Dimensionality of the wavefunction in relative state

[Derek Potter]

A wavefunction of a single particle (ignoring spin etc) is a three dimensional object mapping to 3-D physical space. The wavefunction of two unentangled particles is separable as a product of two independent 3-D wavefunctions. If the particles are entangled, the states cannot be separated, the state of the system needs a 6-D space. In general, 3N dimensions are needed.

However, in the RSF, a system state is considered as a sum, a superposition, of relative states. Each component state is a product of the states of the individual particles. These are individually 3-D states. So it seems to me that RSF and its immediate offspring, MWI, get rid of the non-physicality of the multi-dimensional wavefunction and replace it by independent 3-D wavefunctions, one for each particle. The complexity of the extra dimensions is, of course, replaced by the exponential number of states in the system each with its own coefficient. But the spooky action at a distance goes away and 3-D wavefunctions come back.

Errors in this?

 

[Simon Bridge]

It is possible to have multi-particle wavefunctions. Wavefunctions are properly 3+1D objects.
I think it is an error to place so much importance on the number of dimensions in a mathematical model - or, for that matter, on wavefunctions.
We do lots of stuff to make the maths simpler - but we have to try not to confuse the map for the territory.

 

[bhobba]

However, in the RSF, a system state is considered as a sum, a superposition, of relative states. Each component state is a product of the states of the individual particles.

First I ever heard of that. Can you detail the math?

MW does not change QM - in entangled states what is entangled does not have a state - that's from the very definition of entanglement.

Thanks
Bill

 

[bhobba]

or, for that matter, on wavefunctions.

Just to elaborate a bit further - in QM the state is the fundamental thing - not its representation in a particular basis - which is what a wave-function is. The situation is the same as linear algebra - the vectors are the fundamental thing and what the axioms contain - nor their representations.

Thanks
Bill

 

[Derek Potter]

Just to elaborate a bit further - in QM the state is the fundamental thing - not its representation in a particular basis - which is what a wave-function is. The situation is the same as linear algebra - the vectors are the fundamental thing and what the axioms contain - nor their representations.
Thanks
Bill

edited
Please describe the physical phenomena of entanglement in the momentum basis and explain why Einstein would have thought the correlations were spooky.
I stated the reason for using the position basis in my post.

 

[Derek Potter]

It is possible to have multi-particle wavefunctions. Wavefunctions are properly 3+1D objects.
I think it is an error to place so much importance on the number of dimensions in a mathematical model - or, for that matter, on wavefunctions.
We do lots of stuff to make the maths simpler - but we have to try not to confuse the map for the territory.

In what sense are you using the word "error"? When I use it, I mean something like 3+3 = 2+1+2+1 = 1+1+1+1+1 = 5 where the error is in the third expansion due to dropping a "1". That's an error. I can see no error in discussing whether a 3 gazillion dimensional object can fit into 3-space and, if not, whether it can be re-formulated so that it does.

 

[bhobba]

Please describe the physical phenomena of entanglement in the momentum basis and explain why Einstein would have thought the correlations were spooky. I stated the reason for using the position basis in my post.

Suppose system 1 can be in state |a> and |b>. Suppose system 2 can also be in state |a> and |b>. If system 1 is in state |a>, and system 2 is in state |b>, that is written as |a>|b>. Conversely if system 1 is in state |b> and system 2 is in state |a> that is written as |b>|a>. From the principle of superposition a superposition of those states is also a possible state ie c1|a>|b> + c2|b>|a> is also a possible state. Such a state is by definition called entangled.

The above is a general definition, its specific application to momentum eigenstates is obvious and immediate.

Einstein believed in naive reality, the consequences of which Bell's theorem spells out.

I can't find any mention of a reason for the position basis, but I perhaps am blind. Suppose you spell it out again?

Thanks
Bill

 

[Derek Potter]

Suppose system 1 can be in state |a> and |b>. Suppose system 2 can also be in state |a> and |b>. If system 1 is in state |a>, and system 2 is in state |b>, that is written as |a>|b>. Conversely if system 1 is in state |b> and system 2 is in state |a> that is written as |b>|a>. From the principle of superposition a superposition of those states is also a possible state ie c1|a> + c2|b> is also a possible state. Such a state is by definition called entangled.
The above is a general definition, its specific application to momentum eigenstates is obvious and immediate.

Not to me it's not. In the position basis, there is correlation between (properties of) separated objects. Einstein regarded it as spooky. I see no reason to think the correlations in momentum space would seem spooky. If you can demonstrate some reasonably obvious sense in which locality in momentum space is to be expected from, say, special relativity or some such universal principle, but is actually broken by entanglement, then I will be very grateful and will happily drop the word "physical" from my first sentence. The point would then stand even more generally.

