{not decayed}\rangle+|\text{decayed}\rangle)\otimes |\text{cat alive}\rangle\,\big) &=\\ \frac{1}{\sqrt 2}(|\text{not decayed}\rangle\otimes|\text{cat alive}\rangle+|\text{decayed}\rangle\otimes|\text{cat dead}\rangle) \end{align} So what would be the best way to describe many worlds? Better examples are also welcome.

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  • $\begingroup$ Afaik collapsing into a state is not an operator in QM, it is more like a branch in an if statement, the if statement being like a model for the outside of the system. Or have you found a definition for collapsing into a state? $\endgroup$
    – Emil
    Feb 15 at 12:08
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    $\begingroup$ @Emil In the Copenhagen interpretation it's just a projection opeator: $|\psi\rangle\rightarrow |n\rangle\langle n|\psi\rangle$. But Copenhagen doesn't say anything about the process while many worlds does. I don't know how collapse looks like in many worlds and that's why I'm asking the question. $\endgroup$ Feb 15 at 12:24
  • $\begingroup$ Isn't many worlds saying that the other branches are run in some other world ? $\endgroup$
    – Emil
    Feb 15 at 14:46
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    $\begingroup$ "Many Worlds" is a philosophical intepretation. It does not affect the formalism. $\endgroup$
    – mike stone
    Feb 15 at 15:19
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    $\begingroup$ Part 3 of Sidney Coleman's lecture "Quantum Mechanics in Your Face" gives a nice description of how to think about it. You can read it on arxiv here: arxiv.org/abs/2011.12671 (part 3 starts at the bottom of page 7) or watch it on youtube here youtube.com/watch?v=EtyNMlXN-sw $\endgroup$
    – Andrew
    Feb 15 at 15:21
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MWI is simply standard quantum mechanics without any additional axioms. The Copenhagen interpretation involves making up more machinery (projection onto an eigenstate when we make a measurement). So in bra-ket notation, MWI just looks like standard quantum mechanics.

For Schrodinger's cat, I don't think you're going to get a satisfactory notation using a finite-dimensional vector space. It's an irreversible exponential decay process, and you can't describe such a process using two states. You need decay into an infinite continuum of states.

You might think, OK, then let's work with something like a Schrodinger's cat experiment based on the output of a Stern-Gerlach spectrometer. Then the problem is that you presumably want to work out how the measurement process looks. But again, measurement is an irreversible process, and you can't mock up an irreversible process using a finite-dimensional vector space. Measurement is irreversible precisely because the measuring device has so many degrees of freedom.

So for a really satisfying description using only standard quantum mechanics, you really want the full description in terms of decoherence.

If I understand correctly the many worlds interpretation says the universe is continously splitting into multiple branches

Not really. The stuff about branching is not present at the level of the basic formalism of MWI. Branching is really just a story that we conscious observers tell ourselves in order to make sense of why decoherence prevents us from simultaneously experiencing two different realities, or observing interference between two cat states. If branching was literally going to be present in the formalism, it would be something like a non-Hausdorff manifold, which isn't what any actual formalism of quantum mechanics uses.

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