The preferred basis problem - help

[helenk]

Could someone please tell me what the preferred basis problem is with regards to the Everett/many worlds interpretation?

As I understand it, it refers to the basis which is needed to make macroscopic objects determinate in all worlds. But what does this mean? Does basis refer to an observable i.e. momentum or position, energy or time? If so then presumably we pick a basis when we perform an experiment, so does the problem apply only to “real life” i.e. when there is no instrument to define the basis?

Also, is the question of the preferred basis the same as the question of why we are not aware of more than one world? It seems to be but I cannot understand how picking a basis will imply that we are unaware of other worlds. I imagine that we could be unaware of these other worlds because the branches/our memories of the branches are non-commutable, but I am not sure about this because different observables are non-commutable not different measurements of the same property?

Can someone tell me where I’m going wrong? Thank you :)

 

[Fredrik]

I think the "problem" is that if the terms in the superposition "spin up" + "spin down" represent two different worlds, and the terms in "spin right" + "spin left" don't, then something must be making the up/down basis "special". I don't think this is a problem anymore. Decoherence makes the density matrix of the combined system (the one being measured and its environment) diagonal in some basis, which is determined by the details of the interaction between the system and the environment, and this basis can be considered the "preferred" one. Note however that the decomposition of the omnium (the universe that contains all the worlds) into subsystems can be done in many different ways, and that each decomposition gives us a different way to describe the state of the omnium in terms of "worlds".

Why do we only experience one world? I don't think anyone has a complete answer. I suspect that the question doesn't quite make sense. Suppose that your brain (and in particular your memory) has to go through a series of classical states in order to "experience" something. These classical states are represented by tensor product components in the individual terms of a density matrix, and decoherence correlates them with classical states of the brain's environment. This could make it impossible to define (in other terms) what an experience is before we ask the question of why we don't "experience" superpositions. Instead we might have to take the formation of these correlations as the definition of what we mean by conscious experience, and that would make the question tautological. "Why are the states not uncorrelated in the terms with correlations?"

You might also be interested in this thread and the one I'm linking to in #17.

 

[helenk]

Thanks for your reply, I am reading about decoherence but struggling since I didn’t/don’t know what a basis is. So it does not refer to picking either position or momentum but picking position in relation to another position. Is this right?
(ie picking spin in relation to an axis rather than picking spin (angular momentum) over position).

So am I right in thinking that in the case of a simple lab experiment we pick the basis when we pick which instrument to use? And so the question of decoherence is the question of why this happens in “real life”?

I am at work now so I cannot copy the text, but I am sure that Everett said that we are not aware of other worlds because they are as indifferent to each other as the components of the wavefunction. I am sure that decoherence wasn’t discovered until after Everett’s original paper so what is he getting at here? Sorry if this is not clear I will paste the actual quote when I get home.

Thanks for your help, I’m really trying to get my head around this!

 

[Fredrik]

Thanks for your reply, I am reading about decoherence but struggling since I didn’t/don’t know what a basis is.

Have you studied linear algebra? That's the sort of basis I'm talking about. Link.

So it does not refer to picking either position or momentum but picking position in relation to another position. Is this right?
(ie picking spin in relation to an axis rather than picking spin (angular momentum) over position).

Not really. I could have talked about position vs. momentum instead of the z-component of spin vs. the x-component of spin, but I chose the latter because it's easier to understand a 2-dimensional Hilbert space than an infinite-dimensional one.

So am I right in thinking that in the case of a simple lab experiment we pick the basis when we pick which instrument to use?

Yes.

And so the question of decoherence is the question of why this happens in “real life”?

I'm not sure I understand that question. I suspect that what you would have asked if you had known this stuff already is if decoherence theory explains why interactions between subsystems tend to make the density matrix almost diagonal in some basis. I think the answer to that question is yes, but I haven't really studied decoherence myself. I have only had a quick look at it, just to see what it's about.

I am at work now so I cannot copy the text, but I am sure that Everett said that we are not aware of other worlds because they are as indifferent to each other as the components of the wavefunction. I am sure that decoherence wasn’t discovered until after Everett’s original paper so what is he getting at here? Sorry if this is not clear I will paste the actual quote when I get home.

