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If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density.

• mjburgess 3y ago
  All the people who use thinks like the word "information" in this context are confusing thermodynamic, logical, probabilistic, (+ many others) and equivocating. "Information" is not a physical quantity, and there cant be a "volume" of it. Nor does this have anything to do with real numbers. It is impossible for there to be any system extended in space and time to "zoom infinitely" into a continuous range and hence record an infinite amount of information. No one claims this, and the formulation of physics (entirely on real numbers) does not require it. Rather to say, eg., space is continuous, is to say its unbroken. There is no physical quantity which is becoming infinite.
  • mikk14 3y ago
    Some people would disagree with dismissing information as non physical. For instance: https://scottaaronson.blog/?p=3327 https://scottaaronson.blog/?p=3327 The argument there would be that stuffing an extra bit of information in an information saturated volume would make it collapse into a black hole.
    • mjburgess 3y ago
      It's not entirely clear "Energy" is a physical property either. By physical I mean a causal property of a system which is a basic constituent of reality. For example in E = 1/2mv^2, a particle has kinetic energy in virtue of being matter in motion -- it is motion and matter which are basic. Energy is just a system of accounting which tracks motion in the aggregate over time (with kinetic/potential just being the future/past in the accounting) hence why energy conservation is just a temporal invarience. When making arguments about the physical properties reality has (eg., whether aspects are continuous) you need to be exceptionally clear what your terms mean, and terminology in physics isnt designed for this. There are no "information saturated volumes", this is a series of abstractions piled on top of each other. All the words in this area have quite complex formal definitions that are have quite difficult to unpack semantics, you cannot just go around saying "saturated volumes" -- it is this sort of language which breeds cranks, and pop sci does it with abandon. This entire discussion is a matter of several PhDs, and to be done only well by people with PhDs in the matter (philosophy of physics), or equivalent research. It's not possible to scrap fragaments of what compusci bloggers say and derive much that's likely to be actually correct.
      • pishpash 3y ago
        Nothing in physics is more basic than something else, as there are equivalent formulations in other quantities. Energy and momentum are as real as matter and motion. Matter is bundles of energy exhibiting inertia and motion is just some transformational relationship between phenomena in different areas of space-time. There is nothing "real" about any of this, only what animals like humans have evolved to model directly in their brains.
      • calf 3y ago
        Aaronson is one of the world's top quantum computing scientists, he's a professor at I believe UT Austin. He's also written papers that are basically philosophy of physics. It would be interesting to go over what he has actually said on this topic.
    • mikhailfranco 3y ago
      Yes, the thermodynamic properties of information are well established. Various Hawking-Bekenstein results about black holes relate to information density, especially, shockingly, that information is proportional to surface area, not volume. This makes perfect sense because a black hole has all its incoming matter and energy sprawled, flattened and red-shifted on its horizon (to a distant observer). It can never export its internal state to the outside world, so you might never expect a volume's-worth of states to be exposed. The idea was generalized by 't Hooft to the Holographic Principle, for 2D screens encoding the state of 3D volumes on the other side. However, the full AdS/CFT Correspondence only applies to a certain type of AdS space, not our actual dS space. At the moment, it seems half of theoretical physics doctoral students are trying to extend AdS/CFT to dS space (obviously - strings :) and half of observational astrophysics doctoral students are desperately hoping to show we live in AdS space - LOL
  • n4r9 3y ago
    I suspect that Gisin has a very clear idea of what he means by "information" in this context, having worked for over 40 years at the forefront of theoretical physics with a specialisation in quantum information theory.
    • kergonath 3y ago
      I can link to people who’ve worked their whole life on various fields of Physics who still talk about perpetual motion. I am not saying he is wrong in this specific case, but an appeal to authority is not very convincing.
      • n4r9 3y ago
        I've given your comment quite a lot of thought over the last few days. Maybe too much. At first I was inclined to agree with you that this is an appeal to authority, with the caveat that such appeals do not always constitute a fallacy. For example, if we both agreed that such a Gisin is an expert whose opinion on this topic can be trusted, then his statements are valid evidence for one way or another. But then I realised that the very claim being challenged is whether Gisin knows what he's talking about. Floating his credentials and experience feels like a valid contribution. For what it's worth, back in my PhD days I read several of his papers and saw a couple of his talks at conferences, and can confirm he's one of the leading researchers in the field and is particularly thoughtful and careful in his work.
    • mjburgess 3y ago
      I don't have much issue with Gisin's solution, the idea that the reals are random is one solution to the problem of how to deal with them that I like (since, my meta-issue is whether reality is computable, I say it isnt, and randomness is not computable). There's a problem for people who think reality can be modelled by computable functions of finite inputs: this makes classical physics non-deterministic, because chaos requires infinite precision for determinism. So either you go for "reality is deterministic and continuous, and not computable" or "reality is non-deterministic, and discrete, and not computable" either option in this fork includes properties that offend the minds of the people who want everything to be discrete. I lean towards a preference for determinism & continuity (via, in QM, superdeterminism) since that's trivial to justify on our best physics
• mr_mitm 3y ago
  All observable quantities are eigenvalues of some operator, which are real numbers but discrete. How can they contain infinite amount of information?
  • nyssos 3y ago
    There are operators with continuous spectra. The previous commenter was accidentally half-right, in that the usual intro QM picture where everything lives in L2 really isn't fully rigorous, but this is fairly easy to resolve. The correct setting is a rigged Hilbert space: given an algebra of operators A on a Hilbert space H, let S be the maximal subspace of H such that |sa| is finite for any s in S, a in A. These are your states. Operators in A don't necessarily have eigenvectors in H, but they do have eigenvectors in the space S* of all continuous linear functionals on S. So <x|, for instance, is just the map `psi -> delta_x(psi)`.
