12 ms·

Quantum theory based on real numbers can be experimentally falsified

175 points by SquibblesRedux 5y ago

analog31 5y ago
Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
- PaulDavisThe1st 5y ago
  If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.
  - Koshkin 5y ago
    Unless we live in a computer simulation, of course, in which case we have to content ourselves with the IEEE floats.
    - analog31 5y ago
      God made the IEEE floats, and all the rest is the work of man -- Apologies to Kronecker
  - btilly 5y ago
    You can describe it to within measurement error without any irrational numbers. And with fewer decimal places than you'd probably imagine. See https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... for more.
    - midjji 5y ago
      That often works, but not always, some systems generate sequences of operations which can be symbolically simplified, or equivalently could be exactly computed using reals, but which if computed using any finite precision will fail. A simple example would be solving for the position of a planet in orbit under simple Newtonian gravity, a sufficient number of revolutions latter. For any finite precision the number of orbits can always be increased until failure occurs. Its a cool how few bits of pi are required to compute the circumference of the earth to within an atoms width accuracy. But equally cool how bad it gets if you try the same to compute the position of earth 2^64 years later.
      - btilly 5y ago
        That logic is disingenuous at best. Planetary orbits are chaotic. Long before your imprecision in pi is going to significantly mislead you, shifts in mass due to, for example, earthquakes and weather patterns are going to cause orbits to be impossible to predict. There are theoretical systems where the exact value of pi matters. But no physical system is going to match that, and measurement error is going to quickly exceed calculation errors from pi.
        TheOtherHobbes 5y ago
        Pi is only incidentally a number. Pi is an abstraction that represents a certain relationship. (Actually more of a set of relationships.) If you define pi as a specific constant with limited precision - because "that's all physical systems need" - you lose insights into the web of relationships around it. This is a bad thing and makes many kinds of math harder. It's the conceptual equivalent of lossy data compression. You don't want to do it unless you really, really need to. And if you do it, you need to be aware that you're now using approximations instead of abstractions, and those are not the same thing.
        wruza 5y ago
        But when you remove or take these fluctuations into account, you’re still left with an error. This rational model has no chance to ever be correct computationally, unless you cheat and add more detailed ratio every time you see a loop. Also, how exactly will you define rational pi? Let’s start with 3/1, why go any further. If it doesn’t represent reality (draw a circle and measure it with a string), well, strings have vague length anyway. See where this is going? The bigger problem is analyzis will not work, probably. Can you do analyzis with rationals? E.g. f(x)=x^2 isn’t continuous at f(x)=2/1. I’d theorize that there is a way to do finite calc without giving up on “reals”, by using enough FT coefficients (and packing this complexity into sine waves), but it’s the same sort of cheating probably, unless pi is the “origin” number of all “physical” reals. (I’m not a physicist nor math guy, so one can reframe my ideas as questions instead.)
  - MauranKilom 5y ago
    ...but Planck length!
    - EGreg 5y ago
      Perhaps that’s where quantum uncertainty reaches a threshold where you can’t make time based distinctions anymore.
  - roywiggins 5y ago
    Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.
    - kadoban 5y ago
      Orbits over long stretches of time seem likely to need arbitrarily high amounts of accuracy in pi. Sure you can just pick a rational number close enough for the accuracy you need, but why should the definition of pi need to change based on what you're measuring?
      - roywiggins 5y ago
        Every simulation picks some rational approximation to Pi, because they have to. Either they will run out of time or space or collapse into a black hole before needing more than a finite number of decimal places, so for all plausible purposes we can make do with the first googleplex digits (or whatever) of Pi. I guess my argument is, since you can always just pick a rational approximation to Pi, you cannot prove empirically that we live in a universe where more than a finite number of digits of Pi matter. That is, the mathematical irrationality doesn't really matter, physically speaking, since no experiment could ever prove that every digit in Pi actually contributes to the result. If the universe does have ways to do this, to mix an entire irrational number into a physical outcome, that means hypercomputation is probably possible, since Turing machines definitely can't. https://en.m.wikipedia.org/wiki/Real_computation https://en.m.wikipedia.org/wiki/Real_computation
        zozbot234 5y ago
        > Every simulation picks some rational approximation to Pi, because they have to. Your simulation might need to ask for an increasingly tighter bound on the real value of Pi. You can totally do this with no more than the usual rational numbers, but it's not equivalent to "just picking some rational approximation" and running with it, because what accuracy/precision you pick is outcome-dependent and it's always possible to request more.
  - p2t2p 5y ago