I can't find any mention of a reason for the position basis, but I perhaps am blind. Suppose you spell it out again?
Thanks
Bill

Because the discussion is about whether wavefunctions can be functions of position in real space.

 

[bhobba]

I see no reason to think the correlations in momentum space would seem spooky.

Then, to be blunt, I suggest you think a bit harder. Regardless of what they are entangled in, if they are entangled, measuring one immediately tells you about the other.

Because the discussion is about whether wavefunctions can be functions of position in real space.

I think you are confused between can be and must be.

Thanks
Bill

 

[Derek Potter]

Then, to be blunt, I suggest you think a bit harder. Regardless of what they are entangled in, if they are entangled, measuring one immediately tells you about the other.
Bill

The spookiness lies in the apparent "action at a distance" not in the acquisition of information.

 

[bhobba]

The spookiness lies in the apparent "action at a distance" not in the acquisition of information.

The spookiness lies in the fact knowledge of one particle instantaneously gives knowledge of the other.

Thanks
Bill

 

[Derek Potter]

The spookiness lies in the fact knowledge of one particle instantaneously gives knowledge of the other.
Thanks
Bill

?

 

[bhobba]

?

?

Cognate on the word - action.

Thanks
Bill

 

[Derek Potter]

?
Cognate on the word - action.
Thanks
Bill

And what is spooky about action on a particle which has a different momentum from you?
added - Cognate on the word "distance"!

 

[Derek Potter]

I think discussing the momentum representation in the context of "action at a distance" is as pointless as discussing when a trigger pulse triggers a bomb but insisting on using the frequency domain.

 

[bhobba]

And what is spooky about action on a particle which has a different momentum from you?

For me there is nothing spooky - for Einstein it was.

Action implies something the breaking of entanglement doesn't require - namely some kind of physical link between them, sending information etc. In reality all it is is a correlation.

Thanks
Bill

 

[Derek Potter]

For me there is nothing spooky - for Einstein it was.
Action implies something the breaking of entanglement doesn't require - namely some kind of physical link between them, sending information etc. In reality all it is is a correlation.
Thanks
Bill

The spooky action is not one particle acting on the other but Alice and Bob's detector settings acting on both particles (determining the overall measurement basis). Since Alice and Bob are space-like separated, there is no possible causal link between the settings and the observed correlation. So, yes, "all it is is a correlation". It's a correlation which depends on, but cannot be caused by, Bob and Alice's settings. That's pretty spooky. To me and Einstein. Maybe not to you.

 

[bhobba]

The spooky action is not one particle acting on the other but Alice and Bob's detector settings acting on both particles (determining the overall measurement basis)

I have zero idea why you think its confined to EPR with Alice and Bob.

Thanks
Bill

 

[Derek Potter]

I have zero idea why you think its confined to EPR with Alice and Bob.

No, I'm pretty sure it would work with Jack and Jill too.

 

[atyy]

A wavefunction of a single particle (ignoring spin etc) is a three dimensional object mapping to 3-D physical space. The wavefunction of two unentangled particles is separable as a product of two independent 3-D wavefunctions. If the particles are entangled, the states cannot be separated, the state of the system needs a 6-D space. In general, 3N dimensions are needed.

However, in the RSF, a system state is considered as a sum, a superposition, of relative states. Each component state is a product of the states of the individual particles. These are individually 3-D states. So it seems to me that RSF and its immediate offspring, MWI, get rid of the non-physicality of the multi-dimensional wavefunction and replace it by independent 3-D wavefunctions, one for each particle. The complexity of the extra dimensions is, of course, replaced by the exponential number of states in the system each with its own coefficient. But the spooky action at a distance goes away and 3-D wavefunctions come back.

Errors in this?

I think this is the same idea, in the context of Bohmian Mechanics: http://arxiv.org/abs/1410.3676.

 

[Derek Potter]

I think this is the same idea, in the context of Bohmian Mechanics: http://arxiv.org/abs/1410.3676.

Is it? I don't know anything about BM, but if the paper uses the same decomposition as I was talking about, namely:
Ψ = ∑_ijk a_ijk|u_i>|v_j>|w_k> etc
then I guess it might be the same idea. My question, which has not yet been answered, was much simpler - is this decomposition always possible?

Still, it would certainly be nice to think there can be a synthesis between BM and MW - a BMW perhaps. :) Vorsprung Durch Technik!

