15 ms·

Quantum physics falls apart without imaginary numbers

101 points by wrycoder 3y ago

magicalhippo 3y ago
I didn't have time to read all of it, but it seems to focus a bit much on the imaginary numbers aspect of complex numbers, and how they're all spooky and weird. Reading this abstract[1] and this[2] StackOverflow discussion about the topic, it seems the point is rather more along the lines that complex numbers aren't just plain two-dimensional vectors of real numbers. There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality. Then again, I'm just a programmer. [1]: https://www.nature.com/articles/s41586-021-04160-4 https://www.nature.com/articles/s41586-021-04160-4 [2]: https://physics.stackexchange.com/questions/691623/how-does-this-experiment-rule-out-real-valued-standard-formalism-of-quantum-the https://physics.stackexchange.com/questions/691623/how-does-...
- icapybara 3y ago
  > There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality. I think you couldn't have put it better. I'm just a physicist, though.
retrocryptid 3y ago
One of the first things we were taught in physics was "don't think that imaginary or complex numbers have physical significance. just do the math." And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out.
- sdfghswe 3y ago
  "shut up and calculate", also known as the Feynman approach to quantum mechanics. It's not imprecise. It reproduces experimental results from theory, so it's in fact the most precise approach in existence.
- BrandoElFollito 3y ago
  This is one of the complicated steps in physics (I have a PhD in physics (and forgot everything since)). First you have some math that goes along discovering physics. You split vectors, multiply mass by something and it's fine. Then you have math that helps you with physics. Simple differentials equations that uncover while laws of nature (cooling down speed for instance). This is the golden time for many because you're at this sweet spot where it is exciting but not too hard. Then comes the travel in desert of abstract things you have no idea about and winner why someone hates you by shoving Abel groups down your throat for no reason. Finally comes that sight of relief when you can binding do some maths to end up with a real life solution without too much thinking because you have solid tools. The last part is a bit morally complicated because you have the feeling that you are cheating. Renormalization, I am looking at you. But then I forgot everything because I left academia and my memories may be faulty.
  - nathan_compton 3y ago
    There really isn't anything weird or suspect about renormalization (except the name, perhaps). Read A. Zee's book on Quantum Field Theory.
    - BrandoElFollito 3y ago
      You are certainly right, like I said it was a long time ago. But doing splits and advanced acrobatics to get rid of infinities always felt like a hack (Feynman felt the same do at least I am not alone :))
    - C-x_C-f 3y ago
      > There really isn't anything weird or suspect about renormalization This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.
      - 3y ago
- bmacho 3y ago
  Same holds for negative numbers. There are no negative quantities in physics, negative numbers as quantities only appear if you order your equations wrong. (And one can argue against the other appearances of negative numbers and minus signs.)
  - wrycoder 3y ago
    But, how would one handle positive and negative charge?
    - amelius 3y ago
      You just do the bookkeeping, taking into account that if charges are opposite then they attract, etc.
    - deleted 3y ago
      [deleted]
- nathan_compton 3y ago
  "And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out. " The only thing imprecise about this is "many". Really any formula for an observable of any kind (including probabilities) has to come out to a real number.
  - computerfriend 3y ago
    Not really! Often you get a complex solution and both the real and imaginary components are valid.
    - nathan_compton 3y ago
      A complex solution can be valid but you never measure a complex number. I'd argue that in situations like using complex numbers to simulate time varying electrical activity the ontological status of the imaginary part of the solution is uncontroversial: the complex numbers in that situation have no ontological status at all and what is present is charges. In that case we're simply using the complex numbers as a convenient notation for a variable and its conjugate. In quantum mechanics one is more easily led to wonder about whether the ontological status of the complex numbers in that theory really can be settled so easily.
      - wadd1e 3y ago
        >A complex solution can be valid but you never measure a complex number. I see where you are coming from, and I'm asking this as a genuine question rather than to argue, but what's stopping me from measuring the length and the mass of an object and saying the "length-mass" of it is length + i(mass)? I suppose it isn't useful since complex numbers are not ordered, but aren't "numbers" arbitrary? In measure theory, measures are defined as outputting positive real numbers and +infinity because those happen to align with our intuition about how measures work, but as far as I know, maths(and physics here I guess) does not care about the representation of my quantity which I'm measuring, but it only cares about it's properties.
- C-x_C-f 3y ago
  > don't think that imaginary or complex numbers have physical significance. Yeah I'd say that's the most common approach but I think it's misguided. Complex numbers aren't any less physical than any other number. It just turns out that for historical reasons, it makes sense to define observable quantities using self-adjoint operators (which have real eigenvalues, and the latter are used to measure things like energy). But that doesn't mean the rest is not physical. Just because we can't take a picture of an object in the dark, it doesn't mean the object isn't there when the lights are off.
  - scotty79 3y ago
    I'd say that complex numbers are the only ones that have physical significance. They are what's actually happens in the real world until we disturb it with experiment.
whatshisface 3y ago
This is your regularly scheduled reminder that complex numbers have (real) matrix representations, and what matters in any model is the properties it has not its identity as an object.
