Imaginary Numbers May Be Essential for Describing Reality(quantamagazine.org)
quantamagazine.org
Imaginary Numbers May Be Essential for Describing Reality
https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/
123 comments
One could argue that all numbers are an abstraction; they don't "exist" in any physical form, even integers. Therefore, why is it such a surprise that complex numbers are sometimes also needed to model reality?
I tend to agree with you, despite the famous quotation: "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man")
Imagine going back to Kronecker and their contemporaries and telling them that a spinning top weighs more than a top at rest. They would have laughed you out of the room, but a few years later the arrogant scientific confidence of the 19th century was completely crushed by quantum mechanics and general relativity.
Science is not a body of knowledge, it is a frontier of ignorance. I like to "feel" that complex numbers are part of the fabric of the universe, if only because it would be more beautiful.
"Clouds are not spheres, Mountains are not cones" - Mandelbrot
Science is not a body of knowledge, it is a frontier of ignorance. I like to "feel" that complex numbers are part of the fabric of the universe, if only because it would be more beautiful.
"Clouds are not spheres, Mountains are not cones" - Mandelbrot
A spinning top weighs more than a top at rest? I get that mass increases with speed, but that works in a rotating system as well as a linear system?
At any given instant, velocity is linear. That is, if you take a snapshot of the top, you can draw a straight vector for each particle of the top, indicating the direction of velocity. And, if the top suddenly flew apart, each piece would continue in a straight line, following its velocity at that instant.
Can velocity exist in a snapshot?
Or rephrased: is instantaneous velocity real? Are limits real, or just a mathematical invention?
Well, I think instantaneous velocity is real, and I think I have a convincing argument. Imagine you have a piece of gum stuck to your bike tire. As your tire spins, the piece of gum spins with it, and its velocity is constantly changing. But when the piece of gum flies off your tire, it flies off in a straight line, following the instantaneous velocity at the moment of separation. Seeing as the gum can fly off in a straight line, its instantaneous velocity at that moment must be well-defined.
Well, I think instantaneous velocity is real, and I think I have a convincing argument. Imagine you have a piece of gum stuck to your bike tire. As your tire spins, the piece of gum spins with it, and its velocity is constantly changing. But when the piece of gum flies off your tire, it flies off in a straight line, following the instantaneous velocity at the moment of separation. Seeing as the gum can fly off in a straight line, its instantaneous velocity at that moment must be well-defined.
Both phenomena are a consequence of the equivalence between mass and energy. So since a spinning top has a higher energy than one at rest, it will also have a higher mass. Another example is that an excited atom has a higher mass than the same atom in its ground state.
Relativistic mass is not a well defined abstraction.
My hot take is that this is why it doesn't matter if there are vaccine deniers or flat earthers. The facts speak for themselves and inevitably they'll be wrong again just as the people before them.
Are there any integers in the physical world? To me they all seem in my head.
If I have 3 match sticks, they are each made of billions of molecules, atoms, electrons, quarks, gluons, etc. So there is no integer count, just in my perception.
If I have 3 match sticks, they are each made of billions of molecules, atoms, electrons, quarks, gluons, etc. So there is no integer count, just in my perception.
Perhaps, you would appreciate a similar question. Why does the buck stop at complex numbers, and we don't need quaternions or some other "weirder" field to describe reality?
The answer will have to grapple with both the philosophy of physics and the properties of these various fields.
The answer will have to grapple with both the philosophy of physics and the properties of these various fields.
Another commenter linked this post by Scott Aaronson: https://www.scottaaronson.com/blog/?p=4021
The money quote (which leads a few more paragraphs on the details) is this:
> But, while I remain less than 100% satisfied about “why the complex numbers? why not just the reals?,” there’s one conclusion that my recent circling-back to these questions has made me fully confident about. Namely: quantum mechanics over the quaternions is a flaming garbage fire, which would’ve been rejected at an extremely early stage of God and the angels’ deliberations about how to construct our universe.
The money quote (which leads a few more paragraphs on the details) is this:
> But, while I remain less than 100% satisfied about “why the complex numbers? why not just the reals?,” there’s one conclusion that my recent circling-back to these questions has made me fully confident about. Namely: quantum mechanics over the quaternions is a flaming garbage fire, which would’ve been rejected at an extremely early stage of God and the angels’ deliberations about how to construct our universe.
But elegance is irrelevant. Imaginary numbers are a subset of the quaternions, just as reals are a subset of the imaginaries.
The proper question is not "do I like it" but is it necessary. The answer apparently is No for the quaternions.
The proper question is not "do I like it" but is it necessary. The answer apparently is No for the quaternions.
The referenced argument rests on superluminal communication, not elegance. The quaternions are not even sufficient, unless we believe that superluminal signaling is physical.
Interestingly enough it would be "especially" integers that don't exist physically in mosts scales of reality if physics is to have any mathematical coherence.
