Quantum physics falls apart without imaginary numbers

https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/

by wrycoder • 3 years ago

101 157 3 years ago

I didn't have time to read all of it, but it seems to focus a bit much on the imaginary numbers aspect of complex numbers, and how they're all spooky and weird.

Reading this abstract[1] and this[2] StackOverflow discussion about the topic, it seems the point is rather more along the lines that complex numbers aren't just plain two-dimensional vectors of real numbers. There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality.

Then again, I'm just a programmer.

[1] https://www.nature.com/articles/s41586-021-04160-4

[2] https://physics.stackexchange.com/questions/691623/how-does-...

icapybara• 3 years ago

There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality.

I think you couldn't have put it better. I'm just a physicist, though.

retrocryptid• 3 years ago

One of the first things we were taught in physics was "don't think that imaginary or complex numbers have physical significance. just do the math."

And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out.

sdfghswe• 3 years ago

"shut up and calculate", also known as the Feynman approach to quantum mechanics.

It's not imprecise. It reproduces experimental results from theory, so it's in fact the most precise approach in existence.

BrandoElFollito2• 3 years ago

This is one of the complicated steps in physics (I have a PhD in physics (and forgot everything since)).

First you have some math that goes along discovering physics. You split vectors, multiply mass by something and it's fine.

Then you have math that helps you with physics. Simple differentials equations that uncover while laws of nature (cooling down speed for instance). This is the golden time for many because you're at this sweet spot where it is exciting but not too hard.

Then comes the travel in desert of abstract things you have no idea about and winner why someone hates you by shoving Abel groups down your throat for no reason.

Finally comes that sight of relief when you can binding do some maths to end up with a real life solution without too much thinking because you have solid tools.

The last part is a bit morally complicated because you have the feeling that you are cheating. Renormalization, I am looking at you.

But then I forgot everything because I left academia and my memories may be faulty.

nathan_compton7• 3 years ago

There really isn't anything weird or suspect about renormalization (except the name, perhaps). Read A. Zee's book on Quantum Field Theory.

bmacho• 3 years ago

Same holds for negative numbers. There are no negative quantities in physics, negative numbers as quantities only appear if you order your equations wrong. (And one can argue against the other appearances of negative numbers and minus signs.)

wrycoder• 3 years ago

But, how would one handle positive and negative charge?

nathan_compton7• 3 years ago

"And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out. "

The only thing imprecise about this is "many". Really any formula for an observable of any kind (including probabilities) has to come out to a real number.

computerfriend2• 3 years ago

Not really! Often you get a complex solution and both the real and imaginary components are valid.

C-x_C-f4• 3 years ago

don't think that imaginary or complex numbers have physical significance.

Yeah I'd say that's the most common approach but I think it's misguided. Complex numbers aren't any less physical than any other number. It just turns out that for historical reasons, it makes sense to define observable quantities using self-adjoint operators (which have real eigenvalues, and the latter are used to measure things like energy). But that doesn't mean the rest is not physical. Just because we can't take a picture of an object in the dark, it doesn't mean the object isn't there when the lights are off.

scotty79• 3 years ago

I'd say that complex numbers are the only ones that have physical significance. They are what's actually happens in the real world until we disturb it with experiment.

whatshisface4• 3 years ago

This is your regularly scheduled reminder that complex numbers have (real) matrix representations, and what matters in any model is the properties it has not its identity as an object.

sebzim45003• 3 years ago

Did someone claim otherwise?

whatshisface4• 3 years ago

"Quantum physics falls apart without imaginary numbers."

Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”

contravariant2• 3 years ago

If the title doesn't count then we need to question what it even means for a quantum theory to 'use' imaginary numbers.

Because if real skew symmetric matrices count, then a harmonic oscillator also inescapably uses imaginary numbers.

eigenket3• 3 years ago

This is your regularly scheduled reminder that the author of these papers (https://arxiv.org/abs/2101.10873, https://arxiv.org/abs/2111.15128), which this article is based on know very well that complex numbers have real matrix representations.

What they add which your comment discounts is the locality structure of quantum mechanics, i.e. what happens when you combine multiple quantum systems. Specifically if the state of one system lives in A, and the state of another system lives in B then the state of a combined system lives in the tensor product A ⊗ B.