- sebzim4500 3y ago
  Did someone claim otherwise?
  - whatshisface 3y ago
    "Quantum physics falls apart without imaginary numbers." >Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”
    - sebzim4500 3y ago
      By that logic, it doesn't even need real numbers. Just do everything with cauchy sequences of rationals.
      - whatshisface 3y ago
        But you can't really disagree with that. It's wrong to say that physics "needs" any one thing in particular when you can construct it from other things, and use them instead.
      - deleted 3y ago
        [deleted]
      - jiggawatts 3y ago
        Something I've always wondered is: what is the weakest algebra that could be used to model physics? E.g.: Are nationals sufficient? Integers? Finite integers?
        Koshkin 3y ago
stametseater 3y ago
Well, physics (including classical mechanics) already uses irrational reals, which are pretty spooky themselves. Imaginary numbers don't seem so much worse.
rprenger 3y ago
This Scott Aaronson lecture I really liked is relevant. It's like a "why quantum mechanics probably had to do the weird probability amplitudes (which can be negative and complex) instead of just normal probabilities even without experimental results" lecture: https://www.scottaaronson.com/democritus/lec9.html https://www.scottaaronson.com/democritus/lec9.html
- thechao 3y ago
  I like SA's blog; and, based on that I bought this book (Quantum Computing Since the Time of Democritus). It's expensive and bad. Really mind-numbingly awful. I can't tell if his writing has improved dramatically since he wrote the book, or what. The entire book is done in this tongue-in-cheek pseudo-first-person, chatty, pseudo-Socratic dialogue style. That sort of stuff is fine for, say, a couple of tightly-written pages. But ... not for hundreds of pages. It's a pity, since the information in the book is good.
dboreham 3y ago
Haven't read it, but it's obviously wrong. To expand: any time you read "...magical complex numbers" just mentally replace "complex" with "negative" and then examine how odd the original text now reads. There's nothing fundamentally different between the concept of negative numbers, and complex numbers.
- sebzim4500 3y ago
  "Quantum Physics Falls Apart Without Negative Numbers" sounds a bit obvious, but reasonable IMO.
- deleted 3y ago
  [deleted]
CottonMcKnight 3y ago
I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see those roots in the graph. There is no discontinuity.
- mjhay 3y ago
  The name "imaginary" was due to Descartes and it absolutely was intended as a pejorative, even though they're necessary to algebraically close the reals. Some ancient Greeks, IIRC, were similarly hostile to negative numbers. Of course the "real" numbers have never been controversial despite the whole concept being a lot weirder (and uncomputable), probably because their informal aspects just so happen to line up with everyday intuition.
  - mtlmtlmtlmtl 3y ago
    Irrational numbers have been known about since ancient Greece, but they were in fact controversial back when discovered/invented because they challenged conventional wisdom in Greek mathematics at the time.
  - kgwgk 3y ago
    > Of course the "real" numbers have never been controversial Apparently the existence of irrational numbers was a shock to Pythagoreans. There may be also people unhappy with transcendental numbers.
    - jerf 3y ago
      In 21st century hindsight, being annoyed by irrational numbers seems a bit odd to me. I mean this very much as an opinion. I actually partially understand where they were coming from; it makes a bit more sense than the 21st century perspective would indicate, but still, obviously, not something we'd agree with today. Even from a 21st century perspective, I think that the first "two dimensional number" is always going to freak people out and I can see where it's coming from. Imaginary numbers intrinsically involves leaving numbers that can be used to describe the number of apples you have in your hand, and by the time people get there, they've been pretty darned used to numbers looking like that. Real numbers nominally overshoot that too (you can't really have apples in two hands whose size only differs by 10^(-(10^1000))) but people tend to not have their faces rubbed in this until they get a math degree. Matrices nominally are such numbers too, but they are often presented as shortcuts rather than numbers in and of themselves. Of course in the 21st century now we have a zoo of these representations and the community as a whole is comfortable with it.... but for any given person I still think that first number that isn't something that can be a number of meters or apples is a shock.
tehsauce 3y ago
“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits in nicely with group theory, which organizes all kinds of objects which also have the same properties as numbers.
- nathan_compton 3y ago
  You miss a key part of complex numbers if you think of them as just vectors: they are a field.
  - adammarples 3y ago
    .
  - deleted 3y ago
    [deleted]
  - dr_dshiv 3y ago
    Because they are separate but interacting with real numbers? I don’t understand.
    - tgv 3y ago
      Multiplying vectors differs from multiplying complex numbers.
    - justin_ 3y ago
      He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0] You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity. You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1]. [0] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra https://ncatlab.org/nlab/show/normed+division+algebra
andrew_eu 3y ago
There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with them while studying physics was that, unlike "real" numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don't live in a 1D world. In higher dimensions they rotate and sheer rather than just scale. And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox. [0] https://en.m.wikipedia.org/wiki/Complex_number#History https://en.m.wikipedia.org/wiki/Complex_number#History
- reeboo 3y ago
  Here is fun, and rather short, book on the history of complex numbers that I liked -- https://www.amazon.com/Imaginary-Tale-Princeton-Science-Library/dp/0691169241 https://www.amazon.com/Imaginary-Tale-Princeton-Science-Libr...
