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

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

#21

Post by roywiggins » Wed, Dec 22, 2021, 1:58 AM UTC

katmannthree wrote: post #14

roywiggins wrote: post #10

Earlier quoted context omitted.

Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.

That’s not isomorphic to the discussion at hand though, which refers to modeling systems. Being able to physically construct an object within epsilon does not imply that you won’t run into precision issues when modeling said object at the same degree of precision. 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 to…

It seems quite likely that cutting off Pi at the quintillionth decimal place will not hurt your simulation simply because your knowledge of the initial conditions in the physical universe won't be exact either. If a quintillion digits aren't enough, run your simulation with another quintillion places. Eventually you will match reality to within the accuracy of your ability to measure.

Any issues introduced by using a finite approximation to Pi will eventually be swamped by the uncertainty in the initial conditions. If there's no uncertainty in the initial conditions, there will still be some finite approximation to Pi that will give you results as accurate as you can measure...

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

#22

Post by roywiggins » Wed, Dec 22, 2021, 2:00 AM UTC

kadoban wrote: post #13

roywiggins wrote: post #10

Earlier quoted context omitted.

Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.

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 empirically that we live in a universe where more than a finite number of digits of Pi matter. That is, the mathematical irrationality doesn't really matter, physically speaking, since no experiment could ever prove that every digit in Pi actually contributes to the result.

If the universe does have ways to do this, to mix an entire irrational number into a physical outcome, that means hypercomputation is probably possible, since Turing machines definitely can't.

https://en.m.wikipedia.org/wiki/Real_computation

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

#23

Post by btilly » Wed, Dec 22, 2021, 2:00 AM UTC

midjji wrote: post #11

btilly wrote: post #7

Earlier quoted context omitted.

You can describe it to within measurement error without any irrational numbers. And with fewer decimal places than you'd probably imagine. See https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... for more.

That often works, but not always, some systems generate sequences of operations which can be symbolically simplified, or equivalently could be exactly computed using reals, but which if computed using any finite precision will fail. A simple example would be solving for the position of a planet in orbit under simple Newtonian gravity, a sufficient number of revolutions latter. For any finite precision the number of o…

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 exceed calculation errors from pi.

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

#24

Post by infogulch » Wed, Dec 22, 2021, 2:02 AM UTC

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 along the lines of "things we figured were true for a while because the math just works out so much better this way but we weren't certain enough to call it for sure until now".

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

#25

Post by drran » Wed, Dec 22, 2021, 2:04 AM UTC

midjji wrote: post #11

btilly wrote: post #7

Earlier quoted context omitted.

You can describe it to within measurement error without any irrational numbers. And with fewer decimal places than you'd probably imagine. See https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... for more.

That often works, but not always, some systems generate sequences of operations which can be symbolically simplified, or equivalently could be exactly computed using reals, but which if computed using any finite precision will fail. A simple example would be solving for the position of a planet in orbit under simple Newtonian gravity, a sufficient number of revolutions latter. For any finite precision the number of o…

> 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?

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

#26

Post by katmannthree » Wed, Dec 22, 2021, 2:10 AM UTC

thereisnospork wrote: post #19

katmannthree wrote: post #14

Earlier quoted context omitted.

That’s not isomorphic to the discussion at hand though, which refers to modeling systems. Being able to physically construct an object within epsilon does not imply that you won’t run into precision issues when modeling said object at the same degree of precision. 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 to…

>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 uncertainty can quickly blow past the threshold of utility (e.g. in the case of orbits, your uncertainty could end up being numerically greater than your ability to deal with it [meaning your orbital prediction for N bodies after T time has passed is so imprecise that your craft is not capable of intercepting at all potential states]). This is the power of the real numbers, being able to bypass the accumulation of error in some cases.

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

#27

Post by db48x » Wed, Dec 22, 2021, 2:13 AM UTC

EGreg wrote: post #17

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

No, it means that quantum mechanics is correct, and that you cannot get around the fact that it needs complex numbers. The complex numbers represent what is really going on in the universe better and more accurately than simple ordinary non–complex numbers do.

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

#28

Post by katmannthree » Wed, Dec 22, 2021, 2:16 AM UTC

roywiggins wrote: post #21

katmannthree wrote: post #14

Earlier quoted context omitted.

That’s not isomorphic to the discussion at hand though, which refers to modeling systems. Being able to physically construct an object within epsilon does not imply that you won’t run into precision issues when modeling said object at the same degree of precision. 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 to…

It seems quite likely that cutting off Pi at the quintillionth decimal place will not hurt your simulation simply because your knowledge of the initial conditions in the physical universe won't be exact either. If a quintillion digits aren't enough, run your simulation with another quintillion places. Eventually you will match reality to within the accuracy of your ability to measure. Any issues introduced by using a…

Yes, you can generally find ways to manage life with limited precision. Doing so is useful enough that it’s an entire field (applied mathematics). For many cases though it is a better choice to stick with these particular carefully designed constructs (real, imaginary, and complex numbers) because they are easier to deal with and help keep you from getting bogged down in avoidable numerical tangles. Managing error accumulation is a huge deal for some types of simulations and every little bit counts, to the point of carefully ordering your operations to minimize precision loss with the widest numbers you can afford to use.

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

#29

Post by TheOtherHobbes » Wed, Dec 22, 2021, 2:19 AM UTC

btilly wrote: post #23

midjji wrote: post #11

Earlier quoted context omitted.

That often works, but not always, some systems generate sequences of operations which can be symbolically simplified, or equivalently could be exactly computed using reals, but which if computed using any finite precision will fail. A simple example would be solving for the position of a planet in orbit under simple Newtonian gravity, a sufficient number of revolutions latter. For any finite precision the number of o…

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…

Pi is only incidentally a number. Pi is an abstraction that represents a certain relationship. (Actually more of a set of relationships.)

If you define pi as a specific constant with limited precision - because "that's all physical systems need" - you lose insights into the web of relationships around it.

This is a bad thing and makes many kinds of math harder.

It's the conceptual equivalent of lossy data compression. You don't want to do it unless you really, really need to. And if you do it, you need to be aware that you're now using approximations instead of abstractions, and those are not the same thing.

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

#30

Post by asxd » Wed, Dec 22, 2021, 2:22 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…

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 different predictions in network scenarios comprising independent states and measurements.

Maybe I'm interpreting "real numbers" and "need" differently than the authors, since in my head complex numbers are basically just a structure containing two real numbers and some modified behavior.