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Quantum theory based on real numbers can be experimentally falsified

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Re: Quantum theory based on real numbers can be experimentally falsified

#81

Post by l33tman »

himinlomax wrote: post #77

This reminds me of something that was once linked on HN but that I can't remember enough of to find again. The thesis was that some of the iconic quantum weirdness™ simply disappears when you just dispense with taking the real part (or the norm) of the wave function as a final step and instead just consider the complex value. IIRC this seemed to made the double-slit experiment way more straightforward. Does that ring…

If you don't take the norm, you're essentially not making a "measurement" and then QM is linear and simple. It's also non-predictive of reality :)

Re: Quantum theory based on real numbers can be experimentally falsified

#83

Post by [deleted] »

matt123456789 wrote: post #33
analog31 wrote: post #3

Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.

Most reals are not computable anyway: https://arxiv.org/pdf/math/0411418.pdf

[deleted]

Re: Quantum theory based on real numbers can be experimentally falsified

#84

Post by Pyramus »

katmannthree wrote: post #79
Pyramus wrote: post #73

Earlier quoted context omitted.

I think you misunderstood what parent was saying. There is no evidence the real numbers are based in physical reality. As parent was saying, it doesn't make sense to be able to store infinite information in a single number, or even, say, store all of human knowledge in a single number. Generations of physicists have come to the same conclusion [1] and most professional physicists agree. It's just that (a) the real nu…

I do understand that argument, I just remain unmoved by it. Watch this, I'm about to show you a complete finite representation of an irrational transcendental number: π. That took literally three lines to represent and then an additional half page's worth that I'll skip explaining how to calculate a numerical value to however much precision you have time and space for . Now granted, there is an underlying assumption…

> I'm about to show you a complete finite representation of an irrational transcendental number: π.

You just provided a great argument that π, and many other real numbers, should be part of the 'alternative number system', because they can constructed, or because they represent a finite amount of information. I agree!

> That is literally the point of the reals, that they are an infinitely dense field.

You are arguing that an alternative number system should be 'infinitely dense', and I agree. But take e.g. the finite/constructive reals [1, 2], they are still 'infinitely dense'.

> They were literally created to model reality.

That's exactly my point. Maybe approaching it from the point of view of 'what are the limitations of this model?' is helpful. Also see the discussion in [2].

This is not an argument whether real numbers are useful, a good model, or interesting (there is not doubt they are all three).

[1] https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...

[2] https://news.ycombinator.com/item?id=14080024

Re: Quantum theory based on real numbers can be experimentally falsified

#85

Post by asxd »

Sniffnoy wrote: post #46
asxd wrote: post #30

Earlier quoted context omitted.

This was my first thought on seeing the title. Complex numbers are just vectors with special behavior for some operations, right? I haven't read through the paper, but this statement from the abstract confuses me: > Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make differe…

"Quantum theory based on real numbers" means a specific thing -- quantum mechanics with real amplitudes (and real anything-else-that-would-follow-from-that). It doesn't mean just any way of representing quantum mechanics with real numbers. Of course you can represent quantum mechanics with real numbers, for the reason you say; but for that very reason, that isn't what anyone means by "quantum theory based on real num…

I appreciate you're able to see the quandary. I think the crux of what you said is in your first sentence:

> "Quantum theory based on real numbers" means a specific thing -- quantum mechanics with real amplitudes (and real anything-else-that-would-follow-from-that)

I'm familiar with complex math as far as remedial DSP and electrical engineering goes, so this may be over my head. I'm not sure what a real amplitude is, since generally when I hear "complex number", my head thinks "compact way to represent a frequency, amplitude and phase", all of which are real. It seems like I was reeled in with familiar-sounding terminology that may in fact have a deeper meaning in this context.

Re: Quantum theory based on real numbers can be experimentally falsified

#86

Post by asxd »

itchyjunk wrote: post #40
asxd wrote: post #30

Earlier quoted context omitted.

