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.
> 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.
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.
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.
> There really isn't anything weird or suspect about renormalization
This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.
Like, ok, we get testable answers and they match experiments but also this is _so_ hacky and I can't shake the feeling that one day someone will come along and show that there's some reason why these bad assumptions work out fine. You know, like how "to find the Schwarzschild radius for a black hole of known mass, calculate the radius at which the escape velocity is equal to the speed of light" gives the correct answer even though the theory implied by this method is naive and wrong.
"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.
A complex solution can be valid but you never measure a complex number. I'd argue that in situations like using complex numbers to simulate time varying electrical activity the ontological status of the imaginary part of the solution is uncontroversial: the complex numbers in that situation have no ontological status at all and what is present is charges. In that case we're simply using the complex numbers as a convenient notation for a variable and its conjugate. In quantum mechanics one is more easily led to wonder about whether the ontological status of the complex numbers in that theory really can be settled so easily.
>A complex solution can be valid but you never measure a complex number.
I see where you are coming from, and I'm asking this as a genuine question rather than to argue, but what's stopping me from measuring the length and the mass of an object and saying the "length-mass" of it is length + i(mass)? I suppose it isn't useful since complex numbers are not ordered, but aren't "numbers" arbitrary? In measure theory, measures are defined as outputting positive real numbers and +infinity because those happen to align with our intuition about how measures work, but as far as I know, maths(and physics here I guess) does not care about the representation of my quantity which I'm measuring, but it only cares about it's properties.
Nothing stops you from representing the value that way, but when you go to a meter or lay a yardstick against something, you are measuring a real number (or, at the very least, a number which has no complex character to it). "This many ticks on a ruler" or "this many clicks on a clock."
Well, for one thing, for such quantity to make physical sense, both the real part and the imaginary part should be of the same dimension, e.g. "length." Also, the result of a measurement is supposed to come from (be an eigenvalue of) an observable - an operator, and, on the one hand, I think I'd have a hard time conjuring one up; on the other hand, the eigenvalues are "supposed to be" real anyway! So, no, that doesn't work.
> 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.
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.
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.
But you can't really disagree with that. It's wrong to say that physics "needs" any one thing in particular when you can construct it from other things, and use them instead.
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.
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?
yes, specifically doing things without complex numbers breaks if you require composing systems to work like they do in normal quantum mechanics (composing with the tensor product).
If you drop that assumption about how things compose then you can use something like the real matrix representation where 1 is a 2x2 identity matrix and i is some anti-symmetric 2x2 matrix.
Help a lapsed mathematician out here... the complex numbers can be represented as skew-symmetric 2x2 real matrices, right? And this is an isomorphism? So if the 2x2 real skew-symmetric matrices are "the same" as the complex numbers, how do we end up having this issue that comes up in quantum physics where you can only use the actual complex numbers and not the representation?
Any explanation I can give in a comment here will be pretty flawed, if you want more detail than my previous comment I think your best bet is to read the papers at the arxiv links I posted above.
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
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.
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.
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.
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.
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.
In 21st century hindsight, being annoyed by irrational numbers seems a bit odd to me. I mean this very much as an opinion. I actually partially understand where they were coming from; it makes a bit more sense than the 21st century perspective would indicate, but still, obviously, not something we'd agree with today.
Even from a 21st century perspective, I think that the first "two dimensional number" is always going to freak people out and I can see where it's coming from. Imaginary numbers intrinsically involves leaving numbers that can be used to describe the number of apples you have in your hand, and by the time people get there, they've been pretty darned used to numbers looking like that. Real numbers nominally overshoot that too (you can't really have apples in two hands whose size only differs by 10^(-(10^1000))) but people tend to not have their faces rubbed in this until they get a math degree.
Matrices nominally are such numbers too, but they are often presented as shortcuts rather than numbers in and of themselves.
Of course in the 21st century now we have a zoo of these representations and the community as a whole is comfortable with it.... but for any given person I still think that first number that isn't something that can be a number of meters or apples is a shock.
Plus Pythagoreans had a much more religious attitude towards number than any present-day mathematician. In fact I think it's almost misleading to remember them chiefly for their mathematical contributions (many of which are disputed anyway) while they were first and foremost mystic philosophers.