- fullstackchris 3y ago
  The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imaginary velocity can fit! My intuition leads me to believe that almost some sort of other dimensional effects are at play, and our feeble math just can't accurately describe it. Perhaps it's just the nature of quantum itself, and no special dimensional consideration is needed. It's been a long time since I studied any math or physics...
  - tanseydavid 3y ago
reeboo 3y ago
So does algebra.
- contravariant 3y ago
  And ordinary differential equations.
ars 3y ago
This video by Sabine Hossenfelder is probably more informative on this topic: https://www.youtube.com/watch?v=ALc8CBYOfkw https://www.youtube.com/watch?v=ALc8CBYOfkw (it discusses this paper).
- anthk 3y ago
  Sabine's theories look like a New Age bullshiter parroting "quantumagical" nonsense to sell books, sorry.
deviantbit 3y ago
Just remember that Rene Descartes coined the term Imaginary Numbers, and he also believed in ghosts. Stop calling them imaginary.
- qwertox 3y ago
  They are like negative numbers. You can have 1 apple, but you can not have -1 apples. You can owe +1 apple, and say that you therefore have -1 apples. So -1 does exist as a number, but it is a representation of something that is happening with a positive number. And the "i" is similar to this. Maybe calling it "perpendicular" would have been better suited, because imaginary is really confusing.
sleepyams 3y ago
The utility behind complex numbers (for physicists at least) is really that they are a model for certain algebraic and geometric properties that are together very useful.
patrick451 3y ago
> Later, complex numbers, which are the sum of a real and an imaginary number, gained wide acceptance by mathematicians because of their usefulness for solving complicated mathematical problems. They aren't part of the equations of any fundamental theory of physics, however—except for quantum mechanics. I don't see how this is remotely true. You can't even solve the ODE for an undamped mass-spring system without imaginary numbers. More generally, most of our notion of eigenvalues falls apart if we work over the field of reals rather than complex numbers, and once you lose that, you lose most of linear algebra and with it vast swaths of engineering.
- crazygringo 3y ago
  Could you be more specific, because I was definitely under the impression that the original quote was correct. Where are imaginary numbers required for the undamped mass-spring system? Because a lot of "complex" equations are using complex e^ simply as an alternative to trigonometric functions (for aesthetics or convenience), where there's nothing inherently imaginary whatsoever. The same as much of signal processing. I'm less familiar with using complex numbers in linear algebra, but I know that when I studied it in college we never touched them, so I don't understand how we'd lose most of linear algebra? But I think the point the article is making is that, except for QM, there are no physical instantiations of complex/imaginary values. Rational numbers physically "exist" as a fraction of a distance between two points; real numbers "exist" as actual geometric proportions, and negative numbers "exist" as an opposite direction. But complex/imaginary numbers are just intermediary tools for solving equations (or conveniences to replace trigonometric functions), they don't correspond to anything physical (except, it seems, in QM).
m3kw9 3y ago
That it says “imagination” at first
esalman 3y ago
Well, multiple disciplins including and/or associated with electrical engineering and electronics would fall apart without imaginary numbers.
xchip 3y ago
Congratulations, BTW it also falls apart without the even numbers, same thing for the odd ones.
hackandthink 3y ago
This is not about experimentally falsifying real quantum theory, but nice anyway: Why are amplitudes complex? https://scottaaronson.blog/?p=4021 https://scottaaronson.blog/?p=4021
ordu 3y ago
There is a video depicting the story of a search for a cubic equation solution and invention of complex numbers. Here it is described in a few sentences but really it was a novel, with secrets passed from dying masters to apprentices, duels (mathematical), broken oaths and suchlike. https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo
itvision 3y ago
Websites seem not to work without them as well. The page isn’t redirecting properly An error occurred during a connection to www.scientificamerican.com. This problem can sometimes be caused by disabling or refusing to accept cookies.
xchip 3y ago
You should know that physicists don't talk all day long about quantum mechanics.
neonate 3y ago
https://archive.md/L1SeH https://archive.md/L1SeH
bookofjoe 3y ago
it also falls apart without imaginary superpositions
javajosh 3y ago
Physics and computer science share the feature that it seems easier to start by explaining linear motion (linear programs) but all the really interesting stuff is circular (loops). Complex numbers are the simplest representation for describing and combining rotations in a consistent way. (Other representations like "r theta" are not as simple.) Note also that a world with only monotonic linear motion could not possibly have life or thought or any complex behavior. Rotation is required to model any sort of accretion over time. (note that the typical finite case of "particles in a box" bouncing off the walls is, on average, circular motion too.) See Clifford Algebra for generalization of the complex numbers
2overengineered 3y ago
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odette4 3y ago
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crazygringo 3y ago
I've read the whole article twice now and, at the end of the day, it doesn't seem to actually explain anything at all. It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers. Which makes perfect sense, of course, because all of the rules of complex math are written in terms of real numbers at the end of the day. When CPU's are calculating math operations, it's not like any complex/imaginary bits or bytes are involved. But then the article describes an experimental setup where the results are somehow only consistent with standard/complex QM, and are inconsistent with real QM. But it totally neglects to say how or why. The entire premise behind real QM is that the complex stuff can just be rewritten as real. But this article seems to provide zero explanation, analogy, or intuition whatsoever as to why there's a case where this would ever not be possible. At no point does the article define what it even means to not be reducible to real math. So I can't tell what any of this is supposed to mean at all?
