Quantum Mechanics without Measurement

[DevilsAvocado]

Bell's proof simply fails for nonmeasurable local hidden variables. That doesn't mean they exist, but Bell's proof does not exclude them.

Do you mean completely non-measurable LVH (what use do we have of this? ), or something like my "spinning dices" in post #210?

 

[DevilsAvocado]

Isn't there but a single "pilot wave" that goes through both slits?

You mean like the wavefunction? It will 'automatically' generate a different 'pattern' with one vs. two slits?

 

[atyy]

Do you mean completely non-measurable LVH (what use do we have of this? ), or something like my "spinning dices" in post #210?

"Nonmeasurable" as in "probability distribution cannot be defined over the set". I don't know what use we have of it, but Bell's theorem doesn't exclude local hidden variables over which a probability distribution cannot be defined.

 

[craigi]

Isn't there but a single "pilot wave" that goes through both slits?

That's the normal terminology, but presumably it could be considered a superposition too.

 

[atyy]

This has puzzled me. How is information 'transmitted' between the two 'pilot beams', to make the particle land on the screen in the right place? If a particle is going through one slit, accompanied by only one 'pilot beam', you won't get the correct result, do you?

What I'm trying to say is that there is no Bell inequality violated in the single slit single particle experiment. So if there is nonlocality in this case, it is not proven by Bell's theorem. I can't really construct a case, I was just trying to sketch to stevendaryl what a construction might look like. But mainly the technical thing is no Bell inequality is violated by a single particle.

 

[stevendaryl]

But can the observer be in all frameworks?

I don't know what you mean by being "in" a framework. There might be one framework in which my spatial location is well-defined---I'm either in New York, or I'm in Georgia. In another framework, I might be in a superposition of both locations. I (that is, my body) is in both frameworks. Only one framework is of any use to me, so that's the one I use. But I'm in both of them.

The framework is a choice of which observables have definite values at which times. So I pick the framework that involves whatever observables I'm interested in. It's a subjective choice, it's not part of the physics.

 

[atyy]

I don't know what you mean by being "in" a framework. There might be one framework in which my spatial location is well-defined---I'm either in New York, or I'm in Georgia. In another framework, I might be in a superposition of both locations. I (that is, my body) is in both frameworks. Only one framework is of any use to me, so that's the one I use. But I'm in both of them.

The framework is a choice of which observables have definite values at which times. So I pick the framework that involves whatever observables I'm interested in. It's a subjective choice, it's not part of the physics.

Since you are in both frameworks, the choice of framework which is part of what you are should be in the framework. So in which framework do you make which subjective choice?

 

[stevendaryl]

Do you mean completely non-measurable LVH (what use do we have of this? ), or something like my "spinning dices" in post #210?

In measure theory, you have a set (the sample space, maybe? I forget the terminology) of possibilities. Then you have real numbers (the measures) associated with certain subsets of that set. There is no guarantee that every set of possibilities has an associated measure. (The Banach-Tarskii paradoxical partition of the sphere is an example of the use of nonmeasurable sets). Since Bell's theorem involves probabilities (or correlations, which are defined in terms of probabilities), the terms in the inequality may not be defined if you have nonmeasurable sets. So the proof fails because of a technicality.

 

[DevilsAvocado]

"Nonmeasurable" as in "probability distribution cannot be defined over the set". I don't know what use we have of it, but Bell's theorem doesn't exclude local hidden variables over which a probability distribution cannot be defined.

Wow, this is a crazy world... me just thought you could not find the dar*ed thing.

 

[atyy]

The observer has property X from framework Y and property X' from framework Y'?

I think it has a hard enough time being in one framework, completely at least, nevermind them all.

Can it be partly in all of them? That would be omnipresence.

I suspect this is why Griffiths only claims that CH makes sense for an observer outside the quantum system. But then that seems to me to leave the measurement problem unsolved.

