Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Quantum physics falls apart without imaginary numbers

scientificamerican.com

Post reply on HN
Print view View original

21–30 of 166 posts

Re: Quantum physics falls apart without imaginary numbers

#21

Post by kgwgk »

mjhay wrote: post #14
CottonMcKnight wrote: post #12

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…

> Of course the "real" numbers have never been controversial

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

Re: Quantum physics falls apart without imaginary numbers

#22

Post by adammarples »

nathan_compton wrote: post #16
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…

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

.

Re: Quantum physics falls apart without imaginary numbers

#25

Post by contravariant »

sebzim4500 wrote: post #10
whatshisface wrote: post #5

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.

Did someone claim otherwise?

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

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

Re: Quantum physics falls apart without imaginary numbers

#27

Post by lanstin »

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…

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.

Re: Quantum physics falls apart without imaginary numbers

#29

Post by reeboo »

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…

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

Re: Quantum physics falls apart without imaginary numbers

#30

Post by fullstackchris »

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…

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...

Post reply on HN