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.
Man, "outside of science" is doing a lot of work here. We also don't see quantum mechanics "outside of science," why would they also not be fundamental? The fact of the matter is that there are A LOT of things we can't explain without complex numbers or quantum mechanics, so people believe that they're fundamental. That's not weird.
Yeah, it definitely is. I'm not denying that they're fundamental, I have no reason to doubt that, I just think it's odd that they are so deeply embedded into everything, simple enough to kind of understand the basic idea of, and yet totally separate from everyday experience.
Like, a construction worker may know hundreds of techniques. A painter uses hundreds of colors. A while many people likely only use four operations on real numbers their whole lives. Math is everywhere, essential for engineering, but almost totally hidden from most people.
Probably just a consequence of how we kind of jumped straight from "experience and guesswork" to "Hire a pro if you want to build a barn and let them use computers" though.
real things heating up and down (i.e not an idealized particle but a thing with volume), the motion of a pendulum, electricity flowing through basically anything (same as heating, when you consider actual volume), these are a few of the things I have worked with where I literally have no option but to use imaginary numbers to model them.
Modelling here means predicting what a change would do, for example if I want to make a robot that can balance a stick upright on its hand like you might do with a broom, I use maths related to pendulums to predict the movement of the stick, which require imaginary numbers to show how moving in a certain direction will cause the pendulum to fall (in what direction & how fast). I must emphasise here that I have done these things with real robots, and they do indeed heat/electricity flows/balance as the maths predicts.
Does this imply your brain is doing the same maths, imaginary numbers and all, when you balance a broom on your hand? Is there an alternate mathematical notation that doesn't include such strange unintuitive things? probably, thats for the mathematicians to figure out. But as an engineer, they definately do work.
We used them in the late 80s and early 90s for pretty fractals on home computers; but for SciFi world-building… it can be done, but I don't think you can really get into those worlds as a reader without already being somewhat familiar with the maths.
Reason I think this is having listened to Greg Egan's Dichronauts, in which spacetime is ++-- with all the counterintuitive hyperbolic rotations that this implies.
Other ways to get i into common use besides that, probably gives equally counterintuitive results if you don't already know the maths; for example, instead of a maximum debt you can take out based on your ability to repay, you get a region which, for certain interest and repayment rates, is the Mandelbrot set.
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.
“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.
He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0]
You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity.
You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1].
Exactly. But I want to point out that the field character of the complex numbers is not some incidental quality which happens to distinguish them from 2-vectors. Its absolutely essential to their mathematical character and usage and it also distinguishes them from other complexes we might want to form that behave in a real number like fashion. For instance, there is no way to form a field over the three vectors. In general, one has to give up more and more structure as the dimensions go up.
I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).
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.
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).
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!?”
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.
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...
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?
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.
Except that this is only true when Newton's laws can be reasonably applied at all. In many (most?) quantum situations Newton's laws are completely nonsensical (while, conversely, in most classical situations QM is plain useless).
I am not an expert, but, from listening to physicists and reading popular works, I thought they generally agreed that physics was radically incomplete.
For instance, the two most powerful physical theories, Quantum Mechanics and General Relativity, contradict each other. Quantum mechanics has no explanation for what the collapse of the wave function means (it's really QM + Collapse) and it cannot account for gravity. General relativity, by contrast, assumes continuous space (which is incompatible with quantization) which leads it to predict point singularities (which is incompatible with the uncertainty principle and the Planck limit for physical distance.)
As I said, I am ignorant, but someone more knowledgeable could expand on this.
The possibility of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy. If an electron interacts with such a "classical object", the state of the latter is, generally speaking, altered. The nature and magnitude of this change depend on the state of the electron, and therefore may serve to characterize it quantitatively...
We have defined "apparatus" as a physical object which is governed, with sufficient accuracy, by classical mechanics. Such, for instance, is a body of large enough mass. However, it must not be supposed that apparatus is necessarily macroscopic. Under certain conditions, the part of apparatus may also be taken by an object which is microscopic, since the idea of "with sufficient accuracy" depends on the actual problem proposed.
Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case [correspondence principle], yet at the same time it requires this limiting case for its own formulation.