Gell-Mann and Hartle do try to put the observer in all frameworks, but then they end up with one real history in each framework, and the real histories are not connected, and they have to introduce negative probabilities.

 

[atyy]

Wow, this is a crazy world... me just thought you could not find the dar*ed thing.

BTW, not to be discussed in this forum but if you search strangerep's posts in BTSM he lists some attempts to construct LHV theories in which the LHV cannot have a probability distribution defined over them.

 

[DevilsAvocado]

In measure theory, you have a set (the sample space, maybe? I forget the terminology) of possibilities. Then you have real numbers (the measures) associated with certain subsets of that set. There is no guarantee that every set of possibilities has an associated measure. (The Banach-Tarskii paradoxical partition of the sphere is an example of the use of nonmeasurable sets). Since Bell's theorem involves probabilities (or correlations, which are defined in terms of probabilities), the terms in the inequality may not be defined if you have nonmeasurable sets. So the proof fails because of a technicality.

Ah, thanks! Me brain works now! :thumbs:

 

[craigi]

Since you are in both frameworks, the choice of framework which is part of what you are should be in the framework. So in which framework do you make which subjective choice?

He doesn't need to be completely in any framework. One framework may be a superset of another. So the answer can be one, neither or both. It don't matter.

Frameworks are designed to describe quantum systems. Conceptually, the observer isn't important in this interpretation.

I'd argue that, since there are quantum processes taking place in the brain, it's impossible for the entire observer to exist in a single framework and that this causes no problems for the interpretation, since an observer plays no significant role in a measurement. That really is the realm of the CI and the later Relational Interpretation.

 

[stevendaryl]

Since you are in both frameworks, the choice of framework which is part of what you are should be in the framework. So in which framework do you make which subjective choice?

I don't really understand the question. My brain might have states S1, S2, S3, ... in which I'm thinking about different things. S1 might be the state in which I'm computing probabilities according to framework F1. S2 might be the state in which I'm computing probabilities according to framework F2.

A framework is a choice of which observables have definite values at which times. So if my brain state is an observable, then there might be some frameworks in which I have a definite brain state at each moment, and some in which I'm in a superposition of brain states.

The framework does not determine which brain state I'm in, it only determines the fact that my brain state has a definite value (or not).

 

[DevilsAvocado]

BTW, not to be discussed in this forum but if you search strangerep's posts in BTSM he lists some attempts to construct LHV theories in which the LHV cannot have a probability distribution defined over them.

Thanks atyy, but there is this quite effective "Anyone-Who-Questions-Bell-Vaccine", which is worth about US$1.1 million at the moment... and before anyone is even close receiving the money, too much reading might be wasted time... (no offense)

 

[Doc Al]

You mean like the wavefunction? It will 'automatically' generate a different 'pattern' with one vs. two slits?

That's my understanding.

 

[DevilsAvocado]

Thanks Doc

 

[atyy]

I don't really understand the question. My brain might have states S1, S2, S3, ... in which I'm thinking about different things. S1 might be the state in which I'm computing probabilities according to framework F1. S2 might be the state in which I'm computing probabilities according to framework F2.

A framework is a choice of which observables have definite values at which times. So if my brain state is an observable, then there might be some frameworks in which I have a definite brain state at each moment, and some in which I'm in a superposition of brain states.

The framework does not determine which brain state I'm in, it only determines the fact that my brain state has a definite value (or not).

A brain state S must be defined with respect to a framework. In which framework is your brain in state S1 in which you are computing probabilities according to framework 1? Can your brain be in state S2 in framework 2 in which you are calculating things using framework 1?

 

[stevendaryl]

A brain state S must be defined with respect to a framework.

That doesn't seem at all correct to me.

Let's ridiculously oversimplify and assume that there are a discrete set of things that a person could be thinking about: subject 1, subject 2, etc. There are corresponding brain states $S_1$ in which the person is thinking about subject 1, $S_2$ in which he is thinking about subject 2, etc.