One of the most influential physicists of all time, John Wheeler, was of the philosophical opinion that physical laws must only be describable in terms of a finite integers [1]. This is an ideal that at least some other physicists hope for. Recently, the "It from Qubit" program in physics [2], attempts to understand reality by reducing everything to computation on qubits, and has led to some interesting insights.
[1] Information, physics, quantum: The search for links, John Wheeler 1990
[[2] The Simon's institute has formally funded such a program, though the movement within physics is much larger https://www.simonsfoundation.org/mathematics-physical-scienc...
Edit: fixed citation 1. And removed reference to infinities, as indicated below.
[1] Information, physics, quantum: The search for links, John Wheeler 1990
[[2] The Simon's institute has formally funded such a program, though the movement within physics is much larger https://www.simonsfoundation.org/mathematics-physical-scienc...
Edit: fixed citation 1. And removed reference to infinities, as indicated below.
Do you know if these efforts also only limit the formula describing these Qbit to computable reals?
I think it's a quite interesting crossroad.
Either reals don't exist, and everything is at most turing complete, in which case the universe is detemernistic and predictable, yet there are very hard boundaries to what can be known and proven.
Or reals DO exist in which case the church turing thesis is false, as the universe (with it's operations on reals) is a reified example of a super turing system, in which case we can know a lot more things than in the discrete case, but some of that knowledge can only be grasped intuitively, because the moment you try to articulate it into formulas, it's discretized and thus limited to your sub-universe/brain powerfull turing machine.
I think it's a quite interesting crossroad.
Either reals don't exist, and everything is at most turing complete, in which case the universe is detemernistic and predictable, yet there are very hard boundaries to what can be known and proven.
Or reals DO exist in which case the church turing thesis is false, as the universe (with it's operations on reals) is a reified example of a super turing system, in which case we can know a lot more things than in the discrete case, but some of that knowledge can only be grasped intuitively, because the moment you try to articulate it into formulas, it's discretized and thus limited to your sub-universe/brain powerfull turing machine.
It has been claimed that "we can simulate a BQP machine using efficiently computable entries from the set {−1, −4/5 , − 3/5 , 0, 3/5 , 4/5 , 1}" [1].
So any part of reality that you can reduce to an efficient computation on a quantum computer, admits a description using the rationals only. But I might be out of my depth here.
[1] https://arxiv.org/abs/quant-ph/0003035
So any part of reality that you can reduce to an efficient computation on a quantum computer, admits a description using the rationals only. But I might be out of my depth here.
[1] https://arxiv.org/abs/quant-ph/0003035
Reals are a digital model of analog reality. Is just a model enough for your definition of existence?
There are whole books writen about mathematical objects that turned out to not have existed.
Non-computable reals are also perfectly capable of modeling "analog reality" (because for all intents and purposes, reality could be discrete and finitely describable very small scales).
Non-computable reals are also perfectly capable of modeling "analog reality" (because for all intents and purposes, reality could be discrete and finitely describable very small scales).
>analog reality.
Quantization would like a word with you.
Quantization would like a word with you.
Continuum is not infinity.
You are correct. I only had distant memories. I have edited.
Eh, because of quantisation you could argue only integers exist ;)
Quantised is not synonymous with discretised. The fact that (for example) energy in a simple harmonic oscillator comes in discrete packets doesn't mean that the kinetic energy of a free particle is. Some quantities in quantum mechanics are as continuous as their classical analogues.
Then you learn about Zeeman effect.
The phrase "mosts [sic] scales of reality" is doing an incredible amount of heavy lifting here, though, since the most reliable mathematics describing three of what appear to be the four fundamental forces of nature are indeed quantized.
It's more accurate to say that only integers appear to be relevant in describing the nature of the nuclear forces and electromagnetism/electroweak force.
And it seems that, at least for the moment, we're headed in a direction of quantization of gravity and a Planck unit of time as well.
It's more accurate to say that only integers appear to be relevant in describing the nature of the nuclear forces and electromagnetism/electroweak force.
And it seems that, at least for the moment, we're headed in a direction of quantization of gravity and a Planck unit of time as well.
Quantum mechanics is not a discrete theory in the sense you seem to be suggesting. There is this idea that is is, which comes from the discrete energy levels it predicts for a simple harmonic oscillator, or the hydrogen atom, but quantities like the momentum, kinetic energy, position etc of a quantum particle are completely allowed to vary continuously.
In some sense (because of the wavefunction) quantum mechanics is less discrete than classical mechanics, a qubit system which would be classicaly described by a single bit is described by a state that sits on a continuous manifold (the Bloch sphere).
In some sense (because of the wavefunction) quantum mechanics is less discrete than classical mechanics, a qubit system which would be classicaly described by a single bit is described by a state that sits on a continuous manifold (the Bloch sphere).
