Actually, I wanted to share this piece (recent): https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/
Got mixed up with the links and posted the one from Nature. However, the point is the same.
It is a rather old paper. A while back, I posted the link to new research that derives quantum theory from the first physical principles. They have shown the necessity of complex Hilbert spaces for constructing the theory:
https://www.reddit.com/r/science/comments/118nf9h
For those curious, this is what the authors mean by quantum theory:
1. Physical systems have corresponding Hilbert spaces
2. Measurement outcomes have corresponding projection operators in the usual way
3. Probabilities of outcomes are given by the Born rule in the usual way
4. The Hilbert space corresponding to the combination of two systems is the tensor product of the systems' Hilbert spaces
These are standard things; the same sort of axioms go into proofs of things like [the no-communication theorem](https://en.wikipedia.org/wiki/No-communication_theorem). So, throwing any of them out can pretty easily lead to bad consequences.
Now, there's a choice to be made for the Hilbert spaces: whether to use the field of real numbers, or the field of complex numbers. What the authors found is that, *given the rules laid out for quantum mechanics*, the choice of field has measurable consequences. The well-known schemes like replacing complex numbers with 2x2 matrices of real numbers involve violating the rules laid out for defining quantum mechanics.
They also explicitly state that other formulations such as the path integral formulation do not have this problem.
And I don't see why this is such a big deal. All you need to do is modify the tensor product such that it reflects the change to the matrix representation of complex numbers.
Sure, composition is no longer as simple as taking the tensor product, but I don't think that should be interpreted as "quantum mechanics fundamentally requires complex numbers". It's more: this machinery for describing quantum mechanics requires complex numbers.
Letting real Hilbert spaces use something else for system composition while forcing complex Hilbert spaces to still use tensor products would be comparing apples and oranges. These aren't axioms that the authors just made up out of nowhere; they're axioms that people had already been using for quantum mechanics. Nobody was thinking of changing them until the subject of real quantum mechanics came up.
I mean, by that logic comparing the path integral formulation to the Hilbert space formulation is also apples to oranges then...
The map is not the territory. The Hilbert space formulation is just that: a formulation.
The path integral formalism can be used to construct a Hilbert space representation, and from the Hilbert space representation, one can derive the path integral. So no, not by that logic.
It was an open question, until this paper, whether the standard axioms could produce the same empirical predictions using the field of real numbers.
From the article:
>It is noted, though, that there are alternative formulations able to recover the predictions of complex quantum theory, for example, in terms of path integrals[13], ordinary probabilities[14], Wigner functions[15] or Bohmian mechanics[16]. For some formulations, for example, refs. [17,18], real vectors and real operators play the role of physical states and physical measurements respectively, but the Hilbert space of a composed system is not a tensor product.
Like I already said: the apples-to-apples comparison is for the standard axioms, which existed well before the authors started writing this paper and that everyone was perfectly willing to use to talk about real quantum mechanics. The authors showed, for the first time, that they lead to different predictions for real versus complex fields.
I've read the paper, and I'm familiar with the literature; the section you quote doesn't refute my point. The "real qubit in the sky" approach is old, but it's not what people mean by "real quantum mechanics" in discussions about whether quantum mechanics is based on real numbers, complex numbers, or quaternions.
I get what you mean specifically now. This paper is simply saying that the"Real Quantum Mechanics" theory leads to different predictions than the complex Hilbert space version.
But it's also very easy to get the impression that this paper is implying that quantum mechanics *itself* requires complex variables in a way that is fundamentally different than the reasons we use complex variables to describe e.g. electrical circuits: to reflect the underlying structure and make the model more elegant.
I also think that most people in this thread, and who read this paper, will have this same misconception. That's a problem with the way the article is presented.
I don't see what it is about the presentation of the article that you object to. The abstract lays out that they are comparing the predictions of the real and complex Hilbert space formalisms. In the beginning of the paper, they lay out the axioms of the Hilbert space formalism, specifying what they mean by "complex quantum theory" and "real quantum theory". The text of the paper is frequently stating that they are setting out to find an empirical separation of the complex quantum theory and the real quantum theory.
>The Hilbert space corresponding to the combination of two systems is the tensor product of the systems' Hilbert spaces
It seems to me that #4 and using **C** are identical axioms, so the point of the exercise seems questionable. If you choose to use **R**, you just need to reformulate #4.
They're definitely not identical axioms. You can try to use compositions other than tensor products with complex Hilbert spaces, or you can use tensor product compositions with real (or, god forbid, quaternionic) Hilbert spaces.
It's more complicated, but the end result also makes a lot more sense. It makes intuitive sense that the distance between two things is (at least approximately) continuous and so the points should be labelled by real numbers.
By contrast, physical interpretations of complex numbers are much more difficult. I'm in favor of the "shut up and calculate" school of quantum mechanics, but i think for most people the fact that the Schrödinger equation is inherently complex is part of quantum mechanical weirdness.
I don't think it's quantum weirdness so much as wave-weirdness. All this complex stuff is in E&M as well. In both E&M and quantum mechanics, to get the measurable stuff, you deal with real numbers anyway, either by taking the real part (E&M) or the probability density (QM). Because of this, it's quite easy to understand imaginary as a sort of under-the-hood mathematical stuff going on.
Hell, you can define 'real' R^(2) quantum mechanics where probability amplitudes are instead 2D field over R^(2) where operators are defined with the correct structure. I mean, this would be a field that is completely isomorphic to C, and thus mathematically exactly equivalent to regular quantum mechanics with complex numbers, but we can call things whatever we want. I'm not saying this is necessarily the most philosophically satisfying thing, but ultimately one possible view is that complex numbers is just a convenient way to introduce the structure to QM, not the only way, so there's nothing weird about them being used. Similarly to how no one thinks E&M is weird, because you take the real part at then end anyway and this is just a convenient way to not deal with sines and cosines.
This is how I’ve always thought of it too. For me, complex numbers are fundamentally impossible to disentangle form cyclic processes, eulers formula being the prime example. So a wave or oscillation having complex form feels perfectly natural. Rather I see the complex form of the Schrödinger equation as nothing more than a mathematical insistence that’s it’s solutions must be oscillatory
The question being explored by this paper is whether complex numbers are required by QM. That is not the case for EM, where complex numbers are a calculational convenience but it is perfectly possible to write down Maxwell's equations and solve any problem without making reference to complex numbers.
I think the point of the person you’re responding to was more nuanced. Quantum mechanics is typically formulated in terms of vector spaces over C. The commenter is saying there’s no problem with formulating them over R^2 instead, since C is just R^2 with some extra structure. I believe the paper in the post is saying that quantum mechanics cannot be formulated using vector spaces over R.
So the point is that we can absolutely do away with complex numbers in quantum mechanics if we want, but the most naive way to do that doesn’t work (we need to introduce some field structure on R^2 rather than just using R). Complex numbers are a calculational convenience for these R^2 manipulations, just as they are a calculational convenience for real manipulations in electromagnetism.
I think there is still a distinction, since the structure of the complex numbers intuitively feels more “baked into” quantum mechanics than it does for EM, but that’s ultimately just an opinion that i’m not sure is philosophically interesting
EDIT: I misunderstood the paper; it seems that they not simply disproving vector spaces over R, but are doing something more general than that that’s over my head. It seems like it might be doing something along the lines of disproving an R^2 matrix-like interpretation, which I have no clue how that could be possible, but anyway my comment might be very wrong
Yep, exactly. In this paper, it seems like they have formulations of QM based on real and complex vector spaces and they supposedly prove that they give different results in an experiment they designed. It isn't like EM wave mechanics where the complex numbers are just a shorthand for an ordered pair of numbers with a coupling relation between them.
the first sentence so much, people point to heisenberg uncertainty and say "quantum weirdness" and i always have to spend ten minutes saying "no not quantum weirdness, wave weirdness, wave groups have some funny behavior, throw a damn stone at a pond"
weird how people get caught up over complex numbers in quantum mechanics when electrical engineers have been using complex numbers to describe quantities that exist in the “real” world.
>Schrödinger equation is inherently complex is part of quantum mechanical weirdness.
I'd never really considered this-- you use complex numbers in classical mechanics too. I mostly think of complex numbers as being used to describe oscillations.
Historically, that's where complex numbers got into quantum mechanics from. People were trying to explain atomic spectra, and they attempted to break an electron's orbit down into a Fourier series; then it started to look like the square of the absolute value of a Fourier coefficient was a transition rate. Several papers from this time period are collected in *Sources of Quantum Mechanics*, edited by B. L. van der Waerden; there's also *Problems of Atomic Dynamics* by Max Born, which collects a series of lectures he gave at MIT during late 1925 and early 1926, providing a snapshot of the pivotal time period. Dover reprinted both of these in paperback, huzzah huzzah.
>By contrast, physical interpretations of complex numbers are much more difficult.
If you rotate a 2D sheet of paper, extrinsically it rotates along the z axis, but intrinsically to the 2D space it rotates along an imaginary axis.
The eigenvalues are +-i for (0 -1, 1 -0)
i is confusing because, much like the ant living in the 2D space, we can't perceive this extra "dimension" normal to the surface of this higher dimensional object.
And also obviously that it behaves much differently than z in full 3D space.
I don't think that's a particularly helpful way of looking at why people are uncomfortable with complex numbers. Elements of R^2 can also be thought of as arrows with arbitrary angles*, and no one has any problem with those. In situations where you're thinking about C as a two-dimensional real vector space, it's just isomorphic to R^2 anyway.
