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

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

#31

Post by roywiggins » Wed, Dec 22, 2021, 2:28 AM UTC

katmannthree wrote: post #26

thereisnospork wrote: post #19

Earlier quoted context omitted.

>In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal. The stronger version of the argument is that the length of a steel beam cannot be more precise(-ish) than the radius of an iron atom, so only 10-12 decimal places (in meters) are required to fully describe…

You’d want to generalize that to a beam whose length is a significant portion of the width of the universe (at which point you should also consider relativistic effects, so there’s more math you’ll need to define over the rationals), but even so that’s not addressing the issue at hand which is that you still have to propagate your uncertainty through each calculation. Depending on the function(s) and time steps your…

But you can't actually use real numbers in calculations, you have to use approximations or proceed symbolically as far as possible. Nobody has a real computer[0], so the best you can do is pick a really accurate value for Pi and arrange your calculations as best you can to avoid pathological error propagation.

[0] https://en.m.wikipedia.org/wiki/Real_computation

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

#32

Post by katmannthree » Wed, Dec 22, 2021, 2:37 AM UTC

roywiggins wrote: post #31

katmannthree wrote: post #26

Earlier quoted context omitted.

You’d want to generalize that to a beam whose length is a significant portion of the width of the universe (at which point you should also consider relativistic effects, so there’s more math you’ll need to define over the rationals), but even so that’s not addressing the issue at hand which is that you still have to propagate your uncertainty through each calculation. Depending on the function(s) and time steps your…

But you can't actually use real numbers in calculations, you have to use approximations or proceed symbolically as far as possible. Nobody has a real computer[0], so the best you can do is pick a really accurate value for Pi and arrange your calculations as best you can to avoid pathological error propagation. [0] https://en.m.wikipedia.org/wiki/Real_computation

Right, this is essentially why theoretical and applied mathematics are separate branches. Applied techniques like you described make theory incredibly cumbersome (and, importantly, not better). My argument here is not that applied mathematics is inferior (it’s what my degree is in), just that it’s generally not a good idea to carry applied techniques back into theory.

You start with doing something the most correct way possible on paper and then convert that into the fastest possible method within your allowable bounds on precision and/or convergence. Operational reordering to keep additions in floats with similar exponents is great but you save that concern until it’s time to crunch numbers. When you’re trying to build an entire theory on how something complex works you’ll have a much better time using the available abstractions to manage complexity without getting bogged down in implementation details.

Edit: addressing your point more directly, numerical computation itself must necessarily be done over fixed precision numbers but the tools we use to decide what and how to do that computation come out of theory done over the reals because of those specific properties of the reals. You can make things work over the rationals but the theory is tedious and the results of generally lower utility.

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

#33

Post by matt123456789 » Wed, Dec 22, 2021, 2:42 AM UTC

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

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

#34

Post by zozbot234 » Wed, Dec 22, 2021, 2:43 AM UTC

roywiggins wrote: post #22

kadoban wrote: post #13

Earlier quoted context omitted.

Orbits over long stretches of time seem likely to need arbitrarily high amounts of accuracy in pi. Sure you can just pick a rational number close enough for the accuracy you need, but why should the definition of pi need to change based on what you're measuring?

Every simulation picks some rational approximation to Pi, because they have to. Either they will run out of time or space or collapse into a black hole before needing more than a finite number of decimal places, so for all plausible purposes we can make do with the first googleplex digits (or whatever) of Pi. I guess my argument is, since you can always just pick a rational approximation to Pi, you cannot prove empir…

> Every simulation picks some rational approximation to Pi, because they have to.

Your simulation might need to ask for an increasingly tighter bound on the real value of Pi. You can totally do this with no more than the usual rational numbers, but it's not equivalent to "just picking some rational approximation" and running with it, because what accuracy/precision you pick is outcome-dependent and it's always possible to request more.

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

#35

Post by analog31 » Wed, Dec 22, 2021, 2:45 AM UTC

Koshkin wrote: post #5

PaulDavisThe1st wrote: post #4

Earlier quoted context omitted.

If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.

Unless we live in a computer simulation, of course, in which case we have to content ourselves with the IEEE floats.

God made the IEEE floats, and all the rest is the work of man

-- Apologies to Kronecker

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

#36

Post by sillysaurusx » Wed, Dec 22, 2021, 2:48 AM UTC

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.

You can't. See "Simulating Physics With Computers," linked here: https://twitter.com/theshawwn/status/1394159744139632640

And some thoughts on the limitations with respect to creating AGI: https://twitter.com/theshawwn/status/1446261451061145602

See section 5, "Can quantum systems be probabilistically simulated by a classical computer?"

> The probability that they match is eight-tenths, the probability that they mismatch is plus two-tenths; every physical probability comes out positive. But the original f's are not positive, and therein lies the great difficulty. The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative, and that we do not know, as far as I know, how to simulate. Okay, that's the fundamental problem. I don't know the answer to it, but I wanted to explain that if I try my best to make the equations look as near as possible to what would be imitable by a classical probabilistic computer, I get into trouble.

When I was younger and only slightly more foolish, I wanted to spend a lot of time researching this to see if there was a way around the problem. I quickly realized that perhaps I should focus on adding value in a field that I was good at. :) Maybe one of you can try, since the only alternative is to take Feynman at his word.

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

#37

Post by robbedpeter » Wed, Dec 22, 2021, 3:05 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 think you still "need" the imaginary number, even if you're expressing it as a more granular/verbose set of fundamentals, and I always wonder if there's an order of operations issue, or some syntactic gotcha instead of anything real?

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

#38

Post by roywiggins » Wed, Dec 22, 2021, 3:08 AM UTC

zozbot234 wrote: post #34

roywiggins wrote: post #22

Earlier quoted context omitted.

Every simulation picks some rational approximation to Pi, because they have to. Either they will run out of time or space or collapse into a black hole before needing more than a finite number of decimal places, so for all plausible purposes we can make do with the first googleplex digits (or whatever) of Pi. I guess my argument is, since you can always just pick a rational approximation to Pi, you cannot prove empir…

> Every simulation picks some rational approximation to Pi, because they have to. Your simulation might need to ask for an increasingly tighter bound on the real value of Pi. You can totally do this with no more than the usual rational numbers, but it's not equivalent to "just picking some rational approximation" and running with it, because what accuracy/precision you pick is outcome-dependent and it's always possib…

My point is that there is some finite number of digits your simulation will ever access, whether they're precalculated ahead of time or on done on the fly. If you do it on the fly you can always go back and rerun the computation with the number hardcoded.

There must be some digit after which no computation will ever access, because it will require more negentropy than the entire universe has to even calculate. The digits after that don't matter to the universe.

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

#39

Post by SideQuark » Wed, Dec 22, 2021, 3:09 AM UTC

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…

Once you replace scalars with 2 vectors you aren't basing it on real numbers, you're basing it on operations on 2 vectors.

Of course you can put real numbers at the base of nearly anything, but it the theory operates at a fundamental level on items that are more complex then real numbers, then it's not based on real numbers.

You might as well argue it's based on surreal numbers or Dedekind cuts at that point.

Somewhere Feynman has a quote that qm was the first theory that required complex numbers. Finding and understand that quote should explain this better.

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

#40

Post by itchyjunk » Wed, Dec 22, 2021, 3:14 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…

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 this search for fields led to discovery of quaternion which doesn't have commutativity but is very close to being a field. So I find C as a field to be special (if not a useful distinction).