I didn't mean to paint it as a discrete theory (and in fact I added the shoutout to the Loop Quantum Gravity stuff at the end in part because I was concerned about this interpretation).
However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?
However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?
> However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?
I don't know what this means, but there is this idea that quantum mechanics leads to integer valued stuff because the first models studied were things like the hydrogen atom and the simple harmonic oscillator which both have a natural integer valued observable (the energy). This is not generally the case, however and in quantum mechanics there are many observables which are not integer valued, for example the kinetic energy of a free particle can be any non-negative real.
I don't know what this means, but there is this idea that quantum mechanics leads to integer valued stuff because the first models studied were things like the hydrogen atom and the simple harmonic oscillator which both have a natural integer valued observable (the energy). This is not generally the case, however and in quantum mechanics there are many observables which are not integer valued, for example the kinetic energy of a free particle can be any non-negative real.
To add, perhaps in a very general hand wavy way "quantisation" is something like replacing a commutative algebra A, with a non commutative algebra A_h. I do not necessarily think there is something discrete in this process. In physics functions get sent to operators. States get sent to rays.
It turns out that a lot of interesting physics these operators have physically interesting discrete spectrum.
It turns out that a lot of interesting physics these operators have physically interesting discrete spectrum.
Makes me wonder about all the other things that don’t materially exist yet are nonetheless true and not arbitrary and would be discovered even if humans did not exist.
If you follow the construction of numbers, integers are defined as how many elements are in a set. So integers kind of exist: if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2.
Then you define all the operations between integers as operations of cardinality between set. I really enjoyed that semester when I was getting my degree on Maths, by the way, although I use the knowledge I got there as a curiosity at parties, because it didn't prove useful (not that it is bad).
Then you define all the operations between integers as operations of cardinality between set. I really enjoyed that semester when I was getting my degree on Maths, by the way, although I use the knowledge I got there as a curiosity at parties, because it didn't prove useful (not that it is bad).
> if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2.
The artificiality is high here -- would you trade your bag of "2" apples for someone else's bag of two apples without asking about the size and quality of the apples? While counting numbers are wonderful things for examining things like "sets" it seem to me to be kind of silly to assign any "reality" to them.
Discrete identical things are pretty much nonexistent at human scales. Continuous identical things, however, are numerous, where continuous here is not strict, but practical.
This is not realistic as a foundation for mathematics. But I think the idea of using propositional logic and set theory was a mistake that we are only now starting to correct. Intuitionalists and category theorists (and finitists, frankly) are leading us in a better direction now.
The artificiality is high here -- would you trade your bag of "2" apples for someone else's bag of two apples without asking about the size and quality of the apples? While counting numbers are wonderful things for examining things like "sets" it seem to me to be kind of silly to assign any "reality" to them.
Discrete identical things are pretty much nonexistent at human scales. Continuous identical things, however, are numerous, where continuous here is not strict, but practical.
This is not realistic as a foundation for mathematics. But I think the idea of using propositional logic and set theory was a mistake that we are only now starting to correct. Intuitionalists and category theorists (and finitists, frankly) are leading us in a better direction now.
(I assume you meant natural numbers, as "-1" is not a useful cardinality for a set)
The construction you refer to is due to Frege and Russell but it has its own kinds of subtleties. In particular it does not work in ZFC which we often base upon, so we usually use Peano's axioms instead to define them (depending on your context of course). While they are not all too surprising, I wouldn't call them absolutely obvious.
The construction you refer to is due to Frege and Russell but it has its own kinds of subtleties. In particular it does not work in ZFC which we often base upon, so we usually use Peano's axioms instead to define them (depending on your context of course). While they are not all too surprising, I wouldn't call them absolutely obvious.
Yes, sorry, you are right. Natural numbers, not integeres. My bad.
How do the apples get into the bag?
Do they magically show up?
Or do they transition from not in the bag to in the bag in some unit?
What is the minimum possible unit of Apple change? Is it discrete or continuous?
Maybe a X plank units of apple transitions every second.
Do they magically show up?
Or do they transition from not in the bag to in the bag in some unit?
What is the minimum possible unit of Apple change? Is it discrete or continuous?
Maybe a X plank units of apple transitions every second.
If you split one apple in two, you get two apples?
"Split" means that you have defined division, which implies having rational numbers.
In the beginning there were only Natural numbers.
-1 appeared the moment one decided to define complementary operation to addition. -1 in reality is only a shorthand for "0-1", 0 comes here as neutral element of addition. Addition is well defined on Natural numbers. Subtraction generates Zahl numbers.
Similarly multiplication is well defined on Zahl numbers. It's the complementary operation - division - that generates Quotient/rational numbers.
In the beginning there were only Natural numbers.
-1 appeared the moment one decided to define complementary operation to addition. -1 in reality is only a shorthand for "0-1", 0 comes here as neutral element of addition. Addition is well defined on Natural numbers. Subtraction generates Zahl numbers.