- cubefox 3y ago
  In philosophy of science and mathematics there is actually a famous argument by Hartry Field to the effect that Physics doesn't need to posit the existence of any numbers at all, not even natural numbers: https://academic.oup.com/book/26363 https://academic.oup.com/book/26363 As far as I know (I haven't read the book) he argues for this by logically reconstructing part of Newtonian Mechanics without any numbers (without assuming the Peano axioms) and suggesting that in principle similar things could be done for other theories. Now it would be very surprising if Quantum mechanics would even require the existence of imaginary numbers, since these seem to be much less basic than natural or real numbers. Anyway, I agree that popular articles haven't made it clear what the relevant physicists mean with their thesis. And as a non-physicist it is hard to read the original source.
- jessriedel 3y ago
  You definitely can do quantum mechanics fine without complex numbers. A good article would explain why it becomes much simpler/elegant with them, but apparently this isn't one. Clickbait title.
crazygringo 3y ago
I've always had problems with how complex numbers are taught. The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that's used for geometry or 2D coordinates or anything. Then you progress into all the complex e^ functions where you basically forget about a plane, and it's just a convenient shorthand for math that would otherwise involve a bunch of trigonometric functions. But again, this never provides any actual intuition... it's just convenience. You could still write everything as sin() and cos() etc. I finally felt like I understood complex numbers when I asked myself, how do you create a continuous solution to the function y(x) = (-1)^x? Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...]. But you can work out the math such that it necessarily produces a spiral through them. And it's different from the e^ equations because there's no pi involved. And so rather than thinking of complex numbers as a "plane", it's much better to think of them simply as the inherently spiral motion required to continuously join alternating values. And when I look at how complex numbers are actually used in things with physical correlates -- e.g. signal processing -- the spiral intuition always continues to make sense. Of course, at the end of the day, it's all the same math. But the idea of a continuous spiral path between otherwise discontinuous real numbers has wound up clicking for me in a way that the metaphors of a 2D plane, or of more generalized arbitrary "rotation", never has. That complex numbers are about spiral oscillation, not about planes.
- zuminator 3y ago
  You keep saying spiral but in a 2d spiral the value of |y| would be increasing as well. So if I'm understanding you correctly I think you mean helical? As in: https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259eb66d7438674-pjlq https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259e...
  - dbelford 3y ago
    The Heyser spiral or Heyser corkscrew seems to be a common name for this type of plot. https://www.google.com/search?q=heyser+spiral https://www.google.com/search?q=heyser+spiral And it connects circles, e/euler's formula, and sin/cos is a visually grokkable way.
justinpombrio 3y ago
`i` has a geometric meaning! It's explained by Geometric Algebra: https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Geometric_algebra In three dimensional Geometric Algebra, `i` is defined as the product of three orthogonal unit vectors, `abc`. You have `abc = bca = cab = i` and `acb = cba = bac = -i`. So `i` defines a chirality on the space. David Hesternes has a paper relating that to QM: https://web.archive.org/web/20120406093531/http://www.montgomerycollege.edu/Departments/planet/planet/Numerical_Relativity/Geometric_Algebra/caiqm.pdf https://web.archive.org/web/20120406093531/http://www.montgo...
dimitrios1 3y ago
As a random aside, if anyone wants a fun read, check out An Imaginary Tale: The Story of √-1
ubj 3y ago
My favorite property of "imaginary numbers": i^i = 0.20787957635... (Spoiler alert: it's real!) No, they're not imaginary, and yes they have real-world significance. They represent oscillations in control systems [1]. They're useful for accurately approximating derivatives [2]. I really dislike the term "imaginary" because of how useful they actually are. [1]: https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf [2]: https://mdolab.engin.umich.edu/wiki/guide-complex-step-derivative-approximation https://mdolab.engin.umich.edu/wiki/guide-complex-step-deriv...
hsnewman 3y ago
Quantum physics is a description of reality, not reality itself.
- Koshkin 3y ago
  Thank god - I wouldn't want to see the reality fall apart.
0xBABAD00C 3y ago
Complex/imaginary numbers are just badly named for historical reasons, they represent an objectively central concept in math and physics, and can be derived from axioms of what we expect from a well-behaved number field. For reals, we have: (A) expected properties of addition and multiplication, (B) total order and other order-related nice properties (Dedekind-complete). Any mathematical structure satisfying (A,B) will be equivalent to real numbers. Now if we extend it to get (C) algebraic closure, so that all polynomials have roots, we get the complex numbers.
user8501 3y ago
I had this thought years ago: Real number 1 is 1 OR -1 Imaginary number 1 is 1 AND -1 In other words, when you “break apart” i, you get 1 and -1
throwaway81523 3y ago
I get a redirect loop when I try to view this page. I'll try the archive.md link.
derbOac 3y ago
This reminds me of this paper even though it's probably only tangentially related: https://ieeexplore.ieee.org/abstract/document/6875117 https://ieeexplore.ieee.org/abstract/document/6875117 On the other hand it makes me wonder if there's some deeper truth about the nature of probability representations.

