Quantum physics falls apart without imaginary numbers • Hacker News Board

Re: Quantum physics falls apart without imaginary numbers

#131

Post by SideQuark » Tue, Apr 25, 2023, 4:50 PM UTC

Koshkin wrote: post #79

SideQuark wrote: post #56

Earlier quoted context omitted.

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 ele…

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 QM (and his quote, and your understanding) are nearly 100 years out of date.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.

QM is not special here.

Re: Quantum physics falls apart without imaginary numbers

#132

Post by lacker » Tue, Apr 25, 2023, 5:05 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…

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.

Re: Quantum physics falls apart without imaginary numbers

#133

Post by zuminator » Tue, Apr 25, 2023, 5:08 PM UTC

dbelford wrote: post #130

zuminator wrote: post #125

Earlier quoted context omitted.

You keep saying spiral but in a 2d spiral the value of |y| would be increasing as well. So if I'm understanding you correctly I think you mean helical ? As in: https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259e...

The Heyser spiral or Heyser corkscrew seems to be a common name for this type of plot. https://www.google.com/search?q=heyser+spiral And it connects circles, e/euler's formula, and sin/cos is a visually grokkable way.

Ah, well that's fair! In any case though "spiral" is a pretty general term; I'm really just trying to get a sense of the shape the op was referring to, assumedly not something like an Archimedean spiral. Thank you for that idenfication.

Re: Quantum physics falls apart without imaginary numbers

#134

Post by wadd1e » Tue, Apr 25, 2023, 5:39 PM UTC

nathan_compton wrote: post #129

computerfriend wrote: post #104

Earlier quoted context omitted.

Not really! Often you get a complex solution and both the real and imaginary components are valid.

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 conve…

>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.

Re: Quantum physics falls apart without imaginary numbers

#135

Post by Koshkin » Tue, Apr 25, 2023, 5:43 PM UTC

SideQuark wrote: post #131

Koshkin wrote: post #79

Earlier quoted context omitted.

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 ele…

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…

> all physical theories had to agree

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).

Re: Quantum physics falls apart without imaginary numbers

#136

Post by Koshkin » Tue, Apr 25, 2023, 6:00 PM UTC

TechnicolorByte wrote: post #50

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…

Incredible analogy! The closer is a keeper: > 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).

Re: Quantum physics falls apart without imaginary numbers

#137

Post by nathan_compton » Tue, Apr 25, 2023, 6:09 PM UTC

wadd1e wrote: post #134

nathan_compton wrote: post #129

Earlier quoted context omitted.

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 conve…

>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, mea…

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."

Re: Quantum physics falls apart without imaginary numbers

#138

Post by bollu » Tue, Apr 25, 2023, 6:10 PM UTC

IIAOPSW wrote: post #114

bollu wrote: post #109

Earlier quoted context omitted.

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 an…

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?

Re: Quantum physics falls apart without imaginary numbers

#139

Post by Koshkin » Tue, Apr 25, 2023, 6:16 PM UTC

wadd1e wrote: post #134

nathan_compton wrote: post #129

Earlier quoted context omitted.

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 conve…

>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, mea…

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.

Re: Quantum physics falls apart without imaginary numbers

#140

Post by detrites » Tue, Apr 25, 2023, 6:19 PM UTC

bottom999mottob wrote: post #78

detrites wrote: post #72

Earlier quoted context omitted.

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 par…

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.