Similarly multiplication is well defined on Zahl numbers. It's the complementary operation - division - that generates Quotient/rational numbers.
Technically you have 0 apples, and you have 2 halves of an apple.
"God made the integers; all else is the work of man." Kronecker
Gödel, Escher, Bach (book by Douglas Hofstadter) actually touches on this topic. They are an abstraction, but they are isomorphic to reality. The big news here is that we thought complex numbers have no isomorphism with reality (physics) that couldn't be explained with real numbers only.
https://en.wikipedia.org/wiki/Isomorphism
https://en.wikipedia.org/wiki/Isomorphism
Complex numbers themselves can be explained with real numbers only.
You didn't define what you meant by "explained". You are more then welcome to explain the results of the linked study (once its released) with real numbers only.
https://news.ycombinator.com/item?id=26329883 - like this. AIU the article disproves a specific theory that threw away the imaginary part of wave function or something like that, so the result is kinda expected, but it doesn't mean that complex numbers are indispensable.
That doesn't explain complex numbers with real numbers (matrices are not real numbers). So the point of the article, and my comment still stands.
You can say that about anything, like real quantum mechanics isn't just real numbers, but also functions, equations, operators and a lot of stuff like that.
> Functions, equations, operators
All of those are part of Peano and ZFC axiomatic systems, which were thought to be enough to explain all of physics (albeit not easily). Matrices and imaginary numbers are not part of those formal systems.
Just ask yourself. If you are right, why didn't Schrodinger or the researches that are publishing this study just think of your idea of using matrices?
All of those are part of Peano and ZFC axiomatic systems, which were thought to be enough to explain all of physics (albeit not easily). Matrices and imaginary numbers are not part of those formal systems.
Just ask yourself. If you are right, why didn't Schrodinger or the researches that are publishing this study just think of your idea of using matrices?
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Imaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it.
*complex numbers. But yes you're right. The biggest problem (at least for me) is the name. Because everyone recognizes "real" and "imaginary" as adjectives, they think that's how these words should be understood in the context of numbers. But in truth, they are _proper nouns_, i.e. labels, and nothing more.
As far as the adjective "imaginary" is concerned, all numbers are imaginary.
Really the sooner we stop calling them "real" and "imaginary" the better. They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.
As far as the adjective "imaginary" is concerned, all numbers are imaginary.
Really the sooner we stop calling them "real" and "imaginary" the better. They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.
Mathematicians have largely abandoned the term "imaginary number" and use "complex number" instead. I never used the term "imaginary number" when I was teaching (except to point out that it wasn't very good terminology).
But the old term still survives in some areas. For instance, the real and imaginary parts of a complex number:
But the old term still survives in some areas. For instance, the real and imaginary parts of a complex number:
re (3 + 4 i) = 3 and im (3 + 4 i) = 4.> They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.
Sorry to be blunt, but your comment is horrible. There is far more going on here than you realize. Real numbers have an isomorphism with our reality. We thought we didn't need imaginary numbers to describe reality. They were inteded to be used more as a mathematical tool.
Another good example is euclidean vs non-euclidean geometry. They are both incredibly useful, but the former one was though to be "real" and the latter "imaginary". But it turns, physics is okay with us living in a non-euclidean universe (https://en.wikipedia.org/wiki/Shape_of_the_universe).
Sorry to be blunt, but your comment is horrible. There is far more going on here than you realize. Real numbers have an isomorphism with our reality. We thought we didn't need imaginary numbers to describe reality. They were inteded to be used more as a mathematical tool.
Another good example is euclidean vs non-euclidean geometry. They are both incredibly useful, but the former one was though to be "real" and the latter "imaginary". But it turns, physics is okay with us living in a non-euclidean universe (https://en.wikipedia.org/wiki/Shape_of_the_universe).
> Real numbers have an isomorphism with our reality.
You can probably get a nobel prize by proving this. :)
You can probably get a nobel prize by proving this. :)
Do you have a suggestion for an alternative to "real number"?
Real number are called 实数 in Chinese which is literally `solid number`.
I responded to a similar comment above with a link to what Guass called them.
Continuous numbers?
Imaginary numbers are also continuous, are they not?
So they are, so are many other spaces.
I'd avoid calling complex numbers "numbers" personally. In the context of plain english (and not currently in the context of mathematics) I think of the word "number" as saying that this is somehow a measure of a quantity (We have "a number" of brands of cereal in the store). Complex numbers are not a measure of a quantity.
I'd probably replace the "numbers" part with "field". I don't have a good replacement for the "complex" part, but I think that should change also (because telling people things are complex is almost as bad as telling them that they are imaginary). If no one proposed something better and I was king of the universe I guess I'd just name them after whoever discovered them.
