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In my rudimentary understanding of the many-worlds interpretation of quantum mechanics, one posits the existence of a universal wave function - the state of the entire universe - wherein the various superposition of states never actually collapse, yet appears to collapse to an observer (or macroscopic system) through the mechanism of quantum entanglement.

One says for example that the observer is in any instance measuring precisely one classical state out of a superposition of many by following a certain 'branch'. Or one says that the universe keeps splitting or 'branching' off, which makes intuitive sense in a localized laboratory setting. However, the more I think about it, in particular when I abstract away from the duality between the observer and the observed, the more confused I become.

What is the precise, mathematical definition of a 'branch'? Can it be defined independently of a (classical) observer? If the ontological base is the universal wave function, how are its branches defined, or made sense of?

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    $\begingroup$ Please look at the old related questions, this has been asked every week for years here! $\endgroup$
    – knzhou
    Commented Mar 28, 2020 at 21:57
  • $\begingroup$ I haven't found something applicable to my question in questions posted earlier. $\endgroup$
    – Mellon
    Commented Mar 28, 2020 at 21:59
  • $\begingroup$ I've removed a number of comments that were attempting to answer the question and/or responses to them. Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$
    – David Z
    Commented Mar 28, 2020 at 22:08

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What is the precise, mathematical definition of a 'branch'? Can it be defined independently of a (classical) observer?

The real definition is that distinct branches are exactly those parts of the wavefunction that have no chance of ever interfering with each other again. More precisely, if the wavefunction can be decomposed into $|\psi_1 \rangle + |\psi_2 \rangle$, where $\langle \psi_1 | \psi_2 \rangle = 0$, these are separate branches if the time evolution of the universe will never again make $\langle \psi_1 | \psi_2 \rangle$ nonzero.

The whole point of the philosophy of many worlds is that this definition is completely independent of any external observer, and follows automatically from the evolution of the wavefunction.

In general, a common way to create distinct branches is to entangle systems with many degrees of freedom, in which case no future interference is basically ensured by the second law of thermodynamics. Classical observers are examples of systems with many degrees of freedom, so classical observation does produce branches, but it isn't necessary.

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  • $\begingroup$ Two electron orbitals of a hydrogen atom have this independent property. Any decomposition into an orthonormal eigen function basis has this property. It is a common practice in QM to use such bases. But, this decomposition can happen in many different ways. In this sense, any wave function could be a branch under your definition. $\endgroup$
    – Ponder Stibbons
    Commented Mar 29, 2020 at 6:24
  • $\begingroup$ @PonderStibbons No, not at all. For a hydrogen atom, a wide variety of external perturbations can cause the orbitals to mix. That isn't true for genuine branching. $\endgroup$
    – knzhou
    Commented Mar 29, 2020 at 6:29
  • $\begingroup$ this mixing is not the full picture. In the context (I thought) we are using, QM is linear and so there is no mixing. Any mixing only occurs because some of the degrees of freedom were not accounted for. (Or, using the Born interpretation, discontinuously during measurement). If the full wave function was determined for the atom and the thermal bath, then we would just be back to no mixing. On the other hand if we go to non linear QFT, then all bets are off, IMHO. In that context, I am not sure that the many world interpretation really holds up. Not as originally intended, anyway. $\endgroup$
    – Ponder Stibbons
    Commented Mar 29, 2020 at 10:14
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    $\begingroup$ BTW en.wikipedia.org/wiki/Many-worlds_interpretation#Testability I don't agree with Deutch here, MWI is a philosophical position only. It just says that quantum states are a linear combination (measure) over some topological space. It is no different than the idea is classical probability that since the die can roll 1 through 6 then there must be 6 universes that are selected from when the die is rolled. QM is not required for the MWI. And Newtonian mechanics is not deterministic. Except for observation, Born QM is more deterministic, as it removes chaos from the picture. $\endgroup$
    – Ponder Stibbons
    Commented Mar 29, 2020 at 10:28
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It is not the only way to define it, but I think that the neatest way is as follows.

Consider all the possible universes from start to finish. The state of the observed universe is a complex linear combination over all these universes. However, if two universes are identical up to time t, then we can instead think of them as branching off from each other.

The principle is identical to that of the dictionary storage issue.

"planting" and "planted" are two words.

We can say they are just two words

planting planted

Or we can say they branch

plant -- ing -- ed

This is the essence of the idea of branching in the multi worlds interpretation.

If an observer "goes down one branch" rather than the other they see a wave function that is zero on one of these and unit on the other.

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  • $\begingroup$ By a possible universe, do you mean a possible classical universe? Or possible universal wave functions? I don't think the first case is really coherent, since all observables cannot be determined at once. $\endgroup$
    – Mellon
    Commented Mar 28, 2020 at 23:43
  • $\begingroup$ This only makes sense for a Newtonian spacetime. Relativity ensures that there is no observer-independent way to compare universes up to a time $t$. So this cannot work without introducing an observer in the loop, which opens a large can of worms. $\endgroup$
    – Stéphane Rollandin
    Commented Mar 28, 2020 at 23:53
  • $\begingroup$ @mellon No. Assumption of classical behavior has nothing whatsoever to do with the construction. Whatever definition you use for what you mean by a world, the principle of the construction applies to that. The universal wave functions, in principle, form a vector space, while the space of all universes would be a more general Hilbert Manifold or some other infinite dimensional manifold. An observation in this context would mean something more like an observation strategy, in which the result of one measurement leads to another. $\endgroup$
    – Ponder Stibbons
    Commented Mar 29, 2020 at 6:15
  • $\begingroup$ @StéphaneRollandin given two classical but relativistic universes, you look for a diffeomorphism between one selected patch in each and this gives you a correspondence. Find a maximal such pair of patches that can be thus mapped, and we have the boundary of this patch is a branch surface between the two universes. Different observers will claim the branch is at different places and times, and they will not agree on the simultaneity the of the branching. But, there is no problem of applying this within a relativistic context at all. $\endgroup$
    – Ponder Stibbons
    Commented Mar 29, 2020 at 6:19
  • $\begingroup$ Ok so now time $t$ is replaced by a Cauchy surface, if I understand correctly. This makes branching a very non-local business, which seems to me contradictory to the proclaimed local spirit of MWI. $\endgroup$
    – Stéphane Rollandin
    Commented Mar 29, 2020 at 13:54

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