Given that Lifschitz wrote that before QED, he did not even begin to understand the modern understanding of QM and the electron. Nor did he see any inkling of QMs replacement, (T)QFTs. QM (and his quote, and your understanding) are nearly 100 years out of date.
The entire quote is nonsense - QED (well after Landau wrote his text) shows that the opening sentence is as valid a Asimov book from the period with the wrong number of moons for various planets.
Classical QM was much more classical than modern QM, which has removed a lot of the weasel words used in the above ("large enough mass," "sufficient degree of accuracy," etc. - none of which were defined in Landau's time and all of which have been greatly extended beyond anything he could see).
A trivially simple example is asking why gold is yellow instead of silver like nearby metals. It's a very obvious property, on any mass and sufficient degree of accuracy, but has no classical explanation (since it's due to the interplay of QM and relativity.) There are tons of things like this where your claim (and Landau's handwaving) fail, so no, QM does not approximate classical here, since classical is wrong and QM is right, even at macro scales.
QM also doesn't occupy a very unusual space - all physical theories had to agree with previous knowledge under overlapping domains. Relativity did. Maxwell did. Thermo did. Stat mech did. And ALL of those (there are plenty more) were before QM. And most of those have had more improvements since then, also agreeing with previous theory on overlapping domains, e.g., QFTs have replaced QM for all modern physics and agree on some things, but go vastly beyond what was possible with QM.
You seem to have missed the whole point (which comes after "yet" at the end of the quote). It has nothing to do with approximation. Also, QFT has not brought anything new in terms of solving the measurement problem (if it needs to be solved at all), so the point stands (just as it did "100 years" ago).
You're right - it's likely a made up issue due to human psych, not anything due to physics, thus it's weird you get hung up on it.
> You seem to have missed the whole point (which comes after "yet" at the end of the quote)
Ok, so you agree the first part of the quote is sufficiently incorrect? Let's invalidate the next part.
Your initial claim was "A classical apparatus is part of the QM framework." It is not. QM can be completely defined (and was done so very early on) without any connection to classical, and it took (and is still taking) effort to show that classical things come from it.
As two examples, the Dirac-von Neumann axioms for QM (from which it all can be derived) are from the early 1930s, and have precisely zero mention of need for any classical physics. If you don't believe it, read them, or download von neumann's book and read it. There are subsequently axiomatic forms of QFTs, TQFTs, and so on, none needing anything more than pure math to define. There's a large collection of research over the past ~100 years with groups poking at different axiom sets or arguing if this or that set is complete, still ongoing (e.g., [2]), but AFAIK, there is no big group that claims QM is not based on axioms at this point. Or QFT or TQFTs (which Atiyah spent significant time axiomatizing before he died [3]). QM can be derived from TQFTs.
Care to show me which axiom in TQFTs is the classical apparatus? Say, as opposed to the zillion other math structures that use the same words to define things which coincidentally didn't match physics? (Unless you're Max Tegmark, for which all math is physics, a fringe view but a powerfully thought out one...)
It's nice when they correspond to nature, but there is zero need for nature to define them. They're pure math, and the agreement with nature has led to the entire "It from Bit" or "Unreasonable Effectiveness of Mathematics" views in physics.
As to some really important classical connections that took a long time to derive, Dyson's 1967 proof that matter is stable (which is an incredibly classical observation) under QM is a really neat result [1]. So the classical connections are not needed to state QM, and even historically the connections were interspersed over time, and most were found long after QM was axiomatized.
So, still claim the "yet" phrase is true? If so, how did D&vN make axioms from which all QM derives?
> They're pure math, and the agreement with nature
Sure, as far as the math is concerned, it appears that there is no need for "classical objects." But physics is not (and has never been) "pure math," which is why Landau and Lifshitz, in particular, keep insisting on the importance of understanding the "physical principles" behind the axioms, whatever they are, and the facts of the theory; and so the "agreement with nature" is all but expected; but in order to see that agreement we need to observe, and we can only make observations by looking at "classical objects"; then, to make a connection back to the theory we need to have a way of making the result of the observation directly available to the framework itself - its "physical content" and its math.
Your claim was "A classical apparatus is part of the QM framework."