I'd avoid calling complex numbers "numbers" personally. In the context of plain english (and not currently in the context of mathematics) I think of the word "number" as saying that this is somehow a measure of a quantity (We have "a number" of brands of cereal in the store). Complex numbers are not a measure of a quantity.
I'd probably replace the "numbers" part with "field". I don't have a good replacement for the "complex" part, but I think that should change also (because telling people things are complex is almost as bad as telling them that they are imaginary). If no one proposed something better and I was king of the universe I guess I'd just name them after whoever discovered them.
Would you consider -3 a number? It's certainly not a measure of quantity. A better way of thinking about it is that it's 2 things: a magnitude and a direction. Complex numbers just expand the direction into two dimensions. As long as you accept negative numbers, then you already accept the fact that there can be more than one number of a given size. The complex numbers just contain infinitely many numbers of a given size, instead of only 2.
In plain english, no I don't think I would particularly consider -3 a number. If I say someone gave me a number of horses I don't mean they stole my horse. I think it is the right call to include them in the mathematical definition though. Unlike complex numbers they are almost always useful (they are needed to turn the object into a field, which is needed if you want everyday operations like subtraction to always remain within the set), have a much more intuitive definition of multiplication, are a smaller generalization, etc.
Absolutely agree.
What's your preferred terminology? I like "Forward" and "Reverse" for positive and negative number and "Lateral" for the imaginary unit, since it is just perpendicular to the Forward and Reverse numbers.
What's your preferred terminology? I like "Forward" and "Reverse" for positive and negative number and "Lateral" for the imaginary unit, since it is just perpendicular to the Forward and Reverse numbers.
I just think of them as different axes on a 2D Cartesian plane. That’s basically how they work.
EDIT: similarly, quaternions as a number in 4D, with that extra dimension being handy for avoiding the instability problems when you get near “gimbal lock” in 3D rotations.
EDIT: similarly, quaternions as a number in 4D, with that extra dimension being handy for avoiding the instability problems when you get near “gimbal lock” in 3D rotations.
But the quaternions sort of make the real component special dont they?
That it more an effect of how we use them in computer geometry. Unit quaternions with ijk plus real map to an xyz tilt plus a rotor. But I believe you could pick any such mapping.
On the other hand, dual quaternions do care which is the dual component, since multiplying by complex numbers looks like true rotation, while multiplying by a dual looks like rotation around a point at infinity, aka translation.
So in that case maybe there is something special about the real component.
On the other hand, dual quaternions do care which is the dual component, since multiplying by complex numbers looks like true rotation, while multiplying by a dual looks like rotation around a point at infinity, aka translation.
So in that case maybe there is something special about the real component.
Spooky and strange?
Human and Werewolf.
Spooky Werewolf Algebras on the Monster group. Coming to arxiv this October.
That's pretty close to what Gauss said they should be called:
"If ... +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question"
(https://shitohichiumaya.blogspot.com/2016/10/gausss-quote-fo...)
"If ... +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question"
(https://shitohichiumaya.blogspot.com/2016/10/gausss-quote-fo...)
I also like the name "lateral unit", maybe call the field "planar numbers"?
In quantum physics complex numbers are used to represent circular symmetry, so they are more like modulus and phase.
That's great, but my intuition fails me when I try to consider what it would mean for the answer to some question to be a complex number. It's even worse for quaternions, because I can't even use the spatial analogue there.
I think about it this way: the complex-ness of the number is really just a representational trick. What it means, intuitively, depends on what the question is really asking and why the behaviors of math on complex numbers is well-suited to representing the answer, and that reason might change based on the context.
There is more to it. Vectors typically don't have multiplication defined. Complex numbers form a field and that is a major extra feature. Multiplication and division I'd defined on them.
If this property is not needed one can make do with 2d vectors instead.
There are 4d and 8d analogues as well. Hamilton started an ambitious program to base mathematics of nature on the basis of the 4d ones -- quarternions
If this property is not needed one can make do with 2d vectors instead.
There are 4d and 8d analogues as well. Hamilton started an ambitious program to base mathematics of nature on the basis of the 4d ones -- quarternions
> Imaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it.
Perhaps, but that view makes them seem arbitrary. It makes it seem as if you could just do the same thing in 3d, 4d, etc. and they'd all be equally impacful and meaningful.
Perhaps, but that view makes them seem arbitrary. It makes it seem as if you could just do the same thing in 3d, 4d, etc. and they'd all be equally impacful and meaningful.
Representing complex numbers as 2-dim. real vectors is just one way of constructing complex numbers. It is convenient because you can draw 2-dim. vectors, and it makes complex numbers feel more tangible.
Something like the original motivation (which you might find less "arbitrary") goes like this. We know how to solve some quadratic equations like x^2 = 9: We get x = 3 and x = -3 (though it was a nontrivial thing to realize those negative solutions are there). What about x^2 = 2? We can "see" the solution should be the length of the hypotenuse of a right triangle whose legs are both 1, so maybe encouraged by the geometry we don't feel too bad about having solutions (+/- sqrt(2)) which are, however, not rational numbers.
