5 ms·

At least for now, I'm willing to f'get about issues of "free will". Thanks, I will keep trying to make sense out of Bell's work. I keep getting stuck trying t

At least for now, I'm willing to f'get about issues of "free will". Thanks, I will keep trying to make sense out of Bell's work. I keep getting stuck trying to read quantum mechanics: One place was the claim that the wave functions form a Hilbert space. Nope: As I read in W. Rudin, Real and Complex Analysis, a Hilbert space is a complete inner product space where complete means that every Cauchy convergent sequence is convergent. Well, while the wave functions are likely points in a suitable Hilbert space, they can't be complete, e.g., they can converge to a point in the space, i.e., a function, that is not continuous and, thus, not differentiable in contradiction to the assumption that all wave functions are differentiable. I admit that this is a small point, but I was trying to take quantum mechanics seriously and be careful. Closer to the OP, another place I got stuck was in the approaches of physics to independent and uncorrelated: In probability theory those two are not the same: For two real valued random variables, independence implies uncorrelated. As in W. Feller, in the case of two real valued random variables with joint Gaussian probability density function, uncorrelated implies independence. Generally, however, uncorrelated does not imply independence. Independence is a much stronger property than uncorrelated. Maybe eventually I will figure out what physics means by "uncorrelated", especially for Bell's work. To me, we can take the Ace of Hearts and the Ace of Spades, shuffle them, and deal them out, face down, one each to Bob and Sally. Bob can go a light year away. Sally then looks at her card and knows right away, nothing faster than the speed of light needed, what Bob's card is. We know all the associated probability distributions for Bob and Sally. And we know that as soon as the cards are dealt what each of Bob and Sally have is determined -- so far unknown but still determined. I'm guessing that this Bob-Sally thought experiment may have something to do with entanglement, the EPR (Einstein, Podolsky, Rosen) paradox, "spooky action at a distance", collapse of quantum mechanics wave functions, and Bell's results -- but I need to keep studying.

coderenegade 5y ago
You might be interested in some code that illustrates the problem of hidden variables and the EPR paradox: https://pastebin.com/J4ZUhG8e https://pastebin.com/J4ZUhG8e. The issue is that we can't replicate what QM predicts (and experiments validate) using hidden variables without additional steps or assumptions. For example, in that code, there are a few possible ways we could still produce the QM correlation function with local hidden variables: 1. When we measure the first particle, we alter the second particle to produce the desired correlation. This is the communication loophole. If you move the particles far enough apart, it requires superluminal communication, which GR says is impossible. This is what Einstein was getting at in the EPR paper; if what the Copenhagen interpretation of QM says is true, and the particle is in both states at once, measuring it collapses it to a given state, and the other particle suddenly "knows" what it should be, even though the propagation of information through the universe has a speed limit. 2. We could define an additional hidden variable in [0,1], and do rejection sampling at the detector. Anything that is rejected is not detected, and we only deal with detected particles. This is the detection loophole. I'm not an expert, but this is the only one that really makes sense to me as a good candidate for a hidden variables theory, and I believe it's been ruled out by experiment. 3. We could bias the initial sampling of our hidden variables. The problem is, this requires that we know the setting of the detector ahead of time. As far as I understand it, this is essentially what superdeterminism comes down to: you know the experiment setting ahead of time because everything is completely deterministic, like a movie reel being rolled forwards. You can thus bias the initial sampling step to produce the QM correlation function. Aside from being unfalsifiable, it still leaves open the question of why this would happen. It essentially means that all QM experiments have predetermined outcomes, and for whatever reason, those outcomes are the outcomes we observe. Ultimately, it seems to me that Bell's inequality is more a statement about the fundamental incompatibility of quantum probability with classical mechanics and probability. If QM is correct, and it seems that it is, then you have to give up certain assumptions such as locality.
- naasking 5y ago
  > 3. We could bias the initial sampling of our hidden variables. The problem is, this requires that we know the setting of the detector ahead of time. The standard computational approach to this is to define the distribution as lazily evaluated based on some future state. That's exactly what Sabine is suggesting for a superdeterministic theory. > It essentially means that all QM experiments have predetermined outcomes, and for whatever reason, those outcomes are the outcomes we observe. More or less. This doesn't seem to bother anyone when spin 1/2 particles are all governed by the Dirac equation "for whatever reason", but somehow people really seem to think it matters in this case.
  - coderenegade 5y ago
    I gotcha. It actually seems obvious and fairly reasonable when you put it that way, since it removes the "conspiratorial" element that theories of superdeterminism seem to invoke.
    - naasking 5y ago
      Her recent paper on this is worth a read: Rethinking Superdeterminism, https://www.frontiersin.org/articles/10.3389/fphy.2020.00139/full https://www.frontiersin.org/articles/10.3389/fphy.2020.00139...
lmm 5y ago
> To me, we can take the Ace of Hearts and the Ace of Spades, shuffle them, and deal them out, face down, one each to Bob and Sally. Bob can go a light year away. Sally then looks at her card and knows right away, nothing faster than the speed of light needed, what Bob's card is. > We know all the associated probability distributions for Bob and Sally. And we know that as soon as the cards are dealt what each of Bob and Sally have is determined -- so far unknown but still determined. Right, so that's a hidden variable theory. But the Bell Inequalities give you a set of measurements that can't be explained by hidden variables - one simpler variation is Conway's "free will theorem": Bob and Alice each have something like 29 envelopes, and if he opens one of several sets of 3, then he will always find two black cards and one white card (and Alice will always find the opposite). But there's no way to "statically" assign black and white cards to envelopes so that all of these sets of 3 have that property. So either which cards are in which envelopes really is undetermined until Bob opens a set of 3, or you somehow know which envelopes Bob is going to open at the point where you're putting the cards in the envelopes.
temp438 5y ago
>To me, we can take the Ace of Hearts and the Ace of Spades, shuffle them, and deal them out, face down, one each to Bob and Sally. Bob can go a light year away. Sally then looks at her card and knows right away, nothing faster than the speed of light needed, what Bob's card is. I don't get it. Alice knows it right away only if she knows the rules of the game, electricity in her brain doesn't need to move a light year away. Though, if she's braindamaged, it can take longer than a light year for her to figure that out or may never happen at all.

