Science is not a body of knowledge, it is a frontier of ignorance. I like to "feel" that complex numbers are part of the fabric of the universe, if only because it would be more beautiful.
"Clouds are not spheres, Mountains are not cones" - Mandelbrot
Well, I think instantaneous velocity is real, and I think I have a convincing argument. Imagine you have a piece of gum stuck to your bike tire. As your tire spins, the piece of gum spins with it, and its velocity is constantly changing. But when the piece of gum flies off your tire, it flies off in a straight line, following the instantaneous velocity at the moment of separation. Seeing as the gum can fly off in a straight line, its instantaneous velocity at that moment must be well-defined.
If I have 3 match sticks, they are each made of billions of molecules, atoms, electrons, quarks, gluons, etc. So there is no integer count, just in my perception.
The answer will have to grapple with both the philosophy of physics and the properties of these various fields.
The money quote (which leads a few more paragraphs on the details) is this:
> But, while I remain less than 100% satisfied about “why the complex numbers? why not just the reals?,” there’s one conclusion that my recent circling-back to these questions has made me fully confident about. Namely: quantum mechanics over the quaternions is a flaming garbage fire, which would’ve been rejected at an extremely early stage of God and the angels’ deliberations about how to construct our universe.
The proper question is not "do I like it" but is it necessary. The answer apparently is No for the quaternions.
 Information, physics, quantum: The search for links, John Wheeler 1990
[ The Simon's institute has formally funded such a program, though the movement within physics is much larger https://www.simonsfoundation.org/mathematics-physical-scienc...
Edit: fixed citation 1. And removed reference to infinities, as indicated below.
I think it's a quite interesting crossroad.
Either reals don't exist, and everything is at most turing complete, in which case the universe is detemernistic and predictable, yet there are very hard boundaries to what can be known and proven.
Or reals DO exist in which case the church turing thesis is false, as the universe (with it's operations on reals) is a reified example of a super turing system, in which case we can know a lot more things than in the discrete case, but some of that knowledge can only be grasped intuitively, because the moment you try to articulate it into formulas, it's discretized and thus limited to your sub-universe/brain powerfull turing machine.
So any part of reality that you can reduce to an efficient computation on a quantum computer, admits a description using the rationals only. But I might be out of my depth here.
Non-computable reals are also perfectly capable of modeling "analog reality" (because for all intents and purposes, reality could be discrete and finitely describable very small scales).
Quantization would like a word with you.
It's more accurate to say that only integers appear to be relevant in describing the nature of the nuclear forces and electromagnetism/electroweak force.
And it seems that, at least for the moment, we're headed in a direction of quantization of gravity and a Planck unit of time as well.
In some sense (because of the wavefunction) quantum mechanics is less discrete than classical mechanics, a qubit system which would be classicaly described by a single bit is described by a state that sits on a continuous manifold (the Bloch sphere).
However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?
I don't know what this means, but there is this idea that quantum mechanics leads to integer valued stuff because the first models studied were things like the hydrogen atom and the simple harmonic oscillator which both have a natural integer valued observable (the energy). This is not generally the case, however and in quantum mechanics there are many observables which are not integer valued, for example the kinetic energy of a free particle can be any non-negative real.
It turns out that a lot of interesting physics these operators have physically interesting discrete spectrum.
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).
The artificiality is high here -- would you trade your bag of "2" apples for someone else's bag of two apples without asking about the size and quality of the apples? While counting numbers are wonderful things for examining things like "sets" it seem to me to be kind of silly to assign any "reality" to them.
Discrete identical things are pretty much nonexistent at human scales. Continuous identical things, however, are numerous, where continuous here is not strict, but practical.
This is not realistic as a foundation for mathematics. But I think the idea of using propositional logic and set theory was a mistake that we are only now starting to correct. Intuitionalists and category theorists (and finitists, frankly) are leading us in a better direction now.
The construction you refer to is due to Frege and Russell but it has its own kinds of subtleties. In particular it does not work in ZFC which we often base upon, so we usually use Peano's axioms instead to define them (depending on your context of course). While they are not all too surprising, I wouldn't call them absolutely obvious.
Do they magically show up?
Or do they transition from not in the bag to in the bag in some unit?
What is the minimum possible unit of Apple change? Is it discrete or continuous?
Maybe a X plank units of apple transitions every second.
In the beginning there were only Natural numbers.
-1 appeared the moment one decided to define complementary operation to addition. -1 in reality is only a shorthand for "0-1", 0 comes here as neutral element of addition.
Addition is well defined on Natural numbers. Subtraction generates Zahl numbers.
Similarly multiplication is well defined on Zahl numbers. It's the complementary operation - division - that generates Quotient/rational numbers.
All of those are part of Peano and ZFC axiomatic systems, which were thought to be enough to explain all of physics (albeit not easily). Matrices and imaginary numbers are not part of those formal systems.
Just ask yourself. If you are right, why didn't Schrodinger or the researches that are publishing this study just think of your idea of using matrices?
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.
What's your preferred terminology? I like "Forward" and "Reverse" for positive and negative number and "Lateral" for the imaginary unit, since it is just perpendicular to the Forward and Reverse numbers.