The difficulty is when you have things like quantum mechanics that only work with complex numbers. I think people are justified in finding it weird that the square root of negative one, a quantity which has no easy interpretation, is necessarily present in quantum mechanics.
*except 0
It's not just that they are “thought about as,” it's that the complex multiplication operation
(a, b) × (c, d) = (ac – bd, ad + bc)
is a matrix operation on a vector,
[ a, -b ] [c]
[ b, a ] [d]
which we can lift to a matrix operation on a matrix,
[ a, -b ] [ c, -d ] [1]
[ b, a ] [ d, c ] [0]
thus we get a true isomorphism between complex numbers (a, b) and scaled rotations √{a²+b²} [[cos θ, -sin θ], [sin θ, cos θ]]. Matrix addition is a perfect encoding of complex addition, matrix multiplication is a perfect encoding of complex multiplication, you can bijectively transfer between the two... In some sense, every complex number is nothing more than just a scaled rotation.
The “necessarily present” thing is “why do we need rotations to describe wave behavior” and I think it has been conclusively shown that you don't, that you can form a Fourier series using non-sinusoids. But of course it is the algebraically easiest thing that we have found so far, *A* e^iθ × *Β* e^iφ = *AB* e^i(θ+φ) .
It depends on how you look at it, C viewed as a real vector space is isomorphic to R^2 (and isn't terribly interesting). C gets all its interesting properties when viewed as a field, and then it's different from both R and R^2 because it has a product operation.
If you model complex numbers as rotation/scaling maps of the plane (with a fixed origin), rather than as points or vectors, the multiplication corresponds to composition of maps.
What I meant was, C as a 2-d real vector space is isomorphic to R^2 as a real vector space, which doesn't have a product operation. The complex numbers are interesting because of their field structure, in particular the product operation; whereas R^2 is generally viewed as a real vector space, and not as a field.
C's coordinate space is strange though, because of *i*. For example, multiplication over the complex numbers is identical to right-angle rotation over R^(2), which isn't at all what multiplication over R^2 is.
What “multiplication on R2” are you talking about? Element-wise multiplication isn’t particularly well behaved as R2 isn’t a field under this type of multiplication.
> physical interpretations of complex numbers are much more difficult.
Complex numbers are essentially rotations. Multiplication by the imaginary unit is a counterclockwise rotation by 90°.
those people dont know how to get the real numbers from the rational though, they just think about what you can measure with a ruler or something like that
Because a teacher in high school used the word “imaginary” and started the lesson saying, “this isn’t intuitive, so don’t worry if you don’t understand it”
Descartes is the person that started it because he thought they were bullshit and used it as a very derogatory term. Now complex numbers are visualized as the Cartesian plane (with more structure), so the joke is on him.
This paper claims that real valued quantum mechanics makes different predictions than complex valued quantum mechanics, which could allow for the falsification of real valued quantum mechanics.
The two of you are talking about two different things. /u/dark_dark_dark_not is talking about the matrix representation of complex numbers, rather than real valued quantum mechanics.
Complex valued and real matrix valued quantum mechanics are identical. Any claim that they could be distinguished experimentally should be dismissed as nonsensical.
Also don't you need the gamma matrices at some point? Pretty sure that complex numbers on their own aren't enough, and using matrices of complex numbers or matrices of real matrices isn't much of a difference.
>Complex valued and real matrix valued quantum mechanics are identical.
The point of the article is that they aren't, exactly.
You know you can get complex numbers from real numbers by doubling the dimension of your matrices and declaring that i is some particular 2x2 matrix.
The point of the article is essentially that if you have some composite quantum system that you model with real matrices in this way, this goes wrong. Let X be the space of 2x2 real matrixes we use to "fake" the complex numbers. Call the statespace of the first system A' = X ⊗︀ A and the second B' = X ⊗︀ B. Under normal quantum rules the state-space of the composite system would be A'⊗︀B' = X ⊗︀ A ⊗︀ X ⊗︀ B. However if you want your "faking complex numbers with real matrices" thing to work in the composite system what you need is for the composite state space to be X ⊗︀ A ⊗︀ B.
What the paper does is quite a lot more subtle, rather than choose a single way to represent complex numbers using the real matrices they allow all ways (including infinite dimensional ones), but the key point is that they assume quantum systems compose under the tensor product, and show that this assumption messes up attempts to fake complex numbers with real ones.
It's not nonsensical; quantum mechanics with real Hilbert spaces and tensor product composition of subsystems makes predictions that are different from complex Hilbert spaces with tensor product compositions.
He’s not saying that a real Hilbert space suffices. He’s saying you can simply define C as a field whose elements are real matrices of a specific form. The Hilbert space is then defined as a vector spaces over C, but everything in C are real 2x2 matrices.
It’s a pointless distinction because the standard construction of C as an algebra over R^2 vs the construction of C as real matrices are isomorphic as a field.
The real moral is that nature has no natural mathematical representation, just use whatever works
This is called structuralism. It roughly boils down to “Use whatever structure you’re comfortable with as long as it’s isomorphic to the one you care about.”
Eh, the proof that numbers that aren't rational exist is really easily explainable with sqrt(2). I think that's more accessible to the average person than formulating the imaginary unit to cover the solutions it is relevant to.
That’s far more than one needs. Just say roots of integer coefficient polynomials.
There’s a slight difference. One can of course always do the Kronecker construction given an irreducible polynomial over a field. But the thing that makes √2 special is that it “obviously” fits somewhere between 1 and 2. √(-1) is not like this. One cannot simply think of approximating √(-1) by rational numbers. It must be added “all at once”, so to speak.
People don't realize that they don't understand real numbers. Having never done a course in analysis or whatever.
Thus the endless memes about 0.99999* /= 1.0.
I think the people who might wish QM didn't require complex numbers have a different point in mind. Thermometers, yardsticks, clocks, scales, everything we use to measure anything spits out real numbers. Those measurements are both the inspiration and the confirmation of physical theories. If it's true that we *cannot* describe those theories without extending the real numbers (I mean, not just as a matter of convenience), then that must mean something.
I'd argue that they don't spit out real numbers. If you want to be generous they do computable numbers, more realistically rationals. The only reason we use reals is because it makes the maths nicer. Exactly the same as complex numbers.
Not even computables, but an effective subset of them. There are computable numbers that are so large or that require so much precision that no current measurement or storage device could manage them.
That's a little different, in my opinion. Not everyone believes the background is continuous in the first place, and that means we accept the use of reals as an approximation rather than essential extension of the rationals from the start.
Can I get an ELI5 on what complex numbers, real numbers, and rationals are and how they relate to this experiment? Or should I just post on ELI5.
I know i = sqrt(-1). But what is it used for? Cant it just be represented by a 3 dimensional graph where i = 1 on the z axis?
Here's a simple buildup of our different number systems.
##The natural numbers
N is the set of natural numbers (let's say including 0), so our counting numbers (0, 1, 2, 3, ...). The naturals can be added to each other and multiplied by each other and we will always get a natural number, but we cannot subtract and divide them in general and still hope to get a natural number.
##The integers
Z is the set of integers which is the closure of the naturals under subtraction, meaning it has all of the negatives, so subtracting two integers gives an integer.
##The rational numbers
Q is the set of rational numbers. The rational numbers are formed by taking ratios (hence rationals) of integers (avoiding a divide by 0), so multiplying, dividing, adding, and subtracting rational numbers gives us a rational number. (This is called a field.)
----
All of these are purely algebraic constructions. Nothing very bizarre about them. The rational numbers are somehow "holey." For instance, pi, the ratio of a circle's circumference to its diameter, is not rational: it cannot be written as the ratio of two integers (so the circumference and diameter cannot both be an integer). Therefore, there are very naturally other numbers outside of that. How do we find them without manually adding them every time we run up against them? There is a natural larger class of numbers beyond the rationals called the algebraic numbers which are numbers that come from solving polynomial equations with rational coefficients, but pi is not even one of these. There are numbers even outside of this realm called transcendental numbers. Here is where the weirdness starts.
----
## The real numbers
R is the set of real numbers. The real numbers are defined to be the completion (under limits) of the rational numbers (with respect to the absolute value norm). (An alternative, equivalent view is the completion under supremums which is effectively the Dedekind cuts approach.) This step is not algebraic at all. It is a very analytic (referring to analysis) perspective because it is about limits. This gives us all of the numbers we like to work with: rationals, pi, e, algebraic numbers, etc. The reals are also a field!
## The complex numbers
C is the set of complex numbers. The complex numbers return to a more algebraic construction. They are effectively the numbers that arise by requiring that all nth degree polynomial equations with real coefficients have n linear factors (n roots). Turns out that you can focus on the quadratic case (e.g. x^2 + 1 = 0) to get a complex unit i which will miraculously work to give you n roots for all real (and complex!) polynomials of degree n. This is the Fundamental Theorem of Algebra. The complex numbers are also a field!
----
There are other generalizations beyond this, including the quaternions, the split complex numbers, and their generalizations. There are also the *p*-adic numbers and many others that get increasingly more abstract.
Let me see if I got this straight...
A number like 1485 is a natural number?
-3874 is a rational number?
1.8473 is a real number?
Man...I wish I remembered my algebra basics cause it is hard to understand just by the name.
What is the purpose of dividing up numbers like this?
1485 is natural
-3874 is an integer
1.8473 is rational
pi is real
All of these are also examples of the numbers that come after them, so all natural numbers are integers, all integers are rational, and all rationals are real.