Then there might be an observable, the brain state, which corresponds to an operator $\hat{S}$ that has eigenvalues $\lambda_1, \lambda_2, ...$ and satisfies the
equation:

$\hat{S} | S_j \rangle = \lambda_j |S_j \rangle$

Now, we could make up another operator, $\hat{T}$ that mixes brain states. For example, suppose it works like this:

$\hat{T} |S_j \rangle = \alpha_j |S_{j+1} \rangle + \beta_j |S_{j-1} \rangle$

for some complex constants $\alpha_j$ and $\beta_j$

A framework consists of a choice of observables at particular times. So to simplify, let's consider a single moment of time. So there might be framework $\mathcal{F}_1$ which uses observable $\hat{S}$ at that moment, and framework $\mathcal{F}_2$ which uses observable $\hat{T}$.

So let's consider a brain that is thinking about framework $\mathcal{F}_1$. Maybe that corresponds to brain state $|S_1\rangle$. Maybe a brain that is thinking about framework $\mathcal{F}_2$ is brain state $|S_2\rangle$

So a person in brain state $S_1$ would use framework $\mathcal{F}_1$ and compute such and such a probability that $\hat{S} = \lambda_1$ and would compute such and such a probability that $\hat{S}= \lambda_2$. So within framework $\mathcal{F}_1$, you can analyze the probability that you might have chosen framework $\mathcal{F}_2$ to think about.

The framework does not determine which brain state you are in. The framework determines which observables have definite values. It doesn't determine what those values are.

So there are two different levels of "worlds" in CH: The choice of which framework, and the choice of which history within a framework.

In which framework is your brain in state S1 in which you are computing probabilities according to framework 1?

Well, the way I defined things above, brain states have definite values only in framework $\mathcal{F}_1$. Within that framework, the brain could be in states $|S_1\rangle, |S_2\rangle, |S_3\rangle, etc.$. But the brain cannot be in the state

$\alpha |S_1\rangle + \beta |S_2\rangle$

because the framework $\mathcal{F}_1$ makes brain states definite.

Can your brain be in state S2 in framework 2

The way I've set things up, brain states only have definite values in framework $\mathcal{F}_1$. So it doesn't make sense to talk about having state $|S_2\rangle$ in framework $\mathcal{F}_2$. You can talk about being in some superposition of states,though.

in which you are calculating things using framework 1?

The way I've set things up, having a definite value for the question of "which framework are you using" means having a definite brain state. Only in framework $\mathcal{F}_1$ do you have a definite brain state. So only within framework $\mathcal{F}_1$ can you calculate probabilities according to framework $\mathcal{F}_2$ (or $\mathcal{F}_3$, etc.)

 

[craigi]

Can your brain be in state S2 in framework 2 in which you are calculating things using framework 1?

If it is possible to describe the state of an entire brain in a single framework, then yes.

The vast majority of a brain (the classical part), certainly does fit into a single framework and yes, this can be used to think about other frameworks. You're doing it now! Well... maybe not, but certainly quantum physicists do.

 

[stevendaryl]

Thanks atyy, but there is this quite effective "Anyone-Who-Questions-Bell-Vaccine", which is worth about US$1.1 million at the moment... and before anyone is even close receiving the money, too much reading might be wasted time... (no offense)

You keep interpreting what people are saying as "questioning Bell". Nobody questions the theorem, and nobody questions the predictions of quantum mechanics. The issue is over how to interpret the theorem, and quantum mechanics.

A resolution that uses nonmeasurable sets isn't really going to win any prize money, because constructing a nonmeasurable set is not something you can really do.

 

[atyy]

That doesn't seem at all correct to me.

Let's ridiculously oversimplify and assume that there are a discrete set of things that a person could be thinking about: subject 1, subject 2, etc. There are corresponding brain states $S_1$ in which the person is thinking about subject 1, $S_2$ in which he is thinking about subject 2, etc.