"If ... +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question"
EDIT: similarly, quaternions as a number in 4D, with that extra dimension being handy for avoiding the instability problems when you get near “gimbal lock” in 3D rotations.
On the other hand, dual quaternions do care which is the dual component, since multiplying by complex numbers looks like true rotation, while multiplying by a dual looks like rotation around a point at infinity, aka translation.
So in that case maybe there is something special about the real component.
But the old term still survives in some areas. For instance, the real and imaginary parts of a complex number:
re (3 + 4 i) = 3 and im (3 + 4 i) = 4.
Sorry to be blunt, but your comment is horrible. There is far more going on here than you realize. Real numbers have an isomorphism with our reality. We thought we didn't need imaginary numbers to describe reality. They were inteded to be used more as a mathematical tool.
Another good example is euclidean vs non-euclidean geometry. They are both incredibly useful, but the former one was though to be "real" and the latter "imaginary". But it turns, physics is okay with us living in a non-euclidean universe (https://en.wikipedia.org/wiki/Shape_of_the_universe).
You can probably get a nobel prize by proving this. :)
I'd avoid calling complex numbers "numbers" personally. In the context of plain english (and not currently in the context of mathematics) I think of the word "number" as saying that this is somehow a measure of a quantity (We have "a number" of brands of cereal in the store). Complex numbers are not a measure of a quantity.
I'd probably replace the "numbers" part with "field". I don't have a good replacement for the "complex" part, but I think that should change also (because telling people things are complex is almost as bad as telling them that they are imaginary). If no one proposed something better and I was king of the universe I guess I'd just name them after whoever discovered 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
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.
That's different from extending 1d to 2d, 3d, etc as vectors.
Something like the original motivation (which you might find less "arbitrary") goes like this. We know how to solve some quadratic equations like x^2 = 9: We get x = 3 and x = -3 (though it was a nontrivial thing to realize those negative solutions are there). What about x^2 = 2? We can "see" the solution should be the length of the hypotenuse of a right triangle whose legs are both 1, so maybe encouraged by the geometry we don't feel too bad about having solutions (+/- sqrt(2)) which are, however, not rational numbers.
(Notice that we keep "extending" our idea of "number". Well, if we didn't have feature creep no one would buy the new version.)
Since we now have negative numbers, we ask about solutions to x^2 = -1. The first response is there are "no solutions", because there are no real numbers that square to -1. Rather than give up, we decide to "extend" the real numbers to provide solutions to this equation.
The idea is actually pretty easy if you know some modular arithmetic. For instance, mod 6 arithmetic means that you take the integers and consider things that differ by multiples of 6 to be "the same". As a consequence, 17 = 5, 4 = -14, and (importantly) 6 = 0.
Thinking in the same way, to get solutions to x^2 = -1, we want x^2 + 1 = 0. That's like "6 = 0", so "x^2 + 1" should be our "modulus", but what "number system" do we "mod out of" (instead of the integers, with the integers mod 6)? Since x^2 + 1 is a real polynomial, it looks like our "base number system" should be R[x], the (ring of) polynomials with real coefficients (stuff like -17 x^3 + 1.374 x^2 - 31/42 and so on).
So we should take real polynomials, but agree that two polynomials differing by a multiple of x^2 + 1 are "the same". We conclude that the complex numbers "should be"
R[x] mod <x^2 + 1>
x^3 + 4 x + 1 = (x^3 + x) + (3 x + 1) = x(x^2 + 1) + (3 x + 1)
= 3 x + 1 (mod x^2 + 1) (the multiple of x^2 + 1 is 0)
By this construction, "3 + 4 i" appears as
3 + 4 x (mod x^2 + 1) or (3 + 4 x) + <x^2 + 1>.
You can also construct the complex numbers as a subring of the 2 x 2 real matrices, so "3 + 4 i" appears as
3 [1 0] + 4 [0 1]
[0 1] [-1 0] .
[0 1]^2 = [-1 0] = - [1 0] (like i^2 = -1)
[-1 0] [0 -1] [0 1]
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).
At the very least, he's able to show the problem with QM described using quaternions: superluminal signalling!
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.
There might be ways around it, but I'd wager that anything you come up with can be massaged a bit to make it look like time is an imaginary dimension of space where it's contribution to distance along the spacetime manifold is negative when squared.
What I see from the preprint (notice that the authors have not discussed it with Quanta, because it is under peer-review) and bothers me is that they "add a real qubit" (...) which should be done very carefully because the two real qubits are not independent (they are linked by the hamiltonian).
I have not read the whole preprint, though.
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?
e^(-at)*(A cos(bt) + C sin(bt))
where b is the frequency and a the dampening.
(But you might not be asking for an answer).
Plenty of algebras not thought of as numbers also map onto physical processes usefully. None of this is shocking or new.
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...
> 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.
Have looked over the introduction of the preprint and have some issues with it but am not an expert.
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.
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.
C as C-vector space, which is R (as R-vector space) tensor C (complex dim 1).
R^2 as R-vector space, tensored by C (complex dim 2).
Because when they "add a real qubit", they seem to be adding a degree of freedom which you do not have when you have just one C-qubit.
But -I repeat- I have only looked over the intro.
This bad reputation may have carried up to the 1920’s...
It's called a phase meter. You can buy it on amazon. An oscilloscope also works.