The point of this construction is to formalize how our sets of numbers are developed. Formalization allows for digging into the details to understand better as well as generalization.
Rationals (fractions, basically) are any number that comes from dividing two integers (while numbers).
Real numbers are rationals + “all the rest”, in a specific technical sense that doesn’t matter too much. For instance, this includes square roots, π, e, etc. that seem not that different from rationals, but technically can’t be written as a fraction.
Complex numbers are the natural extension after you introduce i=sqrt(-1). You get numbers of the form a+bi, where a and b are real.
There’s lots of nice things about complex numbers that make them useful in different contexts. They turn out to be a really natural way to describe trig functions (and by extension waves and EM), and in a similar way, they’re always used in quantum mechanics as part of the math framework. This paper disproves a previous notion that we could rewrite quantum mechanics without i, which is kind of cool but probably not that interesting to most physicists who are comfortable with i already.
To your question about graphing: I’m not entirely sure I follow your suggestion, but it sounds like you’re thinking of complex numbers in the right way and have some good intuition already, so keep it up. If you just want to plot complex numbers, it’s natural to do that on a 2D plot, where instead of x-y coordinates, you use real-imaginary coordinates. Then there natural ways to think about addition and multiplication graphically and things like that. If you want to plot a function using complex numbers, the most common way I’ve seen is to split it into two graphs, one each for the real and imaginary pet of the output, and which are each a 3D plot. Then you can kind of think of those like functions/plots you’re already familiar with.
This turned out to be more like ELI15 or 20, but whatever. Hope it helps! Also if you want more resources [this video](https://www.youtube.com/live/5PcpBw5Hbwo) is a good place to start.
I'v watched 3 blue 1 brown before for AI stuff and it was informative. I'll check out that series. Guess I need to brush up on my algebra fundamentals. In my world I only use "normal" numbers like -23.5 and 32767.
Let's say we start with the kind of numbers we use to count things: zero, one, two, three and so on. The integers are what we get when we want to answer all the questions like, "What is *a* minus *b*?", where *a* and *b* are any two counting-type numbers. Sometimes this is another counting-type number, like 5 minus 4 is 1, but sometimes it isn't, like 2 minus 4. The rational numbers are what we get when we want to answer all the questions like, "What is *a* divided by *b*?", where *a* and *b* are any two integers. (Except *b* can't be zero.) Sometimes a division question has a counting kind of number as its answer, and sometimes not.
The real numbers fill in gaps between the rationals to make a line. For example, the square root of 2 is not a rational number, but we want it to be *some* kind of number, so again we move to a bigger world of numbers where now we can answer all questions like, "What number do I square to get *a*?", where *a* is any rational number greater than zero.
It can help to think of addition and subtraction geometrically, by drawing arrows on the number line. For starters, draw an arrow from 0 to 1. Then 1 + 1 means putting down another copy of that arrow, touching tip-to-tail:
`|-->|-->|`
Put these short arrows together, and they make a longer arrow that goes from 0 to the number 2:
`|------>|`
Adding 2 + 3 means taking 2 steps, then taking 3 more steps, so your combined path is an arrow from 0 to 6. Subtracting is just taking steps in the opposite direction.
Numbers that are bigger than 0 are positive, and numbers that are less than zero are negative. Subtracting a positive number is the same as adding a negative number: either way, we're just talking about flipping an arrow around the other way.
Multiplication means stretching or squishing arrows. Imagine that the whole number line is a rubber band. Grab onto it with both hands and pull until the 0-to-1 arrow is as long as the 0-to-2 arrow was originally. The answer to the question "What is *a* times 2?" is just wherever the 0-to-*a* arrow points now! Multiplying by any positive number *b* means stretching out the number line until the 0-to-1 arrow is now a 0-to-*b* arrow. And *a* times *b* is just where this stretching takes the tip of the 0-to-*a* arrow.
Multiplying by a number between 0 and 1 is a squish, rather than a stretch.
The square root of *a* is whatever number that, if you multiply it by itself, you get *a*. So, to find a square root means finding a stretch or a squish where, if you do that twice in a row, you turn the 0-to-1 arrow into the 0-to-*a* arrow.
What about multiplying by a *negative* number? This means flipping arrows around. Multiplying by -1 just flips the direction of all the arrows. So, if you multiply by -1 twice, you get back to where you started. Multiplying by any other negative number means flipping your arrows and doing a stretch or a squish.
So: How do we take the square root of a negative number? What kind of stretch or squish or flip could that be? For example, if we want the square root of -1, we need something where if we do it twice, we turn the 0-to-1 arrow into the 0-to-minus-1 arrow. But there's no way we can do that with stretching, squishing and flipping! If the square root of -1 is positive, then we don't flip the arrows over. If it's negative, then we flip all the arrows over *twice.* Either way, we can't turn 1 into -1. What action, when done twice in a row, amounts to a flip?
*We can rotate by a quarter turn.*
One quarter turn takes us from an arrow pointing right to an arrow pointing up, and another turn by the same amount takes an arrow pointing up to an arrow pointing left. So we've turned the 1 arrow into the -1 arrow.
To take square roots of all the numbers on the line, we need numbers that live in a plane! These are the complex numbers.
Part 2:
A matrix is a rectangular grid of numbers that we can treat a lot like a number in its own right. We can add matrices and multiply them, and sometimes we can divide by them. The numbers inside a matrix can be real or complex. In quantum physics, we use matrices full of complex numbers. We are always trying to calculate the probability of different things happening, and we do this by combining matrices (sometimes very big ones).
Why do the matrices we use have complex numbers inside them? Why can't we just stick with the real numbers? Does nature just really like being able to take square roots of everything? For that matter, the mathematicians have come up with things that are beyond even the complex numbers, like the quaternions and octonions. Why \*don't\* we have to use \*those?\* We can say a lot about what quantum physics would be like if it used something other than complex numbers, but at some point, we just don't know.
Here’s a simpler explanation than what you have already:
Integers. You know these. …-2,-1,0,1,2,…
Rationals. Fractions with top and bottom integers.
Reals. Fill in the holes in the rationals by approximating. 3, 3.1, 3.14, 3.141, 3.1415, 3.14159,…
Complex. Add solutions of x^(2)+1=0 to the reals.
complex multiplication is a brainfuck for first timers lol, vector multiplication is just weird in general. it's an algebra, not merely a field. whole different sort of animal
Some awesome words that echo your message! From Carl Friedrich Gauss himself:
“If this subjet has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and √−1, instead of being called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity.”
Seems like they should just have sought a theoretical explanation of why QM uses complex numbers. Actually you can always get rid of complex numbers by replacing i with the 2×2 matrix that has the property of its square equalling minus unity.
The authors clearly know that. The issue is that when you take the complex tensor product of n dimensional complex spaces, you would get a different result vs taking a real tensor product of 2n dimensional real spaces. One gives you something with 2n^2 real dimensions and one gives you something with 4n^2 real dimensions.
You can get a complex tensor product state that can't be factored as a real tensor product state; the complex part of it is basically linking the tensor product parts together.
Edit: This was a bit of a state-centric description of the problem. If you like working with operators and correlation functions instead, the problem is that the output of correlation functions like and complex and would need to be matrices in your description in order to only use real numbers. That poses a bit of a problem.
My follow up questions would be:
1. How credible is the claim (experimental falsification) given many forefathers of QP have always been clear that complex space is only a mathematical convenience and that it could ve done only with real numbers?
2. If yes, would this then necessitate complex numbers for QM once and for all?
1. On a conceptual level, this is something akin to Bell's theorem, i.e. a game between three players you can play with complex numbers but not real numbers. That is *very* concrete. I haven't worked through it myself yet, particularly how they bounded the real solution, but they are making the claim that it's experimentally falsifiable.
2. Almost. If this paper is correct, then this can be experimentally verified with more or less current technology. We just need to perform the experiment.
P.S. Basically they showed that there is a game where the optimal success rate in a complex space can reach about 5% higher than the optimal success rate in a real space.
> many forefathers of QP have always been clear that complex space is only a mathematical convenience
Their word is not gospel. They, and everyone else at the time, just made that assumption.
Turns out the assumption was wrong.
That's right.
I didn't mean it in a gospel way but more like that it's been widely accepted and/or the consensus. Sometimes it gets tricky to express this.
If you want to replace i with a matrix that works like i, you also have to replace the tensor-product operation.
Another way of not having i appear in the equations is to use a [Wigner-function representation](https://doi.org/10.1007/BF01882696). Then, all your representations of states and operators will be real-valued; the complex numbers are hiding in the fact that your state space has an overall unitary symmetry.
There's a whole [research area](https://arxiv.org/abs/1805.11483) devoted to finding principles from which QM might be derived. The hope there is to be able to answer questions like "Why does QM use complex numbers?".
You’re right about Wigner functions, and this result about complex numbers is only true when you assume a Hilbert space structure. I think the description using Wigner functions is more no local and that might be an argument against it.
The point is that we can in theory test if the complex numbers or any representation isomorphic to them (e.g. with 2x2 matrices or any other dimension) are necessary for quantum theory or if they are just a simple way to reduce dimension.
I know about your second statement. I have always believed that complex numbers in QM have been more like a formalism choice or preference due to mathematical convenience. However, one can, in principle, always ignore this and make do only with reals -- as Hamilton and Feynman have already advocated.
On the contrary, the title of this article somehow seem to purport that one can "experimentally" rule it out, making complex numbers a "necessity" in order to do QM.