Then there might be an observable, the brain state, which corresponds to an operator $\hat{S}$ that has eigenvalues $\lambda_1, \lambda_2, ...$ and satisfies the
equation:

$\hat{S} | S_j \rangle = \lambda_j |S_j \rangle$

Now, we could make up another operator, $\hat{T}$ that mixes brain states. For example, suppose it works like this:

$\hat{T} |S_j \rangle = \alpha_j |S_{j+1} \rangle + \beta_j |S_{j-1} \rangle$

for some complex constants $\alpha_j$ and $\beta_j$

A framework consists of a choice of observables at particular times. So to simplify, let's consider a single moment of time. So there might be framework $\mathcal{F}_1$ which uses observable $\hat{S}$ at that moment, and framework $\mathcal{F}_2$ which uses observable $\hat{T}$.

So let's consider a brain that is thinking about framework $\mathcal{F}_1$. Maybe that corresponds to brain state $|S_1\rangle$. Maybe a brain that is thinking about framework $\mathcal{F}_2$ is brain state $|S_2\rangle$

So a person in brain state $S_1$ would use framework $\mathcal{F}_1$ and compute such and such a probability that $\hat{S} = \lambda_1$ and would compute such and such a probability that $\hat{S}= \lambda_2$. So within framework $\mathcal{F}_1$, you can analyze the probability that you might have chosen framework $\mathcal{F}_2$ to think about.

The framework does not determine which brain state you are in. The framework determines which observables have definite values. It doesn't determine what those values are.

So there are two different levels of "worlds" in CH: The choice of which framework, and the choice of which history within a framework.

So in framework 1, you could be calculating things using framework 2?

 

[craigi]

So in framework 1, you could be calculating things using framework 2?

Yes.

 

[stevendaryl]

So in framework 1, you could be calculating things using framework 2?

Sure. The question "What is the probability of history $H_1$ according to framework $\mathcal{F}_2$?" is a purely mathematical question. I can ask it about any framework and any history.

 

[atyy]

Sure. The question "What is the probability of history $H_1$ according to framework $\mathcal{F}_2$?" is a purely mathematical question. I can ask it about any framework and any history.

But if I am using framework 2, I cannot be using framework 1.

 

[stevendaryl]

But if I am using framework 2, I cannot be using framework 1.

Not at the same time, but at different times, you certainly can. You can calculate using framework 1, and then when you're done, you can calculate using framework 2.

 

[atyy]

Not at the same time, but at different times, you certainly can. You can calculate using framework 1, and then when you're done, you can calculate using framework 2.

But framework 1 says I am using framework 2. I guess there is no inconsistency, since it doesn't say I am using framework 1. But that means if I am using framework 2, I cannot know it (to know it I have to use framework 1)?

 

[stevendaryl]

But framework 1 says I am using framework 2.

I don't agree with that. A framework doesn't say what happens, it says a set of possible things that might happen. A framework is (or determines) a set of possible alternative histories.

So the history "I calculate probabilities using framework A" and "I calculate probabilities using framework B" are two alternative histories in the same framework. An example of a history in a different framework would be "I am in a superposition of using framework A and using framework B"

 

[atyy]

I don't agree with that. A framework doesn't say what happens, it says a set of possible things that might happen. A framework is (or determines) a set of possible alternative histories.

So the history "I calculate probabilities using framework A" and "I calculate probabilities using framework B" are two alternative histories in the same framework. An example of a history in a different framework would be "I am in a superposition of using framework A and using framework B"

Let's say I am using framework 2. Is that statement made in framework 1 or framework 2?

 

[craigi]

This is level of abstraction problem again.

It really doesn't matter which framework your brain is in, so long as you don't try to draw inferences from its quantum state in conjunction with another quantum system.

I don't even know how you'd draw inferences from your own brain's quantum state regardless of any other quantum system.