5 ms·

Uh, the OP frequently mentions "statistical independence", and I am uncomfortable with that: Independence is a property of a set of more than one random variab

Uh, the OP frequently mentions "statistical independence", and I am uncomfortable with that: Independence is a property of a set of more than one random variable and is defined in probability, often used in statistics, but not defined in statistics. There is a polished, elegant, and thorough treatment of such independence, including of uncountably infinite sets of random variables, in J. Neveu, Mathematical Foundations of the Calculus of Probability.

PeterWhittaker 5y ago
In the context of Bell’s Theorem, statistical independence is understood to mean that, if extant, hidden variables are not correlated with how measurements are being performed. Bell’s Theorem is only correct if this assumption holds. Hossenfelder is arguing that the assumption is incorrect: that Bell’s Theorem is incorrect precisely because there ARE hidden variables and that these ARE correlated with measurement settings. She also argues that all mentions of free will in all related discussions are meaningless red herrings that have distracted physicists from properly interpreting Bell’s Theorem and observed violation of Bell’s Inequality. Superdeterminism, arguably misnamed, simply argues that QM is deterministic, where Bell and others have argued it is not.
- graycat 5y ago
  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.
- akvadrako 5y ago
  > Superdeterminism, arguably misnamed, simply argues that QM is deterministic, where Bell and others have argued it is not. No, this is completely wrong. Super-determinism is more than determinism. Super-determinism is about conspiratorial coincidences – so the measurement settings you choose just happen to be the ones which will make it look like the world is quantum. Other quantum interpretations are also deterministic, like many worlds and pilot wave theory. Bell's theorem doesn't assume determinism, but most importantly (1) locality (2) counterfactual definiteness (3) a single classical world (4) no retro-causality and (5) no super-determinism.
  - deleted 5y ago
    [deleted]
  - geertj 5y ago
    I think these consequences are only conspirational if time is an independent variable? What if everything, the entire universe’s past, present, and future ‘happened’ at the same time, as a self consistent solution of some kind of underlying equation or set of rules? Our brain, the seat of our perception of time, is part of the same universe that we’re trying to explain. Maybe time is an artifact of the self referential feedback loop that causes our consciousness.
    - lmm 5y ago
      It's still conspiratorial. You just happened to choose the right particles to measure? It makes a mockery of the idea of trying to do science at all - the experiments you take are already determined, you can't learn anything about what causes have what effects because you can't ever change a causal variable.
      - geertj 5y ago
        Well, I don’t know of any physical law that allows you to choose anything. Whether or not you believe QM is normally deterministic or super deterministic, you have to believe in something outside physics to make all of us not a necessary outcome of an initial state. That doesn’t diminish the pleasures or the reality of consciousness though, or the (illusion) of free will, at least not for me.
- myridium 5y ago
  > Superdeterminism, arguably misnamed, simply argues that QM is deterministic This is incorrect. QM is already deterministic. Where oh where do people get the idea that it is not?
  - bryan0 5y ago
    The born rule for starters.
    - myridium 5y ago
      In QM, evolution is unitary. The Born Rule is an empirical observation that is adequately explained by the decoherence of a quantum system into entangled states of observer, environment, measurement apparatus and the system being measured. E.g. the master equation for a quantum density matrix governs its evolution when the environmental conditions are not completely known or modelled. Classical ensembles or 'decohered' states occur as a consequence. QM is formulated as a deterministic theory. When you try to model an open, interacting system without tracking its environment, you have to sacrifice determinism in that model, because you are failing to track all the information in the system. You settle for probabilistic quantities. Nothing quantum about it either-- same thing happens in classical mechanics. They call this "classical thermodynamics".
      - bryan0 5y ago
        there might be some non-trivial deterministic unitary description via decoherence, but it should not be surprising that many people think that QM is non-deterministic.
kgwgk 5y ago
I guess the whole field of statistical mechanics would make you uncomfortable.
- graycat 5y ago
  What I have seen so far for statistical mechanics is uses of integral calculus notation that has nothing to do with anything in any version of calculus, advanced calculus, measure theory, the Riemann integral, Lebesgue integration, etc.
  - myridium 5y ago
    Well if you're doing fermionic thermal field theory sure (where the `integral' on Grassmann numbers is just some linear operation which satisfies certain properties). But apart from that I thought all the thermodynamic quantities were defined in terms of the partition function, which is a perfectly well-defined (Riemann) integral...