    That is replacing actual reality with math all over again. In the actual reality this ratio is concrete, limited value. I would say that our numbers and approaches are limited that they are unable to precisely define it. It just we are so used to it that no one even thinks about challenging it. And even if they do - good luck finding funding. Now we do have tools that overcome those limitations somewhat - like limits and stuff but doesn’t remove the need for better tools. So I somewhat disagree with both of you - we need new, better numbers that are further from math abstraction and closer to actual reality. Otherwise it is like this story with Poynting vector from Veritasium video - only confuses instead of explaining.
    - drdec 5y ago
      Exactly. The square root of 2 had no more basis in reality than the so-called imaginary number i. Don't confuse the "real" in real numbers with reality, at this point it is just a name.
  - int_19h 5y ago
    If the physical world is entirely quantized, including spacetime itself - which, as I understand, is still considered a valid hypothesis - then wouldn't it be possible to describe it using integers, pretty much by definition?
- katmannthree 5y ago
  Can you? Most likely. Should you? You’ll need to reprove more than a handful of theorems, and for what? What advantage does using rationals instead of reals get you? You might enjoy taking courses in real & complex analysis, the general purpose of which is to impart upon the receiver an understanding of why we’ve constructed those particular number systems and how despite the names they both describe things which are perfectly real in the philosophical sense. Edit: recall that the rationals are simply defined as the set of numbers which can be represented in the form a/b where a is an integer and b is a nonzero integer. This isn’t some deep philosophical tie to an underlying reality, it’s just our first attempt at defining more useful numbers that lie between other useful numbers we already invented (the integers et al). The real and complex numbers are literally just the extension of that process, filling in holes between useful numbers with more numbers until the set is closed (i.e. there are no more holes, every operation between members of the set results in another member of the set). Closure, the real reason we care so much about the reals, is just a surprise tool that helps us later. For anyone who made it this far: if you find any of this interesting you should find a book or lecture on analysis. It’s not particularly difficult and presents deeper insights into the math you likely already learned.
  - Vetch 5y ago
    I enjoyed your comment and agree with all your well put points except one: on the rationals tie with reality. Having implemented exact real computation to better understand reals, I think of a real number as a kind of machine that generates infinite streams. Operations on them instantiate new machines which query their real operands, computating until there's sufficient information to emit a next term of the stream. When the next term needs an infinite amount of information to decide what to spit out next, it results in an "unproductive" infinite loop. Rational numbers are interesting, more realistic, because they always terminate. In the real world, measurement tolerances and physical limits means at some point having to extract a rational. When we work with reals we are really only working with rational approximations or symbols with associated properties and relations. Reals are a powerful and elegant tool to rigorously reason about mathematical spaces and operations on algebraic objects but trying to work with them in reality in their exact form is a fun and visceral lesson on the nature of undecidability. It's hard to go two steps without tripping over a non-terminating loop (such as any operation that starts with an irrational and results in a rational or equality testing in general). This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number).
- deleted 5y ago
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- matt123456789 5y ago
  Most reals are not computable anyway: https://arxiv.org/pdf/math/0411418.pdf https://arxiv.org/pdf/math/0411418.pdf
  - deleted 5y ago
    [deleted]
- sillysaurusx 5y ago
  You can't. See "Simulating Physics With Computers," linked here: https://twitter.com/theshawwn/status/1394159744139632640 https://twitter.com/theshawwn/status/1394159744139632640 And some thoughts on the limitations with respect to creating AGI: https://twitter.com/theshawwn/status/1446261451061145602 https://twitter.com/theshawwn/status/1446261451061145602 See section 5, "Can quantum systems be probabilistically simulated by a classical computer?" > The probability that they match is eight-tenths, the probability that they mismatch is plus two-tenths; every physical probability comes out positive. But the original f's are not positive, and therein lies the great difficulty. The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative, and that we do not know, as far as I know, how to simulate. Okay, that's the fundamental problem. I don't know the answer to it, but I wanted to explain that if I try my best to make the equations look as near as possible to what would be imitable by a classical probabilistic computer, I get into trouble. When I was younger and only slightly more foolish, I wanted to spend a lot of time researching this to see if there was a way around the problem. I quickly realized that perhaps I should focus on adding value in a field that I was good at. :) Maybe one of you can try, since the only alternative is to take Feynman at his word.
  - aeternum 5y ago
    The claim that discretization is not equal to quantization is a key claim and I'm not sure the paper actually proves/shows this. If it turns out that spacetime is discrete and not continuous then we should be able to simulate it. The possible states of some chunk of spacetime (given some maximum/known energy) would be non-infinite and even though it might take us years to calculate each time-step, we could still make progress. I've seen the theory kicked around before, but why couldn't the universe basically consist of plank length sized voxels and a similarly small update timestep? It would still look plenty continuous to us.