101 points by wrycoder 3y ago

I didn't have time to read all of it, but it seems to focus a bit much on the imaginary numbers aspect of complex numbers, and how they're all spooky and weird. Reading this abstract[1] and this[2] StackOverflow discussion about the topic, it seems the point is rather more along the lines that complex numbers aren't just plain two-dimensional vectors of real numbers. There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality. Then again, I'm just a programmer. [1]: https://www.nature.com/articles/s41586-021-04160-4 https://www.nature.com/articles/s41586-021-04160-4 [2]: https://physics.stackexchange.com/questions/691623/how-does-this-experiment-rule-out-real-valued-standard-formalism-of-quantum-the https://physics.stackexchange.com/questions/691623/how-does-...

icapybara 3y ago
> There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality. I think you couldn't have put it better. I'm just a physicist, though.

retrocryptid 3y ago

One of the first things we were taught in physics was "don't think that imaginary or complex numbers have physical significance. just do the math." And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out.

sdfghswe 3y ago
"shut up and calculate", also known as the Feynman approach to quantum mechanics. It's not imprecise. It reproduces experimental results from theory, so it's in fact the most precise approach in existence.
BrandoElFollito 3y ago
This is one of the complicated steps in physics (I have a PhD in physics (and forgot everything since)). First you have some math that goes along discovering physics. You split vectors, multiply mass by something and it's fine. Then you have math that helps you with physics. Simple differentials equations that uncover while laws of nature (cooling down speed for instance). This is the golden time for many because you're at this sweet spot where it is exciting but not too hard. Then comes the travel in desert of abstract things you have no idea about and winner why someone hates you by shoving Abel groups down your throat for no reason. Finally comes that sight of relief when you can binding do some maths to end up with a real life solution without too much thinking because you have solid tools. The last part is a bit morally complicated because you have the feeling that you are cheating. Renormalization, I am looking at you. But then I forgot everything because I left academia and my memories may be faulty.
- nathan_compton 3y ago
  There really isn't anything weird or suspect about renormalization (except the name, perhaps). Read A. Zee's book on Quantum Field Theory.
  - BrandoElFollito 3y ago
    You are certainly right, like I said it was a long time ago. But doing splits and advanced acrobatics to get rid of infinities always felt like a hack (Feynman felt the same do at least I am not alone :))
  - C-x_C-f 3y ago
    > There really isn't anything weird or suspect about renormalization This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.
    - 3y ago
bmacho 3y ago
Same holds for negative numbers. There are no negative quantities in physics, negative numbers as quantities only appear if you order your equations wrong. (And one can argue against the other appearances of negative numbers and minus signs.)
- wrycoder 3y ago
  But, how would one handle positive and negative charge?
  - amelius 3y ago
    You just do the bookkeeping, taking into account that if charges are opposite then they attract, etc.
  - deleted 3y ago
    [deleted]
nathan_compton 3y ago
"And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out. " The only thing imprecise about this is "many". Really any formula for an observable of any kind (including probabilities) has to come out to a real number.
- computerfriend 3y ago
  Not really! Often you get a complex solution and both the real and imaginary components are valid.
  - nathan_compton 3y ago
    A complex solution can be valid but you never measure a complex number. I'd argue that in situations like using complex numbers to simulate time varying electrical activity the ontological status of the imaginary part of the solution is uncontroversial: the complex numbers in that situation have no ontological status at all and what is present is charges. In that case we're simply using the complex numbers as a convenient notation for a variable and its conjugate. In quantum mechanics one is more easily led to wonder about whether the ontological status of the complex numbers in that theory really can be settled so easily.
    - wadd1e 3y ago
      >A complex solution can be valid but you never measure a complex number. I see where you are coming from, and I'm asking this as a genuine question rather than to argue, but what's stopping me from measuring the length and the mass of an object and saying the "length-mass" of it is length + i(mass)? I suppose it isn't useful since complex numbers are not ordered, but aren't "numbers" arbitrary? In measure theory, measures are defined as outputting positive real numbers and +infinity because those happen to align with our intuition about how measures work, but as far as I know, maths(and physics here I guess) does not care about the representation of my quantity which I'm measuring, but it only cares about it's properties.
C-x_C-f 3y ago
> don't think that imaginary or complex numbers have physical significance. Yeah I'd say that's the most common approach but I think it's misguided. Complex numbers aren't any less physical than any other number. It just turns out that for historical reasons, it makes sense to define observable quantities using self-adjoint operators (which have real eigenvalues, and the latter are used to measure things like energy). But that doesn't mean the rest is not physical. Just because we can't take a picture of an object in the dark, it doesn't mean the object isn't there when the lights are off.
- scotty79 3y ago
  I'd say that complex numbers are the only ones that have physical significance. They are what's actually happens in the real world until we disturb it with experiment.

whatshisface 3y ago

This is your regularly scheduled reminder that complex numbers have (real) matrix representations, and what matters in any model is the properties it has not its identity as an object.