If you do the trick where you replace complex numbers by real matrices what you end up is having the state of the first system living in A = X ⊗ A', and the second is in B = X ⊗ B', where A' and B' are real and the X subsystem is the degree of freedom you're using to "fake" the complex numbers.

Then if you try to combine the two systems you end up with a state that lives in A ⊗ B = (X ⊗ A') ⊗ (X ⊗ B') but what you need in order to get the behavior you want is actually for the combined system to live in X ⊗ (A'⊗ B').

What they do is a bit more complicated because they show that all ways to fake complex numbers using real numbers breaks this way of combining systems, but this is the gist.

civilized2• 3 years ago

Thanks, this is very helpful.

When I took undergrad quantum physics, we saw that the Schrödinger equation can be represented without complex numbers. But that's for one particle, and it sounds like you're saying this somehow breaks down when you have multiple systems interacting, due to these tensor products not working as we think they should?

stametseater• 3 years ago

Well, physics (including classical mechanics) already uses irrational reals, which are pretty spooky themselves. Imaginary numbers don't seem so much worse.

rprenger• 3 years ago

This Scott Aaronson lecture I really liked is relevant. It's like a "why quantum mechanics probably had to do the weird probability amplitudes (which can be negative and complex) instead of just normal probabilities even without experimental results" lecture: https://www.scottaaronson.com/democritus/lec9.html

thechao• 3 years ago

I like SA's blog; and, based on that I bought this book (Quantum Computing Since the Time of Democritus). It's expensive and bad. Really mind-numbingly awful. I can't tell if his writing has improved dramatically since he wrote the book, or what. The entire book is done in this tongue-in-cheek pseudo-first-person, chatty, pseudo-Socratic dialogue style. That sort of stuff is fine for, say, a couple of tightly-written pages. But ... not for hundreds of pages. It's a pity, since the information in the book is good.

dboreham• 3 years ago

Haven't read it, but it's obviously wrong.

To expand: any time you read "...magical complex numbers" just mentally replace "complex" with "negative" and then examine how odd the original text now reads. There's nothing fundamentally different between the concept of negative numbers, and complex numbers.

sebzim45003• 3 years ago

"Quantum Physics Falls Apart Without Negative Numbers" sounds a bit obvious, but reasonable IMO.

CottonMcKnight• 3 years ago

I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see those roots in the graph. There is no discontinuity.

mjhay• 3 years ago

The name "imaginary" was due to Descartes and it absolutely was intended as a pejorative, even though they're necessary to algebraically close the reals. Some ancient Greeks, IIRC, were similarly hostile to negative numbers. Of course the "real" numbers have never been controversial despite the whole concept being a lot weirder (and uncomputable), probably because their informal aspects just so happen to line up with everyday intuition.

mtlmtlmtlmtl• 3 years ago

Irrational numbers have been known about since ancient Greece, but they were in fact controversial back when discovered/invented because they challenged conventional wisdom in Greek mathematics at the time.

kgwgk2• 3 years ago

Of course the "real" numbers have never been controversial

Apparently the existence of irrational numbers was a shock to Pythagoreans. There may be also people unhappy with transcendental numbers.

eternityforest2• 3 years ago

Why don't imaginary numbers ever show up "in real life" outside of STEM? It's interesting everyone seems to think they are fundamental to everything, but we don't see them.

In fact we only see plus/minus/times/divide before getting into "You'll probably use a computer for that, and you probably don't need to unless you're an engineer" stuff. Does anything ever happen outside of science that we could use imaginary numbers to understand? Are they just fundamentally outside ordinary experience, or do we not see them because we have workarounds that hide them?

It would be interesting to read a SciFi novel where regular people commonly encounter imaginary numbers and other similarly far from everyday life stuff like calculus.

tokai• 3 years ago

"real" numbers have never been controversial

Some controversy does exist. NJ Wildberger is most famous for not believing in the real numbers.

pcrh• 3 years ago

I have the same attitude towards the use of the term "significant" in statistics, where it has a technical meaning that doesn't correspond to the common language use of the term to mean "important" or "large".

