One could argue that all numbers are an abstraction; they don't "exist" in any physical form, even integers. Therefore, why is it such a surprise that complex numbers are sometimes also needed to model reality?
I tend to agree with you, despite the famous quotation: "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man")
Perhaps, you would appreciate a similar question. Why does the buck stop at complex numbers, and we don't need quaternions or some other "weirder" field to describe reality?
The answer will have to grapple with both the philosophy of physics and the properties of these various fields.
Interestingly enough it would be "especially" integers that don't exist physically in mosts scales of reality if physics is to have any mathematical coherence.
Makes me wonder about all the other things that don’t materially exist yet are nonetheless true and not arbitrary and would be discovered even if humans did not exist.
If you follow the construction of numbers, integers are defined as how many elements are in a set. So integers kind of exist: if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2.
Then you define all the operations between integers as operations of cardinality between set. I really enjoyed that semester when I was getting my degree on Maths, by the way, although I use the knowledge I got there as a curiosity at parties, because it didn't prove useful (not that it is bad).
Gödel, Escher, Bach (book by Douglas Hofstadter) actually touches on this topic. They are an abstraction, but they are isomorphic to reality. The big news here is that we thought complex numbers have no isomorphism with reality (physics) that couldn't be explained with real numbers only.
*complex numbers. But yes you're right. The biggest problem (at least for me) is the name. Because everyone recognizes "real" and "imaginary" as adjectives, they think that's how these words should be understood in the context of numbers. But in truth, they are _proper nouns_, i.e. labels, and nothing more.
As far as the adjective "imaginary" is concerned, all numbers are imaginary.
Really the sooner we stop calling them "real" and "imaginary" the better. They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.
That's great, but my intuition fails me when I try to consider what it would mean for the answer to some question to be a complex number. It's even worse for quaternions, because I can't even use the spatial analogue there.
There is more to it. Vectors typically don't have multiplication defined. Complex numbers form a field and that is a major extra feature. Multiplication and division I'd defined on them.
If this property is not needed one can make do with 2d vectors instead.
There are 4d and 8d analogues as well. Hamilton started an ambitious program to base mathematics of nature on the basis of the 4d ones -- quarternions
> Imaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it.
Perhaps, but that view makes them seem arbitrary. It makes it seem as if you could just do the same thing in 3d, 4d, etc. and they'd all be equally impacful and meaningful.
The thing (which is subtle) is that the dimensionality is 1. Which is, as far as I have read the intro to the preprint cited in the Quanta article, what the authors are mistaken.
Where the analogy to a 2D vector starts to diverge is raising something to a complex power. There's no meaning in raising 3 to the [x,y], but Euler clearly defined it for 3^(xi+y).
Anyone whose ever studied vibrations (e.g. spring mass dampers) already knew this. This is a cool example but it’s not really any more “essential” than macro-level phenomena.
Half the article is explicitly talking about how this fundamentally is different than the phenomena you’re talking about.
Imaginary numbers are merely convenient in the cases you’re describing; you can get the same results without using them. The paper this article is about is claiming that in QM this is fundamentally untrue, and models that don’t use complex numbers cannot match those that do.
Not a physicist, so maybe I took the wrong gist away from the article, but - it seems like what they are saying is that systems that are generally modeled by complex equations have always had real valued equivalents that were known.
So it's not that the complex representation isn't a useful mathematical tool, but that it was an alternate representation that was easier to work with. Can you represent a spring mass dampening with all real valued quantities, even if it's a real piece to work with?
Was going to comment this. Complex numbers are a natural consequence of the basic math axioms and the fact that they are two dimensional (and not 3,4,etc.) is not arbitrary.
Complex numbers encode a specific and essential algebra, which is everywhere in nature. So do quaternions, and so on ad infinitum. Elementary field theories are often simply named by algebraic spaces they span.
Plenty of algebras not thought of as numbers also map onto physical processes usefully. None of this is shocking or new.