PeterWhittaker 5y ago

In the context of Bell’s Theorem, statistical independence is understood to mean that, if extant, hidden variables are not correlated with how measurements are being performed. Bell’s Theorem is only correct if this assumption holds. Hossenfelder is arguing that the assumption is incorrect: that Bell’s Theorem is incorrect precisely because there ARE hidden variables and that these ARE correlated with measurement settings. She also argues that all mentions of free will in all related discussions are meaningless red herrings that have distracted physicists from properly interpreting Bell’s Theorem and observed violation of Bell’s Inequality. Superdeterminism, arguably misnamed, simply argues that QM is deterministic, where Bell and others have argued it is not.

graycat 5y ago
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.
akvadrako 5y ago
> Superdeterminism, arguably misnamed, simply argues that QM is deterministic, where Bell and others have argued it is not. No, this is completely wrong. Super-determinism is more than determinism. Super-determinism is about conspiratorial coincidences – so the measurement settings you choose just happen to be the ones which will make it look like the world is quantum. Other quantum interpretations are also deterministic, like many worlds and pilot wave theory. Bell's theorem doesn't assume determinism, but most importantly (1) locality (2) counterfactual definiteness (3) a single classical world (4) no retro-causality and (5) no super-determinism.
- deleted 5y ago
  [deleted]
- geertj 5y ago
  I think these consequences are only conspirational if time is an independent variable? What if everything, the entire universe’s past, present, and future ‘happened’ at the same time, as a self consistent solution of some kind of underlying equation or set of rules? Our brain, the seat of our perception of time, is part of the same universe that we’re trying to explain. Maybe time is an artifact of the self referential feedback loop that causes our consciousness.
  - lmm 5y ago
    It's still conspiratorial. You just happened to choose the right particles to measure? It makes a mockery of the idea of trying to do science at all - the experiments you take are already determined, you can't learn anything about what causes have what effects because you can't ever change a causal variable.
    - geertj 5y ago
      Well, I don’t know of any physical law that allows you to choose anything. Whether or not you believe QM is normally deterministic or super deterministic, you have to believe in something outside physics to make all of us not a necessary outcome of an initial state. That doesn’t diminish the pleasures or the reality of consciousness though, or the (illusion) of free will, at least not for me.
myridium 5y ago
> Superdeterminism, arguably misnamed, simply argues that QM is deterministic This is incorrect. QM is already deterministic. Where oh where do people get the idea that it is not?
- bryan0 5y ago
  The born rule for starters.
  - myridium 5y ago
    In QM, evolution is unitary. The Born Rule is an empirical observation that is adequately explained by the decoherence of a quantum system into entangled states of observer, environment, measurement apparatus and the system being measured. E.g. the master equation for a quantum density matrix governs its evolution when the environmental conditions are not completely known or modelled. Classical ensembles or 'decohered' states occur as a consequence. QM is formulated as a deterministic theory. When you try to model an open, interacting system without tracking its environment, you have to sacrifice determinism in that model, because you are failing to track all the information in the system. You settle for probabilistic quantities. Nothing quantum about it either-- same thing happens in classical mechanics. They call this "classical thermodynamics".
    - bryan0 5y ago
      there might be some non-trivial deterministic unitary description via decoherence, but it should not be surprising that many people think that QM is non-deterministic.

kgwgk 5y ago

I guess the whole field of statistical mechanics would make you uncomfortable.

graycat 5y ago
What I have seen so far for statistical mechanics is uses of integral calculus notation that has nothing to do with anything in any version of calculus, advanced calculus, measure theory, the Riemann integral, Lebesgue integration, etc.
- myridium 5y ago
  Well if you're doing fermionic thermal field theory sure (where the `integral' on Grassmann numbers is just some linear operation which satisfies certain properties). But apart from that I thought all the thermodynamic quantities were defined in terms of the partition function, which is a perfectly well-defined (Riemann) integral...