Similar to "imaginary" numbers, it's curious how influential/distracting the terms are, though.

tehsauce• 3 years ago

“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits in nicely with group theory, which organizes all kinds of objects which also have the same properties as numbers.

nathan_compton7• 3 years ago

You miss a key part of complex numbers if you think of them as just vectors: they are a field.

adammarples• 3 years ago

dr_dshiv• 3 years ago

Because they are separate but interacting with real numbers? I don’t understand.

mort96• 3 years ago

A vector field?

0xBABAD00C2• 3 years ago

And not just a field, but the algebraic closure of real numbers.

lanstin• 3 years ago

Complex analysis is so much more regular than real analysis differentiability over a two dimensional quantity is so much strong than over a one dimensional quantity that you have much stronger results. Basically, if you know an analytic function in a neighborhood you know it over the entire plane. Plus you have functions like e ^ ( 1 / z ) which is pretty amazing around zero.

rcme• 3 years ago

While what you say is true, I could never intuitively grasp that properties of analytic functions. Like I could read and understand the proofs, as in follow one step to the next, but I could never succinctly describe, intuitively, why one should expect the proofs to hold. Even the most fundamental concepts in complex analysis are more like facts rather than logical deductions (to me).

fsloth• 3 years ago

IMO there is nothing “natural” in interpreting complex numbers as a vector. The fact you get a thing out of them that looks a lot like a vector is one of the stupefying ‘mysteries’ of math which make the discipline so cool.

Complex numbers afaik began as an attempt to solve polynomial equations. They begin from the agreement to invent a number i whose square is -1 so you can solve equations having sqrt(-1) in them.

The jump from sqrt(-1) to plane rotations is to my feeble mind one of the most flabbergastingly unintuitive things in ‘basic’ maths. “A rotation you say? Who ordered that!?”

andrew_eu• 3 years ago

There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane.

The intuition I developed with them while studying physics was that, unlike "real" numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don't live in a 1D world. In higher dimensions they rotate and sheer rather than just scale.

And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox.

[0] https://en.m.wikipedia.org/wiki/Complex_number#History

reeboo2• 3 years ago

Here is fun, and rather short, book on the history of complex numbers that I liked -- https://www.amazon.com/Imaginary-Tale-Princeton-Science-Libr...

fullstackchris• 3 years ago

The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imaginary velocity can fit!

My intuition leads me to believe that almost some sort of other dimensional effects are at play, and our feeble math just can't accurately describe it. Perhaps it's just the nature of quantum itself, and no special dimensional consideration is needed. It's been a long time since I studied any math or physics...

tanseydavid• 3 years ago

Thank you for sharing this concept. I find it very illuminating.

mr_mitm• 3 years ago

This makes no sense whatsoever. Classical momentum is mv, not mv^2. Also, momentum is a vector and generally can't be negative.

Quantum mechanical velocity is complicated as well.

whatshisface4• 3 years ago

Momentum is mv, mv^2 is two times the kinetic energy. (Energy is real in quantum systems as well as classical ones.)

digging• 3 years ago

I thought quantum tunneling was due to uncertainty, allowing the position of a particle to resolve on the other side of a barrier sometimes because it is undefined before the tunneling.

Are we talking about different things? Is one of us way off the mark? Or are these two angles of the same phenomenon?

lanza2• 3 years ago

Classical mechanics are fundamentally wrong. They are low energy approximations to reality. "How did the bal go through the hill when it actually didn't have the momentum to do so" is a nonsense question because the equations of motion you are attempting to use to describe the phenomena are wrong.

You can't and shouldn't try to understand QM from a CM standpoint. If you remember Taylor series expansions, this is like trying to understand `sin(x)` by looking at it's first order Taylor series expansion `x`. Your question about momentum and a potential hill is the same question as "how did the value of sin(x) start decreasing if `x` is linear?" You're using too few terms of the series expansion. The mechanisms of Newton's laws are first order terms of a proper QM solution.

user070223• 3 years ago

That what Carl bender talks about[0] That quantized nature is due to are ability to only measure only real values, which can be a sparse subset(Null set) placed on different sheets of the complex function, like qunatized energy levels of the electron in atoms. He also talk about research showing that exactly what happens when setting an experiment such that there is an interference which I briefly looked at a while back.