I just pointed out it is not, and gave you axioms that define QM, and asked where the classical part is. You ignored, as expected.
>But physics is not (and has never been) "pure math,
This is not clear. I pointed out three pretty solid places that seem to disagree. In fact, a huge chunk of advances over the past 150 years came from where? Pure math. Electron oribital structure - forced by spherical harmonics (we didn't measure the orbits, then write equations - we looked for harmonics, the math game them all, then we experimented and found them all). Spin, forced by the math (relativity + qm had no solutions - to get them forced the representations to have a spin component, forced by pure math, then we found them). Positrons and anti-particles - forced by the math (the equation for electrons had two roots, pure math, at first people ignored the second root, then they decided to look, and found anti-matter). Relativity - forced by the math (the underlying Riemannian geometry is extremely forced - and the "pure math" of it led to a century of predictions, all from the pure math, that later became observed). Noether's theorem underlying all conservation principles - forced by the math (and it's an astounding, very general priciple, devoid of any physical content, but once applied, out pops all sorts of conservation laws, some we had not even observed, all forced by pure math). Quarks, Higgs Boson, nearly all of modern physics - forced by the math, not the other way around.
This list is incredibly extensive, far beyond what is here.
This is why physicists reached the "Unreasonable effectiveness of mathematics" and "It from Bit." More and more of physics is being recast as information theory and computation, since it is becoming more clear that a significant amount of what we call physics and nature is forced by math. Some, like Tegmark as I pointed out, go so far as to claim all math is physics and we just don't see some yet, because our inputs, timeframe, and objervations of multi-verse things are so small. So to claim physics is not math (another of your claims that is too hand wavy and short-sighted to believe) is not demonstrable - it's your belief, one that is slowly but surely giving way to realizing how intertwined they are.
The idea that we simply observe nature than add more terms to model it is long gone. What to look for as pointed to by math, not observation, has led to so many physical discoveries that it is why physicists (and mathematicians, and philosophers) deeply question why is math directing nature, not nature directing math.
Since I provided axioms for QM without your requirement of a "classical apparatus," and you cannot show where in those axioms your "classical apparatus" resides, I guess that part is done now too.
Of course it is. In physics, experiment (observation) remains the sole criterion of truth, no matter what math may be saying. (And I will continue to stress that observations, experiments, and measurements - all are inevitably "classical" by nature, and so in order to draw connections from those back to the underlying physical theory, the latter must include the notion of a classical measurement device.) Also, math is not physics because a mathematical theory includes a lot of scaffolding - things that do not correspond to anything real, and it's a physicist's job to tell the difference.
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.
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.
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?)
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
Thanks for the explanation. In engineering circles (to use an apt word) complex numbers only assist in performing rotation, eg, calculating things such as angle changes or phase changes etc. It sounds to me that it's the same in quantum mechanics.
That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a shorthand trick that permits fast calculations of rotational characteristics.
I'm not sure that they're able to do anything else. A wave is a particle as I understand it, or rather, they are like views into the same thing. But waves have identical characteristics in terms of what is tracked to enable calculations with them.
Eg there's amplitude phase and frequency. Maybe they all have probabilities or are otherwise dynamic, but that's what there at the base level. The complex plane then just takes the role of enabling easier calculation and tracking of state changes.
A wave is a wave whether it is a sine wave on a scope, a wave of pressure through a solid or gas, a light wave, or the path of a particle when viewed as a wave. Or, I'm misunderstanding and there are actually two, or more types of complex numbers.
The main insight of complex numbers is the fact that it relates exponentiation (partial multiplication) to rotation. Take that away then you might as well just use matrix algebra instead.
The idea is that you can “multiply by -1” not just odd or even times, but that you can do so fractionally. You can “flip the direction” partially which corresponds to shortening the original size and adding an “impetus” attribute based on the amount you shortened it by.
The raisin d’être of complex numbers is the derivative theorem of Fourier transforms. In fact, perhaps imaginary numbers should really be renamed “impetus numbers”
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.
Eh, RE: your edit, "multiplication" itself has several different definitions in a vector space. For example, a dot product, outer product, Hadamard product, etc.
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.
> 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.
