# Dimensionality of the wavefunction in relative state

1. Jun 12, 2015

### 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?

2. Jun 12, 2015

### 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.

3. Jun 12, 2015

### Staff: Mentor

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

4. Jun 12, 2015

### Staff: Mentor

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

5. Jun 13, 2015

### Derek Potter

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.

6. Jun 13, 2015

### Derek Potter

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.

Last edited: Jun 13, 2015
7. Jun 13, 2015

### Staff: Mentor

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 cant find any mention of a reason for the position basis, but I perhaps am blind. Suppose you spell it out again?

Thanks
Bill

Last edited: Jun 13, 2015
8. Jun 13, 2015

### Derek Potter

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

9. Jun 13, 2015

### Staff: Mentor

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.

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

Thanks
Bill

10. Jun 13, 2015

### Derek Potter

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

11. Jun 13, 2015

### Staff: Mentor

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

Thanks
Bill

12. Jun 13, 2015

### Derek Potter

???

13. Jun 13, 2015

### Staff: Mentor

????

Cognate on the word - action.

Thanks
Bill

14. Jun 13, 2015

### Derek Potter

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

15. Jun 13, 2015

### 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.

16. Jun 13, 2015

### Staff: Mentor

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

Last edited: Jun 13, 2015
17. Jun 13, 2015

### Derek Potter

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.

18. Jun 13, 2015

### Staff: Mentor

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

Thanks
Bill

19. Jun 13, 2015

### Derek Potter

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

20. Jun 13, 2015

### atyy

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