coderenegade 5y ago

You might be interested in some code that illustrates the problem of hidden variables and the EPR paradox: https://pastebin.com/J4ZUhG8e https://pastebin.com/J4ZUhG8e. The issue is that we can't replicate what QM predicts (and experiments validate) using hidden variables without additional steps or assumptions. For example, in that code, there are a few possible ways we could still produce the QM correlation function with local hidden variables: 1. When we measure the first particle, we alter the second particle to produce the desired correlation. This is the communication loophole. If you move the particles far enough apart, it requires superluminal communication, which GR says is impossible. This is what Einstein was getting at in the EPR paper; if what the Copenhagen interpretation of QM says is true, and the particle is in both states at once, measuring it collapses it to a given state, and the other particle suddenly "knows" what it should be, even though the propagation of information through the universe has a speed limit. 2. We could define an additional hidden variable in [0,1], and do rejection sampling at the detector. Anything that is rejected is not detected, and we only deal with detected particles. This is the detection loophole. I'm not an expert, but this is the only one that really makes sense to me as a good candidate for a hidden variables theory, and I believe it's been ruled out by experiment. 3. We could bias the initial sampling of our hidden variables. The problem is, this requires that we know the setting of the detector ahead of time. As far as I understand it, this is essentially what superdeterminism comes down to: you know the experiment setting ahead of time because everything is completely deterministic, like a movie reel being rolled forwards. You can thus bias the initial sampling step to produce the QM correlation function. Aside from being unfalsifiable, it still leaves open the question of why this would happen. It essentially means that all QM experiments have predetermined outcomes, and for whatever reason, those outcomes are the outcomes we observe. Ultimately, it seems to me that Bell's inequality is more a statement about the fundamental incompatibility of quantum probability with classical mechanics and probability. If QM is correct, and it seems that it is, then you have to give up certain assumptions such as locality.

naasking 5y ago
> 3. We could bias the initial sampling of our hidden variables. The problem is, this requires that we know the setting of the detector ahead of time. The standard computational approach to this is to define the distribution as lazily evaluated based on some future state. That's exactly what Sabine is suggesting for a superdeterministic theory. > It essentially means that all QM experiments have predetermined outcomes, and for whatever reason, those outcomes are the outcomes we observe. More or less. This doesn't seem to bother anyone when spin 1/2 particles are all governed by the Dirac equation "for whatever reason", but somehow people really seem to think it matters in this case.
- coderenegade 5y ago
  I gotcha. It actually seems obvious and fairly reasonable when you put it that way, since it removes the "conspiratorial" element that theories of superdeterminism seem to invoke.
  - naasking 5y ago
    Her recent paper on this is worth a read: Rethinking Superdeterminism, https://www.frontiersin.org/articles/10.3389/fphy.2020.00139/full https://www.frontiersin.org/articles/10.3389/fphy.2020.00139...

lmm 5y ago

> To me, we can take the Ace of Hearts and the Ace of Spades, shuffle them, and deal them out, face down, one each to Bob and Sally. Bob can go a light year away. Sally then looks at her card and knows right away, nothing faster than the speed of light needed, what Bob's card is. > We know all the associated probability distributions for Bob and Sally. And we know that as soon as the cards are dealt what each of Bob and Sally have is determined -- so far unknown but still determined. Right, so that's a hidden variable theory. But the Bell Inequalities give you a set of measurements that can't be explained by hidden variables - one simpler variation is Conway's "free will theorem": Bob and Alice each have something like 29 envelopes, and if he opens one of several sets of 3, then he will always find two black cards and one white card (and Alice will always find the opposite). But there's no way to "statically" assign black and white cards to envelopes so that all of these sets of 3 have that property. So either which cards are in which envelopes really is undetermined until Bob opens a set of 3, or you somehow know which envelopes Bob is going to open at the point where you're putting the cards in the envelopes.

temp438 5y ago