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.
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.
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?
Its simple. Start with your circuit and add one ancila qubit. Then in your set of basis gates (universal for quantum computation), replace every phase shifting operation with a controlled X rotation targeted on that ancila.
For example, lets just use the cliffords plus arbitrary phase rotation {X,Y,Z,H,R(theta)}. X,Z and H are all real. Y is real up to an irrelevant global phase (if you really want to implement it anyway, just do X,Z on the target and then flip the ancila with an additional X). All that's left to handle is R(theta). Map R(theta) into a (real) controlled rotation in X (to wit: C-X(theta)). Thus we have a set of gates, universal for quantum computation, using only real numbers.
If you don't believe in arbitrary rotations X(theta) without intermediate imaginary operations, just pretend I used the T gate instead to extend the Cliffords. Either way you can construct a set which is universal for QC with only real numbers.
Why should it work? Simple. In your math, if you replace every i with |i> your equations are still the same. You've just substituted one squiggle on the page for another that works the same way. The Riemann sphere is just like the Bloch sphere. Complex numbers are a qubit the universe gives you for free.
I am super confused. Why can't I take this purely classical circuit and run it on a classical computer? Somewhere, there should be some blowup into exponential time?
The state space still grows like 2^n in the number of qubits. Again, all this mapping does is rename some variables from "imaginary number" to "degree of freedom on an imaginary system". But, the computer doesn't care what you call your vars.
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.
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.
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.
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.
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.
> 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.
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).
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.
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
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[ 2.1 ms ] story [ 246 ms ] threadReading 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-...
I think you couldn't have put it better. I'm just a physicist, though.
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.
It's not imprecise. It reproduces experimental results from theory, so it's in fact the most precise approach in existence.
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.
But doing splits and advanced acrobatics to get rid of infinities always felt like a hack (Feynman felt the same do at least I am not alone :))
This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.
Like, ok, we get testable answers and they match experiments but also this is _so_ hacky and I can't shake the feeling that one day someone will come along and show that there's some reason why these bad assumptions work out fine. You know, like how "to find the Schwarzschild radius for a black hole of known mass, calculate the radius at which the escape velocity is equal to the speed of light" gives the correct answer even though the theory implied by this method is naive and wrong.
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.
I see where you are coming from, and I'm asking this as a genuine question rather than to argue, but what's stopping me from measuring the length and the mass of an object and saying the "length-mass" of it is length + i(mass)? I suppose it isn't useful since complex numbers are not ordered, but aren't "numbers" arbitrary? In measure theory, measures are defined as outputting positive real numbers and +infinity because those happen to align with our intuition about how measures work, but as far as I know, maths(and physics here I guess) does not care about the representation of my quantity which I'm measuring, but it only cares about it's properties.
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.
>Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”
E.g.: Are nationals sufficient? Integers? Finite integers?
Because if real skew symmetric matrices count, then a harmonic oscillator also inescapably uses imaginary numbers.
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.
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?
If you drop that assumption about how things compose then you can use something like the real matrix representation where 1 is a 2x2 identity matrix and i is some anti-symmetric 2x2 matrix.
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.
Apparently the existence of irrational numbers was a shock to Pythagoreans. There may be also people unhappy with transcendental numbers.
Even from a 21st century perspective, I think that the first "two dimensional number" is always going to freak people out and I can see where it's coming from. Imaginary numbers intrinsically involves leaving numbers that can be used to describe the number of apples you have in your hand, and by the time people get there, they've been pretty darned used to numbers looking like that. Real numbers nominally overshoot that too (you can't really have apples in two hands whose size only differs by 10^(-(10^1000))) but people tend to not have their faces rubbed in this until they get a math degree.
Matrices nominally are such numbers too, but they are often presented as shortcuts rather than numbers in and of themselves.
Of course in the 21st century now we have a zoo of these representations and the community as a whole is comfortable with it.... but for any given person I still think that first number that isn't something that can be a number of meters or apples is a shock.
That’s what Pythagoreans thought about irrational numbers, I guess. You don’t need them to denote a fraction of an apple.
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.
Like, a construction worker may know hundreds of techniques. A painter uses hundreds of colors. A while many people likely only use four operations on real numbers their whole lives. Math is everywhere, essential for engineering, but almost totally hidden from most people.