He's saying that Schrödinger's cat is dead in one world and alive in another. But that ignores the fact that the superposition |atom decayed;cat dead>+|atom not decayed;cat alive> can also be expressed as superpositions in many other ways, with the cat neither dead nor alive in each term.

 

[helenk]

OK, so am I right in saying that Everett did not explain why we are only aware of one world? Because that seems like a very big problem to me and without any suggestion of a resolution doesn't his theory create more problems than it solves (or at least as many)? (I will have to find that quote later).

Also, if i am right in saying that decoherence is analogous to us picking a basis by using certain lab equipment then i can imagine how this could work out of the lab, everything becomes entangled with everything else so they all correspond to the same basis('s?), but doesn't this lead to the question of how anything thing became decohered in the first place? i know I've missed the point but I am not sure why?

Thanks for your help with this :)

 

[helenk]

Everett stated; “all the separate elements of a superposition individually obey the wave equation with complete indifference to the presence or absence ("actuality" or not) of any other elements. This total lack of effect of one branch on another also implies that no observer will ever be aware of any "splitting" process.”
http://www.pbs.org/wgbh/nova/manyworlds/original.html"

Does this refer only to the case when we use an instrument to define the basis? Has Everett just not thought about what happens in more complex systems which require decoherence?

 

[Fredrik]

OK, so am I right in saying that Everett did not explain why we are only aware of one world?

Yes.

Because that seems like a very big problem to me and without any suggestion of a resolution doesn't his theory create more problems than it solves (or at least as many)?

It does create a lot of problems, but it's the simplest and most straightforward way to interpret QM as a description of what actually happens. So it's not easily dismissed. The only option that's any simpler is to assume that QM doesn't tell us what actually happens. (This is the one I prefer).

Also, if i am right in saying that decoherence is analogous to us picking a basis by using certain lab equipment then i can imagine how this could work out of the lab, everything becomes entangled with everything else so they all correspond to the same basis('s?), but doesn't this lead to the question of how anything thing became decohered in the first place? i know I've missed the point but I am not sure why?

All we can do is to subject the system to a particular type of interaction with a part of its environment, but any interaction between the system and its environment will make some basis "preferred" over the others.

Everett stated; “all the separate elements of a superposition individually obey the wave equation with complete indifference to the presence or absence ("actuality" or not) of any other elements. This total lack of effect of one branch on another also implies that no observer will ever be aware of any "splitting" process.”
http://www.pbs.org/wgbh/nova/manyworlds/original.html"

Does this refer only to the case when we use an instrument to define the basis?

No, it applies to all cases.

Has Everett just not thought about what happens in more complex systems which require decoherence?

We need decoherence or something like it, even in the simplest possible case, the spin states of a spin-1/2 particle. The quote is correctly stating that each term evolves independently, but it's ignoring the fact that there are infinitely many ways to express a state vector as a sum of terms. So why should one of these ways define "worlds", but not the others? I think even decoherence doesn't give us a complete answer, but at least it tells us that there's one way to express the state vector as a sum of terms that has a particular significance that the others don't have.

 

[helenk]

Thanks for your reply.

No, it applies to all cases.

Do you mean all cases where a basis is defined? (because of decoherence it is all cases but without it none at all)? Was Everett assuming then that there would be a way to define the basis which he was unaware of? (I thought, he thought, that in the case of a simple experimment the basis would be defined by the instrument).

 

[helenk]

I have some more questions if that is OK, here goes;

It is my understanding that a Schrödinger’s cat is in a superposition of all possible states and this means all possible combinations of its subatomic parts, not just alive and dead.

I am starting to understand that decoherence somehow makes it so that we perceive the cat as existing in two states; dead and alive (since this is the basis we have chosen). We are not aware of perceiving these two states at once, however, because they are separate components of the linear wavefunction.

(am I starting to get this?)

I am thinking of decoherence as being epistemological/subjective in contrast to the (possibly) metaphysical/objective universal wavefunction.

Now, without (the illusion of?) decoherence, the cat would have a non zero probability of being in any subatomic state (i.e. there would be a world for every possible combination of the cats subatomical parts). But to suddenly transform into, say a small dog (made of the same components as the cat), something would have to interact with all of the subatomic parts of the cat at once (or they could just do this randomly in at least one world since it must have a non zero probability).