sebzim4500 3y ago
Did someone claim otherwise?
- whatshisface 3y ago
  "Quantum physics falls apart without imaginary numbers." >Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”
  - sebzim4500 3y ago
    By that logic, it doesn't even need real numbers. Just do everything with cauchy sequences of rationals.
    - whatshisface 3y ago
      But you can't really disagree with that. It's wrong to say that physics "needs" any one thing in particular when you can construct it from other things, and use them instead.
    - deleted 3y ago
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    - jiggawatts 3y ago
      Something I've always wondered is: what is the weakest algebra that could be used to model physics? E.g.: Are nationals sufficient? Integers? Finite integers?
      - Koshkin 3y ago

stametseater 3y ago

Well, physics (including classical mechanics) already uses irrational reals, which are pretty spooky themselves. Imaginary numbers don't seem so much worse.

rprenger 3y ago

This Scott Aaronson lecture I really liked is relevant. It's like a "why quantum mechanics probably had to do the weird probability amplitudes (which can be negative and complex) instead of just normal probabilities even without experimental results" lecture: https://www.scottaaronson.com/democritus/lec9.html https://www.scottaaronson.com/democritus/lec9.html

thechao 3y ago
I like SA's blog; and, based on that I bought this book (Quantum Computing Since the Time of Democritus). It's expensive and bad. Really mind-numbingly awful. I can't tell if his writing has improved dramatically since he wrote the book, or what. The entire book is done in this tongue-in-cheek pseudo-first-person, chatty, pseudo-Socratic dialogue style. That sort of stuff is fine for, say, a couple of tightly-written pages. But ... not for hundreds of pages. It's a pity, since the information in the book is good.

dboreham 3y ago

Haven't read it, but it's obviously wrong. To expand: any time you read "...magical complex numbers" just mentally replace "complex" with "negative" and then examine how odd the original text now reads. There's nothing fundamentally different between the concept of negative numbers, and complex numbers.

sebzim4500 3y ago
"Quantum Physics Falls Apart Without Negative Numbers" sounds a bit obvious, but reasonable IMO.
deleted 3y ago
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CottonMcKnight 3y ago

I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see those roots in the graph. There is no discontinuity.

mjhay 3y ago
The name "imaginary" was due to Descartes and it absolutely was intended as a pejorative, even though they're necessary to algebraically close the reals. Some ancient Greeks, IIRC, were similarly hostile to negative numbers. Of course the "real" numbers have never been controversial despite the whole concept being a lot weirder (and uncomputable), probably because their informal aspects just so happen to line up with everyday intuition.
- mtlmtlmtlmtl 3y ago
  Irrational numbers have been known about since ancient Greece, but they were in fact controversial back when discovered/invented because they challenged conventional wisdom in Greek mathematics at the time.
- kgwgk 3y ago
  > Of course the "real" numbers have never been controversial Apparently the existence of irrational numbers was a shock to Pythagoreans. There may be also people unhappy with transcendental numbers.
  - jerf 3y ago
    In 21st century hindsight, being annoyed by irrational numbers seems a bit odd to me. I mean this very much as an opinion. I actually partially understand where they were coming from; it makes a bit more sense than the 21st century perspective would indicate, but still, obviously, not something we'd agree with today. Even from a 21st century perspective, I think that the first "two dimensional number" is always going to freak people out and I can see where it's coming from. Imaginary numbers intrinsically involves leaving numbers that can be used to describe the number of apples you have in your hand, and by the time people get there, they've been pretty darned used to numbers looking like that. Real numbers nominally overshoot that too (you can't really have apples in two hands whose size only differs by 10^(-(10^1000))) but people tend to not have their faces rubbed in this until they get a math degree. Matrices nominally are such numbers too, but they are often presented as shortcuts rather than numbers in and of themselves. Of course in the 21st century now we have a zoo of these representations and the community as a whole is comfortable with it.... but for any given person I still think that first number that isn't something that can be a number of meters or apples is a shock.

tehsauce 3y ago

“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits in nicely with group theory, which organizes all kinds of objects which also have the same properties as numbers.

nathan_compton 3y ago
You miss a key part of complex numbers if you think of them as just vectors: they are a field.
- adammarples 3y ago
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- deleted 3y ago
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- dr_dshiv 3y ago
  Because they are separate but interacting with real numbers? I don’t understand.
  - tgv 3y ago
    Multiplying vectors differs from multiplying complex numbers.
  - justin_ 3y ago
    He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0] You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity. You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1]. [0] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra https://ncatlab.org/nlab/show/normed+division+algebra

andrew_eu 3y ago

There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with them while studying physics was that, unlike "real" numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don't live in a 1D world. In higher dimensions they rotate and sheer rather than just scale. And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox. [0] https://en.m.wikipedia.org/wiki/Complex_number#History https://en.m.wikipedia.org/wiki/Complex_number#History

reeboo 3y ago
Here is fun, and rather short, book on the history of complex numbers that I liked -- https://www.amazon.com/Imaginary-Tale-Princeton-Science-Library/dp/0691169241 https://www.amazon.com/Imaginary-Tale-Princeton-Science-Libr...
fullstackchris 3y ago
The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imaginary velocity can fit! My intuition leads me to believe that almost some sort of other dimensional effects are at play, and our feeble math just can't accurately describe it. Perhaps it's just the nature of quantum itself, and no special dimensional consideration is needed. It's been a long time since I studied any math or physics...
- tanseydavid 3y ago

reeboo 3y ago

So does algebra.

