Dimensionality of the wavefunction in relative state

[atyy]

That's right. There is no branching in a closed system. The wavefunction of the universe does not branch. The universe does not split. I don't see why that should be a problem.

But by the same token, since there is no "split of the wave function", then there is no "individual 3D wave functions for each particle".

The BMW way is, I don't think, exactly what you have in mind. But if you take the many worlds proposals in which all the branches are present from the start, then you can have that - I think Saunders had one version like that - that's like BMW.

 

[Derek Potter]

But by the same token, since there is no "split of the wave function", then there is no "individual 3D wave functions for each particle".

Why not? The expansion is a mathematical identity, not a physical process. You can leave it as a single wavefunction if you wish. Or you can re-write it expanded in terms of canonical 3D wavefunctions. Or any other basis that you fancy.
As for the branches being there all along, that's fine. A basis state that evolves into a superposition a|u(t)> + b|v)t)> can be written at t = 0 either as a|u(0)> + b|v(0)> or as (a+b)|u(0)> because |u(0)>=|v(0)>. The first has the split even when there is no physical significance to it, the second depicts it as "occurring" once the superposition is present. Since the split is not a physical process it does not matter what we call the state at t=0.

 

[atyy]

Why not? The expansion is a mathematical identity, not a physical process. You can leave it as a single wavefunction if you wish. Or you can re-write it expanded in terms of canonical 3D wavefunctions. Or any other basis that you fancy.
As for the branches being there all along, that's fine. A basis state that evolves into a superposition a|u(t)> + b|v)t)> can be written at t = 0 either as a|u(0)> + b|v(0)> or as (a+b)|u(0)> because |u(0)>=|v(0)>. The first has the split even when there is no physical significance to it, the second depicts it as "occurring" once the superposition is present. Since the split is not a physical process it does not matter what we call the state at t=0.

A branch in MWI is in some sense physical. So one cannot just go back and forth unphysically.

 

[Derek Potter]

A branch in MWI is in some sense physical. So one cannot just go back and forth unphysically.

Why not? The branch structure is not unique, it depends on the basis. Schrödinger's Cat is not merely |alive> and |dead>, it is also |foo> and |bah>. Or |yin> and |yang>. Or simply |cat>. Same creature, same evil experimenter, same state, different bases. Sorry, I just don't get what you're saying.

 

[atyy]

Why not? The branch structure is not unique, it depends on the basis. Schrödinger's Cat is not merely |alive> and |dead>, it is also |foo> and |bah>. Or |yin> and |yang>. Or simply |cat>. Same creature, same evil experimenter, same state, different bases. Sorry, I just don't get what you're saying.

Sure, but that's not what the decoherence form of MWI does. The decoherence is intended to pick a preferred basis.

 

[Derek Potter]

Sure, but that's not what the decoherence form of MWI does. The decoherence is intended to pick a preferred basis.

Of course. And it does. But why would the ability to further expand entangled states into 3D particle states invalidate decoherence theory? Of course with fine-branching it's all too easy to lose track of macroscopic meaning. This is very akin to statistical mechanics - merely defining a low entropy state as one with a small number of microstates is not sufficient, you need to identify the ones that exhibit low entropy at the classical thermodynamic level. And it's not easy. So it is with fine branching in QM: you need to identify or at least create a "measure" of the branches which are particular pointer states. Much easier to start with a more appropriate basis - the pointer states themselves - and work down, showing that a formulation is possible in 3 dimensions. That's non-locality and entanglement and multi-dimensional phase space all subsumed in superposition. Then we can work in the other direction, just retrieving the fact that multi-particle states have a very big basis. This allows us to develop decoherence theory separately from the fine-branch model. Which is what this topic is about :)

 

[atyy]

Much easier to start with a more appropriate basis - the pointer states themselves - and work down, showing that a formulation is possible in 3 dimensions.

But the coarse-grained branches will be superpositions of the fine grained branches. Only the fine grained branches are products of 3D wave functions, so the coarse-grained branches will not be, since they are superpositions of products.

 

[Derek Potter]

But the coarse-grained branches will be superpositions of the fine grained branches.

Exactly. The coarse-grained world of Schrödinger's Cat is a vast collection of fine-grained worlds - worlds where particles have independent 3D wavefunctions. Different bases, different branching, same state. How is that a problem?

 

[atyy]

Exactly. The coarse-grained world of Schrödinger's Cat is a vast collection of fine-grained worlds - worlds where particles have independent 3D wavefunctions. Different bases, different branching, same state. How is that a problem?

That's fine. Then I just don't understand your motivation. Usually people worry about the 3D wave functions when they think something about notions like "physicality" or whatever. In the decoherence version, since the 3D wave functions don't apply (because the coarse-grained branch is a superposition of them), then it would seem that the fine grained "physicality" has been lost.

In case you are wondering, I understand this better in the context of Copenhagen. One can always have the 3D wave functions, since that is essentially how the Hilbert space of the whole system is defined. However, this is not considered physical in the context of Copenhagen, because it is the Born rule that makes things physical, and then we have to square the wave functions and have interference between branches, so the 3D wave functions remain formal, not physical.

 

[Derek Potter]

That's fine. Then I just don't understand your motivation. Usually people worry about the 3D wave functions when they think something about notions like "physicality" or whatever. In the decoherence version, since the 3D wave functions don't apply (because the coarse-grained branch is a superposition of them), then it would seem that the fine grained "physicality" has been lost.

In case you are wondering, I understand this better in the context of Copenhagen. One can always have the 3D wave functions, since that is essentially how the Hilbert space of the whole system is defined. However, this is not considered physical in the context of Copenhagen, because it is the Born rule that makes things physical, and then we have to square the wave functions and have interference between branches, so the 3D wave functions remain formal, not physical.

My motivation is to use a foundation in which 3D wavefunctions are fundamental, as this removes entanglement from the picture whilst retaining reality - albeit as a superposition.

I understand your Copenhagen view. However I do not agree that the Born rule makes things physical. The Born rule just tells us the formula for the density matrix. What makes things physical is some sort of postulate such as "Nothing is real, Man, until we look at it". That is pure metaphysics, superfluous metaphysics at that. And accordingly detestable :)

I do not think interference affects the picture. The 3D functions remain fundamental. Interference requires us to "add and square", not "square and add", which means we have quietly changed basis. Why would we expect the new picture to make sense in terms of particle states when we have just exchanged them for something different? Schrödinger's cat is still a cat whether it's regarded as a collection of atoms or holistically as, well, a cat.

 

[Derek Potter]

Well that seems to have finished the topic off neatly enough. Thanks atyy.