(Notice that we keep "extending" our idea of "number". Well, if we didn't have feature creep no one would buy the new version.)
Since we now have negative numbers, we ask about solutions to x^2 = -1. The first response is there are "no solutions", because there are no real numbers that square to -1. Rather than give up, we decide to "extend" the real numbers to provide solutions to this equation.
The idea is actually pretty easy if you know some modular arithmetic. For instance, mod 6 arithmetic means that you take the integers and consider things that differ by multiples of 6 to be "the same". As a consequence, 17 = 5, 4 = -14, and (importantly) 6 = 0.
Thinking in the same way, to get solutions to x^2 = -1, we want x^2 + 1 = 0. That's like "6 = 0", so "x^2 + 1" should be our "modulus", but what "number system" do we "mod out of" (instead of the integers, with the integers mod 6)? Since x^2 + 1 is a real polynomial, it looks like our "base number system" should be R[x], the (ring of) polynomials with real coefficients (stuff like -17 x^3 + 1.374 x^2 - 31/42 and so on).
So we should take real polynomials, but agree that two polynomials differing by a multiple of x^2 + 1 are "the same". We conclude that the complex numbers "should be"
For instance,
By this construction, "3 + 4 i" appears as
You can also construct the complex numbers as a subring of the 2 x 2 real matrices, so "3 + 4 i" appears as
Something like the original motivation (which you might find less "arbitrary") goes like this. We know how to solve some quadratic equations like x^2 = 9: We get x = 3 and x = -3 (though it was a nontrivial thing to realize those negative solutions are there). What about x^2 = 2? We can "see" the solution should be the length of the hypotenuse of a right triangle whose legs are both 1, so maybe encouraged by the geometry we don't feel too bad about having solutions (+/- sqrt(2)) which are, however, not rational numbers.
(Notice that we keep "extending" our idea of "number". Well, if we didn't have feature creep no one would buy the new version.)
Since we now have negative numbers, we ask about solutions to x^2 = -1. The first response is there are "no solutions", because there are no real numbers that square to -1. Rather than give up, we decide to "extend" the real numbers to provide solutions to this equation.
The idea is actually pretty easy if you know some modular arithmetic. For instance, mod 6 arithmetic means that you take the integers and consider things that differ by multiples of 6 to be "the same". As a consequence, 17 = 5, 4 = -14, and (importantly) 6 = 0.
Thinking in the same way, to get solutions to x^2 = -1, we want x^2 + 1 = 0. That's like "6 = 0", so "x^2 + 1" should be our "modulus", but what "number system" do we "mod out of" (instead of the integers, with the integers mod 6)? Since x^2 + 1 is a real polynomial, it looks like our "base number system" should be R[x], the (ring of) polynomials with real coefficients (stuff like -17 x^3 + 1.374 x^2 - 31/42 and so on).
So we should take real polynomials, but agree that two polynomials differing by a multiple of x^2 + 1 are "the same". We conclude that the complex numbers "should be"
R[x] mod <x^2 + 1>
For instance,
x^3 + 4 x + 1 = (x^3 + x) + (3 x + 1) = x(x^2 + 1) + (3 x + 1)
= 3 x + 1 (mod x^2 + 1) (the multiple of x^2 + 1 is 0)
By this construction, "3 + 4 i" appears as
3 + 4 x (mod x^2 + 1) or (3 + 4 x) + <x^2 + 1>.
You can also construct the complex numbers as a subring of the 2 x 2 real matrices, so "3 + 4 i" appears as
3 [1 0] + 4 [0 1]
[0 1] [-1 0] .
[0 1]^2 = [-1 0] = - [1 0] (like i^2 = -1)
[-1 0] [0 -1] [0 1]
Wonderful explanation, thanks a lot!
You can (almost). You can do the same thing again to get quaternions, and again to get octonions, and so on.
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...
That's different from extending 1d to 2d, 3d, etc as vectors.
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...
That's different from extending 1d to 2d, 3d, etc as vectors.
"And so on"? I thought octonions are where it stops? At least, anything past the octonions is not able to be a normed division algebra, if I understand correctly.
It doesn't stop, but as you said, the next step (sedenions), and all steps after that have non-trivial zero divisors and aren't division algebras.
Exactly. This is what I was hinting at when critical at people who say complex numbers are "just 2d reals with fancy multiplication". It's true, of course, but it makes it seem as if the 2 in "2d" is unimportant/arbitrary.
I know. But these are not as fundamentally important as the complex numbers.
The thing (which is subtle) is that the dimensionality is 1. Which is, as far as I have read the intro to the preprint cited in the Quanta article, what the authors are mistaken.
Algebraically, as vector spaces:
C is the change of base of R when tensored by C
which is not the same as
R^2 as R-vector space tensored by C
(not at all the same).