[0] https://www.youtube.com/watch?v=_Sm7SNlNUOI&list=PLOFVFbzrQ4...

machina_ex_deus• 3 years ago

If you have probability distribution transition function, it makes sense to want to diagonalize it. And you can't do that without algebraic closure of complex numbers. Even a simple transition matrix of a 3 cycle has complex eigenvalues (the 3rd roots of unity).

The stationary probability distribution of some transition matrix is the eigenvectors, so the stationary "probability distribution" of even very simple matrices (like cycle of 3 values) are complex. It's not as magical as they make it out to be.

detrites3• 3 years ago

Maybe my understanding is too limited, but when I had them explained as "rotational" numbers they seemed to reduce to a simple logic shortcut to get signs changing correctly around our arbitrary axes-based coordinate system.

No less useful, but kind of mundane.

Are there other things they fundamentally do, or is everything else rooted in that property? (Or have I just misunderstood them?)

bottom999mottob• 3 years ago

I don't quite understand your question. Imaginary numbers are useful for modeling waves and particles, both foundational things in our universe.

In quantum mechanics, we use complex numbers to describe the behavior of particles. A complex number has two parts: a real part and an imaginary part. The real part represents something we can physically measure, like the position or momentum of a particle. The imaginary part represents something that's a bit harder to grasp - it's related to the probability that the particle will be in a certain state or position.

For example, let's say we're trying to describe the position of an electron in an atom. We can't know for sure where the electron is at any given moment, but we can calculate the probability of finding it in a certain region. The probability is represented by a complex number, with the real part telling us the position and the imaginary part telling us the probability.

Things like Feynman Diagrams model probability distributions of particles and all possible paths they can take from A to B. They allow us to do interesting calculus

TRiG_Ireland• 3 years ago

Or, as someone I know (Gnomon on h2g2) remarked, they're just as real as real numbers, which is to say, not real at all.

dist-epoch• 3 years ago

There is a really good visual and intuitive explanation of complex numbers, and the sub-title is "a tale of numbers that like to turn":

https://acko.net/blog/how-to-fold-a-julia-fractal/

Robotbeat3• 3 years ago

I always just thought of them as a second dimension to the number line. 2D numbers, if you will. That enables rotation as well, of course, as such a thing doesn’t make sense in 1D. And for certain situations this helps resolve ambiguities that would be difficult and messy without this extra dimension.

Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant.

EDIT: the difference between complex numbers and 2D vectors (as I understand it) is just that you can’t really multiply 2 2D vectors together to get another 2D vector (you need to multiply a scalar times the vector to get a vector), but you CAN multiply two complex numbers together to get another complex number.

To do multiplication with 2 vectors you need to introduce new things like the “inner product” (dot product) to produce another vector (as opposed to “outer product” which produces yet ANOTHER kind of thing, a matrix) whereas with complex numbers you can use regular multiplication without introducing new types of multiplication.

tomrod• 3 years ago

Eh, RE: your edit, "multiplication" itself has several different definitions in a vector space. For example, a dot product, outer product, Hadamard product, etc.

computerfriend2• 3 years ago

You've gone up a degree for each example.

The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product.

I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.

lloeki• 3 years ago

Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant.

I always liked that one. Gimbal lock comes in because to be of any practical use you gotta have a reference axis from which you start to rotate, which necessarily creates poles with a singularity. Since you gotta have the axis there's no way around it.

So, how to solve that? Easy: stuff the axis outside of 3d space. It doesn't even matter what the axis is, it's just there to stash the singularity away and you can rotate every which way continuously.

xeonmc2• 3 years ago

For me it’s more intuitive to think of numbers as being either unsigned (magnitude-only) or signed (has magnitude and polarity). It never really made sense to me to accept the concept of negative numbers without also accepting imaginaries.

IIAOPSW3• 3 years ago

Quantum physics would not fall apart. In quantum computing you can always replace the imaginary component at the cost of just one extra qubit in your circuit. The real part maps to the |0> state on your ancila, the imaginary to the |1> component. In terms of quantum mechanics as a theory, this implies you can always do away with the imaginary part at the cost of introducing just one fictitious two-level degree of freedom, like idk call it "spin" or something I guess. Oh shit.