 

[Derek Potter]

Ich denke du meinst "The Ultimate Driving Experience" nicht "Vorsprung Durch Technik", Dummkopf.

 

[atyy]

Anyway, yes the decomposition of a wavfunction the way you want is always possible in non-relativistic quantum mechanics. In fact, that is how the Hilbert space of the full system is defined. First we have the basis functions for one particle. Then the basis functions for many particles is the tensor product of the one-particle basis functions.

The part of your proposal I'm not sure about is that it seems to be the finest possible branching, but usually in deocherence we only have a few branches compared to the size of the system, because we need a large number of particles to define the environment and apparatus etc.

However, maybe it will work it the form of MWI in which the Schmidt basis is always chosen. I believe Zeh suggested this.

 

[Derek Potter]

Anyway, yes the decomposition of a wavfunction the way you want is always possible in non-relativistic quantum mechanics. In fact, that is how the Hilbert space of the full system is defined. First we have the basis functions for one particle. Then the basis functions for many particles is the tensor product of the one-particle basis functions.

Excellent!

The part of your proposal I'm not sure about is that it seems to be the finest possible branching, but usually in deocherence we only have a few branches compared to the size of the system, because we need a large number of particles to define the environment and apparatus etc.
However, maybe it will work it the form of MWI in which the Schmidt basis is always chosen. I believe Zeh suggested this.

I had to look up the Schmidt basis and I can't say I got very far. But I think you must be referring to the way decoherence can be expressed in terms of the entanglement of the system and the environment - as a summation of products again.

It's certainly the finest possible branching*: this allows representation of *any* wavefunction (ignoring spin) in 3-space., which is what I'm interested in in this thread. Presumably the expansion |u>|v>|w> etc could easily include |environment> - it's just adds another index (rather a big one) amongst the i, j, k etc. That would leave the environment just "adding decoherence" but otherwise not modeled. Alternatively, the |environment> term could be expanded, making the environment part of the 3D system. Does that sound right?

* added - Well fine enough for my purposes. Not sure about the maths of branching a bit more, say to one dimension :) That would be seriously deviating from the topic.

 

[atyy]

It's certainly the finest possible branching*: this allows representation of *any* wavefunction (ignoring spin) in 3-space., which is what I'm interested in in this thread. Presumably the expansion |u>|v>|w> etc could easily include |environment> - it's just adds another index (rather a big one) amongst the i, j, k etc. That would leave the environment just "adding decoherence" but otherwise not modeled. Alternatively, the |environment> term could be expanded, making the environment part of the 3D system. Does that sound right?

I don't think you can expand the environment this way without creating more branches, because the environment is massively entangled.

 

[Derek Potter]

I don't think you can expand the environment this way without creating more branches, because the environment is massively entangled.

That just means that the |u>|v>|w>|environment> states are in a massive superposition: ∑_ijkE Obviously |u>|v>|w>|environment> is a pure state but |u>|v>|w> is not; in fact it's an improper mixed one. But that's harmless.

 

[atyy]

That just means that the |u>|v>|w>|environment> states are in a massive superposition: ∑_ijkE Obviously |u>|v>|w>|environment> is a pure state but |u>|v>|w> is not; in fact it's an improper mixed one. But that's harmless.

It is not harmless. To give every particle its own 3D wave function is to expand out the wave function fully, which means the number of branches will be huge. Write it out and see. You can try something simpler like spin up or down (which is like each particle having two possible positions). Then the game is to write the wave function in sums of products of individual spin up and spin down basis vectors, which is 2^N branches, where N is the number of spins

 

[Derek Potter]

Why is a huge number of branches a problem? Once we have seen how the expansion can be interpreted in 3 dimensional space, we are quite free to go back to using 3N dimensional phase space or r^3N dimensional state space as we please. The number (volume) of states or branches is the same.

 

[atyy]

Why is a huge number of branches a problem? Once we have seen how the expansion can be interpreted in 3 dimensional space, we are quite free to go back to using 3N dimensional phase space or r^3N dimensional state space as we please. The number (volume) of states or branches is the same.

The huge number of branches is a problem for MWI with decoherence, because there is no environment to decohere, if we are using the finest possible split. It shouldn't be a problem for BMW.

I don't think you are free to go back. If you went back all the way, there wouldn't even be any branches, just the evolving wave function.

 

[Derek Potter]

The huge number of branches is a problem for MWI with decoherence, because there is no environment to decohere, if we are using the finest possible split. It shouldn't be a problem for BMW

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.

 

[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 "occuring" 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 "occuring" 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. Schrodinger'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. Schrodinger'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.