Algebraically, as vector spaces:
C is the change of base of R when tensored by C
which is not the same as
R^2 as R-vector space tensored by C
(not at all the same).
Where the analogy to a 2D vector starts to diverge is raising something to a complex power. There's no meaning in raising 3 to the [x,y], but Euler clearly defined it for 3^(xi+y).
The necessity of complex numbers in quantum mechanics was also discussed a while ago in a blog post by Scott Aaronson: https://www.scottaaronson.com/blog/?p=4021.
At the very least, he's able to show the problem with QM described using quaternions: superluminal signalling!
At the very least, he's able to show the problem with QM described using quaternions: superluminal signalling!
Anyone whose ever studied vibrations (e.g. spring mass dampers) already knew this. This is a cool example but it’s not really any more “essential” than macro-level phenomena.
Half the article is explicitly talking about how this fundamentally is different than the phenomena you’re talking about.
Imaginary numbers are merely convenient in the cases you’re describing; you can get the same results without using them. The paper this article is about is claiming that in QM this is fundamentally untrue, and models that don’t use complex numbers cannot match those that do.
Imaginary numbers are merely convenient in the cases you’re describing; you can get the same results without using them. The paper this article is about is claiming that in QM this is fundamentally untrue, and models that don’t use complex numbers cannot match those that do.
They do match. The thing is that the description is much more complex, nothing else.
What I see from the preprint (notice that the authors have not discussed it with Quanta, because it is under peer-review) and bothers me is that they "add a real qubit" (...) which should be done very carefully because the two real qubits are not independent (they are linked by the hamiltonian).
I have not read the whole preprint, though.
What I see from the preprint (notice that the authors have not discussed it with Quanta, because it is under peer-review) and bothers me is that they "add a real qubit" (...) which should be done very carefully because the two real qubits are not independent (they are linked by the hamiltonian).
I have not read the whole preprint, though.
To a certain extent, you could probably say the same about the time term in the flat spacetime metric in relativity.
There might be ways around it, but I'd wager that anything you come up with can be massaged a bit to make it look like time is an imaginary dimension of space where it's contribution to distance along the spacetime manifold is negative when squared.
There might be ways around it, but I'd wager that anything you come up with can be massaged a bit to make it look like time is an imaginary dimension of space where it's contribution to distance along the spacetime manifold is negative when squared.
Not a physicist, so maybe I took the wrong gist away from the article, but - it seems like what they are saying is that systems that are generally modeled by complex equations have always had real valued equivalents that were known.
So it's not that the complex representation isn't a useful mathematical tool, but that it was an alternate representation that was easier to work with. Can you represent a spring mass dampening with all real valued quantities, even if it's a real piece to work with?
So it's not that the complex representation isn't a useful mathematical tool, but that it was an alternate representation that was easier to work with. Can you represent a spring mass dampening with all real valued quantities, even if it's a real piece to work with?
Sorry because I do not understand your wording very well (English is not my FL), so the answer to your question is yes, you can:
e^(-at)*(A cos(bt) + C sin(bt))
where b is the frequency and a the dampening.
(But you might not be asking for an answer).
e^(-at)*(A cos(bt) + C sin(bt))
where b is the frequency and a the dampening.
(But you might not be asking for an answer).
Was going to comment this. Complex numbers are a natural consequence of the basic math axioms and the fact that they are two dimensional (and not 3,4,etc.) is not arbitrary.
Complex numbers encode a specific and essential algebra, which is everywhere in nature. So do quaternions, and so on ad infinitum. Elementary field theories are often simply named by algebraic spaces they span.
Plenty of algebras not thought of as numbers also map onto physical processes usefully. None of this is shocking or new.
Plenty of algebras not thought of as numbers also map onto physical processes usefully. None of this is shocking or new.
Roger Penrose devoted more than half of his doorstopper of a book "Road to Reality" to complex numbers in which he uses Riemann surfaces to visualize the complex-number fabric of Reality™:
https://openlibrary.org/works/OL3474173W/The_Road_to_Reality...
I was interested in plotting these Riemann surfaces and wrote a little article about it sometime ago: https://honeybee.freeddns.org/Visualizing%2520roots%252C%252...
I was interested in plotting these Riemann surfaces and wrote a little article about it sometime ago: https://honeybee.freeddns.org/Visualizing%2520roots%252C%252...
I thought we already knew this. Euler's Identity is used in electrical engineering.
That can be reformulated using only real numbers, similar to some math the article mentions for quantum mechanics:
> Any property of complex numbers can be captured by combinations of real numbers plus new rules to keep them in line, opening up the mathematical possibility of an all-real version of quantum mechanics.
> Any property of complex numbers can be captured by combinations of real numbers plus new rules to keep them in line, opening up the mathematical possibility of an all-real version of quantum mechanics.