    - sillysaurusx 5y ago
      It’s because of the anisotropy. Suppose the universe was a regular grid of voxels. You could run experiments to prove that. I don’t understand the details, but that was Feynman’s counter argument. (See the “messenger lectures” series on YouTube.) If the grid isn’t regular, you run into other problems. But iirc at one point Feynman was toying with the idea that the grid might be randomly distributed.
- nocturnial 5y ago
  Maybe you're interested in this paper: Discretization of the Bloch sphere, fractal invariant sets and Bell’s theorem https://royalsocietypublishing.org/doi/10.1098/rspa.2019.0350 https://royalsocietypublishing.org/doi/10.1098/rspa.2019.035... There also a video where he explains the paper: https://www.youtube.com/watch?v=YglT09Korr0&t=2700s https://www.youtube.com/watch?v=YglT09Korr0&t=2700s There are some consequences by using Q instead of C that can be experimentally tested.
doubleunplussed 5y ago
Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc. Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the time. Dirac figured he needed some 4×4 matrices in order to be able to take the square root of some operator at some point, which is how his equation predicted the existence of antimatter (because it implied the wavefunction had to be a vector with more components - some of the other components turned out to be antimatter). But complex numbers happen to have the right properties that at least we can do without matrices and vectors for the lowest-level quantities in quantum mechanics: the state amplitudes or the values of wavefunctions of spinless, non-relativistic particles. This article is saying that you can't do away with these properties at the lowest level of quantum mechanics, whether or not you actually use complex numbers to represent them.
- deleted 5y ago
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- midjji 5y ago
  Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.
  - kevin_thibedeau 5y ago
    'i' is a 90-degree rotation. It isn't hidden at all. Similarly, everywhere a complex exponential shows up, there's also some spinny/rotational thing happening.
    - gfody 5y ago
      you can always imagine another dimension, with all the symmetry of every dimension so far plus some novel one
EGreg 5y ago
Okay so for us laypeople - what does this mean? Quantum theory is wrong? :-)
- infogulch 5y ago
  No, the opposite. Some might 'hope' that you could represent everything in QM with simpler Real numbers, and avoid Complex numbers (or equivalent formulations that include rotation symmetries, as other commenters point out). But the title and article claim that any such hope is falsifiably dashed. Now, the use of Complex numbers to represent QM is basically standard for like half a century, so this result is more along the lines of "things we figured were true for a while because the math just works out so much better this way but we weren't certain enough to call it for sure until now".
  - MarkusQ 5y ago
    Actually, despite the title, the claim in the paper is that real-only-QM could be falsified, not that it has been. As far as I can tell, no one has actually done the experiment to check. Logically, it's just as possible that complex-QM could fall. There's a certain amount of hubris in seeing a case where there are two theories that agree in all known cases except one we haven't checked, and assuming "Well that must mean the one we thought of first is right!"
    - infogulch 5y ago
      > real-only-QM could be falsified, not that it has been Thanks for the clarification!
- db48x 5y ago
  No, it means that quantum mechanics is correct, and that you cannot get around the fact that it needs complex numbers. The complex numbers represent what is really going on in the universe better and more accurately than simple ordinary non–complex numbers do.
- steve76 5y ago
daxfohl 5y ago
I wonder if this can also be used to disprove the possibility of any QM based on quaternions?
- alubeixu 5y ago
  Most likely yes. The difficult part about such a claim is just having a quaternionic QM theory that is composable and interesting.
oofbey 5y ago
It’s quite strange that the abstract implies Einstein is a founder of QM. More the opposite I think. Einstein remained deeply skeptical of many fundamental aspects of QM - believing in hidden variable theory through his famous statement “God does not play dice with the universe” which was only proven false after his death through experimental measurements of the Bell inequalities. This paper describes another set of inequalities similar to the Bell inequalities, but testing whether QM requires complex numbers or not, instead of whether there are hidden variables that can explain things like superposition. The statement in the abstract I’m referring to is this: > This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural.[3] The reference [3] is to a letter by Einstein. Maybe I’m nitpicking words, but scientists tend to spend a lot of time getting the abstracts to their papers right. Any ideas why they would write it this way?
- subroutine 5y ago
  Einstein is considered a founder (1 of 3) of QM because he was the first to described light as quanta, and won the Nobel prize for it. It's ok to be skeptical, even of your own hypotheses/theories.
- fsh 5y ago
  Reference [3] is a letter from Schrödinger to Lorentz. I would also argue that Einstein had one of the deepest understandings of QM at the time. Everyone knew that the theory is weird, but with the EPR paper Einstein managed to pinpoint quite well where this weirdness shows up. It was also quite reasonable to assume that local hidden variables exist 40 years before a violation of Bell's inequality could be experimentally demonstrated.
- himinlomax 5y ago
  Einstein famously helped solve the ultraviolet catastrophe by "inventing" photons as the mechanism for the quantization of light proposed by Planck. He even got the Nobel prize for it. So he was very much a founder of quantum mechanic.
