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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

#61

Post by aeternum »

sillysaurusx wrote: post #36
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 mism…

The claim that discretization is not equal to quantization is a key claim and I'm not sure the paper actually proves/shows this.

If it turns out that spacetime is discrete and not continuous then we should be able to simulate it. The possible states of some chunk of spacetime (given some maximum/known energy) would be non-infinite and even though it might take us years to calculate each time-step, we could still make progress.

I've seen the theory kicked around before, but why couldn't the universe basically consist of plank length sized voxels and a similarly small update timestep? It would still look plenty continuous to us.

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

#62

Post by bopbeepboop »

User23 wrote: post #53
Vetch wrote: post #41

Earlier quoted context omitted.

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.

No post body was provided.

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

#63

Post by wruza »

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?

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.

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

#64

Post by Rerarom »

resters wrote: post #60
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.

So you would like them to not be representable?

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

#65

Post by subroutine »

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…

Einstein is considered a founder (1 of 3) of QM because he was the first to described light as quanta, and won the Nobel prize for it. It's ok to be skeptical, even of your own hypotheses/theories.

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

#66

Post by GoblinSlayer »

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.

Rotational symmetry a straightforward consequence of the wave equation.

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

#67

Post by fsh »

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…

Reference [3] is a letter from Schrödinger to Lorentz. I would also argue that Einstein had one of the deepest understandings of QM at the time. Everyone knew that the theory is weird, but with the EPR paper Einstein managed to pinpoint quite well where this weirdness shows up. It was also quite reasonable to assume that local hidden variables exist 40 years before a violation of Bell's inequality could be experimentally demonstrated.

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

#68

Post by selestify »

crdrost wrote: post #49
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…

So not having yet read through OP I am not terribly surprised that this is true and I can kind of give a quick sketch in terms of a QM game that I want everyone to know, called Betrayal. The idea is that it's a collaborative game for three people, you are trying to work together to beat the rules of the game. Meanwhile the rules are trying to set you up so that one of the people betrays the other two. In 3 relativist…

I wish I understood enough QM to understand this comment! Any suggestions for where I might learn about GHZ states and the like?

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

#69

Post by rnhmjoj »

I'm surprised the article doesn't mention phases or the idea of projective Hilbert spaces. The QM formulation it gives is well known ambiguous: the condition =1 does not determine φ, because it's also satisfied by any other state ψ=exp(iλ)φ with λ real. So, the state of a physical system is really described by the ray (equivalent class) of all state vectors differing by a phase.

I wonder if this has any consequence on their reasoning and conclusions.

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

#70

Post by kgwgk »

mst_moonshine wrote: post #48
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…

Agreed. So what you need is the 'complex structure' behind rather than just 'complex numbers'. Any form of representations (numbers, matrices, and so on) should correspond to a unique structure. The question why the complex structure emerges in quantum mechanics is more interesting.

Skilling and Knuth have some interesting papers on the subject:

The Symmetrical Foundation of Measure, Probability, and Quantum Theories https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800...

The ABC of Physics https://www.mdpi.com/2673-9984/3/1/9/htm

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