contravariant 3y ago
And ordinary differential equations.

ars 3y ago

This video by Sabine Hossenfelder is probably more informative on this topic: https://www.youtube.com/watch?v=ALc8CBYOfkw https://www.youtube.com/watch?v=ALc8CBYOfkw (it discusses this paper).

anthk 3y ago
Sabine's theories look like a New Age bullshiter parroting "quantumagical" nonsense to sell books, sorry.

deviantbit 3y ago

Just remember that Rene Descartes coined the term Imaginary Numbers, and he also believed in ghosts. Stop calling them imaginary.

qwertox 3y ago
They are like negative numbers. You can have 1 apple, but you can not have -1 apples. You can owe +1 apple, and say that you therefore have -1 apples. So -1 does exist as a number, but it is a representation of something that is happening with a positive number. And the "i" is similar to this. Maybe calling it "perpendicular" would have been better suited, because imaginary is really confusing.

sleepyams 3y ago

The utility behind complex numbers (for physicists at least) is really that they are a model for certain algebraic and geometric properties that are together very useful.

patrick451 3y ago

> Later, complex numbers, which are the sum of a real and an imaginary number, gained wide acceptance by mathematicians because of their usefulness for solving complicated mathematical problems. They aren't part of the equations of any fundamental theory of physics, however—except for quantum mechanics. I don't see how this is remotely true. You can't even solve the ODE for an undamped mass-spring system without imaginary numbers. More generally, most of our notion of eigenvalues falls apart if we work over the field of reals rather than complex numbers, and once you lose that, you lose most of linear algebra and with it vast swaths of engineering.

crazygringo 3y ago
Could you be more specific, because I was definitely under the impression that the original quote was correct. Where are imaginary numbers required for the undamped mass-spring system? Because a lot of "complex" equations are using complex e^ simply as an alternative to trigonometric functions (for aesthetics or convenience), where there's nothing inherently imaginary whatsoever. The same as much of signal processing. I'm less familiar with using complex numbers in linear algebra, but I know that when I studied it in college we never touched them, so I don't understand how we'd lose most of linear algebra? But I think the point the article is making is that, except for QM, there are no physical instantiations of complex/imaginary values. Rational numbers physically "exist" as a fraction of a distance between two points; real numbers "exist" as actual geometric proportions, and negative numbers "exist" as an opposite direction. But complex/imaginary numbers are just intermediary tools for solving equations (or conveniences to replace trigonometric functions), they don't correspond to anything physical (except, it seems, in QM).

m3kw9 3y ago

That it says “imagination” at first

esalman 3y ago

Well, multiple disciplins including and/or associated with electrical engineering and electronics would fall apart without imaginary numbers.

xchip 3y ago

Congratulations, BTW it also falls apart without the even numbers, same thing for the odd ones.

hackandthink 3y ago

This is not about experimentally falsifying real quantum theory, but nice anyway: Why are amplitudes complex? https://scottaaronson.blog/?p=4021 https://scottaaronson.blog/?p=4021

ordu 3y ago

There is a video depicting the story of a search for a cubic equation solution and invention of complex numbers. Here it is described in a few sentences but really it was a novel, with secrets passed from dying masters to apprentices, duels (mathematical), broken oaths and suchlike. https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo

itvision 3y ago

Websites seem not to work without them as well. The page isn’t redirecting properly An error occurred during a connection to www.scientificamerican.com. This problem can sometimes be caused by disabling or refusing to accept cookies.

xchip 3y ago

You should know that physicists don't talk all day long about quantum mechanics.

neonate 3y ago

https://archive.md/L1SeH https://archive.md/L1SeH

bookofjoe 3y ago

it also falls apart without imaginary superpositions

javajosh 3y ago

Physics and computer science share the feature that it seems easier to start by explaining linear motion (linear programs) but all the really interesting stuff is circular (loops). Complex numbers are the simplest representation for describing and combining rotations in a consistent way. (Other representations like "r theta" are not as simple.) Note also that a world with only monotonic linear motion could not possibly have life or thought or any complex behavior. Rotation is required to model any sort of accretion over time. (note that the typical finite case of "particles in a box" bouncing off the walls is, on average, circular motion too.) See Clifford Algebra for generalization of the complex numbers

2overengineered 3y ago

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odette4 3y ago

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crazygringo 3y ago

I've read the whole article twice now and, at the end of the day, it doesn't seem to actually explain anything at all. It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers. Which makes perfect sense, of course, because all of the rules of complex math are written in terms of real numbers at the end of the day. When CPU's are calculating math operations, it's not like any complex/imaginary bits or bytes are involved. But then the article describes an experimental setup where the results are somehow only consistent with standard/complex QM, and are inconsistent with real QM. But it totally neglects to say how or why. The entire premise behind real QM is that the complex stuff can just be rewritten as real. But this article seems to provide zero explanation, analogy, or intuition whatsoever as to why there's a case where this would ever not be possible. At no point does the article define what it even means to not be reducible to real math. So I can't tell what any of this is supposed to mean at all?