Put another way, complex numbers are a qubit the universe gives you for free.

bollu2• 3 years ago

I do not believe this. What will one do about intermetidate computations which can have complex coefficients? In general, you'd need some way to change the gates/ unitary matrices themselves to be purely real. So you'd need to find an isomorpism from U(n) into a subgroup of SO(poly(n)) for this claim to work. Why does such an isomorphism exist?

dxuh• 3 years ago

I often wonder about how much mystery and skepticism would surround them if they were simply called "complete numbers" instead. Of course that's not a great name either, but it's vaguely motivated them being algebraically closed and most importantly it's a neutral, or even slightly positive name.

MengerSponge• 3 years ago

I prefer "subtle" over "imaginary"

albrewer• 3 years ago

These[0] two[1] videos are probably the best overview of imaginary numbers I've ever come across. The whole channel is a treasure trove of the history behind math, notation, and conventions used in physics.

[0] https://www.youtube.com/watch?v=CdwxpSInhvU

[1] https://www.youtube.com/watch?v=M12CJIuX8D4

amelius2• 3 years ago

I guess people felt the same when negative numbers were introduced.

lacker• 3 years ago

Whether a physical theory can be consistent without them is an interesting question

I agree, it is interesting, and the answer is yes, a physical theory can be completely consistent while never using complex numbers.

If you don't want to use imaginary numbers, you can avoid them for almost anything by using matrices instead. Use 2x2 matrices instead of complex numbers, and instead of 1, use the identity matrix:

1 0

0 1

Instead of i, use a matrix that corresponds to a 90 degree rotation:

0 -1

1 0

This means that i * i = -1, and so pretty much everything else will just work the same way as it does with complex numbers. Adding, multiplying, calculus, all the same. No need for complex numbers if you don't like them.

I do like complex numbers though. They are just a bit more concise and convenient than using matrices.

winter_blue• 3 years ago

Can complex numbers be seen as just a tuple of length 2, containing two real numbers?

reeboo2• 3 years ago

So does algebra.

contravariant2• 3 years ago

And ordinary differential equations.

ars• 3 years ago

This video by Sabine Hossenfelder is probably more informative on this topic: https://www.youtube.com/watch?v=ALc8CBYOfkw (it discusses this paper).

anthk• 3 years ago

Sabine's theories look like a New Age bullshiter parroting "quantumagical" nonsense to sell books, sorry.

deviantbit• 3 years ago

Just remember that Rene Descartes coined the term Imaginary Numbers, and he also believed in ghosts.

Stop calling them imaginary.

qwertox• 3 years ago

They are like negative numbers. You can have 1 apple, but you can not have -1 apples. You can owe +1 apple, and say that you therefore have -1 apples. So -1 does exist as a number, but it is a representation of something that is happening with a positive number. And the "i" is similar to this. Maybe calling it "perpendicular" would have been better suited, because imaginary is really confusing.

sleepyams• 3 years ago

The utility behind complex numbers (for physicists at least) is really that they are a model for certain algebraic and geometric properties that are together very useful.

patrick451• 3 years ago

Later, complex numbers, which are the sum of a real and an imaginary number, gained wide acceptance by mathematicians because of their usefulness for solving complicated mathematical problems. They aren't part of the equations of any fundamental theory of physics, however—except for quantum mechanics.

I don't see how this is remotely true. You can't even solve the ODE for an undamped mass-spring system without imaginary numbers. More generally, most of our notion of eigenvalues falls apart if we work over the field of reals rather than complex numbers, and once you lose that, you lose most of linear algebra and with it vast swaths of engineering.

crazygringo3• 3 years ago

Could you be more specific, because I was definitely under the impression that the original quote was correct.

Where are imaginary numbers required for the undamped mass-spring system? Because a lot of "complex" equations are using complex e^ simply as an alternative to trigonometric functions (for aesthetics or convenience), where there's nothing inherently imaginary whatsoever. The same as much of signal processing.

