Quantum physics falls apart without imaginary numbers • Hacker News Board

Re: Quantum physics falls apart without imaginary numbers

#91

Post by IIAOPSW » Tue, Apr 25, 2023, 2:08 PM UTC

andrew_eu wrote: post #15

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 the…

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.

Re: Quantum physics falls apart without imaginary numbers

#92

Post by crazygringo » Tue, Apr 25, 2023, 2:16 PM UTC

patrick451 wrote: post #34

> 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 imag…

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).

Re: Quantum physics falls apart without imaginary numbers

#93

Post by thechao » Tue, Apr 25, 2023, 2:17 PM UTC

rprenger wrote: post #7

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.

Re: Quantum physics falls apart without imaginary numbers

#94

Post by p_j_w » Tue, Apr 25, 2023, 2:19 PM UTC

eternityforest wrote: post #59

mjhay wrote: post #14

Earlier quoted context omitted.

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…

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 num…

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.

Re: Quantum physics falls apart without imaginary numbers

#95

Post by computerfriend » Tue, Apr 25, 2023, 2:20 PM UTC

Robotbeat wrote: post #82

andrew_eu wrote: post #15

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 the…

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. G…

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.

Re: Quantum physics falls apart without imaginary numbers

#96

Post by wrycoder » Tue, Apr 25, 2023, 2:21 PM UTC

bmacho wrote: post #48

retrocryptid wrote: post #3

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.

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.)

But, how would one handle positive and negative charge?

Re: Quantum physics falls apart without imaginary numbers

#97

Post by thatnerd » Tue, Apr 25, 2023, 2:21 PM UTC

C-x_C-f wrote: post #70

nathan_compton wrote: post #17

Earlier quoted context omitted.

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

> 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.

That's always been my take too.

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.

Re: Quantum physics falls apart without imaginary numbers

#98

Post by fsloth » Tue, Apr 25, 2023, 2:22 PM UTC

tehsauce wrote: post #13

“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…

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!?”

Re: Quantum physics falls apart without imaginary numbers

#99

Post by RugnirViking » Tue, Apr 25, 2023, 2:23 PM UTC

eternityforest wrote: post #59

mjhay wrote: post #14

Earlier quoted context omitted.

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…

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 num…

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.

Re: Quantum physics falls apart without imaginary numbers

#100

Post by dxuh » Tue, Apr 25, 2023, 2:24 PM UTC

andrew_eu wrote: post #15

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 the…

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.