cubefox 3y ago
In philosophy of science and mathematics there is actually a famous argument by Hartry Field to the effect that Physics doesn't need to posit the existence of any numbers at all, not even natural numbers: https://academic.oup.com/book/26363 https://academic.oup.com/book/26363 As far as I know (I haven't read the book) he argues for this by logically reconstructing part of Newtonian Mechanics without any numbers (without assuming the Peano axioms) and suggesting that in principle similar things could be done for other theories. Now it would be very surprising if Quantum mechanics would even require the existence of imaginary numbers, since these seem to be much less basic than natural or real numbers. Anyway, I agree that popular articles haven't made it clear what the relevant physicists mean with their thesis. And as a non-physicist it is hard to read the original source.
jessriedel 3y ago
You definitely can do quantum mechanics fine without complex numbers. A good article would explain why it becomes much simpler/elegant with them, but apparently this isn't one. Clickbait title.

crazygringo 3y ago

I've always had problems with how complex numbers are taught. The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that's used for geometry or 2D coordinates or anything. Then you progress into all the complex e^ functions where you basically forget about a plane, and it's just a convenient shorthand for math that would otherwise involve a bunch of trigonometric functions. But again, this never provides any actual intuition... it's just convenience. You could still write everything as sin() and cos() etc. I finally felt like I understood complex numbers when I asked myself, how do you create a continuous solution to the function y(x) = (-1)^x? Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...]. But you can work out the math such that it necessarily produces a spiral through them. And it's different from the e^ equations because there's no pi involved. And so rather than thinking of complex numbers as a "plane", it's much better to think of them simply as the inherently spiral motion required to continuously join alternating values. And when I look at how complex numbers are actually used in things with physical correlates -- e.g. signal processing -- the spiral intuition always continues to make sense. Of course, at the end of the day, it's all the same math. But the idea of a continuous spiral path between otherwise discontinuous real numbers has wound up clicking for me in a way that the metaphors of a 2D plane, or of more generalized arbitrary "rotation", never has. That complex numbers are about spiral oscillation, not about planes.

zuminator 3y ago
You keep saying spiral but in a 2d spiral the value of |y| would be increasing as well. So if I'm understanding you correctly I think you mean helical? As in: https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259eb66d7438674-pjlq https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259e...
- dbelford 3y ago
  The Heyser spiral or Heyser corkscrew seems to be a common name for this type of plot. https://www.google.com/search?q=heyser+spiral https://www.google.com/search?q=heyser+spiral And it connects circles, e/euler's formula, and sin/cos is a visually grokkable way.

justinpombrio 3y ago

`i` has a geometric meaning! It's explained by Geometric Algebra: https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Geometric_algebra In three dimensional Geometric Algebra, `i` is defined as the product of three orthogonal unit vectors, `abc`. You have `abc = bca = cab = i` and `acb = cba = bac = -i`. So `i` defines a chirality on the space. David Hesternes has a paper relating that to QM: https://web.archive.org/web/20120406093531/http://www.montgomerycollege.edu/Departments/planet/planet/Numerical_Relativity/Geometric_Algebra/caiqm.pdf https://web.archive.org/web/20120406093531/http://www.montgo...

dimitrios1 3y ago

As a random aside, if anyone wants a fun read, check out An Imaginary Tale: The Story of √-1

ubj 3y ago

My favorite property of "imaginary numbers": i^i = 0.20787957635... (Spoiler alert: it's real!) No, they're not imaginary, and yes they have real-world significance. They represent oscillations in control systems [1]. They're useful for accurately approximating derivatives [2]. I really dislike the term "imaginary" because of how useful they actually are. [1]: https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf [2]: https://mdolab.engin.umich.edu/wiki/guide-complex-step-derivative-approximation https://mdolab.engin.umich.edu/wiki/guide-complex-step-deriv...

hsnewman 3y ago

Quantum physics is a description of reality, not reality itself.

Koshkin 3y ago
Thank god - I wouldn't want to see the reality fall apart.

0xBABAD00C 3y ago

Complex/imaginary numbers are just badly named for historical reasons, they represent an objectively central concept in math and physics, and can be derived from axioms of what we expect from a well-behaved number field. For reals, we have: (A) expected properties of addition and multiplication, (B) total order and other order-related nice properties (Dedekind-complete). Any mathematical structure satisfying (A,B) will be equivalent to real numbers. Now if we extend it to get (C) algebraic closure, so that all polynomials have roots, we get the complex numbers.

user8501 3y ago

I had this thought years ago: Real number 1 is 1 OR -1 Imaginary number 1 is 1 AND -1 In other words, when you “break apart” i, you get 1 and -1

throwaway81523 3y ago

I get a redirect loop when I try to view this page. I'll try the archive.md link.

derbOac 3y ago

This reminds me of this paper even though it's probably only tangentially related: https://ieeexplore.ieee.org/abstract/document/6875117 https://ieeexplore.ieee.org/abstract/document/6875117 On the other hand it makes me wonder if there's some deeper truth about the nature of probability representations.