- lamontcg 5y ago
rnhmjoj 5y ago
I'm surprised the article doesn't mention phases or the idea of projective Hilbert spaces. The QM formulation it gives is well known ambiguous: the condition <φ|φ>=1 does not determine φ, because it's also satisfied by any other state ψ=exp(iλ)φ with λ real. So, the state of a physical system is really described by the ray (equivalent class) of all state vectors differing by a phase. I wonder if this has any consequence on their reasoning and conclusions.
- alubeixu 5y ago
  You can work in projective space or with any representative in the full space, or keeping track of global phases, it really doesn't make any difference. But all physicists are aware of this afaik. As you say, it is well known, so no need to mention it.
himinlomax 5y ago
This reminds me of something that was once linked on HN but that I can't remember enough of to find again. The thesis was that some of the iconic quantum weirdness™ simply disappears when you just dispense with taking the real part (or the norm) of the wave function as a final step and instead just consider the complex value. IIRC this seemed to made the double-slit experiment way more straightforward. Does that ring a bell to anyone?
- l33tman 5y ago
  If you don't take the norm, you're essentially not making a "measurement" and then QM is linear and simple. It's also non-predictive of reality :)
scotty79 5y ago
Actually real physical experiments don't deal with real numbers. Every result of every experiment was a rational number. Real numbers in physics are no less bizzare than complex numbers.
magicalhippo 5y ago
Lucien Hardy wrote a paper[1] showing how one "naturally" ends up with a quantum theory by demanding a few reasonable axioms. The paper also goes into how this implies complex numbers (and rules out quaternions). Scott Aaronson has a more accessible (and humorous) article on it here[2]. Entanglement has been shown to be intimately linked to this[3] result, which is interesting given the experimental evidence[4] for entanglement. Not my field, but I found this interesting at least. [1]: https://arxiv.org/abs/quant-ph/0101012 https://arxiv.org/abs/quant-ph/0101012 [2]: https://www.scottaaronson.com/democritus/lec9.html https://www.scottaaronson.com/democritus/lec9.html [3]: https://arxiv.org/abs/0911.0695 https://arxiv.org/abs/0911.0695 [4]: https://en.wikipedia.org/wiki/Quantum_entanglement#Notable_experimental_results_proving_quantum_entanglement https://en.wikipedia.org/wiki/Quantum_entanglement#Notable_e...
- mr_mitm 5y ago
  > x^n + y^n = z^n > There's a cute little fact -- unfortunately I won't have time to prove it in class -- that the above equation has nontrivial integer solutions when n=1 or n=2, but not for any larger integers n. I love this type of humour.
  - seeekr 5y ago
    Care to explain? My maths are failing me.
    - profquail 5y ago
      https://en.wikipedia.org/wiki/Fermat's_Last_Theorem https://en.wikipedia.org/wiki/Fermat's_Last_Theorem
      - magicalhippo 5y ago
        Which is extra funny as the actual proof[1] is so involved, roughly ~130 pages as published and spanning multiple very complex mathematical subjects. [1]: https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_...
tzs 5y ago
> Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural. I wonder if the use of complex numbers in QM theories to describe a real world that only needed real numbers inspired Asimov's 1942 short story "The Imaginary"? In that story psychology has been developed into a hard science. In some third rate college on some backwater planet some first year psychology students were doing a lab where they ran some animals through sequences of stimuli and observing the reactions and verifying they matched what the math said should happen. One of the animals fell asleep, which was not what was supposed to happen. It was reproducible and very specific. You run through that exact sequence of stimuli, and as soon as you hit the last one it falls asleep. Vary the order and it doesn't sleep. Vary the timing by even a tiny amount, no sleep. Word of this got back to the galactic federation's leading psychologist. Think the Einstein of psychology. He utterly could not explain it. Eventually though he came up with equations that worked, but they involved imaginary numbers. When applying these equations to the specific stimulus sequence all the imaginary quantities squared or cancelled out and you ended up with a real result, which was that the animal would sleep. This was controversial and caused quite an uproar in psychological circles, and while the leading psychologist was away dealing with that a couple of his students found a case where the imaginary numbers did not get squared or cancelled out. The predicted real world reaction to the stimulus sequence involved an imaginary number, and they have no idea what the heck that even means. They try it, and what it means turns out to be that some kind of slowly expanding radiation field gets created around the animal that kills other life that spends too long in the field. The top psychologist is called back and is able to calculate further stimuli that will stop the expansion. That works and catastrophe is averted. Asimov in 1942 would certainly have been aware of QM and the whole "complex number theory to make real number predictions" aspect of it.
johnp271 5y ago
There is a nice 10 minute discussion about this paper by physicist Sabine Hossenfelder on her "Science without the gobbelydgook" youtube series (which I recommend). She considers the existential questions regarding necessity of complex numbers a "super-niche nerd fight". You can find the video here https://www.youtube.com/watch?v=ALc8CBYOfkw&t=78s https://www.youtube.com/watch?v=ALc8CBYOfkw&t=78s.
sssss_99999 5y ago
sssss_99999 5y ago
sssss_99999 5y ago
sssss_99999 5y ago

Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.

PaulDavisThe1st 5y ago
If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.
- Koshkin 5y ago
  Unless we live in a computer simulation, of course, in which case we have to content ourselves with the IEEE floats.
  - analog31 5y ago
    God made the IEEE floats, and all the rest is the work of man -- Apologies to Kronecker
- btilly 5y ago
  You can describe it to within measurement error without any irrational numbers. And with fewer decimal places than you'd probably imagine. See https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... for more.
  - midjji 5y ago
    That often works, but not always, some systems generate sequences of operations which can be symbolically simplified, or equivalently could be exactly computed using reals, but which if computed using any finite precision will fail. A simple example would be solving for the position of a planet in orbit under simple Newtonian gravity, a sufficient number of revolutions latter. For any finite precision the number of orbits can always be increased until failure occurs. Its a cool how few bits of pi are required to compute the circumference of the earth to within an atoms width accuracy. But equally cool how bad it gets if you try the same to compute the position of earth 2^64 years later.
    - btilly 5y ago
      That logic is disingenuous at best. Planetary orbits are chaotic. Long before your imprecision in pi is going to significantly mislead you, shifts in mass due to, for example, earthquakes and weather patterns are going to cause orbits to be impossible to predict. There are theoretical systems where the exact value of pi matters. But no physical system is going to match that, and measurement error is going to quickly exceed calculation errors from pi.
      - TheOtherHobbes 5y ago
        Pi is only incidentally a number. Pi is an abstraction that represents a certain relationship. (Actually more of a set of relationships.) If you define pi as a specific constant with limited precision - because "that's all physical systems need" - you lose insights into the web of relationships around it. This is a bad thing and makes many kinds of math harder. It's the conceptual equivalent of lossy data compression. You don't want to do it unless you really, really need to. And if you do it, you need to be aware that you're now using approximations instead of abstractions, and those are not the same thing.
      - wruza 5y ago
        But when you remove or take these fluctuations into account, you’re still left with an error. This rational model has no chance to ever be correct computationally, unless you cheat and add more detailed ratio every time you see a loop. Also, how exactly will you define rational pi? Let’s start with 3/1, why go any further. If it doesn’t represent reality (draw a circle and measure it with a string), well, strings have vague length anyway. See where this is going? The bigger problem is analyzis will not work, probably. Can you do analyzis with rationals? E.g. f(x)=x^2 isn’t continuous at f(x)=2/1. I’d theorize that there is a way to do finite calc without giving up on “reals”, by using enough FT coefficients (and packing this complexity into sine waves), but it’s the same sort of cheating probably, unless pi is the “origin” number of all “physical” reals. (I’m not a physicist nor math guy, so one can reframe my ideas as questions instead.)
- MauranKilom 5y ago
  ...but Planck length!
  - EGreg 5y ago
    Perhaps that’s where quantum uncertainty reaches a threshold where you can’t make time based distinctions anymore.
- roywiggins 5y ago
  Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.
  - kadoban 5y ago
    Orbits over long stretches of time seem likely to need arbitrarily high amounts of accuracy in pi. Sure you can just pick a rational number close enough for the accuracy you need, but why should the definition of pi need to change based on what you're measuring?
    - roywiggins 5y ago
      Every simulation picks some rational approximation to Pi, because they have to. Either they will run out of time or space or collapse into a black hole before needing more than a finite number of decimal places, so for all plausible purposes we can make do with the first googleplex digits (or whatever) of Pi. I guess my argument is, since you can always just pick a rational approximation to Pi, you cannot prove empirically that we live in a universe where more than a finite number of digits of Pi matter. That is, the mathematical irrationality doesn't really matter, physically speaking, since no experiment could ever prove that every digit in Pi actually contributes to the result. If the universe does have ways to do this, to mix an entire irrational number into a physical outcome, that means hypercomputation is probably possible, since Turing machines definitely can't. https://en.m.wikipedia.org/wiki/Real_computation https://en.m.wikipedia.org/wiki/Real_computation
      - zozbot234 5y ago
        > Every simulation picks some rational approximation to Pi, because they have to. Your simulation might need to ask for an increasingly tighter bound on the real value of Pi. You can totally do this with no more than the usual rational numbers, but it's not equivalent to "just picking some rational approximation" and running with it, because what accuracy/precision you pick is outcome-dependent and it's always possible to request more.
- p2t2p 5y ago
  That is replacing actual reality with math all over again. In the actual reality this ratio is concrete, limited value. I would say that our numbers and approaches are limited that they are unable to precisely define it. It just we are so used to it that no one even thinks about challenging it. And even if they do - good luck finding funding. Now we do have tools that overcome those limitations somewhat - like limits and stuff but doesn’t remove the need for better tools. So I somewhat disagree with both of you - we need new, better numbers that are further from math abstraction and closer to actual reality. Otherwise it is like this story with Poynting vector from Veritasium video - only confuses instead of explaining.
  - drdec 5y ago
    Exactly. The square root of 2 had no more basis in reality than the so-called imaginary number i. Don't confuse the "real" in real numbers with reality, at this point it is just a name.
