Quantum theory based on real numbers can be experimentally falsified • Hacker News Board

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

#121

Post by wruza » Thu, Dec 23, 2021, 1:00 AM UTC

midjji wrote: post #114

wruza wrote: post #59

Earlier quoted context omitted.

But when you remove or take these fluctuations into account, you’re still left with an error. This rational model has no chance to ever be correct computationally, unless you cheat and add more detailed ratio every time you see a loop. Also, how exactly will you define rational pi? Let’s start with 3/1, why go any further. If it doesn’t represent reality (draw a circle and measure it with a string), well, strings hav…

What you are proposing is essentially using a different number basis, and the simplest applicable one would be a basis which expresses numbers using whole numbers{...,-1,0,1,2...} times pi. This allows you to exactly express pi as just 1, and all such using only whole numbers which is nice. It also unfortunately means that simple things like y= 1_{decimal} x would have to be expressed using the relative to basis tran…

I though about a slightly different thing: Fourier series with rational coefficients, which I believe is not the same as simply Q times pi. Sin(x) has nothing to do with pi, apart from the fact it becomes 0 at pi times n, which seems irrelevant. It was based on an assumption that the nature is all-waves, so these components would pack our “synthetic, infinite” R (which we try to get rid of itt) naturally into these waves and then the computation would be left with just coefficients and their relations. The problem of iterations disappears, because instead of collecting a decimal error, we would collect more and more sines, as if they were fundamental, like sort of ‘i’, but many of them. But if you speak and think only sines, then the problem of unfolding them doesn’t exist. sin 3/4x + 1/7 sin 1/3x would be a simplest answer (an “anyternion”), probably written as (1,3/4)+(1/7,1/3) because these are the irreducible resulting wave parameters and sin is just an implied concept.

Anyway, it’s a stupid layman’s theory. I’m pretty sure that a parallel/on-demand digit computation in R-based models is much easier than trying to get rid of philosophically infinite things which in practice do not matter that much.

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

#122

Post by Sniffnoy » Thu, Dec 23, 2021, 3:21 AM UTC

asxd wrote: post #85

Sniffnoy wrote: post #46

Earlier quoted context omitted.

"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,…

"Amplitudes" here refers to the thing that QM uses instead of probabilities, not any other sort of amplitude.

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

#123

Post by dreamcompiler » Thu, Dec 23, 2021, 5:06 AM UTC

kgwgk wrote: post #103

phkahler wrote: post #99

Earlier quoted context omitted.

>> complex numbers are exceedingly handy for wave functions, as any EE can attest. Because they encode phase information. Also because they come about in the solution of differential equations. Physicists often talk about amplitudes, but I never hear them talk about phase. There was one paper that I can't find, complete with a diagram that suggested (to me) that phase was determining quite a bit.

A wave function describes - usually at least - the quantum state of an isolated quantum system. The phase has no physical meaning. The relative phase between wave functions could mean something... but not if the systems are isolated.

Phase doesn't really matter much in EE for an isolated sinusoid either. But when you compare the phase of a transmitted signal to a local reference, or you compare the phases of the sinusoids in an FFT to one another, or you're trying to synchronize the Texas electrical grid to the rest of the country, phase means a lot. Phases are almost always only useful in a relative sense.

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

#124

Post by rstuart4133 » Thu, Dec 23, 2021, 6:18 AM UTC

asxd wrote: post #30

doubleunplussed wrote: post #2

Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc. Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the…

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…

I'm no an expert in complex numbers, or quantum theory or a heavy user of them. But I am a computer programmer.

To a computer programmer, the fundamental difference between complex numbers is they can express infinite repetition succinctly. Now I try to write down what that means precisely, it's hard. An example of the effect is the polynomial for sin() is infinite, or you can express it as sin(x) = (e^(ix) - e^(-ix))/2i using complex numbers.

To a computer programmer who makes his living from writing down formulas, the difference between having to write an infinite amount of code to express a concept or just use 20 characters above is profound. Different people find their profundity in different places, I guess - but this the sort of thing that drives us programmers to create entire new computer languages.

That's not to say the primary observation I see being made here is wrong. That observation is that there complex numbers bring nothing really new to the table. You can do everything they do in other ways, say with matrices and a few extra rules. And indeed, mathematicians have come up with numerous other ways of expressing iteration, ∫ and Σ springing to mind.

But nonetheless, the hyperoperations (counting, addition, multiplication, exponentiation, ...) are special. They are most heavily used mathematical operations, by a huge margin. They have one job - to capture the operation we programming nerds call iteration. But when restricted to real numbers, (I'm no mathematician, so this is conjecture), they can't capture self similarity. The result of function expressed as a finite number of hyperoperations operating on reals always flys off to infinity, or asymptotically approaches a constant, or is undefined over part of it's domain. When you add complex numbers you get another possibility: a possibly intricate pattern repeated infinitely.

So yes, complex numbers are basically just a structure containing two real numbers and some modified behaviour, and these is nothing special about that. But there is something profound in the way they allow the planets favourite mathematical operations to finitely express infinitely more behaviours - with no more lexical overhead.