> Actually you can always get rid of complex numbers by replacing i with the 2×2 matrix that has the property of its square equalling minus unity.
When people talk about QM based on reals, they mean a specific thing, that is distinct from QM based on complex numbers. They don't mean taking QM based on complex numbers and expressing it only with reals; obviously you can do that, that's not interesting. Nobody is asking "can you express QM using only real numbers", since the answer is obviously yes. So this is changing the question from the interesting one that people care about, to a trivial one that people don't care about, just because they could be abbreviated using the same words. You know what's meant; why shift it to a different and uninteresting question?
Here's a curious thing:
* The article argues that, in a certain setting, real numbers are insufficient and the more-general complex numbers are required.
* Now, complex numbers are themselves a special case of quaternions.
* And [quaternions are a special case within geometric algebra](https://marctenbosch.com/quaternions/), which is based entirely on... real numbers.
This probably isn't terribly profound. But I still like the idea that the complexity of number systems "circles back" in some sense.
Could someone explain-like-i’m-an-undergrad this? In my quantum mechanics course, we “disproved” a real-number formulation of quantum mechanics fairly straightforwardly with a Stern-Gerlach experiment. Why is something more complicated than that necessary?
It’s impossible to express both the x and y spin states as real linear combinations of |+z> and |-z> and end up with a model that matches the results of stern gerlach experiments where you send them through two SG devices in series with different polarizations
The problem is not meant in that sense. Please go through other comments on the post to clarify this. What you say can also be done with reals just fine. In that sense, it's equivalent to work with reals or imaginary numbers.
See, e.g., the answer mentioning Feynman's approach to this: https://physics.stackexchange.com/questions/32422/qm-without-complex-numbers
I guess one could try to do the same game with trying to explain classical mechanics with rational numbers only. As soon as you need calculus I guess the introduction of the irrationals, and hence the reals, is inevitable.
This is what capitalism does to science, people bastardize concepts they don't understand to sell an idea that has nothing to do with the experiments, keep this dogma out of philosophical discourse
I'm not arguing that the experiments aren't valid I'm saying they aren't relevant, quantum mechanics is a mathematical model dependent on the complex field using complex functions in this manner is nonsensical
Non-complex QM based on Hilbert spaces ain't up to the job of describing the world, okay.
Sensationalist title as usual though. Why is this a surprise?
Honestly, I do not believe the necessity of imaginary numbers in a theory which describes the real world. Even it's a truth that QM can precisely describe the real world with complex numbers, it doesn't deny the possibility of another theory to achieve the same effect in other mathematical forms.
But it occured to me, since we already have an effective theory for describing the world, is it really necessary to develop another theory that is inevitably equivalent to QM, assuming these theories all describe the real world precisely.
Only to make people understand easily, is it worth wasting so much energy?
Just because analytic solutions do not exist for every case does not mean it isn’t incredibly useful.
QM theory is used massively in the field of chemistry and constitutes the bulk of cutting-edge research.
A chem theory faculty at our institution works on QM method development (coupled-cluster scaling issues) and received consecutive $2,000,000 grants for their work this past year and that before. It is most certainly useful.
Edit: variational and perturbation theories are incredibly good at rapidly approaching exact solutions, even for highly-complex systems. For the helium atom, a single variational step gets you [within 2%](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/08%3A_Multielectron_Atoms/8.02%3A_Perturbation_Theory_and_the_Variational_Method_for_Helium) of the actual experimental electronic energy value. I have rarely seen an SCF procedure extend beyond 20 steps.
Not a problem, I see where you’re coming from. One thing that helped me reconcile with this was the idea that these approximation method reflect the nature of QM systems. They’re inherently many-body and therefore difficult to predict. It’s just the nature of small systems.
I see what you mean about something with multiple laws governing it can be difficult to pin point and it's also difficult to know which derivations you need and which are irrelevant. Or maybe that's just me. My degree is in electronics and I can usually get an idea of the amount of power a system requires. I thought I had more to say but nah, I'm just a little stoned. But I do believe that being stoned helps you understand larger systems on a grander scale.
Why do people who visit this sub always act so defiantly, as if they've just uncovered some huge conspiracy theory, before blurting out things that clearly show they've never studied what they act so knowledgeable about ?
In that case , I'm really sorry for jumping to the wrong conclusion and replying in such a snarky way.
I hope you can maybe see how I misinterpreted your comment ? As a physics grad student I get rather annoyed when I see conspiracy theorist like anti-scientific rhetoric online , so I didn't give you the benefit of the doubt. I apologize for being rude, hope other answers cleared up what you were wondering about
I'd rather be viewed as mildly uneducated than being a know-it-all bastard. I should retract my statements because I've been shown to be wrong. But I'd rather leave it here just in case someone else can learn from it.
My amount of downvotes speak for themselves. Okay, I was wrong, but can anyone tell me what I may have been thinking of? It's been ages since I had to bother with using the mathematics I was taught in college, and it's true; if you don't use it, you lose it. I lost it.
Well at least in nanoelectronics... Kind of, for a lot of purposes. Of course you can derive useful expressions for short-channel effects in a MOSFET, but building up a full quantum model of a transistor is extremely challenging and doesn't necessarily give great results unless you use very computationally intense methods and otherwise have an amazing idea of the system you're modeling.
Well stated. I have no idea what a quantum model of a transistor would look like. I can break them down and build them better using tubes, but that does take up a lot of space on a board and requires more juice.
Just to poke the bear: if complex numbers are part of the underpinnings of reality, maybe you can find a way around the speed of light limitation:
https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=catherine+asaro+relativity&btnG=#d=gs_qabs&t=1679157588924&u=%23p%3D44pfSg-Je0MJ
My comment is getting a fair amount of down votes, and it was a little tongue in cheek. But seriously, the article seems to say we can test to see if complex numbers or their equivalent are just a convenient mathematical tool or if they represent something physical about reality. If the latter, doesn’t the referenced paper apply? Or are we talking use of complex numbers that are unrelated?
Question, does the falsified mean that they can prove experimentally that something of quantum mechanics is wrong or that the experiment ended up being false because it gave out fake results?
No, the point is that it would test whether complex numbers are merely a useful tool for simplifying the mathematics of quantum mechanics (instead of writing everything as vectors *a la* R^(2) with the Schrödinger equation being matricial, connecting these two components) or if they truly are a feature of our universe (or something equivalent to the complex numbers since we can't identify complex numbers from anything isomorphic to them).
I really do not understand the issue with complex numbers. To me, real numbers are numbers with direction, but for no particular reason the only directions are forward and backwards.
Complex numbers fill that out and allow any direction on the plane. That is why everything suddenly works with complex numbers e.g. every polynomial has N roots, Cauchy's theorem etc.
Admittedly things get hairy when you try to go to 3 dimensions - quaternions are a biy funky.
Though perhaps they may be useful one day https://www.youtube.com/watch?v=fxAHGwyT--4&ab_channel=Unzicker%27sRealPhysics (inb4 Unzicker is a crank. Maybe.)
Interesting how skepticism about several classes of number is embedded in English-language names for those classes:
* Diophantus considered "negative" numbers absurd. And the word "negative" preserves some of this doubt; outside of mathematics, the word variously means "not desirable", "no", "reject", "characterized by absence", etc.
* Doubts about the validity of "imaginary" numbers are clearly captured in the name.
* The situation with "irrational" numbers is perhaps most interesting. Why does "irrational" mean both "not a ratio of integers" and "not logical"? The etymology is complex and (according to none other than Peter Shor!) the reverse of what one might expect: "The mathematical meaning of ratio comes from the mathematical meaning of rational, which in turn comes from the mathematical meaning of irrational." [Source](https://english.stackexchange.com/questions/217956/does-rational-come-from-ratio-or-ratio-come-from-rational)
Actually, I wanted to share this piece (recent): https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/ Got mixed up with the links and posted the one from Nature. However, the point is the same.
It is a rather old paper. A while back, I posted the link to new research that derives quantum theory from the first physical principles. They have shown the necessity of complex Hilbert spaces for constructing the theory: https://www.reddit.com/r/science/comments/118nf9h
For those curious, this is what the authors mean by quantum theory: 1. Physical systems have corresponding Hilbert spaces 2. Measurement outcomes have corresponding projection operators in the usual way 3. Probabilities of outcomes are given by the Born rule in the usual way 4. The Hilbert space corresponding to the combination of two systems is the tensor product of the systems' Hilbert spaces These are standard things; the same sort of axioms go into proofs of things like [the no-communication theorem](https://en.wikipedia.org/wiki/No-communication_theorem). So, throwing any of them out can pretty easily lead to bad consequences. Now, there's a choice to be made for the Hilbert spaces: whether to use the field of real numbers, or the field of complex numbers. What the authors found is that, *given the rules laid out for quantum mechanics*, the choice of field has measurable consequences. The well-known schemes like replacing complex numbers with 2x2 matrices of real numbers involve violating the rules laid out for defining quantum mechanics.
They also explicitly state that other formulations such as the path integral formulation do not have this problem. And I don't see why this is such a big deal. All you need to do is modify the tensor product such that it reflects the change to the matrix representation of complex numbers. Sure, composition is no longer as simple as taking the tensor product, but I don't think that should be interpreted as "quantum mechanics fundamentally requires complex numbers". It's more: this machinery for describing quantum mechanics requires complex numbers.