This was my first thought on seeing the title. Complex numbers are just vectors with special behavior for some operations, right? I haven't read through the paper, but this statement from the abstract confuses me: > Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make differe…

Although I like the complex numbers and two dimensional real numbers being compared and contrasted (yes, R^2 with vector multiplication and and complex number can be thought of as representing the same thing), I think this way of thinking misses that we also have a field of complex numbers. Point being R^2 isn't a field but C is. If i remember, you don't get any other fields past this. No R^3,..,R^n. I also think thi…

"Field" is a term I've heard come up again and again since college, in engineering-adjacent math. Over the years, I've occasionally looked it up and tried to understand the importance, but I've never found any literature that made much sense to my admittedly short-sighted mind. I'm putting this bluntly (and sincerely), but this seems like a decent time to ask a question I should have asked long ago during college: What is a field and why do we care?

Re: Quantum theory based on real numbers can be experimentally falsified

#87

Post by magicalhippo »

Lucien Hardy wrote a paper[1] showing how one "naturally" ends up with a quantum theory by demanding a few reasonable axioms. The paper also goes into how this implies complex numbers (and rules out quaternions).

Scott Aaronson has a more accessible (and humorous) article on it here[2].

Entanglement has been shown to be intimately linked to this[3] result, which is interesting given the experimental evidence[4] for entanglement.

Not my field, but I found this interesting at least.

[1]: https://arxiv.org/abs/quant-ph/0101012

[2]: https://www.scottaaronson.com/democritus/lec9.html

[3]: https://arxiv.org/abs/0911.0695

[4]: https://en.wikipedia.org/wiki/Quantum_entanglement#Notable_e...

Re: Quantum theory based on real numbers can be experimentally falsified

#88

Post by mr_mitm »

magicalhippo wrote: post #87

Lucien Hardy wrote a paper[1] showing how one "naturally" ends up with a quantum theory by demanding a few reasonable axioms. The paper also goes into how this implies complex numbers (and rules out quaternions). Scott Aaronson has a more accessible (and humorous) article on it here[2]. Entanglement has been shown to be intimately linked to this[3] result, which is interesting given the experimental evidence[4] for e…

> x^n + y^n = z^n

> There's a cute little fact -- unfortunately I won't have time to prove it in class -- that the above equation has nontrivial integer solutions when n=1 or n=2, but not for any larger integers n.

I love this type of humour.

Re: Quantum theory based on real numbers can be experimentally falsified

#89

Post by a9h74j »

a9h74j wrote: post #45
asxd wrote: post #30

Earlier quoted context omitted.

This was my first thought on seeing the title. Complex numbers are just vectors with special behavior for some operations, right? I haven't read through the paper, but this statement from the abstract confuses me: > Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make differe…

> Complex numbers are just vectors with special behavior for some operations, right? Decades since I took Complex Analysis, but: Not if you care about poles, zeros, residuals, cuts, conformal mappings, multiple layers of some sort overlaying the same point on the complex plane. It goes way beyond "declaring two variables" vs. "declaring a struct containing two variables." Not just a representation issue.

EDIT: To add a simple analogy: Is a parabola only x^2 -- just another polynominal -- or is a parabola a conic section first and foremost?

polynominal in x == "flatland" view

Re: Quantum theory based on real numbers can be experimentally falsified

#90

Post by seeekr »

mr_mitm wrote: post #88
magicalhippo wrote: post #87

Lucien Hardy wrote a paper[1] showing how one "naturally" ends up with a quantum theory by demanding a few reasonable axioms. The paper also goes into how this implies complex numbers (and rules out quaternions). Scott Aaronson has a more accessible (and humorous) article on it here[2]. Entanglement has been shown to be intimately linked to this[3] result, which is interesting given the experimental evidence[4] for e…

> x^n + y^n = z^n > There's a cute little fact -- unfortunately I won't have time to prove it in class -- that the above equation has nontrivial integer solutions when n=1 or n=2, but not for any larger integers n. I love this type of humour.

Care to explain? My maths are failing me.
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