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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

#71

Post by C-x_C-f »

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.

> 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.

Re: Quantum physics falls apart without imaginary numbers

#72

Post by detrites »

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…

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?)

Re: Quantum physics falls apart without imaginary numbers

#73

Post by C-x_C-f »

mort96 wrote: post #58
nathan_compton wrote: post #16

Earlier quoted context omitted.

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

A vector field?

It's a different concept [0], regrettably the word "field" is vastly overworked in math (and physics)

[0] https://en.wikipedia.org/wiki/Field_(mathematics)

Re: Quantum physics falls apart without imaginary numbers

#74

Post by C-x_C-f »

jerf wrote: post #33
kgwgk wrote: post #21

Earlier quoted context omitted.

> 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.

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…

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.

Re: Quantum physics falls apart without imaginary numbers

#75

Post by TRiG_Ireland »

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…

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

Re: Quantum physics falls apart without imaginary numbers

#76

Post by scotty79 »

C-x_C-f wrote: post #71
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.

> 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)…

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.

Re: Quantum physics falls apart without imaginary numbers

#77

Post by Zuider »

SideQuark wrote: post #56
Koshkin wrote: post #52

Earlier quoted context omitted.

> 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?

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.

Re: Quantum physics falls apart without imaginary numbers

#78

Post by bottom999mottob »

detrites wrote: post #72
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…

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

Re: Quantum physics falls apart without imaginary numbers

#79

Post by Koshkin »

SideQuark wrote: post #56
Koshkin wrote: post #52

Earlier quoted context omitted.

> 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?

From Quantum Mechanics by Landau and Lifshitz:

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.

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