I'm less familiar with using complex numbers in linear algebra, but I know that when I studied it in college we never touched them, so I don't understand how we'd lose most of linear algebra?

But I think the point the article is making is that, except for QM, there are no physical instantiations of complex/imaginary values. Rational numbers physically "exist" as a fraction of a distance between two points; real numbers "exist" as actual geometric proportions, and negative numbers "exist" as an opposite direction. But complex/imaginary numbers are just intermediary tools for solving equations (or conveniences to replace trigonometric functions), they don't correspond to anything physical (except, it seems, in QM).

m3kw9• 3 years ago

That it says “imagination” at first

esalman• 3 years ago

Well, multiple disciplins including and/or associated with electrical engineering and electronics would fall apart without imaginary numbers.

xchip2• 3 years ago

Congratulations, BTW it also falls apart without the even numbers, same thing for the odd ones.

hackandthink• 3 years ago

This is not about experimentally falsifying real quantum theory, but nice anyway:

Why are amplitudes complex?

https://scottaaronson.blog/?p=4021

ordu• 3 years ago

There is a video depicting the story of a search for a cubic equation solution and invention of complex numbers. Here it is described in a few sentences but really it was a novel, with secrets passed from dying masters to apprentices, duels (mathematical), broken oaths and suchlike.

https://www.youtube.com/watch?v=cUzklzVXJwo

itvision• 3 years ago

Websites seem not to work without them as well.

The page isn’t redirecting properly

An error occurred during a connection to www.scientificamerican.com.

    This problem can sometimes be caused by disabling or refusing to accept cookies.

xchip2• 3 years ago

You should know that physicists don't talk all day long about quantum mechanics.

neonate• 3 years ago

https://archive.md/L1SeH

bookofjoe• 3 years ago

it also falls apart without imaginary superpositions

javajosh• 3 years ago

Physics and computer science share the feature that it seems easier to start by explaining linear motion (linear programs) but all the really interesting stuff is circular (loops). Complex numbers are the simplest representation for describing and combining rotations in a consistent way. (Other representations like "r theta" are not as simple.)

Note also that a world with only monotonic linear motion could not possibly have life or thought or any complex behavior. Rotation is required to model any sort of accretion over time. (note that the typical finite case of "particles in a box" bouncing off the walls is, on average, circular motion too.)

See Clifford Algebra for generalization of the complex numbers

crazygringo3• 3 years ago

I've read the whole article twice now and, at the end of the day, it doesn't seem to actually explain anything at all.

It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers.

Which makes perfect sense, of course, because all of the rules of complex math are written in terms of real numbers at the end of the day. When CPU's are calculating math operations, it's not like any complex/imaginary bits or bytes are involved.

But then the article describes an experimental setup where the results are somehow only consistent with standard/complex QM, and are inconsistent with real QM.

But it totally neglects to say how or why. The entire premise behind real QM is that the complex stuff can just be rewritten as real. But this article seems to provide zero explanation, analogy, or intuition whatsoever as to why there's a case where this would ever not be possible. At no point does the article define what it even means to not be reducible to real math.

So I can't tell what any of this is supposed to mean at all?

cubefox• 3 years ago

In philosophy of science and mathematics there is actually a famous argument by Hartry Field to the effect that Physics doesn't need to posit the existence of any numbers at all, not even natural numbers:

https://academic.oup.com/book/26363

As far as I know (I haven't read the book) he argues for this by logically reconstructing part of Newtonian Mechanics without any numbers (without assuming the Peano axioms) and suggesting that in principle similar things could be done for other theories.

Now it would be very surprising if Quantum mechanics would even require the existence of imaginary numbers, since these seem to be much less basic than natural or real numbers.

Anyway, I agree that popular articles haven't made it clear what the relevant physicists mean with their thesis. And as a non-physicist it is hard to read the original source.

jessriedel• 3 years ago

You definitely can do quantum mechanics fine without complex numbers. A good article would explain why it becomes much simpler/elegant with them, but apparently this isn't one. Clickbait title.

Strilanc• 3 years ago

So I can't tell what any of this is supposed to mean at all?