- int_19h 5y ago
  If the physical world is entirely quantized, including spacetime itself - which, as I understand, is still considered a valid hypothesis - then wouldn't it be possible to describe it using integers, pretty much by definition?
katmannthree 5y ago
Can you? Most likely. Should you? You’ll need to reprove more than a handful of theorems, and for what? What advantage does using rationals instead of reals get you? You might enjoy taking courses in real & complex analysis, the general purpose of which is to impart upon the receiver an understanding of why we’ve constructed those particular number systems and how despite the names they both describe things which are perfectly real in the philosophical sense. Edit: recall that the rationals are simply defined as the set of numbers which can be represented in the form a/b where a is an integer and b is a nonzero integer. This isn’t some deep philosophical tie to an underlying reality, it’s just our first attempt at defining more useful numbers that lie between other useful numbers we already invented (the integers et al). The real and complex numbers are literally just the extension of that process, filling in holes between useful numbers with more numbers until the set is closed (i.e. there are no more holes, every operation between members of the set results in another member of the set). Closure, the real reason we care so much about the reals, is just a surprise tool that helps us later. For anyone who made it this far: if you find any of this interesting you should find a book or lecture on analysis. It’s not particularly difficult and presents deeper insights into the math you likely already learned.
- Vetch 5y ago
  I enjoyed your comment and agree with all your well put points except one: on the rationals tie with reality. Having implemented exact real computation to better understand reals, I think of a real number as a kind of machine that generates infinite streams. Operations on them instantiate new machines which query their real operands, computating until there's sufficient information to emit a next term of the stream. When the next term needs an infinite amount of information to decide what to spit out next, it results in an "unproductive" infinite loop. Rational numbers are interesting, more realistic, because they always terminate. In the real world, measurement tolerances and physical limits means at some point having to extract a rational. When we work with reals we are really only working with rational approximations or symbols with associated properties and relations. Reals are a powerful and elegant tool to rigorously reason about mathematical spaces and operations on algebraic objects but trying to work with them in reality in their exact form is a fun and visceral lesson on the nature of undecidability. It's hard to go two steps without tripping over a non-terminating loop (such as any operation that starts with an irrational and results in a rational or equality testing in general). This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number).
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matt123456789 5y ago
Most reals are not computable anyway: https://arxiv.org/pdf/math/0411418.pdf https://arxiv.org/pdf/math/0411418.pdf
- deleted 5y ago
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sillysaurusx 5y ago
You can't. See "Simulating Physics With Computers," linked here: https://twitter.com/theshawwn/status/1394159744139632640 https://twitter.com/theshawwn/status/1394159744139632640 And some thoughts on the limitations with respect to creating AGI: https://twitter.com/theshawwn/status/1446261451061145602 https://twitter.com/theshawwn/status/1446261451061145602 See section 5, "Can quantum systems be probabilistically simulated by a classical computer?" > The probability that they match is eight-tenths, the probability that they mismatch is plus two-tenths; every physical probability comes out positive. But the original f's are not positive, and therein lies the great difficulty. The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative, and that we do not know, as far as I know, how to simulate. Okay, that's the fundamental problem. I don't know the answer to it, but I wanted to explain that if I try my best to make the equations look as near as possible to what would be imitable by a classical probabilistic computer, I get into trouble. When I was younger and only slightly more foolish, I wanted to spend a lot of time researching this to see if there was a way around the problem. I quickly realized that perhaps I should focus on adding value in a field that I was good at. :) Maybe one of you can try, since the only alternative is to take Feynman at his word.
- aeternum 5y ago
  The claim that discretization is not equal to quantization is a key claim and I'm not sure the paper actually proves/shows this. If it turns out that spacetime is discrete and not continuous then we should be able to simulate it. The possible states of some chunk of spacetime (given some maximum/known energy) would be non-infinite and even though it might take us years to calculate each time-step, we could still make progress. I've seen the theory kicked around before, but why couldn't the universe basically consist of plank length sized voxels and a similarly small update timestep? It would still look plenty continuous to us.
  - sillysaurusx 5y ago
    It’s because of the anisotropy. Suppose the universe was a regular grid of voxels. You could run experiments to prove that. I don’t understand the details, but that was Feynman’s counter argument. (See the “messenger lectures” series on YouTube.) If the grid isn’t regular, you run into other problems. But iirc at one point Feynman was toying with the idea that the grid might be randomly distributed.
nocturnial 5y ago
Maybe you're interested in this paper: Discretization of the Bloch sphere, fractal invariant sets and Bell’s theorem https://royalsocietypublishing.org/doi/10.1098/rspa.2019.0350 https://royalsocietypublishing.org/doi/10.1098/rspa.2019.035... There also a video where he explains the paper: https://www.youtube.com/watch?v=YglT09Korr0&t=2700s https://www.youtube.com/watch?v=YglT09Korr0&t=2700s There are some consequences by using Q instead of C that can be experimentally tested.