Yes but they are claiming that the link between the real and imaginary parts of the solutions to the Schrodinger Equation are "more linked" than just by the hamiltonian.
Have looked over the introduction of the preprint and have some issues with it but am not an expert.
Have looked over the introduction of the preprint and have some issues with it but am not an expert.
But that is only for convenience. You can do everything with real trigonometry, just that it’s a bit more cumbersome. The imaginary numbers don’t correspond to anything in physical reality in electrical engineering.
The issue is not the "imaginary numbers not real etc!!" fluff. Maths is a tool to model reality and perhaps imaginary/real is a bad name. The interesting question why is C the correct choice of mathematical field to do "quantum mechanics" over -- what are the physical reasons this is needed.
You do wonder what exactly is meant by "doing quantum mechanics". I assume the start of the story is going from functions over C, quantising and getting some Hilbert space of operators/ getting wavefunctions.
Naively I would think that "quantum mechanics over the reals" might be less interesting somehow.
Consider the difference between doing complex analysis and real analysis. C loses total ordering as a field, but then gains additional nice properties over R. Quaternions however are non commutative which is a bit strange. Complex differentiable functions are more restrictive than functions on RxR (need to satisfy a stronger property). At some level you would think as models of reality you might need one structure over the other.
You do wonder what exactly is meant by "doing quantum mechanics". I assume the start of the story is going from functions over C, quantising and getting some Hilbert space of operators/ getting wavefunctions.
Naively I would think that "quantum mechanics over the reals" might be less interesting somehow.
Consider the difference between doing complex analysis and real analysis. C loses total ordering as a field, but then gains additional nice properties over R. Quaternions however are non commutative which is a bit strange. Complex differentiable functions are more restrictive than functions on RxR (need to satisfy a stronger property). At some level you would think as models of reality you might need one structure over the other.
Imho, the presence of i in the schrodinger equation has a simple origin. The equation describes evolution of a wave and waves can be conveniently described with amplitude and phase components bundled together. With such description a wave can be moved or shifted by multiplying it by a complex number. Otoh, if the same wave is described by real numbers only, each number meaning the displacement from the null state, then shifting the wave pattern becomes a non trivial operation.
Nonsense. Just use a [[1 0] [0 -1]] matrix and it's kin and do away with "essential" imaginary numbers.
The paper is describing a test of a particular alternative to quantum mechanics called "real quantum mechanics", its rather more specific than you are suggesting
https://arxiv.org/abs/2101.10873
I know a few of the people on that preprint, and I'm pretty certain they all know you can embed the complex numbers in the real 2x2 matrices ;)
As an aside I'm pretty sure the matrix you need is [[0,-1],[1,0]], which squares to -I, unless you don't want to use matrix multiplication in your construction.
https://arxiv.org/abs/2101.10873
I know a few of the people on that preprint, and I'm pretty certain they all know you can embed the complex numbers in the real 2x2 matrices ;)
As an aside I'm pretty sure the matrix you need is [[0,-1],[1,0]], which squares to -I, unless you don't want to use matrix multiplication in your construction.
as I was writing that matrix out from memory I was thinking there was something funny... Thank you for correcting me.
I have read the introduction (only cursorily). What bothers me is that they may be conflating
C as C-vector space, which is R (as R-vector space) tensor C (complex dim 1).
with
R^2 as R-vector space, tensored by C (complex dim 2).
Because when they "add a real qubit", they seem to be adding a degree of freedom which you do not have when you have just one C-qubit.
But -I repeat- I have only looked over the intro.
C as C-vector space, which is R (as R-vector space) tensor C (complex dim 1).
with
R^2 as R-vector space, tensored by C (complex dim 2).
Because when they "add a real qubit", they seem to be adding a degree of freedom which you do not have when you have just one C-qubit.
But -I repeat- I have only looked over the intro.
No. You've just reinvented them. Writing something differently and giving it a different name doesn't do away with that something.
That's the point, in mathematics you can model the same object in many different ways save for trivial transformations.
I would like to know why Schrodinger was so upset by the presence of i in his equation. It isn't clear to me why all the hate for i? It's not like it makes the math more complex: we're talking about quantum mechanics math!
There was initial backlash against complex numbers in the math community. i was deemed "imaginary" IIRC by its detractors, for the same reason the primeval atom was dubbed the "Big Bang": mockery, to accelerate its dismissal.
This bad reputation may have carried up to the 1920’s...
This bad reputation may have carried up to the 1920’s...
Information doesn’t exist without physics. So you could argue that math does exist in physical form (neutrons, bits on a hard disk, sound waves, printed paper etc.)
> No instrument has ever returned a reading with an i.
It's called a phase meter. You can buy it on amazon. An oscilloscope also works.
It's called a phase meter. You can buy it on amazon. An oscilloscope also works.
Huh? But it's literally mathematical definition that a complex number is a pair of real numbers. Spinors are more interesting.
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