Quantum mechanics is defined by three axioms:

1. States are unit vectors. Vectors whose 2-norm is 1.

2. Operations preserve the 2-norm. They are described by unitary matrices.

3. Systems are combined using the tensor product. If system A has state u and system B has state v, then the combined system (A, B) has state u⊗v. If you apply operation x to system A and operation y to system B, the operation on the combined system is x⊗y.

What this paper proved is that, if you use these axioms but limit yourself to unit vectors and unitary matrices with real entries, you cannot explain some experiments. This is surprising because a complex number can be thought of as just a pair of real numbers. For example, in numpy, you can turn any complex ndarray into a real ndarray by making an ndarray with one additional index of length 2, like this:

    def complex_to_real(complex_ndarray):
        real_ndarray = np.zeroes(shape=(*complex_ndarray.shape, 2), dtype=np.float64)
        real_ndarray[..., 0] = np.real(complex_ndarray)
        real_ndarray[..., 1] = np.imag(complex_ndarray)
        return real_ndarray

These real ndarrays can represent all states and operations that the complex ndarrays could. So how could they possibly fail to explain any experiment?

The problem is axiom (3), where systems are combined using the tensor product. The issue is that complex_to_real(A ⊗ B) has one more index than A⊗B, but complex_to_real(A)⊗complex_to_real(B) has two additional indices because you gained one real-vs-imaginary index from A and also one from B. The tensor product doesn't understand that these indices should be merged, instead of concatenated. Using complex numbers tweaks the definition of the tensor product so that it does merge these indices. The experiment is basically a way of checking that you contracted those extra indices instead of keeping them both.

So really this result is not about complex numbers vs real numbers. It's about how the states of quantum systems are combined. If you want to use real numbers, you can't use the normal tensor product; you have to use a modified one that understands every system has a special real-vs-imaginary index and that when combining systems you must contract these indices together. Complex numbers just happen to have a tensor product that packages this contract-one-index functionality nicely.

crazygringo3• 3 years ago

I've always had problems with how complex numbers are taught.

The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that's used for geometry or 2D coordinates or anything.

Then you progress into all the complex e^ functions where you basically forget about a plane, and it's just a convenient shorthand for math that would otherwise involve a bunch of trigonometric functions. But again, this never provides any actual intuition... it's just convenience. You could still write everything as sin() and cos() etc.

I finally felt like I understood complex numbers when I asked myself, how do you create a continuous solution to the function y(x) = (-1)^x? Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...]. But you can work out the math such that it necessarily produces a spiral through them. And it's different from the e^ equations because there's no pi involved.

And so rather than thinking of complex numbers as a "plane", it's much better to think of them simply as the inherently spiral motion required to continuously join alternating values. And when I look at how complex numbers are actually used in things with physical correlates -- e.g. signal processing -- the spiral intuition always continues to make sense.

Of course, at the end of the day, it's all the same math. But the idea of a continuous spiral path between otherwise discontinuous real numbers has wound up clicking for me in a way that the metaphors of a 2D plane, or of more generalized arbitrary "rotation", never has. That complex numbers are about spiral oscillation, not about planes.

zuminator2• 3 years ago

You keep saying spiral but in a 2d spiral the value of |y| would be increasing as well. So if I'm understanding you correctly I think you mean helical? As in:

https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259e...

dbelford• 3 years ago

The Heyser spiral or Heyser corkscrew seems to be a common name for this type of plot. https://www.google.com/search?q=heyser+spiral

And it connects circles, e/euler's formula, and sin/cos is a visually grokkable way.

knome• 3 years ago

If you should be interested, 3Blue1Brown has a video primer on complex numbers that includes describing multiplications by i as rotations on the plane, showing why a plane is useful for its representation. I always enjoy videos from that channel. https://www.youtube.com/watch?v=5PcpBw5Hbwo

djsavvy• 3 years ago

Back when I did olympiad math we would sometimes solve classical geometry problems with a "complex bash" --- you assign points complex coordinates and then do some algebra to show that the desired property must hold true.

feoren• 3 years ago

I find this "spiral" concept showing up in other cases as well, and I agree that it's under-rated (or under-studied) as a way to understand the complex numbers. By thinking about periodic points on a complex spiral, you can extend many useful properties of polynomials to those with rational coefficients. We can begin to talk about the roots of equations like x^(3/2) + x^(1/2) = 0 and we find results that very naturally extend from roots of similar "standard" polynomials. We can see that x^(1/2) = 0 has half as many roots as x^1 = 0, and x^(1/4) = 0 has half again as many roots, even if they all "project" down to 1 -- their periodicity on the spiral is 4pi and 8pi instead of the standard 2pi. I feel like there's a lot going on there that hasn't been explored.

gowld• 3 years ago

Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...].

y = cos(pi * (x-1)). No complex numbers needed.