Probably just a consequence of how we kind of jumped straight from "experience and guesswork" to "Hire a pro if you want to build a barn and let them use computers" though.
Modelling here means predicting what a change would do, for example if I want to make a robot that can balance a stick upright on its hand like you might do with a broom, I use maths related to pendulums to predict the movement of the stick, which require imaginary numbers to show how moving in a certain direction will cause the pendulum to fall (in what direction & how fast). I must emphasise here that I have done these things with real robots, and they do indeed heat/electricity flows/balance as the maths predicts.
Does this imply your brain is doing the same maths, imaginary numbers and all, when you balance a broom on your hand? Is there an alternate mathematical notation that doesn't include such strange unintuitive things? probably, thats for the mathematicians to figure out. But as an engineer, they definately do work.
Reason I think this is having listened to Greg Egan's Dichronauts, in which spacetime is ++-- with all the counterintuitive hyperbolic rotations that this implies.
Other ways to get i into common use besides that, probably gives equally counterintuitive results if you don't already know the maths; for example, instead of a maximum debt you can take out based on your ability to repay, you get a region which, for certain interest and repayment rates, is the Mandelbrot set.
Some controversy does exist. NJ Wildberger is most famous for not believing in the real numbers.
Similar to "imaginary" numbers, it's curious how influential/distracting the terms are, though.
You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity.
You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1].
[0] https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra
I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).
[0] https://en.wikipedia.org/wiki/Field_(mathematics)
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!?”
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
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...
Quantum mechanical velocity is complicated as well.
Are we talking about different things? Is one of us way off the mark? Or are these two angles of the same phenomenon?
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.
> The mechanisms of Newton’s laws are first order terms of a proper QM solution.
All physical theories are “fundamentally wrong.”
> You can’t
A classical apparatus is part of the QM framework.
Ergo: the commenter doesn’t know what they are talking about.
> A classical apparatus is part of the QM framework
This sounds like nonsense. Care to elaborate, preferably with a link to a good source?
For instance, the two most powerful physical theories, Quantum Mechanics and General Relativity, contradict each other. Quantum mechanics has no explanation for what the collapse of the wave function means (it's really QM + Collapse) and it cannot account for gravity. General relativity, by contrast, assumes continuous space (which is incompatible with quantization) which leads it to predict point singularities (which is incompatible with the uncertainty principle and the Planck limit for physical distance.)
As I said, I am ignorant, but someone more knowledgeable could expand on this.
The possibility of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy. If an electron interacts with such a "classical object", the state of the latter is, generally speaking, altered. The nature and magnitude of this change depend on the state of the electron, and therefore may serve to characterize it quantitatively...
We have defined "apparatus" as a physical object which is governed, with sufficient accuracy, by classical mechanics. Such, for instance, is a body of large enough mass. However, it must not be supposed that apparatus is necessarily macroscopic. Under certain conditions, the part of apparatus may also be taken by an object which is microscopic, since the idea of "with sufficient accuracy" depends on the actual problem proposed.
Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case [correspondence principle], yet at the same time it requires this limiting case for its own formulation.
The entire quote is nonsense - QED (well after Landau wrote his text) shows that the opening sentence is as valid a Asimov book from the period with the wrong number of moons for various planets.
Classical QM was much more classical than modern QM, which has removed a lot of the weasel words used in the above ("large enough mass," "sufficient degree of accuracy," etc. - none of which were defined in Landau's time and all of which have been greatly extended beyond anything he could see).
A trivially simple example is asking why gold is yellow instead of silver like nearby metals. It's a very obvious property, on any mass and sufficient degree of accuracy, but has no classical explanation (since it's due to the interplay of QM and relativity.) There are tons of things like this where your claim (and Landau's handwaving) fail, so no, QM does not approximate classical here, since classical is wrong and QM is right, even at macro scales.
QM also doesn't occupy a very unusual space - all physical theories had to agree with previous knowledge under overlapping domains. Relativity did. Maxwell did. Thermo did. Stat mech did. And ALL of those (there are plenty more) were before QM. And most of those have had more improvements since then, also agreeing with previous theory on overlapping domains, e.g., QFTs have replaced QM for all modern physics and agree on some things, but go vastly beyond what was possible with QM.