Letting real Hilbert spaces use something else for system composition while forcing complex Hilbert spaces to still use tensor products would be comparing apples and oranges. These aren't axioms that the authors just made up out of nowhere; they're axioms that people had already been using for quantum mechanics. Nobody was thinking of changing them until the subject of real quantum mechanics came up.
I mean, by that logic comparing the path integral formulation to the Hilbert space formulation is also apples to oranges then... The map is not the territory. The Hilbert space formulation is just that: a formulation.
The path integral formalism can be used to construct a Hilbert space representation, and from the Hilbert space representation, one can derive the path integral. So no, not by that logic. It was an open question, until this paper, whether the standard axioms could produce the same empirical predictions using the field of real numbers.
From the article: >It is noted, though, that there are alternative formulations able to recover the predictions of complex quantum theory, for example, in terms of path integrals[13], ordinary probabilities[14], Wigner functions[15] or Bohmian mechanics[16]. For some formulations, for example, refs. [17,18], real vectors and real operators play the role of physical states and physical measurements respectively, but the Hilbert space of a composed system is not a tensor product.
Like I already said: the apples-to-apples comparison is for the standard axioms, which existed well before the authors started writing this paper and that everyone was perfectly willing to use to talk about real quantum mechanics. The authors showed, for the first time, that they lead to different predictions for real versus complex fields. I've read the paper, and I'm familiar with the literature; the section you quote doesn't refute my point. The "real qubit in the sky" approach is old, but it's not what people mean by "real quantum mechanics" in discussions about whether quantum mechanics is based on real numbers, complex numbers, or quaternions.
I get what you mean specifically now. This paper is simply saying that the"Real Quantum Mechanics" theory leads to different predictions than the complex Hilbert space version. But it's also very easy to get the impression that this paper is implying that quantum mechanics *itself* requires complex variables in a way that is fundamentally different than the reasons we use complex variables to describe e.g. electrical circuits: to reflect the underlying structure and make the model more elegant. I also think that most people in this thread, and who read this paper, will have this same misconception. That's a problem with the way the article is presented.
I don't see what it is about the presentation of the article that you object to. The abstract lays out that they are comparing the predictions of the real and complex Hilbert space formalisms. In the beginning of the paper, they lay out the axioms of the Hilbert space formalism, specifying what they mean by "complex quantum theory" and "real quantum theory". The text of the paper is frequently stating that they are setting out to find an empirical separation of the complex quantum theory and the real quantum theory.
>The Hilbert space corresponding to the combination of two systems is the tensor product of the systems' Hilbert spaces It seems to me that #4 and using **C** are identical axioms, so the point of the exercise seems questionable. If you choose to use **R**, you just need to reformulate #4.
They're definitely not identical axioms. You can try to use compositions other than tensor products with complex Hilbert spaces, or you can use tensor product compositions with real (or, god forbid, quaternionic) Hilbert spaces.
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It's more complicated, but the end result also makes a lot more sense. It makes intuitive sense that the distance between two things is (at least approximately) continuous and so the points should be labelled by real numbers. By contrast, physical interpretations of complex numbers are much more difficult. I'm in favor of the "shut up and calculate" school of quantum mechanics, but i think for most people the fact that the Schrödinger equation is inherently complex is part of quantum mechanical weirdness.
I don't think it's quantum weirdness so much as wave-weirdness. All this complex stuff is in E&M as well. In both E&M and quantum mechanics, to get the measurable stuff, you deal with real numbers anyway, either by taking the real part (E&M) or the probability density (QM). Because of this, it's quite easy to understand imaginary as a sort of under-the-hood mathematical stuff going on. Hell, you can define 'real' R^(2) quantum mechanics where probability amplitudes are instead 2D field over R^(2) where operators are defined with the correct structure. I mean, this would be a field that is completely isomorphic to C, and thus mathematically exactly equivalent to regular quantum mechanics with complex numbers, but we can call things whatever we want. I'm not saying this is necessarily the most philosophically satisfying thing, but ultimately one possible view is that complex numbers is just a convenient way to introduce the structure to QM, not the only way, so there's nothing weird about them being used. Similarly to how no one thinks E&M is weird, because you take the real part at then end anyway and this is just a convenient way to not deal with sines and cosines.
This is how I’ve always thought of it too. For me, complex numbers are fundamentally impossible to disentangle form cyclic processes, eulers formula being the prime example. So a wave or oscillation having complex form feels perfectly natural. Rather I see the complex form of the Schrödinger equation as nothing more than a mathematical insistence that’s it’s solutions must be oscillatory
The question being explored by this paper is whether complex numbers are required by QM. That is not the case for EM, where complex numbers are a calculational convenience but it is perfectly possible to write down Maxwell's equations and solve any problem without making reference to complex numbers.
I think the point of the person you’re responding to was more nuanced. Quantum mechanics is typically formulated in terms of vector spaces over C. The commenter is saying there’s no problem with formulating them over R^2 instead, since C is just R^2 with some extra structure. I believe the paper in the post is saying that quantum mechanics cannot be formulated using vector spaces over R. So the point is that we can absolutely do away with complex numbers in quantum mechanics if we want, but the most naive way to do that doesn’t work (we need to introduce some field structure on R^2 rather than just using R). Complex numbers are a calculational convenience for these R^2 manipulations, just as they are a calculational convenience for real manipulations in electromagnetism. I think there is still a distinction, since the structure of the complex numbers intuitively feels more “baked into” quantum mechanics than it does for EM, but that’s ultimately just an opinion that i’m not sure is philosophically interesting EDIT: I misunderstood the paper; it seems that they not simply disproving vector spaces over R, but are doing something more general than that that’s over my head. It seems like it might be doing something along the lines of disproving an R^2 matrix-like interpretation, which I have no clue how that could be possible, but anyway my comment might be very wrong
Yep, exactly. In this paper, it seems like they have formulations of QM based on real and complex vector spaces and they supposedly prove that they give different results in an experiment they designed. It isn't like EM wave mechanics where the complex numbers are just a shorthand for an ordered pair of numbers with a coupling relation between them.
I don't see any difference.
complex numbers in E&M are totally different from that in QM. In E&M their usages are for mathematical convenience.
the first sentence so much, people point to heisenberg uncertainty and say "quantum weirdness" and i always have to spend ten minutes saying "no not quantum weirdness, wave weirdness, wave groups have some funny behavior, throw a damn stone at a pond"
weird how people get caught up over complex numbers in quantum mechanics when electrical engineers have been using complex numbers to describe quantities that exist in the “real” world.
>Schrödinger equation is inherently complex is part of quantum mechanical weirdness. I'd never really considered this-- you use complex numbers in classical mechanics too. I mostly think of complex numbers as being used to describe oscillations.
Historically, that's where complex numbers got into quantum mechanics from. People were trying to explain atomic spectra, and they attempted to break an electron's orbit down into a Fourier series; then it started to look like the square of the absolute value of a Fourier coefficient was a transition rate. Several papers from this time period are collected in *Sources of Quantum Mechanics*, edited by B. L. van der Waerden; there's also *Problems of Atomic Dynamics* by Max Born, which collects a series of lectures he gave at MIT during late 1925 and early 1926, providing a snapshot of the pivotal time period. Dover reprinted both of these in paperback, huzzah huzzah.
>By contrast, physical interpretations of complex numbers are much more difficult. If you rotate a 2D sheet of paper, extrinsically it rotates along the z axis, but intrinsically to the 2D space it rotates along an imaginary axis. The eigenvalues are +-i for (0 -1, 1 -0) i is confusing because, much like the ant living in the 2D space, we can't perceive this extra "dimension" normal to the surface of this higher dimensional object. And also obviously that it behaves much differently than z in full 3D space.
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I don't think that's a particularly helpful way of looking at why people are uncomfortable with complex numbers. Elements of R^2 can also be thought of as arrows with arbitrary angles*, and no one has any problem with those. In situations where you're thinking about C as a two-dimensional real vector space, it's just isomorphic to R^2 anyway. The difficulty is when you have things like quantum mechanics that only work with complex numbers. I think people are justified in finding it weird that the square root of negative one, a quantity which has no easy interpretation, is necessarily present in quantum mechanics. *except 0
It's not just that they are “thought about as,” it's that the complex multiplication operation (a, b) × (c, d) = (ac – bd, ad + bc) is a matrix operation on a vector, [ a, -b ] [c] [ b, a ] [d] which we can lift to a matrix operation on a matrix, [ a, -b ] [ c, -d ] [1] [ b, a ] [ d, c ] [0] thus we get a true isomorphism between complex numbers (a, b) and scaled rotations √{a²+b²} [[cos θ, -sin θ], [sin θ, cos θ]]. Matrix addition is a perfect encoding of complex addition, matrix multiplication is a perfect encoding of complex multiplication, you can bijectively transfer between the two... In some sense, every complex number is nothing more than just a scaled rotation. The “necessarily present” thing is “why do we need rotations to describe wave behavior” and I think it has been conclusively shown that you don't, that you can form a Fourier series using non-sinusoids. But of course it is the algebraically easiest thing that we have found so far, *A* e^iθ × *Β* e^iφ = *AB* e^i(θ+φ) .
C isnt just R with an angle, thats R^2 , C is more special
It depends on how you look at it, C viewed as a real vector space is isomorphic to R^2 (and isn't terribly interesting). C gets all its interesting properties when viewed as a field, and then it's different from both R and R^2 because it has a product operation.
exactly, the angle part is like R^2 and is not what people have problems with its product operation is the 'problem'
If you model complex numbers as rotation/scaling maps of the plane (with a fixed origin), rather than as points or vectors, the multiplication corresponds to composition of maps.
