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Quantum theory based on real numbers can be experimentally falsified

nature.com

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

#51

Post by int_19h »

PaulDavisThe1st wrote: post #4
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.

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.

If the physical world is entirely quantized, including spacetime itself - which, as I understand, is still considered a valid hypothesis - then wouldn't it be possible to describe it using integers, pretty much by definition?

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

#52

Post by kevin_thibedeau »

midjji wrote: post #8
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…

Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.

'i' is a 90-degree rotation. It isn't hidden at all. Similarly, everywhere a complex exponential shows up, there's also some spinny/rotational thing happening.

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

#53

Post by User23 »

Vetch wrote: post #41
katmannthree wrote: post #12

Earlier quoted context omitted.

Can you? Most likely. Should you? You’ll need to reprove more than a handful of theorems, and for what? What advantage does using rationals instead of reals get you? You might enjoy taking courses in real & complex analysis, the general purpose of which is to impart upon the receiver an understanding of why we’ve constructed those particular number systems and how despite the names they both describe things which are…

I enjoyed your comment and agree with all your well put points except one: on the rationals tie with reality. Having implemented exact real computation to better understand reals, I think of a real number as a kind of machine that generates infinite streams. Operations on them instantiate new machines which query their real operands, computating until there's sufficient information to emit a next term of the stream.…

Or, put tersely, most reals aren’t computable. The implications of that observation are not entirely clear or distinct.

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

#54

Post by gfody »

kevin_thibedeau wrote: post #52
midjji wrote: post #8

Earlier quoted context omitted.

Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.

'i' is a 90-degree rotation. It isn't hidden at all. Similarly, everywhere a complex exponential shows up, there's also some spinny/rotational thing happening.

you can always imagine another dimension, with all the symmetry of every dimension so far plus some novel one

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

#55

Post by btilly »

drran wrote: post #25
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…

> 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?

Sorry, in 2^64 years the Sun will have become a red giant, spent a billion years as that, then turned into a white dwarf. The fate of the Earth by that point is...probably not good.

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

#56

Post by gigatexal »

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 is super clear. Thanks for clarifying.

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

#57

Post by dreamcompiler »

midjji wrote: post #8
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…

Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.

The other side of that same coin is that we're describing wave functions of probability (with interference between probabilities) and complex numbers are exceedingly handy for wave functions, as any EE can attest.

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

#58

Post by oofbey »

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 inequalities similar to the Bell inequalities, but testing whether QM requires complex numbers or not, instead of whether there are hidden variables that can explain things like superposition.

The statement in the abstract I’m referring to is this:

> This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural.[3]

The reference [3] is to a letter by Einstein. Maybe I’m nitpicking words, but scientists tend to spend a lot of time getting the abstracts to their papers right. Any ideas why they would write it this way?

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

#59

Post by wruza »

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…

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 have vague length anyway. See where this is going?

The bigger problem is analyzis will not work, probably. Can you do analyzis with rationals? E.g. f(x)=x^2 isn’t continuous at f(x)=2/1.

I’d theorize that there is a way to do finite calc without giving up on “reals”, by using enough FT coefficients (and packing this complexity into sine waves), but it’s the same sort of cheating probably, unless pi is the “origin” number of all “physical” reals.

(I’m not a physicist nor math guy, so one can reframe my ideas as questions instead.)

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

#60

Post by resters »

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…

You describe the one thing I don't like about complex numbers, that people often don't realize that the same things can be represented fully using other mathematical objects. Complex numbers are basically syntactic sugar for that more general type of object.
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