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Quantum physics falls apart without imaginary numbers

scientificamerican.com

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Re: Quantum physics falls apart without imaginary numbers

#51

Post by nathan_compton »

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.

"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.

Re: Quantum physics falls apart without imaginary numbers

#52

Post by Koshkin »

lanza wrote: post #42
fullstackchris wrote: post #30

Earlier quoted context omitted.

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 imag…

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…

> Classical mechanics are fundamentally wrong.

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.

Re: Quantum physics falls apart without imaginary numbers

#53

Post by ordu »

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

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

Re: Quantum physics falls apart without imaginary numbers

#55

Post by dr_dshiv »

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.

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

Re: Quantum physics falls apart without imaginary numbers

#56

Post by SideQuark »

Koshkin wrote: post #52
lanza wrote: post #42

Earlier quoted context omitted.

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…

> Classical mechanics are fundamentally wrong. 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.

It is not known if all physical theories are fundamentally wrong.

> A classical apparatus is part of the QM framework

This sounds like nonsense. Care to elaborate, preferably with a link to a good source?

Re: Quantum physics falls apart without imaginary numbers

#57

Post by pcrh »

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.

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.

Re: Quantum physics falls apart without imaginary numbers

#58

Post by mort96 »

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.

A vector field?

Re: Quantum physics falls apart without imaginary numbers

#59

Post by eternityforest »

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…

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.

Re: Quantum physics falls apart without imaginary numbers

#60

Post by jiggawatts »

sebzim4500 wrote: post #31
whatshisface wrote: post #20

Earlier quoted context omitted.

"Quantum physics falls apart without imaginary numbers." > Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”

By that logic, it doesn't even need real numbers. Just do everything with cauchy sequences of rationals.

Something I've always wondered is: what is the weakest algebra that could be used to model physics?

E.g.: Are nationals sufficient? Integers? Finite integers?

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