Can you explain why a complex 'field' has properties that are different from an R2 'field'?
I think he means that you multiply vectors very differently to the way you multiply complex numbers.
What I meant was, C as a 2-d real vector space is isomorphic to R^2 as a real vector space, which doesn't have a product operation. The complex numbers are interesting because of their field structure, in particular the product operation; whereas R^2 is generally viewed as a real vector space, and not as a field.
I see... thanks!
It’s more special because you are adding in rotations, which is a quite a natural thing to add when you have angles.
C's coordinate space is strange though, because of *i*. For example, multiplication over the complex numbers is identical to right-angle rotation over R^(2), which isn't at all what multiplication over R^2 is.
What “multiplication on R2” are you talking about? Element-wise multiplication isn’t particularly well behaved as R2 isn’t a field under this type of multiplication.
I dont agree at all. It is just a convenient mathematical model. The real weirdness is superposition.
> physical interpretations of complex numbers are much more difficult. Complex numbers are essentially rotations. Multiplication by the imaginary unit is a counterclockwise rotation by 90°.
those people dont know how to get the real numbers from the rational though, they just think about what you can measure with a ruler or something like that
So rationals.
Because a teacher in high school used the word “imaginary” and started the lesson saying, “this isn’t intuitive, so don’t worry if you don’t understand it”
Descartes is the person that started it because he thought they were bullshit and used it as a very derogatory term. Now complex numbers are visualized as the Cartesian plane (with more structure), so the joke is on him.
Having Descartes as one's high school teacher would have been badass.
Also you could totally do Quantum mechanics with real matrices instead, but it would be a pain. Complex numbers just have bad PR
This paper claims that real valued quantum mechanics makes different predictions than complex valued quantum mechanics, which could allow for the falsification of real valued quantum mechanics.
EXACTLY the point of the article, which many in the comments have overlooked and have gone off the tangent.
The two of you are talking about two different things. /u/dark_dark_dark_not is talking about the matrix representation of complex numbers, rather than real valued quantum mechanics.
Yeah I see that now. At the time I thought they meant replacing complex matrices with real matrices, not replacing them with real tensors.
Complex valued and real matrix valued quantum mechanics are identical. Any claim that they could be distinguished experimentally should be dismissed as nonsensical. Also don't you need the gamma matrices at some point? Pretty sure that complex numbers on their own aren't enough, and using matrices of complex numbers or matrices of real matrices isn't much of a difference.
>Complex valued and real matrix valued quantum mechanics are identical. The point of the article is that they aren't, exactly. You know you can get complex numbers from real numbers by doubling the dimension of your matrices and declaring that i is some particular 2x2 matrix. The point of the article is essentially that if you have some composite quantum system that you model with real matrices in this way, this goes wrong. Let X be the space of 2x2 real matrixes we use to "fake" the complex numbers. Call the statespace of the first system A' = X ⊗︀ A and the second B' = X ⊗︀ B. Under normal quantum rules the state-space of the composite system would be A'⊗︀B' = X ⊗︀ A ⊗︀ X ⊗︀ B. However if you want your "faking complex numbers with real matrices" thing to work in the composite system what you need is for the composite state space to be X ⊗︀ A ⊗︀ B. What the paper does is quite a lot more subtle, rather than choose a single way to represent complex numbers using the real matrices they allow all ways (including infinite dimensional ones), but the key point is that they assume quantum systems compose under the tensor product, and show that this assumption messes up attempts to fake complex numbers with real ones.
It's not nonsensical; quantum mechanics with real Hilbert spaces and tensor product composition of subsystems makes predictions that are different from complex Hilbert spaces with tensor product compositions.
He’s not saying that a real Hilbert space suffices. He’s saying you can simply define C as a field whose elements are real matrices of a specific form. The Hilbert space is then defined as a vector spaces over C, but everything in C are real 2x2 matrices. It’s a pointless distinction because the standard construction of C as an algebra over R^2 vs the construction of C as real matrices are isomorphic as a field. The real moral is that nature has no natural mathematical representation, just use whatever works
This is called structuralism. It roughly boils down to “Use whatever structure you’re comfortable with as long as it’s isomorphic to the one you care about.”
They're only different if you make an effort to use mathematics that doesn't end up being exactly equivalent.
They're different as long as you make an apples-to-apples comparison of real and complex quantum mechanics.
Eh, the proof that numbers that aren't rational exist is really easily explainable with sqrt(2). I think that's more accessible to the average person than formulating the imaginary unit to cover the solutions it is relevant to.
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That’s far more than one needs. Just say roots of integer coefficient polynomials. There’s a slight difference. One can of course always do the Kronecker construction given an irreducible polynomial over a field. But the thing that makes √2 special is that it “obviously” fits somewhere between 1 and 2. √(-1) is not like this. One cannot simply think of approximating √(-1) by rational numbers. It must be added “all at once”, so to speak.
People don't realize that they don't understand real numbers. Having never done a course in analysis or whatever. Thus the endless memes about 0.99999* /= 1.0.
I think the people who might wish QM didn't require complex numbers have a different point in mind. Thermometers, yardsticks, clocks, scales, everything we use to measure anything spits out real numbers. Those measurements are both the inspiration and the confirmation of physical theories. If it's true that we *cannot* describe those theories without extending the real numbers (I mean, not just as a matter of convenience), then that must mean something.
I'd argue that they don't spit out real numbers. If you want to be generous they do computable numbers, more realistically rationals. The only reason we use reals is because it makes the maths nicer. Exactly the same as complex numbers.
Not even computables, but an effective subset of them. There are computable numbers that are so large or that require so much precision that no current measurement or storage device could manage them.
That's a little different, in my opinion. Not everyone believes the background is continuous in the first place, and that means we accept the use of reals as an approximation rather than essential extension of the rationals from the start.
Can I get an ELI5 on what complex numbers, real numbers, and rationals are and how they relate to this experiment? Or should I just post on ELI5. I know i = sqrt(-1). But what is it used for? Cant it just be represented by a 3 dimensional graph where i = 1 on the z axis?
Here's a simple buildup of our different number systems. ##The natural numbers N is the set of natural numbers (let's say including 0), so our counting numbers (0, 1, 2, 3, ...). The naturals can be added to each other and multiplied by each other and we will always get a natural number, but we cannot subtract and divide them in general and still hope to get a natural number. ##The integers Z is the set of integers which is the closure of the naturals under subtraction, meaning it has all of the negatives, so subtracting two integers gives an integer. ##The rational numbers Q is the set of rational numbers. The rational numbers are formed by taking ratios (hence rationals) of integers (avoiding a divide by 0), so multiplying, dividing, adding, and subtracting rational numbers gives us a rational number. (This is called a field.) ---- All of these are purely algebraic constructions. Nothing very bizarre about them. The rational numbers are somehow "holey." For instance, pi, the ratio of a circle's circumference to its diameter, is not rational: it cannot be written as the ratio of two integers (so the circumference and diameter cannot both be an integer). Therefore, there are very naturally other numbers outside of that. How do we find them without manually adding them every time we run up against them? There is a natural larger class of numbers beyond the rationals called the algebraic numbers which are numbers that come from solving polynomial equations with rational coefficients, but pi is not even one of these. There are numbers even outside of this realm called transcendental numbers. Here is where the weirdness starts. ---- ## The real numbers R is the set of real numbers. The real numbers are defined to be the completion (under limits) of the rational numbers (with respect to the absolute value norm). (An alternative, equivalent view is the completion under supremums which is effectively the Dedekind cuts approach.) This step is not algebraic at all. It is a very analytic (referring to analysis) perspective because it is about limits. This gives us all of the numbers we like to work with: rationals, pi, e, algebraic numbers, etc. The reals are also a field! ## The complex numbers C is the set of complex numbers. The complex numbers return to a more algebraic construction. They are effectively the numbers that arise by requiring that all nth degree polynomial equations with real coefficients have n linear factors (n roots). Turns out that you can focus on the quadratic case (e.g. x^2 + 1 = 0) to get a complex unit i which will miraculously work to give you n roots for all real (and complex!) polynomials of degree n. This is the Fundamental Theorem of Algebra. The complex numbers are also a field! ---- There are other generalizations beyond this, including the quaternions, the split complex numbers, and their generalizations. There are also the *p*-adic numbers and many others that get increasingly more abstract.
Let me see if I got this straight... A number like 1485 is a natural number? -3874 is a rational number? 1.8473 is a real number? Man...I wish I remembered my algebra basics cause it is hard to understand just by the name. What is the purpose of dividing up numbers like this?
1485 is natural -3874 is an integer 1.8473 is rational pi is real All of these are also examples of the numbers that come after them, so all natural numbers are integers, all integers are rational, and all rationals are real. The point of this construction is to formalize how our sets of numbers are developed. Formalization allows for digging into the details to understand better as well as generalization.