QM is not special here.
You seem to have missed the whole point (which comes after "yet" at the end of the quote). It has nothing to do with approximation. Also, QFT has not brought anything new in terms of solving the measurement problem (if it needs to be solved at all), so the point stands (just as it did "100 years" ago).
You're right - it's likely a made up issue due to human psych, not anything due to physics, thus it's weird you get hung up on it.
> You seem to have missed the whole point (which comes after "yet" at the end of the quote)
Ok, so you agree the first part of the quote is sufficiently incorrect? Let's invalidate the next part.
Your initial claim was "A classical apparatus is part of the QM framework." It is not. QM can be completely defined (and was done so very early on) without any connection to classical, and it took (and is still taking) effort to show that classical things come from it.
As two examples, the Dirac-von Neumann axioms for QM (from which it all can be derived) are from the early 1930s, and have precisely zero mention of need for any classical physics. If you don't believe it, read them, or download von neumann's book and read it. There are subsequently axiomatic forms of QFTs, TQFTs, and so on, none needing anything more than pure math to define. There's a large collection of research over the past ~100 years with groups poking at different axiom sets or arguing if this or that set is complete, still ongoing (e.g., [2]), but AFAIK, there is no big group that claims QM is not based on axioms at this point. Or QFT or TQFTs (which Atiyah spent significant time axiomatizing before he died [3]). QM can be derived from TQFTs.
Care to show me which axiom in TQFTs is the classical apparatus? Say, as opposed to the zillion other math structures that use the same words to define things which coincidentally didn't match physics? (Unless you're Max Tegmark, for which all math is physics, a fringe view but a powerfully thought out one...)
It's nice when they correspond to nature, but there is zero need for nature to define them. They're pure math, and the agreement with nature has led to the entire "It from Bit" or "Unreasonable Effectiveness of Mathematics" views in physics.
As to some really important classical connections that took a long time to derive, Dyson's 1967 proof that matter is stable (which is an incredibly classical observation) under QM is a really neat result [1]. So the classical connections are not needed to state QM, and even historically the connections were interspersed over time, and most were found long after QM was axiomatized.
So, still claim the "yet" phrase is true? If so, how did D&vN make axioms from which all QM derives?
[1] https://fisherp.scripts.mit.edu/wordpress/wp-content/uploads...
[2] https://link.springer.com/article/10.1007/s10701-008-9230-4
[3] https://en.wikipedia.org/wiki/Topological_quantum_field_theo...
Sure, as far as the math is concerned, it appears that there is no need for "classical objects." But physics is not (and has never been) "pure math," which is why Landau and Lifshitz, in particular, keep insisting on the importance of understanding the "physical principles" behind the axioms, whatever they are, and the facts of the theory; and so the "agreement with nature" is all but expected; but in order to see that agreement we need to observe, and we can only make observations by looking at "classical objects"; then, to make a connection back to the theory we need to have a way of making the result of the observation directly available to the framework itself - its "physical content" and its math.
I just pointed out it is not, and gave you axioms that define QM, and asked where the classical part is. You ignored, as expected.
>But physics is not (and has never been) "pure math,
This is not clear. I pointed out three pretty solid places that seem to disagree. In fact, a huge chunk of advances over the past 150 years came from where? Pure math. Electron oribital structure - forced by spherical harmonics (we didn't measure the orbits, then write equations - we looked for harmonics, the math game them all, then we experimented and found them all). Spin, forced by the math (relativity + qm had no solutions - to get them forced the representations to have a spin component, forced by pure math, then we found them). Positrons and anti-particles - forced by the math (the equation for electrons had two roots, pure math, at first people ignored the second root, then they decided to look, and found anti-matter). Relativity - forced by the math (the underlying Riemannian geometry is extremely forced - and the "pure math" of it led to a century of predictions, all from the pure math, that later became observed). Noether's theorem underlying all conservation principles - forced by the math (and it's an astounding, very general priciple, devoid of any physical content, but once applied, out pops all sorts of conservation laws, some we had not even observed, all forced by pure math). Quarks, Higgs Boson, nearly all of modern physics - forced by the math, not the other way around.