If we assume that quantum randomness has no effect on consciousness/conscious decisions and we take a many minds approach, then we could say that anything (large dead cat, large live cat, live dog etc etc) as well as anyone who is like us but different (say our evil twin) must be from an improbable world where all of these “leaps” happened at once, or they must have “branched” much earlier and lead different lives completely.

Am I right in thinking that decoherence removes the non zero probability of us turning into one of the things above (dead cat, evil twin etc) or just the illusion of this? And am I right to think that we do not observe the worlds that “branched” earlier because decoherence occurred when they branched?

 

[Fredrik]

Do you mean all cases where a basis is defined? (because of decoherence it is all cases but without it none at all)?

I really mean the Everett quote holds for all state vectors, in any situation, regardless of which basis we use to express it. That's why it has a "preferred basis problem".

There are always infinitely many bases to choose from. What decoherence does is (among other things) to single out one of them as "special".

Was Everett assuming then that there would be a way to define the basis which he was unaware of? (I thought, he thought, that in the case of a simple experimment the basis would be defined by the instrument).

I don't know much about what Everett thought, but what you said sounds good to me.

It is my understanding that a Schrödinger’s cat is in a superposition of all possible states and this means all possible combinations of its subatomic parts, not just alive and dead.

"Alive" and "dead" are the relevant states (or classes of states) when we think of the cat as one subsystem, but there are many different ways to decompose a system into subsystems. You can also take component parts of the cat to be subsystems.

We are not aware of perceiving these two states at once, however, because they are separate components of the linear wavefunction.

Instead of expressing the state as |dead> + |alive>, you could express it as |X>+|Y>, where |X>=|dead> + |alive> and |Y>=|dead> - |alive>, so it's clear that it's not sufficient that |dead> and |alive> appear in two different terms.

I am starting to understand that decoherence somehow makes it so that we perceive the cat as existing in two states; dead and alive (since this is the basis we have chosen).

I don't think decoherence does that. What it does is to make sure that the combined system you+cat isn't in a superposition of |happy>|alive> and |sad>|dead>. It makes sure that you+cat can be approximately described as being either in the state |happy>|alive> or in the state |sad>|dead>.

I don't think this really explains why we perceive the cat as either dead or alive.

I am thinking of decoherence as being epistemological/subjective in contrast to the (possibly) metaphysical/objective universal wavefunction.

Decoherence isn't subjective, but the choice of how to decompose the omnium into subsystems is more than just subjective, it's arbitrary.

Now, without (the illusion of?) decoherence, the cat would have a non zero probability of being in any subatomic state (i.e. there would be a world for every possible combination of the cats subatomical parts). But to suddenly transform into, say a small dog (made of the same components as the cat), something would have to interact with all of the subatomic parts of the cat at once (or they could just do this randomly in at least one world since it must have a non zero probability).

This sort of stuff gets really complicated. I could be way off here, but I think the answer is that there's always a basis in which the cat is described the same way we would describe a small dog, but no stable records of the dog will form in its environment. For example, no memories in physicist's brains.

I'm glad you asked this, because my attempt to answer has made one thing a bit clearer in my mind: The reason why we should prefer the basis selected by the interaction between the subsystems. I've been saying (in the other threads) that we define the "worlds" as correlations between subsystems that appear when the time evolution of the density matrix is expressed in terms of the preferred basis, without really understanding why. I think I do now. It's not that it would be wrong to describe the terms of the density matrix expressed in another basis as "worlds". It's just that the states of the system wouldn't get correlated with states of the environment that can be thought of as stable records (memories) of what just happened to the system. It's hard to explain what I mean, so maybe this doesn't make any sense to you, but the gist of what I'm saying is that we're ignoring these worlds because they can't contain conscious observers.

 

[helenk]

This is all making a bit more sense now, I will have to keep thinking about it. thanks for trying to help :)

 

[helenk]

Is the preferred basis problem only a problem for mixed states? Is Everett referring to a pure state and claiming we are unaware of other worlds due to the linearity of the waveequation, something which is not relevant to a mixed state?