    • eigenket 3y ago
      I take minor issue with the phrase "correct" here. Thats one way you can do things but its also works completely fine to not do that. Another way of setting these things up has your states be honest elements of L2, and says observables are just POVMs (i.e. maps from a space of measurable sets to positive operators which obey some natural restrictions like additivity). Then given a measurable subset A of the spectrum of some operator the Born-rule probability is just given by an inner product like < phi | P phi> where P is the projector you get if you integrate the spectral measure of the operator over A. This has the advantage of not having any funky "rigged" states suddenly appearing in your calculations and is also exactly how we deal with non-projective measurements in finite dimensional quantum mechanics. See here, for example https://en.wikipedia.org/wiki/POVM https://en.wikipedia.org/wiki/POVM
      • nyssos 3y ago
        Sure, I should have been clearer: a rigged Hilbert space is the right setting for bras and kets. You can also get rid of them entirely. In my experience QM classes unfortunately tend to split the difference by slinging around suggestive nonsense like \int_{x} |x><x|.
• mikhailfranco 3y ago
  I am very sympathetic to Gisin and his cause, but he does not propose any sensible resolution. By the way, not a fault, and no blame for him. Pointing out logical deficiencies always comes before a satisfying solution, and he is to be praised for his insight. There are many interesting ways to probe this problem.... here's one: Say I tell you to imagine a circle, an ideal Platonic circle in a Cartesian coordinate system (real coordinates, first uneasiness). Let's ignore translation, so it is centered at (0,0). I tell you the radius. Can you imagine the circle with Plato? Model the circle? Reproduce the circle? Do you need pi? Does the circle include or encode pi? But pi is has infinite information. Perhaps all you need is the square root function? But that's also an infinite Taylor series expansion. You can plot and recreate the circle to any precision if you have a square root function. The series will only need to run to the required precision. The circle will always be granular, depending on the number of terms you use in pi, or the square root function. Yeah, right, obvious, so why is that a problem? What if I tell you the circle is the physical manifestation of equipotentials of a stationary charge (say, nucleus), or mass (say Earth), with inverse square law - so basically a geometric fall-off with range determined by spatial (circular, spherical) considerations. What is the force at some distant point? Do you need pi? Do you need square root function? Or reciprocals? How does the other charge or mass feel the 56,323rd decimal place of the force due to the potential? Maybe it doesn't, because by the time it has felt the second decimal place, time has moved on, the charges/masses have moved on, and the nuance of what would've/should've been felt in a never changing universe are never experienced. There is a modified differential equation that relates various time derivatives to precision of experienced forces (this almost sounds like relativity :) The discrete explanation with photons goes like this: the force is produced by radiating photons. They automatically encode the geometric expansion as inverse square law, because of their pathways, no need for pi, or sqrt functions. But that is statistical, the accuracy is only as good as the number of photons that can arrive from the source. The circular/spherical nature of the force only emerges over time, as photons arrive and act. The accuracy of smooth circularity and inverse square only establishes itself over time... Elapsed time affects experienced precision - hmmm, interesting. How would you quantify such a thing, where time changes the precision of what you feel? Well, the other obvious example is the Heisenberg Uncertainty Principle. This is just a simple and obvious example of Fourier Analysis for any theory based on a linear wave equation. It almost doesn't need stating, and if it must have a name, it is certainly Fourier, not Heisenberg. Anyway, any math/physics/engineering student knows Fourier to their core, and it gives a nice solution to the information problem: coordinates may be real-valued degrees of freedom, but there is no way to mathematically or physically resolve all coordinates and their derivatives to infinite precision. It's just not possible, even if the underlying equations/reality maintain the fiction of real-valuedness. Fourier combines time, waves, amplitude, velocity (momentum, etc.) with a specific expression for possible information. A picture is worth a thousand words at this point, just look at a wave-packet, it's obvious. Fourier is a masterwork, and vastly underappreciated as a fundamental limit on knowledge, in a real world sitting on smooth continuous waves. So Fourier sets limits on knowledge, even in the wavy world of the smooth continuum. Of course, I do not believe in the smooth continuum anyway, but Fourier is my wingman to fight the real-infinitists on their own smooth turf.
• waveBidder 3y ago
  But we never know any quantity to full precision, so it's not like we get infinite bits about any given quantity.
  • Koshkin 3y ago
    That depends on the unit of measure. We know that the charge of the electron is exactly 1, with the appropriately chosen unit.
• kergonath 3y ago
  > If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. Yeah. As they said, it’s computer science leaking out. It can be misleading to reason about entropy, which is the relevant physical concept, as if it were strictly equivalent to information as computer scientists understand it. Entropy works perfectly fine with continuous densities of states and real numbers.
• codethief 3y ago
  As the sibling hinted at, Gisin's statement is quite sloppy and – at the very least – confuses "definite" "information"¹ (a given real number) with "uncertain" "information" (entropy), at least if you follow the definition of entropy by the book: The probability distribution for an observable that takes on (exactly) the value of a given real number with probability 1 has entropy 0. That being said, Gisin's approach is still interesting and his results can still be valid. But he starts with the assumption that real (irrational) numbers are unphysical, i.e. that – in a sense – our observable from above can actually only take on certain (rational) values, and then he derives certain predictions from that. ¹) Putting "information" in quotation marks here because no one really knows what it is.