This list is incredibly extensive, far beyond what is here.
This is why physicists reached the "Unreasonable effectiveness of mathematics" and "It from Bit." More and more of physics is being recast as information theory and computation, since it is becoming more clear that a significant amount of what we call physics and nature is forced by math. Some, like Tegmark as I pointed out, go so far as to claim all math is physics and we just don't see some yet, because our inputs, timeframe, and objervations of multi-verse things are so small. So to claim physics is not math (another of your claims that is too hand wavy and short-sighted to believe) is not demonstrable - it's your belief, one that is slowly but surely giving way to realizing how intertwined they are.
The idea that we simply observe nature than add more terms to model it is long gone. What to look for as pointed to by math, not observation, has led to so many physical discoveries that it is why physicists (and mathematicians, and philosophers) deeply question why is math directing nature, not nature directing math.
Since I provided axioms for QM without your requirement of a "classical apparatus," and you cannot show where in those axioms your "classical apparatus" resides, I guess that part is done now too.
Of course it is. In physics, experiment (observation) remains the sole criterion of truth, no matter what math may be saying. (And I will continue to stress that observations, experiments, and measurements - all are inevitably "classical" by nature, and so in order to draw connections from those back to the underlying physical theory, the latter must include the notion of a classical measurement device.) Also, math is not physics because a mathematical theory includes a lot of scaffolding - things that do not correspond to anything real, and it's a physicist's job to tell the difference.
[0] https://www.youtube.com/watch?v=_Sm7SNlNUOI&list=PLOFVFbzrQ4...
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.
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?)
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
That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a shorthand trick that permits fast calculations of rotational characteristics.
I'm not sure that they're able to do anything else. A wave is a particle as I understand it, or rather, they are like views into the same thing. But waves have identical characteristics in terms of what is tracked to enable calculations with them.
Eg there's amplitude phase and frequency. Maybe they all have probabilities or are otherwise dynamic, but that's what there at the base level. The complex plane then just takes the role of enabling easier calculation and tracking of state changes.
A wave is a wave whether it is a sine wave on a scope, a wave of pressure through a solid or gas, a light wave, or the path of a particle when viewed as a wave. Or, I'm misunderstanding and there are actually two, or more types of complex numbers.
The idea is that you can “multiply by -1” not just odd or even times, but that you can do so fractionally. You can “flip the direction” partially which corresponds to shortening the original size and adding an “impetus” attribute based on the amount you shortened it by.
The raisin d’être of complex numbers is the derivative theorem of Fourier transforms. In fact, perhaps imaginary numbers should really be renamed “impetus numbers”
So I guess the fractionality is used by quantum mechanics as a simple convenience that allows probabilities to stay close to related attributes.
https://acko.net/blog/how-to-fold-a-julia-fractal/
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.
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.
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.
Put another way, complex numbers are a qubit the universe gives you for free.
For example, lets just use the cliffords plus arbitrary phase rotation {X,Y,Z,H,R(theta)}. X,Z and H are all real. Y is real up to an irrelevant global phase (if you really want to implement it anyway, just do X,Z on the target and then flip the ancila with an additional X). All that's left to handle is R(theta). Map R(theta) into a (real) controlled rotation in X (to wit: C-X(theta)). Thus we have a set of gates, universal for quantum computation, using only real numbers.
If you don't believe in arbitrary rotations X(theta) without intermediate imaginary operations, just pretend I used the T gate instead to extend the Cliffords. Either way you can construct a set which is universal for QC with only real numbers.
Why should it work? Simple. In your math, if you replace every i with |i> your equations are still the same. You've just substituted one squiggle on the page for another that works the same way. The Riemann sphere is just like the Bloch sphere. Complex numbers are a qubit the universe gives you for free.
[0]: https://www.youtube.com/watch?v=CdwxpSInhvU
[1]:https://www.youtube.com/watch?v=M12CJIuX8D4
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.
Stop calling them imaginary.
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.
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).
Why are amplitudes complex?
https://scottaaronson.blog/?p=4021
https://www.youtube.com/watch?v=cUzklzVXJwo
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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