doubleunplussed 5y ago

Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc. Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the time. Dirac figured he needed some 4×4 matrices in order to be able to take the square root of some operator at some point, which is how his equation predicted the existence of antimatter (because it implied the wavefunction had to be a vector with more components - some of the other components turned out to be antimatter). But complex numbers happen to have the right properties that at least we can do without matrices and vectors for the lowest-level quantities in quantum mechanics: the state amplitudes or the values of wavefunctions of spinless, non-relativistic particles. This article is saying that you can't do away with these properties at the lowest level of quantum mechanics, whether or not you actually use complex numbers to represent them.

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midjji 5y ago
Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.
- kevin_thibedeau 5y ago
  'i' is a 90-degree rotation. It isn't hidden at all. Similarly, everywhere a complex exponential shows up, there's also some spinny/rotational thing happening.
  - gfody 5y ago
    you can always imagine another dimension, with all the symmetry of every dimension so far plus some novel one

EGreg 5y ago

Okay so for us laypeople - what does this mean? Quantum theory is wrong? :-)

infogulch 5y ago
No, the opposite. Some might 'hope' that you could represent everything in QM with simpler Real numbers, and avoid Complex numbers (or equivalent formulations that include rotation symmetries, as other commenters point out). But the title and article claim that any such hope is falsifiably dashed. Now, the use of Complex numbers to represent QM is basically standard for like half a century, so this result is more along the lines of "things we figured were true for a while because the math just works out so much better this way but we weren't certain enough to call it for sure until now".
- MarkusQ 5y ago
  Actually, despite the title, the claim in the paper is that real-only-QM could be falsified, not that it has been. As far as I can tell, no one has actually done the experiment to check. Logically, it's just as possible that complex-QM could fall. There's a certain amount of hubris in seeing a case where there are two theories that agree in all known cases except one we haven't checked, and assuming "Well that must mean the one we thought of first is right!"
  - infogulch 5y ago
    > real-only-QM could be falsified, not that it has been Thanks for the clarification!
db48x 5y ago
No, it means that quantum mechanics is correct, and that you cannot get around the fact that it needs complex numbers. The complex numbers represent what is really going on in the universe better and more accurately than simple ordinary non–complex numbers do.
steve76 5y ago

daxfohl 5y ago

I wonder if this can also be used to disprove the possibility of any QM based on quaternions?

alubeixu 5y ago
Most likely yes. The difficult part about such a claim is just having a quaternionic QM theory that is composable and interesting.

oofbey 5y ago

It’s quite strange that the abstract implies Einstein is a founder of QM. More the opposite I think. Einstein remained deeply skeptical of many fundamental aspects of QM - believing in hidden variable theory through his famous statement “God does not play dice with the universe” which was only proven false after his death through experimental measurements of the Bell inequalities. This paper describes another set of inequalities similar to the Bell inequalities, but testing whether QM requires complex numbers or not, instead of whether there are hidden variables that can explain things like superposition. The statement in the abstract I’m referring to is this: > This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural.[3] The reference [3] is to a letter by Einstein. Maybe I’m nitpicking words, but scientists tend to spend a lot of time getting the abstracts to their papers right. Any ideas why they would write it this way?

subroutine 5y ago
Einstein is considered a founder (1 of 3) of QM because he was the first to described light as quanta, and won the Nobel prize for it. It's ok to be skeptical, even of your own hypotheses/theories.
fsh 5y ago
Reference [3] is a letter from Schrödinger to Lorentz. I would also argue that Einstein had one of the deepest understandings of QM at the time. Everyone knew that the theory is weird, but with the EPR paper Einstein managed to pinpoint quite well where this weirdness shows up. It was also quite reasonable to assume that local hidden variables exist 40 years before a violation of Bell's inequality could be experimentally demonstrated.
himinlomax 5y ago
Einstein famously helped solve the ultraviolet catastrophe by "inventing" photons as the mechanism for the quantization of light proposed by Planck. He even got the Nobel prize for it. So he was very much a founder of quantum mechanic.
lamontcg 5y ago

rnhmjoj 5y ago

I'm surprised the article doesn't mention phases or the idea of projective Hilbert spaces. The QM formulation it gives is well known ambiguous: the condition <φ|φ>=1 does not determine φ, because it's also satisfied by any other state ψ=exp(iλ)φ with λ real. So, the state of a physical system is really described by the ray (equivalent class) of all state vectors differing by a phase. I wonder if this has any consequence on their reasoning and conclusions.