Roger Penrose devoted more than half of his doorstopper of a book "Road to Reality" to complex numbers in which he uses Riemann surfaces to visualize the complex-number fabric of Reality™:
https://openlibrary.org/works/OL3474173W/The_Road_to_Reality...
That can be reformulated using only real numbers, similar to some math the article mentions for quantum mechanics:
> Any property of complex numbers can be captured by combinations of real numbers plus new rules to keep them in line, opening up the mathematical possibility of an all-real version of quantum mechanics.
Yes but they are claiming that the link between the real and imaginary parts of the solutions to the Schrodinger Equation are "more linked" than just by the hamiltonian.
Have looked over the introduction of the preprint and have some issues with it but am not an expert.
But that is only for convenience. You can do everything with real trigonometry, just that it’s a bit more cumbersome. The imaginary numbers don’t correspond to anything in physical reality in electrical engineering.
The issue is not the "imaginary numbers not real etc!!" fluff. Maths is a tool to model reality and perhaps imaginary/real is a bad name. The interesting question why is C the correct choice of mathematical field to do "quantum mechanics" over -- what are the physical reasons this is needed.
You do wonder what exactly is meant by "doing quantum mechanics". I assume the start of the story is going from functions over C, quantising and getting some Hilbert space of operators/ getting wavefunctions.
Naively I would think that "quantum mechanics over the reals" might be less interesting somehow.
Consider the difference between doing complex analysis and real analysis. C loses total ordering as a field, but then gains additional nice properties over R. Quaternions however are non commutative which is a bit strange. Complex differentiable functions are more restrictive than functions on RxR (need to satisfy a stronger property). At some level you would think as models of reality you might need one structure over the other.
Imho, the presence of i in the schrodinger equation has a simple origin. The equation describes evolution of a wave and waves can be conveniently described with amplitude and phase components bundled together. With such description a wave can be moved or shifted by multiplying it by a complex number. Otoh, if the same wave is described by real numbers only, each number meaning the displacement from the null state, then shifting the wave pattern becomes a non trivial operation.
The paper is describing a test of a particular alternative to quantum mechanics called "real quantum mechanics", its rather more specific than you are suggesting
I know a few of the people on that preprint, and I'm pretty certain they all know you can embed the complex numbers in the real 2x2 matrices ;)
As an aside I'm pretty sure the matrix you need is [[0,-1],[1,0]], which squares to -I, unless you don't want to use matrix multiplication in your construction.
I would like to know why Schrodinger was so upset by the presence of i in his equation. It isn't clear to me why all the hate for i? It's not like it makes the math more complex: we're talking about quantum mechanics math!
There was initial backlash against complex numbers in the math community. i was deemed "imaginary" IIRC by its detractors, for the same reason the primeval atom was dubbed the "Big Bang": mockery, to accelerate its dismissal.
This bad reputation may have carried up to the 1920’s...
Information doesn’t exist without physics. So you could argue that math does exist in physical form (neutrons, bits on a hard disk, sound waves, printed paper etc.)
[+] [-] Xophmeister|5 years ago|reply
[+] [-] bloak|5 years ago|reply
[+] [-] abdullahkhalids|5 years ago|reply
The answer will have to grapple with both the philosophy of physics and the properties of these various fields.
[+] [-] j-pb|5 years ago|reply
[+] [-] CyberRabbi|5 years ago|reply
[+] [-] 101008|5 years ago|reply
Then you define all the operations between integers as operations of cardinality between set. I really enjoyed that semester when I was getting my degree on Maths, by the way, although I use the knowledge I got there as a curiosity at parties, because it didn't prove useful (not that it is bad).
[+] [-] ko27|5 years ago|reply
https://en.wikipedia.org/wiki/Isomorphism
[+] [-] unknown|5 years ago|reply
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[+] [-] Ambolia|5 years ago|reply
[+] [-] injb|5 years ago|reply
As far as the adjective "imaginary" is concerned, all numbers are imaginary.
Really the sooner we stop calling them "real" and "imaginary" the better. They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.