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

#125

Post by lamontcg » Thu, Dec 23, 2021, 6:58 AM UTC

oofbey wrote: post #58

It’s quite strange that the abstract implies Einstein is a founder of QM. More the opposite I think. Einstein remained deeply skeptical of many fundamental aspects of QM - believing in hidden variable theory through his famous statement “God does not play dice with the universe” which was only proven false after his death through experimental measurements of the Bell inequalities. This paper describes another set of…

You've got a naive way of looking at Einstein's skepticism towards the foundations of QM. He wasn't like one of the modern day quacks trying to disprove QM or SR/GR. He helped to build QM and got his Nobel prize for the photoelectric effect which showed that light was quantized and essentially discovered the photon. His argument with QM was that it had to be incomplete in the same sense that Galilean relativity was correct but incomplete, and he didn't accept that physics could be non-local. By applying that principle he proposed the EPR paradox which led to Bell's inequality and the Aspect experiment. He bet wrong on the outcome of that experiment, but it wouldn't have happened without his theoretical work in first proposing the experiment. The fact that he guessed the outcome of the experiment wrong is almost irrelevant, although in our culture of boosting the profile of people who get lucky and guess correctly that is probably difficult to see.

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

#126

Post by drran » Thu, Dec 23, 2021, 9:50 AM UTC

wruza wrote: post #63

drran wrote: post #25

Earlier quoted context omitted.

> But equally cool how bad it gets if you try the same to compute the position of earth 2^64 years later. In 2^64 years, Earth will be inside of a larger body. How many digits of Pi you need to predict that?

In 2^64 years, Earth will be inside of a larger body How do you know that in the first place? Maybe with rationals science it “will not”, so there is “no problem” at all.

We are falling into Great Attractor, which is also falling into Shapley Attractor: https://www.youtube.com/watch?v=PEAY7LNhdxM

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

#127

Post by MarkusQ » Thu, Dec 23, 2021, 4:07 PM UTC

infogulch wrote: post #24

EGreg wrote: post #17

Okay so for us laypeople - what does this mean? Quantum theory is wrong? :-)

No, the opposite. Some might 'hope' that you could represent everything in QM with simpler Real numbers, and avoid Complex numbers (or equivalent formulations that include rotation symmetries, as other commenters point out). But the title and article claim that any such hope is falsifiably dashed. Now, the use of Complex numbers to represent QM is basically standard for like half a century, so this result is more alo…

Actually, despite the title, the claim in the paper is that real-only-QM could be falsified, not that it has been. As far as I can tell, no one has actually done the experiment to check. Logically, it's just as possible that complex-QM could fall. There's a certain amount of hubris in seeing a case where there are two theories that agree in all known cases except one we haven't checked, and assuming "Well that must mean the one we thought of first is right!"

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

#128

Post by amai » Thu, Dec 23, 2021, 7:59 PM UTC

zarzavat wrote: post #116

amai wrote: post #115

Earlier quoted context omitted.

"That these two are the same thing is somewhat of a fluke (it doesn't work in higher dimensions)." Can you elaborate on this? What is an algebraic closure of the reals in higher dimensions?

The complex numbers are the closure regardless of dimension. When I was writing that I was thinking of the Quaternions, which are the 4 dimensional analog of the complex numbers, in 2^N dimensions this is the Cayley Dickson construction. The fluke is this: Euclidean space of dimension N has N(N-1)/2 rotational dimensions. If you plug 2 into that you get 2x1/2 which is 1 dimension. i.e. the rotations in 2D space look…

But couldn't one use two unit quaternions to describe rotations in 4d space? An antisymmetric matrix in 4 dimensions has N(N-1)/2 = 6 independent variables. But each unit quaternion has three independent variables, so two of them would be enough to describe rotations in 4d.

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

#129

Post by infogulch » Thu, Dec 23, 2021, 10:42 PM UTC

MarkusQ wrote: post #127

infogulch wrote: post #24

Earlier quoted context omitted.

No, the opposite. Some might 'hope' that you could represent everything in QM with simpler Real numbers, and avoid Complex numbers (or equivalent formulations that include rotation symmetries, as other commenters point out). But the title and article claim that any such hope is falsifiably dashed. Now, the use of Complex numbers to represent QM is basically standard for like half a century, so this result is more alo…

Actually, despite the title, the claim in the paper is that real-only-QM could be falsified, not that it has been. As far as I can tell, no one has actually done the experiment to check. Logically, it's just as possible that complex-QM could fall. There's a certain amount of hubris in seeing a case where there are two theories that agree in all known cases except one we haven't checked, and assuming "Well that must m…

> real-only-QM could be falsified, not that it has been

Thanks for the clarification!

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

#130

Post by btilly » Sun, Dec 26, 2021, 8:25 PM UTC

wruza wrote: post #59

btilly wrote: post #23

Earlier quoted context omitted.

That logic is disingenuous at best. Planetary orbits are chaotic. Long before your imprecision in pi is going to significantly mislead you, shifts in mass due to, for example, earthquakes and weather patterns are going to cause orbits to be impossible to predict. There are theoretical systems where the exact value of pi matters. But no physical system is going to match that, and measurement error is going to quickly…

But when you remove or take these fluctuations into account, you’re still left with an error. This rational model has no chance to ever be correct computationally, unless you cheat and add more detailed ratio every time you see a loop. Also, how exactly will you define rational pi? Let’s start with 3/1, why go any further. If it doesn’t represent reality (draw a circle and measure it with a string), well, strings hav…

I am a math guy. There is no way to actually do these calculations in practice that does not at some point boil down to something done on a computer to finite precision. For example if you want to plot planetary orbits you should plug in your most precise available measurements, use Runge-Kutta, and produce rational approximations.

There is no way to test our theories in practice that doesn't at some point involve comparing this finite precision prediction with a finite precision observation.

Pontificating about the failures of a discrete approximation to be able to be computationally accurate in a continuous world may be fun armchair philosophy, but CANNOT be useful scientifically. Because we can only measure and work with finite precision approximations to that hypothesized continuous reality.