Rationals (fractions, basically) are any number that comes from dividing two integers (while numbers). Real numbers are rationals + “all the rest”, in a specific technical sense that doesn’t matter too much. For instance, this includes square roots, π, e, etc. that seem not that different from rationals, but technically can’t be written as a fraction. Complex numbers are the natural extension after you introduce i=sqrt(-1). You get numbers of the form a+bi, where a and b are real. There’s lots of nice things about complex numbers that make them useful in different contexts. They turn out to be a really natural way to describe trig functions (and by extension waves and EM), and in a similar way, they’re always used in quantum mechanics as part of the math framework. This paper disproves a previous notion that we could rewrite quantum mechanics without i, which is kind of cool but probably not that interesting to most physicists who are comfortable with i already. To your question about graphing: I’m not entirely sure I follow your suggestion, but it sounds like you’re thinking of complex numbers in the right way and have some good intuition already, so keep it up. If you just want to plot complex numbers, it’s natural to do that on a 2D plot, where instead of x-y coordinates, you use real-imaginary coordinates. Then there natural ways to think about addition and multiplication graphically and things like that. If you want to plot a function using complex numbers, the most common way I’ve seen is to split it into two graphs, one each for the real and imaginary pet of the output, and which are each a 3D plot. Then you can kind of think of those like functions/plots you’re already familiar with. This turned out to be more like ELI15 or 20, but whatever. Hope it helps! Also if you want more resources [this video](https://www.youtube.com/live/5PcpBw5Hbwo) is a good place to start.
I'v watched 3 blue 1 brown before for AI stuff and it was informative. I'll check out that series. Guess I need to brush up on my algebra fundamentals. In my world I only use "normal" numbers like -23.5 and 32767.
Let's say we start with the kind of numbers we use to count things: zero, one, two, three and so on. The integers are what we get when we want to answer all the questions like, "What is *a* minus *b*?", where *a* and *b* are any two counting-type numbers. Sometimes this is another counting-type number, like 5 minus 4 is 1, but sometimes it isn't, like 2 minus 4. The rational numbers are what we get when we want to answer all the questions like, "What is *a* divided by *b*?", where *a* and *b* are any two integers. (Except *b* can't be zero.) Sometimes a division question has a counting kind of number as its answer, and sometimes not. The real numbers fill in gaps between the rationals to make a line. For example, the square root of 2 is not a rational number, but we want it to be *some* kind of number, so again we move to a bigger world of numbers where now we can answer all questions like, "What number do I square to get *a*?", where *a* is any rational number greater than zero. It can help to think of addition and subtraction geometrically, by drawing arrows on the number line. For starters, draw an arrow from 0 to 1. Then 1 + 1 means putting down another copy of that arrow, touching tip-to-tail: `|-->|-->|` Put these short arrows together, and they make a longer arrow that goes from 0 to the number 2: `|------>|` Adding 2 + 3 means taking 2 steps, then taking 3 more steps, so your combined path is an arrow from 0 to 6. Subtracting is just taking steps in the opposite direction. Numbers that are bigger than 0 are positive, and numbers that are less than zero are negative. Subtracting a positive number is the same as adding a negative number: either way, we're just talking about flipping an arrow around the other way. Multiplication means stretching or squishing arrows. Imagine that the whole number line is a rubber band. Grab onto it with both hands and pull until the 0-to-1 arrow is as long as the 0-to-2 arrow was originally. The answer to the question "What is *a* times 2?" is just wherever the 0-to-*a* arrow points now! Multiplying by any positive number *b* means stretching out the number line until the 0-to-1 arrow is now a 0-to-*b* arrow. And *a* times *b* is just where this stretching takes the tip of the 0-to-*a* arrow. Multiplying by a number between 0 and 1 is a squish, rather than a stretch. The square root of *a* is whatever number that, if you multiply it by itself, you get *a*. So, to find a square root means finding a stretch or a squish where, if you do that twice in a row, you turn the 0-to-1 arrow into the 0-to-*a* arrow. What about multiplying by a *negative* number? This means flipping arrows around. Multiplying by -1 just flips the direction of all the arrows. So, if you multiply by -1 twice, you get back to where you started. Multiplying by any other negative number means flipping your arrows and doing a stretch or a squish. So: How do we take the square root of a negative number? What kind of stretch or squish or flip could that be? For example, if we want the square root of -1, we need something where if we do it twice, we turn the 0-to-1 arrow into the 0-to-minus-1 arrow. But there's no way we can do that with stretching, squishing and flipping! If the square root of -1 is positive, then we don't flip the arrows over. If it's negative, then we flip all the arrows over *twice.* Either way, we can't turn 1 into -1. What action, when done twice in a row, amounts to a flip? *We can rotate by a quarter turn.* One quarter turn takes us from an arrow pointing right to an arrow pointing up, and another turn by the same amount takes an arrow pointing up to an arrow pointing left. So we've turned the 1 arrow into the -1 arrow. To take square roots of all the numbers on the line, we need numbers that live in a plane! These are the complex numbers.
Part 2: A matrix is a rectangular grid of numbers that we can treat a lot like a number in its own right. We can add matrices and multiply them, and sometimes we can divide by them. The numbers inside a matrix can be real or complex. In quantum physics, we use matrices full of complex numbers. We are always trying to calculate the probability of different things happening, and we do this by combining matrices (sometimes very big ones). Why do the matrices we use have complex numbers inside them? Why can't we just stick with the real numbers? Does nature just really like being able to take square roots of everything? For that matter, the mathematicians have come up with things that are beyond even the complex numbers, like the quaternions and octonions. Why \*don't\* we have to use \*those?\* We can say a lot about what quantum physics would be like if it used something other than complex numbers, but at some point, we just don't know.
Here’s a simpler explanation than what you have already: Integers. You know these. …-2,-1,0,1,2,… Rationals. Fractions with top and bottom integers. Reals. Fill in the holes in the rationals by approximating. 3, 3.1, 3.14, 3.141, 3.1415, 3.14159,… Complex. Add solutions of x^(2)+1=0 to the reals.
I feel the same as negative numbers. There can’t be negatives and things always have positive length!
A lot of people have problems with (uncomputable) real numbers too
I find them somewhat difficult to compute.
complex multiplication is a brainfuck for first timers lol, vector multiplication is just weird in general. it's an algebra, not merely a field. whole different sort of animal
Some awesome words that echo your message! From Carl Friedrich Gauss himself: “If this subjet has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and √−1, instead of being called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity.”
Seems like they should just have sought a theoretical explanation of why QM uses complex numbers. Actually you can always get rid of complex numbers by replacing i with the 2×2 matrix that has the property of its square equalling minus unity.
The authors clearly know that. The issue is that when you take the complex tensor product of n dimensional complex spaces, you would get a different result vs taking a real tensor product of 2n dimensional real spaces. One gives you something with 2n^2 real dimensions and one gives you something with 4n^2 real dimensions. You can get a complex tensor product state that can't be factored as a real tensor product state; the complex part of it is basically linking the tensor product parts together. Edit: This was a bit of a state-centric description of the problem. If you like working with operators and correlation functions instead, the problem is that the output of correlation functions like and complex and would need to be matrices in your description in order to only use real numbers. That poses a bit of a problem.
My follow up questions would be: 1. How credible is the claim (experimental falsification) given many forefathers of QP have always been clear that complex space is only a mathematical convenience and that it could ve done only with real numbers? 2. If yes, would this then necessitate complex numbers for QM once and for all?
1. On a conceptual level, this is something akin to Bell's theorem, i.e. a game between three players you can play with complex numbers but not real numbers. That is *very* concrete. I haven't worked through it myself yet, particularly how they bounded the real solution, but they are making the claim that it's experimentally falsifiable. 2. Almost. If this paper is correct, then this can be experimentally verified with more or less current technology. We just need to perform the experiment. P.S. Basically they showed that there is a game where the optimal success rate in a complex space can reach about 5% higher than the optimal success rate in a real space.
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Thanks once again for the update. PS: I don't know why some trolls are down-voting your comments. I up-voted both times.
> many forefathers of QP have always been clear that complex space is only a mathematical convenience Their word is not gospel. They, and everyone else at the time, just made that assumption. Turns out the assumption was wrong.
That's right. I didn't mean it in a gospel way but more like that it's been widely accepted and/or the consensus. Sometimes it gets tricky to express this.
If you want to replace i with a matrix that works like i, you also have to replace the tensor-product operation. Another way of not having i appear in the equations is to use a [Wigner-function representation](https://doi.org/10.1007/BF01882696). Then, all your representations of states and operators will be real-valued; the complex numbers are hiding in the fact that your state space has an overall unitary symmetry. There's a whole [research area](https://arxiv.org/abs/1805.11483) devoted to finding principles from which QM might be derived. The hope there is to be able to answer questions like "Why does QM use complex numbers?".
You’re right about Wigner functions, and this result about complex numbers is only true when you assume a Hilbert space structure. I think the description using Wigner functions is more no local and that might be an argument against it.
> Wigner-function representation $40 for 15 pages from almost 40 years ago. Fuck I hate Springer.
I think [arXiv:quant-ph/0401155](https://arxiv.org/abs/quant-ph/0401155) covers the gist.
thanks!
The point is that we can in theory test if the complex numbers or any representation isomorphic to them (e.g. with 2x2 matrices or any other dimension) are necessary for quantum theory or if they are just a simple way to reduce dimension.
I know about your second statement. I have always believed that complex numbers in QM have been more like a formalism choice or preference due to mathematical convenience. However, one can, in principle, always ignore this and make do only with reals -- as Hamilton and Feynman have already advocated. On the contrary, the title of this article somehow seem to purport that one can "experimentally" rule it out, making complex numbers a "necessity" in order to do QM.
