Imaginary Numbers May Be Essential for Describing Reality

https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/

by theafh • 5 years ago

64 118 5 years ago

Ambolia• 5 years ago

Imaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it.

injb4• 5 years ago

*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.

janderson3• 5 years ago

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.

jonnycomputer• 5 years ago

Do you have a suggestion for an alternative to "real number"?

GoblinSlayer10• 5 years ago

In quantum physics complex numbers are used to represent circular symmetry, so they are more like modulus and phase.

ko275• 5 years ago

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).

bikenaga2• 5 years ago

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:

  re (3 + 4 i) = 3  and  im (3 + 4 i) = 4.

denkquer• 5 years ago

Imaginary numbers are 1d. complex are 2d.

simonh• 5 years ago

There are 2d values that aren't imaginary numbers.

yakubin• 5 years ago

And real numbers are ∞-d, over rationals.

rimiform• 5 years ago

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.

munchbunny• 5 years ago

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.

srean• 5 years ago

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

gspr4• 5 years ago

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.

leereeves3• 5 years ago

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.

bikenaga2• 5 years ago

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"

  R[x]  mod  <x^2 + 1>

where the "<x^2 + 1>" means "all (polynomial) multiples of 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)

(To agree that some things should be regarded as "the same" is to construct an equivalence relation, and the "bags" of "same things" are equivalence classes. I'm omitting the details from abstract algebra, but this construction is called "forming the quotient ring R[x]/<x^2 + 1>".)

By this construction, "3 + 4 i" appears as

  3 + 4 x (mod x^2 + 1)  or  (3 + 4 x) + <x^2 + 1>.

In the construction of the complex numbers via 2-dim. real vectors, "3 + 4 i" appears as (3, 4).

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] .

Notice that

  [0  1]^2 = [-1 0] = - [1 0]   (like i^2 = -1)
  [-1 0]     [0 -1]     [0 1]

So those are three ways of constructing the complex numbers, and each has pros and cons. No one of them "is" the complex numbers. "The" complex numbers are really an isomorphism class of rings, and we're just giving different representatives from that class.

SavantIdiot2• 5 years ago

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).

pfortuny6• 5 years ago

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).

elil17• 5 years ago

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.

Armisael16• 5 years ago

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.

pfortuny6• 5 years ago

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.

Enginerrrd• 5 years ago

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.

CyberRabbi2• 5 years ago

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.

ihunter2839• 5 years ago

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?

pfortuny6• 5 years ago

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).

Xophmeister• 5 years ago

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?

j-pb3• 5 years ago

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.

zeroonetwothree• 5 years ago

Eh, because of quantisation you could argue only integers exist ;)

jMyles2• 5 years ago

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.

abdullahkhalids4• 5 years ago

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.

CyberRabbi2• 5 years ago

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.

1010082• 5 years ago

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).

random314• 5 years ago

"God made the integers; all else is the work of man." Kronecker

germanier• 5 years ago

(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.

GoblinSlayer10• 5 years ago

If you split one apple in two, you get two apples?

andrewla• 5 years ago

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.

nullserver• 5 years ago

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.

abdullahkhalids4• 5 years ago

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.

Twisol2• 5 years ago

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.

bloak• 5 years ago

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")

michelpp• 5 years ago

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

diroussel• 5 years ago

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.

ko275• 5 years ago

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

GoblinSlayer10• 5 years ago

Complex numbers themselves can be explained with real numbers only.

lizknope• 5 years ago

I thought we already knew this. Euler's Identity is used in electrical engineering.

pfortuny6• 5 years ago

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.

leereeves3• 5 years ago

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.

Ma8ee2• 5 years ago

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.

dnautics2• 5 years ago

Nonsense. Just use a [[1 0] [0 -1]] matrix and it's kin and do away with "essential" imaginary numbers.

eigenket5• 5 years ago

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.

dnautics2• 5 years ago

as I was writing that matrix out from memory I was thinking there was something funny... Thank you for correcting me.

pfortuny6• 5 years ago

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.

gspr4• 5 years ago

No. You've just reinvented them. Writing something differently and giving it a different name doesn't do away with that something.

GoblinSlayer10• 5 years ago

That's the point, in mathematics you can model the same object in many different ways save for trivial transformations.

GoblinSlayer10• 5 years ago

Huh? But it's literally mathematical definition that a complex number is a pair of real numbers. Spinors are more interesting.

spekcular• 5 years ago

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!

cableclasper• 5 years ago

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...

SavantIdiot2• 5 years ago

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!

pygy_• 5 years ago

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...

frongpik• 5 years ago

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.

mbrodersen• 5 years ago

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.)

foxes2• 5 years ago

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.

alpineidyll3• 5 years ago

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.

siproprio• 5 years ago

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.

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