alubeixu 5y ago
You can work in projective space or with any representative in the full space, or keeping track of global phases, it really doesn't make any difference. But all physicists are aware of this afaik. As you say, it is well known, so no need to mention it.

himinlomax 5y ago

This reminds me of something that was once linked on HN but that I can't remember enough of to find again. The thesis was that some of the iconic quantum weirdness™ simply disappears when you just dispense with taking the real part (or the norm) of the wave function as a final step and instead just consider the complex value. IIRC this seemed to made the double-slit experiment way more straightforward. Does that ring a bell to anyone?

l33tman 5y ago
If you don't take the norm, you're essentially not making a "measurement" and then QM is linear and simple. It's also non-predictive of reality :)

scotty79 5y ago

Actually real physical experiments don't deal with real numbers. Every result of every experiment was a rational number. Real numbers in physics are no less bizzare than complex numbers.

magicalhippo 5y ago

Lucien Hardy wrote a paper[1] showing how one "naturally" ends up with a quantum theory by demanding a few reasonable axioms. The paper also goes into how this implies complex numbers (and rules out quaternions). Scott Aaronson has a more accessible (and humorous) article on it here[2]. Entanglement has been shown to be intimately linked to this[3] result, which is interesting given the experimental evidence[4] for entanglement. Not my field, but I found this interesting at least. [1]: https://arxiv.org/abs/quant-ph/0101012 https://arxiv.org/abs/quant-ph/0101012 [2]: https://www.scottaaronson.com/democritus/lec9.html https://www.scottaaronson.com/democritus/lec9.html [3]: https://arxiv.org/abs/0911.0695 https://arxiv.org/abs/0911.0695 [4]: https://en.wikipedia.org/wiki/Quantum_entanglement#Notable_experimental_results_proving_quantum_entanglement https://en.wikipedia.org/wiki/Quantum_entanglement#Notable_e...

mr_mitm 5y ago
> x^n + y^n = z^n > There's a cute little fact -- unfortunately I won't have time to prove it in class -- that the above equation has nontrivial integer solutions when n=1 or n=2, but not for any larger integers n. I love this type of humour.
- seeekr 5y ago
  Care to explain? My maths are failing me.
  - profquail 5y ago
    https://en.wikipedia.org/wiki/Fermat's_Last_Theorem https://en.wikipedia.org/wiki/Fermat's_Last_Theorem
    - magicalhippo 5y ago
      Which is extra funny as the actual proof[1] is so involved, roughly ~130 pages as published and spanning multiple very complex mathematical subjects. [1]: https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_...

tzs 5y ago

> Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural. I wonder if the use of complex numbers in QM theories to describe a real world that only needed real numbers inspired Asimov's 1942 short story "The Imaginary"? In that story psychology has been developed into a hard science. In some third rate college on some backwater planet some first year psychology students were doing a lab where they ran some animals through sequences of stimuli and observing the reactions and verifying they matched what the math said should happen. One of the animals fell asleep, which was not what was supposed to happen. It was reproducible and very specific. You run through that exact sequence of stimuli, and as soon as you hit the last one it falls asleep. Vary the order and it doesn't sleep. Vary the timing by even a tiny amount, no sleep. Word of this got back to the galactic federation's leading psychologist. Think the Einstein of psychology. He utterly could not explain it. Eventually though he came up with equations that worked, but they involved imaginary numbers. When applying these equations to the specific stimulus sequence all the imaginary quantities squared or cancelled out and you ended up with a real result, which was that the animal would sleep. This was controversial and caused quite an uproar in psychological circles, and while the leading psychologist was away dealing with that a couple of his students found a case where the imaginary numbers did not get squared or cancelled out. The predicted real world reaction to the stimulus sequence involved an imaginary number, and they have no idea what the heck that even means. They try it, and what it means turns out to be that some kind of slowly expanding radiation field gets created around the animal that kills other life that spends too long in the field. The top psychologist is called back and is able to calculate further stimuli that will stop the expansion. That works and catastrophe is averted. Asimov in 1942 would certainly have been aware of QM and the whole "complex number theory to make real number predictions" aspect of it.

johnp271 5y ago

There is a nice 10 minute discussion about this paper by physicist Sabine Hossenfelder on her "Science without the gobbelydgook" youtube series (which I recommend). She considers the existential questions regarding necessity of complex numbers a "super-niche nerd fight". You can find the video here https://www.youtube.com/watch?v=ALc8CBYOfkw&t=78s https://www.youtube.com/watch?v=ALc8CBYOfkw&t=78s.

sssss_99999 5y ago