Does MWI Adequately Explain Observer Branching in Quantum Mechanics?

[PeterDonis]

yet they are not in superposition

You're misstating it. The state $a \vert + \rangle + b \vert - \rangle$ is the superposition of $+$ and $-$ that we have been talking about. The state $b \vert + \rangle - a \vert - \rangle$ is also a superposition (a different superposition) of $+$ and $-$. And all of this depends on our having chosen $+$ and $-$ as a basis.

If, instead, we choose the states $a \vert + \rangle + b \vert - \rangle$ and $b \vert + \rangle - a \vert - \rangle$ as a basis (which we can since they are orthogonal and any state in the Hilbert space can be expressed as a linear combination of them), then the states $+$ and $-$ are now superpositions, whereas our new basis states are not. So yes, whether or not a state is a superposition is basis dependent, as Sabine says.

However, the choice of basis is not arbitrary once you specify a particular measurement. For example, in our spin example, we specified that we were measuring spin about the axis for which $+$ and $-$ are the eigenstates. That means the $+$ and $-$ basis is picked out by the physics of the measurement we are making; the choice of basis is no longer arbitrary. So the fact that the state $a \vert + \rangle + b \vert - \rangle$ is a superposition in this basis is now physically relevant, since it affects how the state of system + measuring device (+ environment once we bring that in) will evolve in time, in contrast to the way everything would evolve in time if the particle we were measuring were in the state $+$ (not a superposition in this basis).

Or, we could have chosen a different spin measurement, one for which the eigenstates were $a \vert + \rangle + b \vert - \rangle$ and $b \vert + \rangle - a \vert - \rangle$. Then a particle in the state $a \vert + \rangle + b \vert - \rangle$ would not be in a superposition, and everything would evolve in time the way you would expect if you measure a particle that is in an eigenstate of the measurement (i.e., no "branching" from the MWI point of view). We could still write things in the $+$ and $-$ basis, as I just did, but the measurement operator would look more complicated in this basis (the matrix would not be diagonal), and you would have to do a more complicated calculation to confirm what I just said about how things would evolve in time (whereas the calculation in the basis of eigenstates of the measurement is trivial).

We are rapidly approaching the point where I am going to close this thread as we have beaten this subject to death as much as we can in a "B" level thread. As I've said before, you really need to spend some time working through a basic QM textbook (and learning linear algebra, since all this that I've been saying about superpositions and basis and measurement operators is linear algebra 101).

 

[PeterDonis]

let's say one basis is the cat is in state of dead-and-aliveness and another basis is in state of dead-minus-alive (or others). Why are these basis not in superposition?

Because, as Sabine says, you can pick any pair of orthogonal states you like as a basis (note that we are treating the cat as if it were a spin-1/2 particle, which of course it isn't, but that complication is usually glossed over in discussions of this sort--although it really shouldn't be, see below). But, as I noted in my previous post, once you start talking about making measurements on the system, the choice of basis is no longer arbitrary--you can pick any basis you like, but only one basis will correspond to the basis of eigenstates of the measurement you are making. In the case of the cat, the alive/dead measurement is the one we know how to physically realize (or at least sort of realize--see below). We don't know how to physically realize a measurement which has "alive plus dead" and "alive minus dead" as its eigenstates.

(And, since the cat is not a spin-1/2 particle but a huge conglomeration of something like $10^{25}$ atoms, all of which are interacting, and which is interacting continuously with its environment, it's not clear that there even is a measurement that has "alive plus dead" and "alive minus dead" as eigenstates. That's because it's not clear that there even is a single quantum measurement that has "alive" and "dead" as eigenstates! The "alive state" of the cat is not a single quantum state; it's a huge subspace of the cat's Hilbert space, and the "dead state" is another huge subspace that we are assuming is disjoint from the first one. But we don't actually know that that's the case. We don't have a foolproof "alive/dead meter" that we can attach to a cat that goes one way if the cat is alive and another way if the cat is dead, the way we can unambiguously measure the spin of a spin-1/2 particle. When we say that a cat is "alive" or "dead", that is a judgment based on what amounts to a huge number of quantum measurements that has no useful relationship to the kind of clean, idealized measurement we make on the spin of a single spin-1/2 particle. So it's unfortunate that those two cases are so often conflated in discussions of this sort.)

 

[Denis]

Decoherence is necessary, together with locality, for solving the basis selection problem, which is crucial to "branching".

Which is one of the problems why MWI does not make sense.

Decoherence itself is a quite meaningful method to study various quantum systems, in particular greater ones which have some interaction with the environment which cannot be completely suppressed.

But this already presupposes some subdivision of the world into some parts, in particular the system to be studied and the environment. Without such a subdivision given, decoherence does not make sense. To apply decoherence, MWI has to presuppose something it officially does not presuppose. At least, I'm not aware what are these subdivisions into systems, environments and observers we need even before we can define the worlds - given that, as claimed, branching (and therefore the resulting branches) are not yet defined.