Yes, you can make a 3-D spiral / helix by mapping from R to C, but (a) that's only when your domain is R; ignoring when the domain is C.

The spiral doesn't help you with a simple task like solving this: 0 = x^2 +1

A helix is a time-based view of drawing a circle, in the plane.

Your favorite use of complex numbers isn't the only use of complex numbers.

https://brilliant.org/wiki/complex-numbers-in-geometry/

https://web.evanchen.cc/handouts/cmplx/en-cmplx.pdf

User23• 3 years ago

Are you familiar with Visual Complex Analysis?

Your spiral motion intuition is really cool and complements the amplitwist model that book introduced me to nicely.

On a random tangent, one of the reasons why I love Common Lisp is that it has functions like CIS[1] in its library. That got me realizing you can literally view SIN and COS as CAR and CDR for what's essentially a cons of non-Complex numbers. I don't claim there's any benefit to doing so, but it's not wrong.

[1] http://clhs.lisp.se/Body/f_cis.htm

andrepd• 3 years ago

i is a number defined to have i² = -1. Everything else follows from that. The complex plane arises naturally as a useful representation of complex numbers, it is not the starting point.

I do not understand at all that stuff about "spiral path between otherwise discontinuous real numbers", but I suspect you may be making things more complicated than they need.

justinpombrio• 3 years ago

`i` has a geometric meaning! It's explained by Geometric Algebra: https://en.wikipedia.org/wiki/Geometric_algebra

In three dimensional Geometric Algebra, `i` is defined as the product of three orthogonal unit vectors, `abc`. You have `abc = bca = cab = i` and `acb = cba = bac = -i`. So `i` defines a chirality on the space.

David Hesternes has a paper relating that to QM: https://web.archive.org/web/20120406093531/http://www.montgo...

dimitrios1• 3 years ago

As a random aside, if anyone wants a fun read, check out An Imaginary Tale: The Story of √-1

ubj• 3 years ago

My favorite property of "imaginary numbers":

i^i = 0.20787957635... (Spoiler alert: it's real!)

No, they're not imaginary, and yes they have real-world significance. They represent oscillations in control systems [1]. They're useful for accurately approximating derivatives [2]. I really dislike the term "imaginary" because of how useful they actually are.

[1] https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf

[2] https://mdolab.engin.umich.edu/wiki/guide-complex-step-deriv...

hsnewman• 3 years ago

Quantum physics is a description of reality, not reality itself.

Koshkin9• 3 years ago

Thank god - I wouldn't want to see the reality fall apart.

0xBABAD00C2• 3 years ago

Complex/imaginary numbers are just badly named for historical reasons, they represent an objectively central concept in math and physics, and can be derived from axioms of what we expect from a well-behaved number field. For reals, we have: (A) expected properties of addition and multiplication, (B) total order and other order-related nice properties (Dedekind-complete). Any mathematical structure satisfying (A,B) will be equivalent to real numbers. Now if we extend it to get (C) algebraic closure, so that all polynomials have roots, we get the complex numbers.

user8501• 3 years ago

I had this thought years ago:

Real number 1 is 1 OR -1 Imaginary number 1 is 1 AND -1

In other words, when you “break apart” i, you get 1 and -1

throwaway81523• 3 years ago

I get a redirect loop when I try to view this page. I'll try the archive.md link.

derbOac• 3 years ago

This reminds me of this paper even though it's probably only tangentially related:

https://ieeexplore.ieee.org/abstract/document/6875117

On the other hand it makes me wonder if there's some deeper truth about the nature of probability representations.

Quantum physics falls apart without imaginary numbers

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