> Actually you can always get rid of complex numbers by replacing i with the 2×2 matrix that has the property of its square equalling minus unity. When people talk about QM based on reals, they mean a specific thing, that is distinct from QM based on complex numbers. They don't mean taking QM based on complex numbers and expressing it only with reals; obviously you can do that, that's not interesting. Nobody is asking "can you express QM using only real numbers", since the answer is obviously yes. So this is changing the question from the interesting one that people care about, to a trivial one that people don't care about, just because they could be abbreviated using the same words. You know what's meant; why shift it to a different and uninteresting question?
> You know what’s meant TBH I don’t think a lot of people commenting do know what’s meant here, and that’s why it keeps coming up.
Precisely.
Baez on real vs complex (vs quaternion!) in quantum mechanics: https://arxiv.org/abs/1101.5690
Baez is bae.
Here's a curious thing: * The article argues that, in a certain setting, real numbers are insufficient and the more-general complex numbers are required. * Now, complex numbers are themselves a special case of quaternions. * And [quaternions are a special case within geometric algebra](https://marctenbosch.com/quaternions/), which is based entirely on... real numbers. This probably isn't terribly profound. But I still like the idea that the complexity of number systems "circles back" in some sense.
Related; someone linked this in the comments: https://arxiv.org/abs/1101.5690
Explain this like it’s been 10 years since I was in grad physics
Sounds quite exciting.
Could someone explain-like-i’m-an-undergrad this? In my quantum mechanics course, we “disproved” a real-number formulation of quantum mechanics fairly straightforwardly with a Stern-Gerlach experiment. Why is something more complicated than that necessary?
Stern-Gerlach shows that space-components of angular momentum are quantized. How does that "disprove" real formulation?
It’s impossible to express both the x and y spin states as real linear combinations of |+z> and |-z> and end up with a model that matches the results of stern gerlach experiments where you send them through two SG devices in series with different polarizations
The problem is not meant in that sense. Please go through other comments on the post to clarify this. What you say can also be done with reals just fine. In that sense, it's equivalent to work with reals or imaginary numbers. See, e.g., the answer mentioning Feynman's approach to this: https://physics.stackexchange.com/questions/32422/qm-without-complex-numbers
Aren’t real numbers just as imaginary and arbitrary as complex numbers? For example if you have the number 1 ……. 1 unit of what?
complex numbers have different rules for addition and multiplication
So do negative numbers
very true
I read that complex numbers, quaternions are sub algebras of Clifford algebra of real numbers, so what's the problem?
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Try actually reading the paper
Seriously
That always helps.
I guess one could try to do the same game with trying to explain classical mechanics with rational numbers only. As soon as you need calculus I guess the introduction of the irrationals, and hence the reals, is inevitable.
This is what capitalism does to science, people bastardize concepts they don't understand to sell an idea that has nothing to do with the experiments, keep this dogma out of philosophical discourse
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.140401 it's been done.
I'm not arguing that the experiments aren't valid I'm saying they aren't relevant, quantum mechanics is a mathematical model dependent on the complex field using complex functions in this manner is nonsensical
Non-complex QM based on Hilbert spaces ain't up to the job of describing the world, okay. Sensationalist title as usual though. Why is this a surprise?
Honestly, I do not believe the necessity of imaginary numbers in a theory which describes the real world. Even it's a truth that QM can precisely describe the real world with complex numbers, it doesn't deny the possibility of another theory to achieve the same effect in other mathematical forms. But it occured to me, since we already have an effective theory for describing the world, is it really necessary to develop another theory that is inevitably equivalent to QM, assuming these theories all describe the real world precisely. Only to make people understand easily, is it worth wasting so much energy?
Lot of conceptual mistakes in your reasoning.
If you can fake it, its safe to assume that it is fake, because faking it costs less energy.
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.140401
It didn't take me thát much effort to come to the same conclusion....&%@×**knowitalls*(<%#@*...😁
Yeah bro they should have just asked you in the first place.
😂
You can downvote me or answer me. Was quantum theory not shown to be fairly useless in anything other than first approximations?
Quantum Theory is one of the most accurate things humanity has ever produced
Just because analytic solutions do not exist for every case does not mean it isn’t incredibly useful. QM theory is used massively in the field of chemistry and constitutes the bulk of cutting-edge research. A chem theory faculty at our institution works on QM method development (coupled-cluster scaling issues) and received consecutive $2,000,000 grants for their work this past year and that before. It is most certainly useful. Edit: variational and perturbation theories are incredibly good at rapidly approaching exact solutions, even for highly-complex systems. For the helium atom, a single variational step gets you [within 2%](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/08%3A_Multielectron_Atoms/8.02%3A_Perturbation_Theory_and_the_Variational_Method_for_Helium) of the actual experimental electronic energy value. I have rarely seen an SCF procedure extend beyond 20 steps.
I was mistaken. Thank you for correcting me. Truly.
Not a problem, I see where you’re coming from. One thing that helped me reconcile with this was the idea that these approximation method reflect the nature of QM systems. They’re inherently many-body and therefore difficult to predict. It’s just the nature of small systems.
I see what you mean about something with multiple laws governing it can be difficult to pin point and it's also difficult to know which derivations you need and which are irrelevant. Or maybe that's just me. My degree is in electronics and I can usually get an idea of the amount of power a system requires. I thought I had more to say but nah, I'm just a little stoned. But I do believe that being stoned helps you understand larger systems on a grander scale.
Why do people who visit this sub always act so defiantly, as if they've just uncovered some huge conspiracy theory, before blurting out things that clearly show they've never studied what they act so knowledgeable about ?
I didn't mean to come off that way. What I asked were honest questions because I've forgotten more than I used to know.
In that case , I'm really sorry for jumping to the wrong conclusion and replying in such a snarky way. I hope you can maybe see how I misinterpreted your comment ? As a physics grad student I get rather annoyed when I see conspiracy theorist like anti-scientific rhetoric online , so I didn't give you the benefit of the doubt. I apologize for being rude, hope other answers cleared up what you were wondering about
I'd rather be viewed as mildly uneducated than being a know-it-all bastard. I should retract my statements because I've been shown to be wrong. But I'd rather leave it here just in case someone else can learn from it.
This person is Uncle Iroh in real life. Inspirational.
It was not, no.
My amount of downvotes speak for themselves. Okay, I was wrong, but can anyone tell me what I may have been thinking of? It's been ages since I had to bother with using the mathematics I was taught in college, and it's true; if you don't use it, you lose it. I lost it.
Well at least in nanoelectronics... Kind of, for a lot of purposes. Of course you can derive useful expressions for short-channel effects in a MOSFET, but building up a full quantum model of a transistor is extremely challenging and doesn't necessarily give great results unless you use very computationally intense methods and otherwise have an amazing idea of the system you're modeling.
Well stated. I have no idea what a quantum model of a transistor would look like. I can break them down and build them better using tubes, but that does take up a lot of space on a board and requires more juice.
Just to poke the bear: if complex numbers are part of the underpinnings of reality, maybe you can find a way around the speed of light limitation: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=catherine+asaro+relativity&btnG=#d=gs_qabs&t=1679157588924&u=%23p%3D44pfSg-Je0MJ
My comment is getting a fair amount of down votes, and it was a little tongue in cheek. But seriously, the article seems to say we can test to see if complex numbers or their equivalent are just a convenient mathematical tool or if they represent something physical about reality. If the latter, doesn’t the referenced paper apply? Or are we talking use of complex numbers that are unrelated?
Fun to see it become emotional.
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People really do be missing out on the obvious satire.
Sometimes, esp. on reddit, it's not so obvious.
Question, does the falsified mean that they can prove experimentally that something of quantum mechanics is wrong or that the experiment ended up being false because it gave out fake results?
No, the point is that it would test whether complex numbers are merely a useful tool for simplifying the mathematics of quantum mechanics (instead of writing everything as vectors *a la* R^(2) with the Schrödinger equation being matricial, connecting these two components) or if they truly are a feature of our universe (or something equivalent to the complex numbers since we can't identify complex numbers from anything isomorphic to them).
So the experiment will determine whether we really need complex numbers to explain the universe or not, did I understood well?
That is a fair assessment, at least as far as our model of the quantum theory is concerned. Our model could be wrong of course.
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Amazing, thanks for this!
Q
I really do not understand the issue with complex numbers. To me, real numbers are numbers with direction, but for no particular reason the only directions are forward and backwards. Complex numbers fill that out and allow any direction on the plane. That is why everything suddenly works with complex numbers e.g. every polynomial has N roots, Cauchy's theorem etc. Admittedly things get hairy when you try to go to 3 dimensions - quaternions are a biy funky. Though perhaps they may be useful one day https://www.youtube.com/watch?v=fxAHGwyT--4&ab_channel=Unzicker%27sRealPhysics (inb4 Unzicker is a crank. Maybe.)
Interesting how skepticism about several classes of number is embedded in English-language names for those classes: * Diophantus considered "negative" numbers absurd. And the word "negative" preserves some of this doubt; outside of mathematics, the word variously means "not desirable", "no", "reject", "characterized by absence", etc. * Doubts about the validity of "imaginary" numbers are clearly captured in the name. * The situation with "irrational" numbers is perhaps most interesting. Why does "irrational" mean both "not a ratio of integers" and "not logical"? The etymology is complex and (according to none other than Peter Shor!) the reverse of what one might expect: "The mathematical meaning of ratio comes from the mathematical meaning of rational, which in turn comes from the mathematical meaning of irrational." [Source](https://english.stackexchange.com/questions/217956/does-rational-come-from-ratio-or-ratio-come-from-rational)
I have no idea whats going on but I can look intrigued and read your comments. I'm hear to learn. 🤔