[+] [-] rimiform|5 years ago|reply
[+] [-] srean|5 years ago|reply
If this property is not needed one can make do with 2d vectors instead.
There are 4d and 8d analogues as well. Hamilton started an ambitious program to base mathematics of nature on the basis of the 4d ones -- quarternions
[+] [-] gspr|5 years ago|reply
Perhaps, but that view makes them seem arbitrary. It makes it seem as if you could just do the same thing in 3d, 4d, etc. and they'd all be equally impacful and meaningful.
[+] [-] pfortuny|5 years ago|reply
Algebraically, as vector spaces:
C is the change of base of R when tensored by C
which is not the same as
R^2 as R-vector space tensored by C
(not at all the same).
[+] [-] SavantIdiot|5 years ago|reply
[+] [-] denkquer|5 years ago|reply
[+] [-] darig|5 years ago|reply
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[+] [-] spekcular|5 years ago|reply
At the very least, he's able to show the problem with QM described using quaternions: superluminal signalling!
[+] [-] elil17|5 years ago|reply
[+] [-] Armisael16|5 years ago|reply
Imaginary numbers are merely convenient in the cases you’re describing; you can get the same results without using them. The paper this article is about is claiming that in QM this is fundamentally untrue, and models that don’t use complex numbers cannot match those that do.
[+] [-] ihunter2839|5 years ago|reply
So it's not that the complex representation isn't a useful mathematical tool, but that it was an alternate representation that was easier to work with. Can you represent a spring mass dampening with all real valued quantities, even if it's a real piece to work with?
[+] [-] CyberRabbi|5 years ago|reply
[+] [-] alpineidyll3|5 years ago|reply
Plenty of algebras not thought of as numbers also map onto physical processes usefully. None of this is shocking or new.
[+] [-] cableclasper|5 years ago|reply
I was interested in plotting these Riemann surfaces and wrote a little article about it sometime ago: https://honeybee.freeddns.org/Visualizing%2520roots%252C%252...
[+] [-] lizknope|5 years ago|reply
[+] [-] leereeves|5 years ago|reply
> Any property of complex numbers can be captured by combinations of real numbers plus new rules to keep them in line, opening up the mathematical possibility of an all-real version of quantum mechanics.
[+] [-] pfortuny|5 years ago|reply
Have looked over the introduction of the preprint and have some issues with it but am not an expert.
[+] [-] Ma8ee|5 years ago|reply
[+] [-] foxes|5 years ago|reply
You do wonder what exactly is meant by "doing quantum mechanics". I assume the start of the story is going from functions over C, quantising and getting some Hilbert space of operators/ getting wavefunctions.
Naively I would think that "quantum mechanics over the reals" might be less interesting somehow.
Consider the difference between doing complex analysis and real analysis. C loses total ordering as a field, but then gains additional nice properties over R. Quaternions however are non commutative which is a bit strange. Complex differentiable functions are more restrictive than functions on RxR (need to satisfy a stronger property). At some level you would think as models of reality you might need one structure over the other.
[+] [-] frongpik|5 years ago|reply
[+] [-] dnautics|5 years ago|reply
[+] [-] eigenket|5 years ago|reply
https://arxiv.org/abs/2101.10873
I know a few of the people on that preprint, and I'm pretty certain they all know you can embed the complex numbers in the real 2x2 matrices ;)
As an aside I'm pretty sure the matrix you need is [[0,-1],[1,0]], which squares to -I, unless you don't want to use matrix multiplication in your construction.
[+] [-] gspr|5 years ago|reply
[+] [-] SavantIdiot|5 years ago|reply
[+] [-] pygy_|5 years ago|reply
This bad reputation may have carried up to the 1920’s...
[+] [-] mbrodersen|5 years ago|reply
[+] [-] siproprio|5 years ago|reply
It's called a phase meter. You can buy it on amazon. An oscilloscope also works.
[+] [-] GoblinSlayer|5 years ago|reply
[+] [-] unknown|5 years ago|reply
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[+] [-] godelzilla|5 years ago|reply
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